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Fujihira, Takeo. "Hamiltonian Hopf bifurcation with symmetry." Thesis, Imperial College London, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.444087.
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Harlim, John. "Codimension three Hopf and cusp bifurcation." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2001. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/MQ58343.pdf.
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Salih, Rizgar Haji. "Hopf bifurcation and centre bifurcation in three dimensional Lotka-Volterra systems." Thesis, University of Plymouth, 2015. http://hdl.handle.net/10026.1/3504.
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This thesis presents a study of the centre bifurcation and chaotic behaviour of three dimensional Lotka-Volterra systems. In two dimensional systems, Christopher (2005) considered a simple computational approach to estimate the cyclicity bifurcating from the centre. We generalized the technique to estimate the cyclicity of the centre in three dimensional systems. A lower bounds is given for the cyclicity of a hopf point in the three dimensional Lotka-Volterra systems via centre bifurcations. Sufficient conditions for the existence of a centre are obtained via the Darboux method using inverse Jacobi multiplier functions. For a given centre, the cyclicity is bounded from below by considering the linear parts of the corresponding Liapunov quantities of the perturbed system. Although the number obtained is not new, the technique is fast and can easily be adapted to other systems. The same technique is applied to estimate the cyclicity of a three dimensional system with a plane of singularities. As a result, eight limit cycles are shown to bifurcate from the centre by considering the quadratic parts of the corresponding Liapunov quantities of the perturbed system. This thesis also examines the chaotic behaviour of three dimensional Lotka-Volterra systems. For studying the chaotic behaviour, a geometric method is used. We construct an example of a three dimensional Lotka-Volterra system with a saddle-focus critical point of Shilnikov type as well as a loop. A construction of the heteroclinic cycle that joins the critical point with two other critical points of type planar saddle and axial saddle is undertaken. Furthermore, the local behaviour of trajectories in a small neighbourhood of the critical points is investigated. The dynamics of the Poincare map around the heteroclinic cycle can exhibit chaos by demonstrating the existence of a horseshoe map. The proof uses a Shilnikov-type structure adapted to the geometry of these systems. For a good understanding of the global dynamics of the system, the behaviour at infinity is also examined. This helps us to draw the global phase portrait of the system. The last part of this thesis is devoted to a study of the zero-Hopf bifurcation of the three dimensional Lotka-Volterra systems. Explicit conditions for the existence of two first integrals for the system and a line of singularity with zero eigenvalue are given. We characteristic the parameters for which a zero-Hopf equilibrium point takes place at any points on the line. We prove that there are three 3-parameter families exhibiting such equilibria. First order of averaging theory is also applied but we show that it gives no information about the possible periodic orbits bifurcating from the zero-Hopf equilibria.
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Pacha, Andújar Juan Ramón. "On the quasiperiodic hamiltonian andronov-hopf bifurcation." Doctoral thesis, Universitat Politècnica de Catalunya, 2002. http://hdl.handle.net/10803/5830.
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Aquest treball es situa dintre del marc dels sistemes dinàmics hamiltonians de tres graus de llibertat. Allà considerem famílies d'òrbites periòdiques amb una transició estable-complex inestable: sigui L el paràmetre que descriu la família i suposarem que per a valors del paràmetre més petits que un cert valor crític, L', els multiplicadors característics de les òrbites periòdiques corresponents hi són sobre el cercle unitat, quan L=L' aquests col·lisionen per parelles conjugades (òrbita ressonant o crítica) i per L > L', abandonen el cercle unitat cap al pla complex (col·lisió de Krein amb signatura oposada). El canvi d'estabilitat subseqüent es descriu a la literatura com "transició estable a complex inestable". Tanmateix, a partir d'estudis numèrics sobre certes aplicacions simplèctiques (n'esmentarem D. Pfenniger, Astron. Astrophys. 150, 97-111, 1985), és coneguda l'aparició (sota condicions d'incommensurabilitat) de fenòmens de bifurcació quasi-periòdica, en particular, el desplegament de famílies de tors 2-dimensionals. A més aquesta bifurcació s'assembla a la (clàssica) bifurcació d'Andronov-Hopf, en el sentit de què hi sorgeixen objectes linealment estables (tors-2D el·líptics) "al voltant" d'objectes inestables de dimensionalitat més baixa (òrbites periòdiques), i recíprocament, n'apareixen tors inestables (hiperbòlics) "al voltant" d'òrbites periòdiques linealment estables.Nostre objectiu és entendre la dinàmica local en un entorn de l'òrbita periòdica ressonant per tal de provar, analíticament, l'existència dels tors invariants bifurcats segons l'esquema descrit dalt. Això el portem a terme mitjançant l'anàlisi següent:
(i) Primer de tot obtenim d'una manera constructiva (això és, donant algorismes) una forma normal ressonant en un entorn de l'òrbita periòdica crítica. Aquesta forma normal la portem fins a qualsevol ordre arbitrari r. Així doncs, mostrem que el hamiltonià inicial es pot posar com la suma de la forma normal (integrable) més una resta no integrable. A partir d'aquí, podem estudiar la dinàmica de la forma normal, prescindint dels altres termes i, amb aquest tractament (formal) del problema, som capaços d'identificar els paràmetres que governen tant l'existència de la bifurcació com la seva tipologia (directa, inversa). Cal, remarcar que el que es fa fins aquí, no és només un procés qualitatiu, ja que a més ens permet derivar parametritzacions molt acurades dels tors no pertorbats.
(ii) A continuació, calculem acotacions "òptimes" per a la resta. D'aquesta manera, esperem provar que un bon nombre de tors (en sentit de la mesura) es preserven quan s'afegeix la pertorbació.
(iii) Finalment, apliquem mètodes KAM per establir que la majoria (veure comentari dalt) dels tors bifurcats sobreviuen. Aquests mètodes es basen en la construcció d'un esquema de convergència quadràtica capaç de contrarestar l'efecte dels petits divisors que apareixen quan s'aplica teoria de pertorbacions per trobar solucions quasi-periòdiques. En el nostre cas, a més, resulta que alguna de les condicions "típiques" que s'imposen sobre les freqüències (intrínseques i normals) dels tors no pertorbats, no estan ben definides per als tors bifurcats, de manera que ens ha calgut desenvolupar un tractament més específic.
keywords: Bifurcation problems, perturbations, normal forms, small divisors, KAM theory.
Classificació AMS: 37J20, 37J25, 37J40
This work is placed into the context of the three-degree of freedom Hamiltonian systems, where we consider families of periodic orbits undergoing transitions stable-complex unstable. More precisely: Let L be the parameter of the family and assuming that, for values of L smaller than some critical value say, L', the characteristic multipliers of the periodic orbits lie on the unit circle, when L=L' they colllide pairwise (critical or resonant periodic orbit) and, for L > L' leave the unit circle towards the complex plane (Krein collision with opposite signature).
From numerical studies on some concrete symplectic maps (for instance, D. Pfennniger, Astron. Astrophys. 150, 97-111, 1985) it is known the rising (under certain irrationality conditions), of quasi-periodic bifurcation phenomena, in particular, the appearance of unfolded 2D invariant tori families. Moreover, the bifurcation takes place in a way that resembles the classical Andronov-Hopf one, in the sense that either stable invariant objects (elliptic tori) unfold "around" linear unstable periodic orbits, or conversely, unstable invariant structures (hyperbolic tori) appear "surrounding" stable periodic orbits.
Our objective is, thus, to understand the (local) dynamics in a neighbourhood of the critical periodic orbit well enough to prove analytically, the existence of such quasi-periodic solutions together with the bifurcation pattern described above. This is carried out through three steps:
(i) First, we derive, in a constructive way (i. e., giving algorithms), a resonant normal form around the critical periodic orbit up to any arbitrary order r. Whence, we show that the initial raw Hamiltonian can be casted --through a symplectic change--, into an integrable part, the normal form itself, plus a (non-integrable) remainder. From here, one can study the dynamics of the normal form, skipping the remainder off. As a result of this (formal) approach, we are able to indentify the parameters governing both, the presence of the bifurcation and its type (direct, inverse). We remark that this is not a merely qualitative process for, in addition, accurate parametrizations of the bifurcated families of invariant tori are derived in this way.
(ii) Beyond the formal approach, we compute "optimal" bounds for the remainder of the normal form, so one expects to prove the preservation of a higher (in the measure sense) number of invariant tori --than, indeed, with a less sharp estimates--.
(iii) Finally, we apply KAM methods to establish the persistence of (most, in the measure sense) of the bifurcated invariant tori. These methods involve the design of a suitable quadratic convergent scheme, able to overcome the effect of the small divisors appearing in perturbation techniques when one looks for quasi-periodic solutions. In this case though, some of the "typical" conditions that one imposes on the frequencies (intrinsic and normal) of the unperturbed invariant tori do not work, due to the proximity to parabolic tori, so one is bound to sketch specific tricks.
keywords: Bifurcation problems, perturbations, normal forms, small divisors, KAM theory
AMS classification: 37J20, 37J25, 37J40
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Lee, Yoon-Mee. "Hopf Bifurcation in a Parabolic Free Boundary Problem." DigitalCommons@USU, 1992. https://digitalcommons.usu.edu/etd/7138.
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We deal with a free boundary problem for a nonlinear parabolic equation, which includes a parameter in the free boundary condition. This type of system has been used in models of ecological systems, in chemical reactor theory and other kinds of propagation phenomena involving reactions and diffusion.
The main purpose of this dissertation is to show the global existence, uniqueness of solutions and that a Hopf bifurcation occurs at a critical value of the parameter r. The existence and uniqueness of the solution for this problem are shown by finding an equivalent regular free boundary problem to which existence results can be applied. We then show that as the bifurcation parameter r decreases and passes through a critical value rc, the stationary solution loses stability and a stable periodic solution appears. Several figures have been included, which illustrate this transistion. The pascal source program used in the numerical simulation is included in an appendix.
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Dupuis, Étienne. "De l'existence d'hypertores près d'une bifurcation de Hopf-Hopf avec résonance 1:2." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp03/MQ57112.pdf.
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Bateman, Craig A. "Hopf bifurcation analysis for depth control of submersible vehicles." Thesis, Monterey, Calif. : Springfield, Va. : Naval Postgraduate School ; Available from National Technical Information Service, 1993. http://handle.dtic.mil/100.2/ADA276213.
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Alolyan, Ibraheem. "Global minimization of Hopf bifurcation surfaces with application to nematic electroconvection." Access citation, abstract and download form; downloadable file 7.34 Mb, 2004. http://wwwlib.umi.com/dissertations/fullcit/3131651.
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Lewis, Gregory M. "Double Hopf bifurcations in two geophysical fluid dynamics models." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp03/NQ48653.pdf.
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Haaf, Hermann. "Existence of periodic travelling waves to reaction-diffusion equations with excitable-oscillatory kinetics." Thesis, University of Warwick, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.387347.
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Wulf, Volker. "Numerical analysis of delay differential equations undergoing a Hopf bifurcation." Thesis, University of Liverpool, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.367052.
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Zhang, Yanyan. "Periodic Forcing of a System near a Hopf Bifurcation Point." The Ohio State University, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=osu1291174795.
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Jones, Steven R. "Hopf Bifurcations and Horseshoes Especially Applied to the Brusselator." Diss., CLICK HERE for online access, 2005. http://contentdm.lib.byu.edu/ETD/image/etd825.pdf.
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Lepine, Francoise. "Pitchfork and Hopf bifurcation threshold in stochastic equations with delayed feedback." Thesis, McGill University, 2009. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=32364.
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The bifurcation diagram of a model nonlinear Langevin equation with delay is obtained. A finite delay implies non-Markovian behavior, and is the subject of current interest as such equations model delayed feedback loops in many settings, including the study of cell regulation. We show that the bifurcation remains sharp, both in the ranges of direct and oscillatory instabilities. Below threshold, the stationary distribution function is a delta function at the trivial state x = 0 despite the delay. At threshold, the stationary distribution function becomes a power law p(x) ∼ x ^a with −1 < a < 0, where a = −1 at threshold and monotonously increases with increasing value of the control parameter. Unlike the case without delay, the bifurcation threshold is shifted by fluctuations, and the shift scales linearly with the noise intensity D.Le diagramme de bifurcation d' une équation de Langevin non linéaire retardée est obtenu. Un retard fini implique un processus non markovien. De telles équations permettent de modéliser des boucles d'autorégulation retardées dans de nombreux systèmes, incluant la régulation cellulaire. Nous démontrons que la bifurcation demeure aigüe dans les cas d'instabilités stationnaires et oscillatoires. Sous le seuil de bifurcation, la fonction de distribution de probabilité stationnaire est une fonction delta à la solution triviale x = 0, malgré la presence du retard. Au seuil de bifurcation, la distribution stationnaire devient une loi de puissance p(x) ∼ x ^a avec −1 < a < 0, où a = −1 au seuil and croît de facon monotone avec le paramètre de contrôle. Contrairement au cas non-retardé, le seuil de bifurcation est déplacé par les fluctuations, et ce déplacement augmente linéairement avec l'intensité du bruit D.
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Gaivão, José Pedro. "Exponentially small splitting of invariant manifolds near a Hamiltonian-Hopf bifurcation." Thesis, University of Warwick, 2010. http://wrap.warwick.ac.uk/4534/.
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Consider an analytic two-degrees of freedom Hamiltonian system with an equilibrium point that undergoes a Hamiltonian-Hopf bifurcation, i.e., the eigenvalues of the linearized system at the equilibrium change from complex ±β ±iα (α,β > 0) for ε > 0 to pure imaginary ±iω1 and ±iω2 (ω1 ≠ ω2 ≠ 0) for ε < 0. At ε = 0 the equilibrium has a pair of doubled pure imaginary eigenvalues. Depending on the sign of a certain coefficient of the normal form there are two main bifurcation scenarios. In one of these (the stable case), two dimensional stable and unstable manifolds of the equilibrium shrink and disappear as ε → 0+. At any order of the normal form the stable and unstable manifolds coincide and the invariant manifolds are indistinguishable using classical perturbation theory. In particular, Melnikov’s method is not capable to evaluate the splitting. In this thesis we have addressed the problem of measuring the splitting of these manifolds for small values of the bifurcation parameter ε. We have estimated the size of the splitting which depends on a singular way from the bifurcation parameter. In order to measure the splitting we have introduced an homoclinic invariant ωε which extends the Lazutkin’s homoclinic invariant defined for area-preserving maps. The main result of this thesis is an asymptotic formula for the homoclinic invariant. Assuming reversibility, we have proved that there is a symmetric homoclinic orbit such that its homoclinic invariant can be estimated as follows, ωε = ±2e−πα/2β (ω0 + O(ε1−μ)). where μ > 0 is arbitrarily small and ω0 is known as the Stokes constant. This asymptotic formula implies that the splitting is exponentially small (with respect to ε). When ω0 ≠ 0 then the invariant manifolds intersect transversely. The Stokes constant ω0 is defined for the Hamiltonian at the moment of bifurcation only. We also prove that it does not vanish identically. Finally, we apply our methods to study homoclinic solutions in the Swift-Hohenberg equation. Our results show the existence of multi-pulse homoclinic solutions and a small scale chaos.
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Chifan, Iustina. "Hopf Bifurcation Analysis for a Variant of the Logistic Equation with Delays." Thesis, Université d'Ottawa / University of Ottawa, 2020. http://hdl.handle.net/10393/40504.
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This thesis contains some results on the behavior of a delay differential equation (DDE) with two delays, at a Hopf bifurcation, for the nonzero equilibrium, using the growth rate, r, as bifurcation parameter. This DDE is a model for population growth, incorporating a maturation delay, and a second delay in the harvesting term. Considering a Taylor expansion of the non-dimensionalized model, we find a region of stability for the nonzero equilibrium, after which we find a pair of ODEs which help define the flow on the center manifold. We then find an expression for the first Lypapunov coefficient, which changes sign, so we also find the second Lyapunov coefficient, allowing us to predict multi-stability in the model. Numerical simulations provide examples of the behavior expected. For a similar model with one delay (PMC model), we prove the Hopf bifurcation at the nonzero equilibrium is always supercritical.
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Arakawa, Vinicius Augusto Takahashi [UNESP]. "Um estudo de bifurcações de codimensão dois de campos de vetores." Universidade Estadual Paulista (UNESP), 2008. http://hdl.handle.net/11449/94243.
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Nesse trabalho são apresentados alguns resultados importantes sobre bifurcações de codimensão dois de campos de vetores. O resultado principal dessa dissertação e o teorema que d a o diagrama de bifurcação e os retratos de fase da Bifurcação de Bogdanov-Takens. Para a demonstracão são usadas algumas técnicas basicas de Sistemas Dinâmicos e Teoria das Singularidades, tais como Integrais Abelianas, desdobramentos de Sistemas Hamiltonianos, desdobramentos versais, Teorema de Preparação de Malgrange, entre outros. Outra importante bifurcação clássica apresentada e a Bifurca cão do tipo Hopf-Zero, quando a matriz Jacobiana possui um autovalor simples nulo e um par de autovalores imagin arios puros. Foram usadas algumas hipóteses que garantem propriedades de simetria do sistema, dentre elas, assumiuse que o sistema era revers vel. Assim como na Bifurcação de Bogdanov-Takens, foram apresentados o diagrama de bifurcao e os retratos de fase da Bifurcação Hopf-zero bifurcação reversível. As técnicas usadas para esse estudo foram a forma normal de Belitskii e o método do Blow-up polar.
In this work is presented some important results about codimension two bifurcations of vector elds. The main result of this work is the theorem that gives the local bifurcation diagram and the phase portraits of the Bogdanov-Takens bifurcation. In order to give the proof, some classic tools in Dynamical System and Singularities Theory are used, such as Abelian Integral, versal deformation, Hamiltonian Systems, Malgrange Preparation Theorem, etc. Another classic bifurcation phenomena, known as the Hopf-Zero bifurcation, when the Jacobian matrix has a simple zero and a pair of purely imaginary eigenvalues, is presented. In here, is added the hypothesis that the system is reversible, which gives some symmetry in the problem. Like in Bogdanov-Takens bifurcation, the bifurcation diagram and the local phase portraits of the reversible Hopf-zero bifurcation were presented. The main techniques used are the Belitskii theory to nd a normal forms and the polar Blow-up method.
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Langlois, Mélanie. "Étude de bifurcation de Hopf-Hopf avec résonance 1:2 pour un système de deux neurones couplés avec délais." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp03/MQ57127.pdf.
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Arakawa, Vinicius Augusto Takahashi. "Um estudo de bifurcações de codimensão dois de campos de vetores /." São José do Rio Preto : [s.n.], 2008. http://hdl.handle.net/11449/94243.
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Orientador: Claudio Aguinaldo BuzziBanca: João Carlos da Rocha Medrado
Banca: Luciana de Fátima Martins
Resumo: Nesse trabalho são apresentados alguns resultados importantes sobre bifurcações de codimensão dois de campos de vetores. O resultado principal dessa dissertação e o teorema que d a o diagrama de bifurcação e os retratos de fase da Bifurcação de Bogdanov-Takens. Para a demonstracão são usadas algumas técnicas basicas de Sistemas Dinâmicos e Teoria das Singularidades, tais como Integrais Abelianas, desdobramentos de Sistemas Hamiltonianos, desdobramentos versais, Teorema de Preparação de Malgrange, entre outros. Outra importante bifurcação clássica apresentada e a Bifurca cão do tipo Hopf-Zero, quando a matriz Jacobiana possui um autovalor simples nulo e um par de autovalores imagin arios puros. Foram usadas algumas hipóteses que garantem propriedades de simetria do sistema, dentre elas, assumiuse que o sistema era revers vel. Assim como na Bifurcação de Bogdanov-Takens, foram apresentados o diagrama de bifurcao e os retratos de fase da Bifurcação Hopf-zero bifurcação reversível. As técnicas usadas para esse estudo foram a forma normal de Belitskii e o método do Blow-up polar.
Abstract: In this work is presented some important results about codimension two bifurcations of vector elds. The main result of this work is the theorem that gives the local bifurcation diagram and the phase portraits of the Bogdanov-Takens bifurcation. In order to give the proof, some classic tools in Dynamical System and Singularities Theory are used, such as Abelian Integral, versal deformation, Hamiltonian Systems, Malgrange Preparation Theorem, etc. Another classic bifurcation phenomena, known as the Hopf-Zero bifurcation, when the Jacobian matrix has a simple zero and a pair of purely imaginary eigenvalues, is presented. In here, is added the hypothesis that the system is reversible, which gives some symmetry in the problem. Like in Bogdanov-Takens bifurcation, the bifurcation diagram and the local phase portraits of the reversible Hopf-zero bifurcation were presented. The main techniques used are the Belitskii theory to nd a normal forms and the polar Blow-up method.
Mestre
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TRI, ABDELJALIL POTIER FERRY MICHEL. "METHODES ASYMPTOTIQUES NUMERIQUES POUR LES FLUIDES VISQUEUX INCOMPRESSIBLES ET LA BIFURCATION DE HOPF /." [S.l.] : [s.n.], 1996. ftp://ftp.scd.univ-metz.fr/pub/Theses/1996/Tri.Abdeljalil.SMZ9651.pdf.
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Chan, David. "A normal form approach to non-resonant and resonant Hopf bifurcation from relative equilibria." Thesis, University of Surrey, 2006. http://epubs.surrey.ac.uk/843715/.
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The main aim of this thesis has been to investigate the distinction between a nonresonant and resonant Hopf bifurcation from relative equilibria. Resonant and nonresonant Hopf bifurcations from relative equilibria posed in two spatial dimensions, in systems with Euclidean SE(2) symmetry, have been extensively studied in the context of spiral waves in a plane and are now well understood. We initially investigate Hopf bifurcations from relative equilibria posed in systems with compact S0(3) symmetry where S0(3) is the group of rotations in three dimensions/on a sphere. Unlike the SE(2) case the skew product equations cannot be solved directly and we use the normal form theory due to Fiedler and Turaev to simplify these systems. We show that the normal form theory resolves the nonresonant case, but not the resonant case. New methods developed in this thesis combined with the normal form theory resolves the resonant case. We find that the resonant Hopf bifurcation produces a motion which is strikingly different from the nonresonant case. By considering the geometric properties of the underlying relative equilibrium we give a definition of resonance directly related to the Hopf bifurcation phenomena. This yields conditions for the occurrence of a resonant Hopf bifurcation from a relative equilibrium in a system with general compact symmetry. By extending the approach used to resolve the S0(3) case we then solve the skew product equations for a general compact symmetry group. The specific case of a Hopf bifurcation from a relative equilibria with symmetry of an irregular dodecahedron is also considered. Furthermore we look at nonresonant and resonant Hopf Bifurcations from relative equilibria posed in a system with noncompact Euclidean symmetry in three spatial dimensions.
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Sathaye, Archana S. "BIFDE: a numerical software package for the hopf bifurcation problem in functional differential equations." Thesis, Virginia Polytechnic Institute and State University, 1986. http://hdl.handle.net/10919/101145.
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A software package has been written to compute the Hopf bifurcation structure in functional differential equations. The package is modular, and consists of several routines which perform one or more tasks. In conjunction with the routines available in this package, the user is required to provide a few routines which describe the specific system under analysis. Three example systems (from epidemiology, biochemistry and aerospace engineering) have been analyzed to illustrate the use of this package.M.S.
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BENSAADE, MY EL HASSAN POTIER-FERRY M. "METHODES ASYMPTOTIQUES-NUMERIQUES POUR LE CALCUL DE BIFURCATION DE HOPF ET DE SOLUTIONS PERIODIQUES /." [S.l.] : [s.n.], 1995. ftp://ftp.scd.univ-metz.fr/pub/Theses/1995/Ben_Saadi.Hassan.SMZ9547.pdf.
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Bramburger, Jason. "Steady State/Hopf Interactions in the Van Der Pol Oscillator with Delayed Feedback." Thèse, Université d'Ottawa / University of Ottawa, 2013. http://hdl.handle.net/10393/24325.
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In this thesis we consider the traditional Van der Pol Oscillator with a forcing dependent on a delay in feedback. The delay is taken to be a nonlinear function of both position and velocity which gives rise to many different types of bifurcations. In particular, we study the Zero-Hopf bifurcation that takes place at certain parameter values using methods of centre manifold reduction of DDEs and normal form theory. We present numerical simulations that have been accurately predicted by the phase portraits in the Zero-Hopf bifurcation to confirm our numerical results and provide a physical understanding of the oscillator with the delay in feedback.
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Turan, Mehmet. "New Classes Of Differential Equations And Bifurcation Of Discontinuous Cycles." Phd thesis, METU, 2009. http://etd.lib.metu.edu.tr/upload/3/12610747/index.pdf.
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In this thesis, we introduce two new classes of differential equations, which essentially extend, in several directions, impulsive differential equations and equations on
time scales. Basics of the theory for quasilinear systems are discussed, and particular results are obtained so that further investigations of the theory are guaranteed. Applications of the newly-introduced systems are shown through a center manifold theorem, and further, Hopf bifurcation Theorem is proved for a three-dimensional discontinuous dynamical system.
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Ver, Hulst Henri. "Marrying the physics of critical oscillators with traveling-wave models of the cochlea." Electronic Thesis or Diss., Université Paris sciences et lettres, 2024. http://www.theses.fr/2024UPSLS048.
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Les propriétés mécaniques non linéaires de la cochlée ont été reconnues comme signatures d'un processus actif d’amplification par des oscillateurs “critiques”, c’est-à-dire opérant à un point d'instabilité oscillatoire—une bifurcation de Hopf. Dans ce cadre, chaque oscillateur est réglé à une fréquence qui dépend de la position dans la cochlée selon la tonotopie de cet organe auditif. Un oscillateur critique stimulé à sa fréquence caractéristique représente six ordres de grandeurs du stimulus en seulement deux ordres de grandeur de sa réponse, ce qui correspond à une loi de puissance d'exposant 1/3. Cette propriété, générique, décrit la non-linéarité compressive associée à l'amplification cochléaire. Cependant, la pertinence du concept d'oscillation critique comme principe l'amplification cochléaire est remise en question par trois observations. Premièrement, les exposants des lois de puissance mesurées dans la cochlée peuvent dévier du comportement générique d'un oscillateur critique. Deuxièmement, la sensibilité maximale d’un oscillateur critique est inversement proportionnelle à la bande passante de l’oscillateur, résultant en un produit "gain-bande passante" indépendant du niveau de stimulation. La cochlée ne suit pas cette règle : le produit gain-bande passante diminue avec le niveau de stimulation sonore. Enfin, la réponse de la cochlée à un bruit blanc d’intensité variable diffère de celle d’un oscillateur critique. Le concept d’oscillation critique peut-il être réconcilié avec ces trois propriétés de la phénoménologie cochléaire ?Dans ma thèse, j'ai abordé cette question en intégrant une distribution tonotopique oscillateurs critiques dans un modèle d'onde progressive de la cochlée. Ce modèle est non linéaire et a pris en compte l'aspect bidimensionnel de l'hydrodynamique, le couplage viscoélastique longitudinal entre les oscillateurs, et le pompage d'énergie par les oscillateurs dans l'onde. Il a été résolu numériquement dans le domaine temporel en s’appuyant sur le formalisme de la fonction de Green. Le modèle a produit, avec un jeu de paramètres fixés et sur une large gamme de niveaux sonores, une famille de réponses en fonction de la fréquence et du niveau sonore correspondant à celles mesurées dans la cochlée. Le succès du modèle repose sur un phénomène de focalisation de la pression du fluide induit par l'hydrodynamique 2D ainsi que sur l’accumulation d’énergie au cours de la propagation des ondes. Ceci conduit à un produit gain-bande passante qui diminue lorsque le niveau sonore croît, tout en préservant, sans y correspondre exactement, le comportement générique de la loi de puissance des oscillateurs critiques. De plus, le modèle a pu reproduire la correspondance entre les réponses à un son pur et au bruit blanc mesurées dans la cochlée.Associer la physique nonlinéaire des oscillateurs critiques à un modèle d'onde progressive de la cochlée permet donc d’expliquer la non-linéarité compressive de l'amplification cochléaire, tout en garantissant que la bande passante de l’amplificateur reste large et varie peu avec le niveau sonore. Mes résultats renforcent le concept d'oscillation critique comme principe physique de l'amplification cochléaire, en soulignant l'importance du couplage des oscillateurs critiques via l’onde progressiveThe nonlinear mechanical properties of the cochlea have been recognized as signatures of active amplification by critical oscillators—active dynamical systems that each operate at a critical point of oscillatory instability called a Hopf bifurcation. Within this framework, each oscillator is tuned to a distinct frequency according to the tonotopic map of the cochlea. When stimulated at its characteristic frequency, a single critical oscillator compresses a millionfold increase of the input level into precisely a hundredfold increase of the response, corresponding to a one-third power-law. This generic property approximates the compressive nonlinearity associated with cochlear amplification at intermediate sound-pressure levels. However, the relevance of critical oscillation as the physical basis of cochlear amplification is challenged by three observations. First, measured level functions can deviate from the generic power-law behavior of a single critical oscillator. Second, a critical oscillator evinces a maximal sensitivity that is inversely proportional to the bandwidth of its frequency tuning, resulting in a “gain-bandwidth” product that does not vary with stimulus level. The cochlea does not work by this rule: the gain-bandwidth product increases with decreasing sound-pressure levels, so that sensitivity to low sound levels is high but tuning is relatively broad. Finally, both in the cochlea and in a critical oscillator, the frequency-tuning curve of the response to single tones of varying frequency but given sound-pressure level can be matched by a tuning curve obtained all at once in response to white noise at an equivalent white-noise level. The equivalent white-noise level required to match the tuning curves increases linearly with the single-tone level in the cochlea, while it increases as a power law of exponent 2/3 for a critical oscillator. The cochlea thus appears to violate fundamental requirements for a principle of cochlear amplification based on critical oscillators, questioning the relevance of criticality for capturing the physical properties of cochlear amplification.In my thesis work, I tackled this question by integrating tonopically-distributed critical oscillators in a traveling-wave model of the cochlea. This nonlinear model, which accounted for two-dimensional hydrodynamics, longitudinal coupling between oscillators and net energy pumping by the oscillators into the wave, was solved numerically in the time domain using the Green function formalism. The model produced, with a single set of parameters and over a broad range of input levels, a family of tuning curves that could match those measured in the cochlea. The success of the description relied on pressure focusing afforded by 2D hydrodynamics as well as accumulation of energy gain as traveling waves progressed from the base toward the characteristic place of maximal response in the modeled cochlea. Specifically, I found that the gain-bandwidth product decreased at increasing levels, while preserving, but not precisely, the generic power-law behavior of critical oscillators. Moreover, the model was in turn able to reproduce the relation between single-tone and white-noise responses observed in the cochlea.Marrying the physics of critical oscillators with a traveling wave model of the cochlea could thus account for the compressive nonlinearity underlying cochlear amplification, while ensuring that the bandwidth of the tuning curves remained relatively broad and varied little with sound level. Altogether, my results strengthen the concept of critical oscillation as the physical basis of nonlinear cochlear amplification, while emphasizing the importance and benefits of coupling tonotopically organized critical oscillators via traveling waves
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Wiser, Justin Allen. "Harmonic Resonance Dynamics of the Periodically Forced Hopf Oscillator." The Ohio State University, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=osu1373380266.
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Barbosa, Pricila da Silva. "Bifurcação de Poincaré-Andronov-Hopf para difeomorfismos do plano." Universidade de São Paulo, 2010. http://www.teses.usp.br/teses/disponiveis/45/45132/tde-08062010-123725/.
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O objetivo principal deste trabalho é apresentar uma exposição detalhada do Teorema de Poincaré-Andronov-Hopf para uma família de transformações do plano. Apresentaremos também uma aplicação a um sistema dinâmico que modela a evolução do preço e excesso de demanda em um mercado constituído por uma única mercadoria.The main purpose of this work is to present a detailed exposition of the Poincaré-Andronov-Hopf Theorem for a family of transformations in the plane. We also present an application to a dynamical system modelling the evolution of the price and the excess demand in a single asset market.
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Ben, Saadi My El Hassan. "Méthodes asymptotiques-numériques pour le calcul de bifurcations de Hopf et de solutions périodiques." Metz, 1995. http://docnum.univ-lorraine.fr/public/UPV-M/Theses/1995/Ben_Saadi.Hassan.SMZ9547.pdf.
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Dans ce travail, nous avons présenté une étude sur les équations différentielles ordinaires admettant des solutions périodiques ou des points de bifurcations de Hopf. Pour cette étude, nous avons appliqué des techniques approximatives dans l'esprit des méthodes asymptotiques-numériques qui n'avaient été appliquées jusqu'à présent qu'en statique. Nous avons commencé notre test sur des équations différentielles conservatives ou dissipatives à un seul degré de liberté. Le domaine de validité d'une représentation en séries entières des solutions périodiques est toujours limité par le rayon de convergence. Grâce aux techniques discutées (approximants de Padé, technique de projection et transformation d'Euler), on a pu augmenter ce domaine de validité à une valeur très élevée. Dans la deuxième partie nous nous sommes intéressés à la détection des points de bifurcation de Hopf par des algorithmes attachés à la méthode asymptotique-numérique. Ces points sont détectés alors au moyen d'un problème linéaire et perturbé dépendant de deux paramètres réels, et qui se prête bien à la résolution par les techniques de développements en séries entières. On introduit un indicateur de bifurcation qui est ensuite calculé par des séries entières de deux variables. Ensuite, nous avons caractérisé les points de bifurcation de Hopf à partir de cet indicateur. On a également montré que l'indicateur est en réalité une fraction rationnelle de ces paramètres. Les séries peuvent donc être remplacées par des approximants de Padé et conduire à la valeur exacte de l'indicateur. On a également montré que des stratégies réduites, c'est-à-dire des stratégies qui utilisent moins de termes dans la série, permettaient aussi de déterminer le point de bifurcation de Hopf. Dans cette thèse, l'efficacité de ces procédures a été testée sur des problèmes à petit nombre de degrés de liberté. Les applications à des problèmes à grand nombre de liberté font l'objet d'autres thèses à MetzIn this work, we have presented a study on the ordinary differential equations which have periodic solutions or Hopf bifurcation points. For this study, we have applied an asymptotic-numerical methods that have been applied up to now only in static. We have started our test on the conservative differential equations or dissipative ones which have one degree of freedom. The domain of validity of the representation by power series is limited by a raduis of convergence. By use of the techniques discuted (approximants of Padé, projection technique and transformation of Euler), we have extended this domain up to a large value. In the second part, we have detected the Hopf bifurcation points by an asymptotic numerical algorithm. So, these points are detected through a perturbed and linear problem which depends on two real parameters. Indeed, we have introduced an Hopf bifurcation index which is expanded firstly into power series of two parameters. Then, we have caracterized the Hopf bifurcation points from this index. Since, we have showed that the index is a rational fraction. So, the series can be replaced by the approximants of Padé which lead to the exact value of the index. We have also showed that the "reduced strategies", i. E, the approximants of Padé which replace the series truncated at inferior orders, permit also to detect the Hopf bifurcation points. The efficiency, of these procedures is tested on the problems with small number of degrees of freedom. The applications on the systems with great number of degrees of freedom are the aim of others thesis in Metz
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Tri, Abdeljalil. "Méthodes asymptotiques numériques pour les fluides visqueux incompressibles et la détection de la bifurcation de Hopf." Metz, 1996. http://docnum.univ-lorraine.fr/public/UPV-M/Theses/1996/Tri.Abdeljalil.SMZ9651.pdf.
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Les méthodes de perturbation sont depuis longtemps un moyen efficace de résoudre certaines classes de problèmes non-linéaires dans divers domaines scientifiques. Ces méthodes sont souvent appliquées dans un cadre purement analytique, en se limitant au calcul de quelques termes seulement. Depuis plusieurs années, nous nous attachons à montrer que le couplage d'une technique de perturbation et d'une méthode d'éléments finis peut conduire à des méthodes numériques extrêmement fiables et robustes pour certaines catégories de problèmes non-linéaires. Dans ce travail, nous appliquons ces techniques pour le calcul des branches de solutions stationnaires des équations de Navier-Stokes. Nous abordons aussi le problème de la détection des bifurcations stationnaires et de la bifurcation de HopfPerturbation methods (asymptotic expansions) are usually considered as powerful methods for solving many kinds of non-linear problems. However, these methods are very often apllied in a purely analytic framework, and the calculation is limited to the first few terms of the series. Since a few years, we have shown that the combination of perturbation techniques and finite element method can lead to a robust numerical method for some categories of non-linear problems. In this thesis, we aplly these techniques to compute branches of stationary solutions of Navier-Stokes equations and to detect stationary and Hopf bifurcation
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Carbone, Vera Lucia. "Existência e bifurcações de soluções periódicas da equação de Wright." Universidade de São Paulo, 1999. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-07022001-135507/.
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Este trabalho é concernente a periodicidade na equação de Wright. Provaremos a existência de soluções periódicas não constantes, explorando o conceito de ejetividade de um teorema de ponto fixo. Além disso, provamos a existência de uma seqüência infinita de Bifurcação de Hopf.This work is concerned with periodicity in the Wright's equation. We prove the existence of nonconstant periodic solutions by exploiting the ejectivity concept in a theorem of fixed point. Furthemore, we prove the existence of an infinite sequence of Hopf Bifurcations.
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Mondraǵon, Palomino Octavio. "A numerical study of the effects of multiplicative noise on a supercritical delay induced Hopf bifurcation in a gene expression model /." Thesis, McGill University, 2006. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=101627.
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In the context of gene expression, we proposed a nonlinear stochastic delay differential equation as a mathematical model to study the effects of extrinsic noise on a delay induced Hopf bifurcation. We envisaged its direct numerical resolution. Following the example of the noisy oscillator, we first solved a linearized version of the equation, close to the Hopf bifurcation. The numerical scheme used is a combination of a standard algorithm to solve a deterministic delay differential equation and a stochastic Euler scheme. From our calculations we verified that the deterministic behaviour is fully recovered. For the stochastic case, we found that our solution is qualitatively accurate, in the sense that the noise induced shift in the critical value a, follows the trend the known analytic results predict. However, our numerical solution systematically overestimates the value of the shift. This is explained because the accuracy in the numerical estimation of the decay rate of a solution towards the stationary state value is a function of the control parameter a. We believe the mismatch between the numerical solution and the analytic results is due to a lack of convergence of our scheme, rather than to lack of accuracy. As our numerical scheme is an hybrid, the convergence problem can be improved, both at the deterministic and at the stochastic parts of the scheme. In this work we left our numerical results on the nonlinear case out, because before proceeding to the investigation of the nonlinear equation, the convergence must be assured in the linear case.
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Bekhoucha, Ferhat. "Dynamique non linéaire des poutres en composite en mouvement de rotation." Thesis, Lorient, 2015. http://www.theses.fr/2015LORIS389.
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Le travail présenté dans ce manuscrit est une contribution à l’étude des vibrations non-linéaires des poutres isotropes et en composite, en mouvement de rotation. Le modèle mathématique utilisé est basé sur la formulation intrinsèque et géométriquement exacte de Hodges, dédiée au traitement des poutres ayant des grands déplacements et de petites déformations. La résolution est faite dans le domaine fréquentiel suite à une discrétisation spatio-temporelle, en utilisant l’approximation de Galerkin et la méthode de l’équilibrage harmonique, avec des conditions aux limites correspondantes aux poutres encastrées-libres. Le systéme dynamique final est traité par des méthodes de continuation : la méthode asymptotique numérique et la méthode pseudo-longueur d’arc. Des algorithmes basés sur ces méthodes de continuation ont été développés et une étude comparative de convergence a été menée. Cette étude a cerné les aspects : statique, analyse modale linéaire, vibrations libres non-linéaires et les vibrations forcées non-linéaires des poutres rotatives. Ces algorithmes de continuations ont été testés pour le calculs des courbes de réponse sur des cas traités dans la littérature. La résonance interne et la stabilité des solutions obtenues sont étudiéesThe work presented in this manuscript is a contribution to the non-linear vibrations of the isotropic beams and composite rotating beams study. The mathematical model used is based on the intrinsic formulation and geometrically exact of Hodges, developped for beams subjected to large displacements and small deformations. The resolution is done in the frequency domain after a spatial-temporal dicretisation, by using the Galerkin approximation and the the harmonic balance method, with boundary conditions corresponding to the clamped-free. The final dynamic system is treated by continuation methods : asymptotic numerical method and the pseudo-arc length method, whose algorithms based on these continuation methods were developed and a convergence study was carried out. This study surround the aspects : statics, linear modal analysis, non-linear free vibrations and the non-linear forced vibrations of the rotating beams. These continuation algorithms were tested for the response curves calculations on cases elaborated in the literature. Internal resonance and the stability of the solutions obtained are studied
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Hong, Christian I. "Mathematical Modeling of Circadian Rhythms in Drosophila melanogaster." Thesis, Virginia Tech, 1999. http://hdl.handle.net/10919/42168.
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Circadian rhythms are periodic physiological cycles that recur about every 24 hours, by means of which organisms integrate their physiology and behavior to the daily cycle of light and temperature imposed by the rotation of the earth. Circadian derives from the Latin word circa "about" and dies "day". Circadian rhythms have three noteworthy properties. They are endogenous, that is, they persist in the absence of external cues (in an environment of constant light intensity, temperature, etc.). Secondly, they are temperature compensated, that is, the nearly 24 hour period of the endogenous oscillator is remarkably independent of ambient temperature. Finally, they are phase shifted by light. The circadian rhythm can be either advanced or delayed by applying a pulse of light in constant darkness. Consequently, the circadian rhythm will synchronize to a periodic light-dark cycle, provided the period of the driving stimulus is not too far from the period of the endogenous rhythm.
A window on the molecular mechanism of 24-hour rhythms was opened by the identification of circadian rhythm mutants and their cognate genes in Drosophila, Neurospora, and now in other organisms. Since Konopka and Benzer first discovered the period mutant in Drosophila in 1971 (Konopka and Benzer, 1971), there have been remarkable developments. Currently, the consensus opinion of molecular geneticists is that the 24-hour period arises from a negative feedback loop controlling the transcription of clock genes. However, a better understanding of this mechanism requires an approach that integrates both mathematical and molecular biology. From the recent discoveries in molecular biology and through a mathematical approach, we propose that the mechanism of circadian rhythm is based upon the combination of both negative and positive feedback.Master of Science
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Nadrowski, Björn. "Propriétés mécaniques de touffes ciliaires actives." Paris 7, 2004. http://www.theses.fr/2004PA077198.
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Silva, Vinicius Barros da. "Bifurcação de Hopf e formas normais : uma nova abordagem para sistemas dinâmicos /." Rio Claro, 2018. http://hdl.handle.net/11449/180496.
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Orientador: Edson Denis LeonelResumo: Este estudo objetiva provar que sistemas dinâmicos de dimensão N, de codimensão um e satisfazendo as condições do teorema da bifurcação de Hopf, podem ser expressos em uma forma analítica simplificada que preserva a topologia do espaço de fases da configuração original, na vizinhança do ponto de equilíbrio. A esta forma simplificada é atribuído o nome de forma normal. Para tanto, foi utilizado a teoria da variedade central, necessária para reduzir a dimensão de sistemas à sua variedade bidimensional, e o teorema das formas normais, utilizando-se como método para determinar a forma simplificada da variedade central associada aos sistemas dinâmicos, atendendo as condições do teorema da bifurcação de Hopf. A partir da análise dos resultados aqui encontrados foi possível construir a prova matemática de que sistemas de dimensão N, atendendo as condições do teorema de Hopf, podem ser reescritos em uma expressão analítica geral e simplificada. Enfim, através deste estudo foi possível resumir todos os resultados aqui obtidos em um teorema geral que, além de reduzir a custosa tarefa de obtenção de formas normais, abrange sistemas N-dimensionais com ocorrência da bifurcação de Hopf.
Abstract: In this work we prove the following: consider a N-dimensional system that is reduced to its center manifold. If it is proved the system satisfies the conditions of Hopf bifurcation theorem, then the original system of differential equations is rewritten in a simpler analytical expression that preserves the phase space topology. This last is also known as the normal form. The center manifold is used to derive a reduced order expression, and the normal form theory is applied to simplify the form of the dynamics on the center manifold. The key results here allow constructing a general mathematical proof for the normal form of N-dimensional systems reduced to its center manifold. In the class of dynamical systems under Hopf bifurcations, the present work reduces the work done to obtain normal forms.
Mestre
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Smith, Todd Blanton. "Variational embedded solitons, and traveling wavetrains generated by generalized Hopf bifurcations, in some NLPDE systems." Doctoral diss., University of Central Florida, 2011. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/5042.
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In this Ph.D. thesis, we study regular and embedded solitons and generalized and degenerate Hopf bifurcations. These two areas of work are seperate and independent from each other. First, variational methods are employed to generate families of both regular and embedded solitary wave solutions for a generalized Pochhammer PDE and a generalized microstructure PDE that are currently of great interest. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the family of the trial functions). Thus, the residual is calculated, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that only the parameter regimes for the existence of solitary waves had previously been analyzed for the microstructure PDE considered here, the results obtained here are both new and timely. Second, we consider generalized and degenerate Hopf bifurcations in three different models: i. a predator-prey model with general predator death rate and prey birth rate terms, ii. a laser-diode system, and iii. traveling-wave solutions of twospecies predator-prey/reaction-diffusion equations with arbitrary nonlinear/reaction terms. For specific choices of the nonlinear terms, the quasi-periodic orbit in the post-bifurcation regime is constructed for each system using the method of multiple scales, and its stability is analyzed via the corresponding normal form obtained by reducing the system down to the center manifold. The resulting predictions for the post-bifurcation dynamics provide an organizing framework for the variety of possible behaviors.; These predictions are verified and supplemented by numerical simulations, including the computation of power spectra, autocorrelation functions, and fractal dimensions as appropriate for the periodic and quasiperiodic attractors, attractors at infinity, as well as bounded chaotic attractors obtained in various cases. The dynamics obtained in the three systems is contrasted and explained on the basis of the bifurcations occurring in each. For instance, while the two predator-prey models yield a variety of behaviors in the post-bifurcation regime, the laser-diode evinces extremely stable quasiperiodic solutions over a wide range of parameters, which is very desirable for robust operation of the system in oscillator mode.ID: 029809927; System requirements: World Wide Web browser and PDF reader.; Mode of access: World Wide Web.; Thesis (Ph.D.)--University of Central Florida, 2011.; Includes bibliographical references (p. 121-129).
Ph.D.
Doctorate
Mathematics
Sciences
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Vieira, Ailton Luiz. "Bifurcação de Hopf em um modelo para a dinâmica do vírus varicela-zoster." Universidade Federal de Viçosa, 2011. http://locus.ufv.br/handle/123456789/4908.
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Previous issue date: 2011-02-21This paper proposes a system of differential equations composed of five ordinary nonlinear equations engaged in a structure based on the SIR model of Kermack and McKendrick 1927, which aims to describe the dynamics of varicella-zoster virus in human populations. Analysis of its equilibrium points we find the emergence of a Hopf bifurcation. Mirrored in article Bifurcation analysis of model for the biological control of Sotomayor et al., through the Hopf analysis of the conditions of non-degeneracy and transversality, we guarantee the appearance of a periodic orbit.
Este trabalho propõe um sistema de equações diferenciais ordinárias composto por cinco equações não lineares acopladas, numa estrutura baseada no modelo SIR de Kermack e Mckendrick 1927, que visa descrever a dinâmica do vírus varicela-zoster na população de humanos. Da análise de seus pontos de equilíbrio verificamos o surgimento de uma bifurcação de Hopf. Espelhados no artigo Bifurcation analysis of a model for biological control de Sotomayor et al., por meio da análise das condições de Hopf, de não degenerescência e de transversalidade, garantimos o aparecimento de uma órbita periódica.
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Boullu, Lois. "Étude d’équations à retard appliquées à la régulation de la production de plaquettes sanguines." Thesis, Lyon, 2018. http://www.theses.fr/2018LYSE1239/document.
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L’objectif de cette thèse est d’étudier, à l’aide de modèles mathématiques, le mécanisme de régulation qui permet au corps de maintenir une quantité optimale de plaquettes sanguines. Le premier chapitre présente le contexte biologique et mathématique. Dans un second chapitre, un modèle pour la mégacaryopoïèse est introduit qui suppose une régulation ponctuelle par le nombre de plaquettes du taux de différentiation des cellules souches vers la lignée mégacaryocytaire et du nombre de plaquettes produites par mégacaryocyte. Nous montrons que la dynamique de ce modèle est régie par une équation différentielle à retard x'(t) = -?x(t)+f(x(t))g(x(t-t)), et nous obtenons ensuite de nouvelles conditions suffisantes pour la stabilité et l’oscillation des solutions de cette équation. Dans le troisième chapitre, nous analysons un second modèle pour la mégacaryopoïèse qui considère cette fois-ci une régulation opérée en continu uniquement via la vitesse de maturation des mégacaryoblastes. L’analyse de stabilité nécessite d’adapter un cadre pré-existant aux cas où le paramètre de bifurcation n’est pas le retard, et permet de montrer que l’augmentation du taux de mort des mégacaryoblastes conduit à l’apparition de solutions périodiques, en accord avec les observations cliniques de la thrombopénie cyclique amégacaryocytaire. Le dernier chapitre est consacré l’analyse de stabilité d’une équation différentielle à deux retards qui apparait notamment dans le cadre de la mégacaryopoïèse lorsque l’on considère que les plaquettes ont une durée de vie limitéeThe object of this thesis is the study, using mathematical models, of the regulation mechanism maintaining an optimal quantity of blood platelets. The first chapter presents the biological and mathematical context of the thesis. In a second chapter, we introduce a model for megakaryopoiesis assuming a regulation by the platelet quantity of both the differentiation rate of stem cells to the platelet cell line and the amount of platelets produced by each megakaryocyte. We show that the dynamic of this model corresponds to a delay differential equation x'(t) = -?x(t) + f(x(t))g(x(t - t)), and we obtain for this equation new sufficient conditions for stability and for the oscillation of solutions. In a third chapter, we analyze a second model for megakaryopoiesis in which the regulation is continuous through the maturation speed of megakaryocyte progenitors. The stability analysis requires to adapt a pre-existing framework to problems where the bifurcation parameter is not the delay, and allows to show that increasing the death rate of megakaryocyte progenitors leads to the onset of periodic solutions, in agreement with clinical observation of amegakaryocytic cyclical thrombocytopenia. The last chapter covers a differential equation with two delays that appears among others in a model of platelet production which considers that platelet death can both age-independent and age-dependent
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Mohan, Anant. "Nonlinear Investigation of the Use of Controllable Primary Suspensions to Improve Hunting in Railway Vehicles." Thesis, Virginia Tech, 2003. http://hdl.handle.net/10919/33740.
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Hunting is a very common instability exhibited by rail vehicles operating at high speeds. The hunting phenomenon is a self excited lateral oscillation that is produced by the forward speed of the vehicle and the wheel-rail interactive forces that result from the conicity of the wheel-rail contours and the friction-creep characteristics of the wheel-rail contact geometry. Hunting can lead to severe ride discomfort and eventual physical damage to wheels and rails.
A comprehensive study of the lateral stability of a single wheelset, a single truck, and the complete rail vehicle has been performed. This study investigates bifurcation phenomenon and limit cycles in rail vehicle dynamics. Sensitivity of the critical hunting velocity to various primary and secondary stiffness and damping parameters has been examined.
This research assumes the rail vehicle to be moving on a smooth, level, and tangential track, and all parts of the rail vehicle to be rigid. Sources of nonlinearities in the rail vehicle model are the nonlinear wheel-rail profile, the friction-creep characteristics of the wheel-rail contact geometry, and the nonlinear vehicle suspension characteristics. This work takes both single-point and two-point wheel-rail contact conditions into account.
The results of the lateral stability study indicate that the critical velocity of the rail vehicle is most sensitive to the primary longitudinal stiffness. A method has been developed to eliminate hunting behavior in rail vehicles by increasing the critical velocity of hunting beyond the operational speed range. This method involves the semi-active control of the primary longitudinal stiffness using the wheelset yaw displacement. This approach is seen to considerably increase the critical hunting velocity.Master of Science
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Daudois, Ludovic. "Prédiction des cycles limites d'oscillations pour une structure tridimensionnelle soumise à un écoulement fluide en présence d'une non linéarité structurale." Châtenay-Malabry, Ecole centrale de Paris, 2004. http://www.theses.fr/2004ECAP0953.
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Pour les structures aéronautiques, la prise en compte de non linéarités structurales de type jeu (par exemple les liaisons aile gouverne) dans les équations aéroélastiques fait apparaître des mouvements oscillatoires appelés cycles limites. La détermination de ces mouvements (amplitudes, fréquences d'oscillations) est nécessaire pour l'étude en fatigue des structures concernées. Dans ce travail sont développés des outils de modélisation et des méthodes numériques permettant d'évaluer le domaine de stabilité d'un système aéroélastique pour une structure non linéaire tridimensionnelle soumise à un écoulement fluide. Cette méthodologie repose sur l'introduction d'une base de Ritz issue du système mécanique, et permet d'une part de réduire la taille du système mécanique, et d'autre part de développer un modèle d'état pour le calcul des forces aérodynamiques dans le domaine temporel. Cette formulation permet d'exprimer le système couplé fluide structure sous la forme d'un système différentiel de dimension réduite du premier ordre. La pression génératrice de l'écoulement est choisie comme paramètre de bifurcation. A partir des points de bifurcation de Hopf, des branches de cycles limites stables ou instables peuvent être ainsi déterminées par des méthodes numériques reposant sur une méthode de continuation de Keller. Pour un système couplé de faible dimension, les différentes méthodes donnent des résultats satisfaisants. Pour le cas d'une structure tridimensionnelle en présence de non linéarités de type jeu, des solutions périodiques et chaotiques sont détectées.
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Liu, Shouzong. "AGE-STRUCTURED PREDATOR-PREY MODELS." OpenSIUC, 2018. https://opensiuc.lib.siu.edu/dissertations/1577.
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In this thesis, we study the population dynamics of predator-prey interactions described by mathematical models with age/stage structures. We first consider fixed development times for predators and prey and develop a stage-structured predator-prey model with Holling type II functional response. The analysis shows that the threshold dynamics holds. That is, the predator-extinction equilibrium is globally stable if the net reproductive number of the predator $\mathcal{R}_0$ is less than $1$, while the predator population persists if $\mathcal{R}_0$ is greater than $1$. Numerical simulations are carried out to demonstrate and extend our theoretical results. A general maturation function for predators is then assumed, and an age-structured predator-prey model with no age structure for prey is formulated. Conditions for the existence and local stabilities of equilibria are obtained. The global stability of the predator-extinction equilibrium is proved by constructing a Lyapunov functional. Finally, we consider a special case of the maturation function discussed before. More specifically, we assume that the development times of predators follow a shifted Gamma distribution and then transfer the previous model into a system of differential-integral equations. We consider the existence and local stabilities of equilibria. Conditions for existence of Hopf bifurcation are given when the shape parameters of Gamma distributions are $1$ and $2$.
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Slimani, Safia. "Système dynamique stochastique de certains modèles proies-prédateurs et applications." Thesis, Normandie, 2018. http://www.theses.fr/2018NORMR123/document.
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Ce travail est consacré à l’étude de la dynamique d’un système proie-prédateur de type Leslie-Gower défini par un système d’équations différentielles ordinaires (EDO) ou d’équations différentielles stochastiques (EDS), ou par des systèmes couplés d’EDO ou d’EDS. L’objectif principal est de faire l’analyse mathématique et la simulation numérique des modèles construits. Cette thèse est divisée en deux parties : La première partie est consacrée à un système proie-prédateur où les proies utilisent un refuge, le modèle est donné par un système d’équations différentielles ordinaires ou d’équations différentielles stochastiques. Le but de cette partie est d’étudier l’impact du refuge ainsi que la perturbation stochastique sur le comportement des solutions du système. Dans la deuxième partie, nous considérons un système proie-prédateur couplé en réseau. Il s’agit d’étudier comment des couplages plus ou moins forts entre plusieurs systèmes affectent l’existence et la position des points d’équilibre, et la stabilité de ces systèmesThis work is devoted to the study of the dynamics of a predator-prey system of Leslie-Gower type defined by a system of ordinary differential equations (EDO) or stochastic differential equations (EDS), or by coupled systems of EDO or EDS. The main objective is to do mathematical analysis and numerical simulation of the models built. This thesis is divided into two parts : The first part is dedicated to a predator-prey system where the prey uses a refuge, the model is given by a system of ordinary differential equations or stochastic differential equations. The purpose of this part is to study the impact of the refuge as well as the stochastic perturbation on the behavior of the solutions of the system. In the second part, we consider a networked predator-prey system. We show that symmetric couplings speed up the convergence to a stationary distribution
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Singh, Neeta. "EPIDEMIOLOGICAL MODELS FOR MUTATING PATHOGENS WITH TEMPORARY IMMUNITY." Doctoral diss., University of Central Florida, 2006. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/3005.
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Significant progress has been made in understanding different scenarios for disease transmissions and behavior of epidemics in recent years. A considerable amount of work has been done in modeling the dynamics of diseases by systems of ordinary differential equations. But there are very few mathematical models that deal with the genetic mutations of a pathogen. In-fact, not much has been done to model the dynamics of mutations of pathogen explaining its effort to escape the host's immune defense system after it has infected the host. In this dissertation we develop an SIR model with variable infection age for the transmission of a pathogen that can mutate in the host to produce a second infectious mutant strain. We assume that there is a period of temporary immunity in the model. A temporary immunity period along with variable infection age leads to an integro-differential-difference model. Previous efforts on incorporating delays in epidemic models have mainly concentrated on inclusion of latency periods (this assumes that the force of infection at a present time is determined by the number of infectives in the past). We begin with reviewing some basic models. These basic models are the building blocks for the later, more detailed models. Next we consider the model for mutation of pathogen and discuss its implications. Finally, we improve this model for mutation of pathogen by incorporating delay induced by temporary immunity. We examine the influence of delay as we establish the existence, and derive the explicit forms of disease-free, boundary and endemic equilibriums. We will also investigate the local stability of each of these equilibriums. The possibility of Hopf bifurcation using delay as the bifurcation parameter is studied using both analytical and numerical solutions.Ph.D.
Department of Mathematics
Arts and Sciences
Mathematics
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Júnior, Josaphat Ricardo Ribeiro Gouveia. "Bifurcações da região de estabilidade induzidas por bifurcações locais do tipo Hopf." Universidade de São Paulo, 2015. http://www.teses.usp.br/teses/disponiveis/18/18154/tde-02072015-142327/.
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Pontos de equilíbrio assintoticamente estáveis de sistemas dinâmicos não lineares geralmente não são globalmente estáveis. Na maioria dos casos, há um subconjunto de condições iniciais, chamada região de estabilidade (ou área de atração), cujas trajetórias tendem ao ponto de equilíbrio quando o tempo tende ao infinito. Devido à importância das regiões de estabilidade em aplicações, e motivado principalmente pelo problema de analise de estabilidade transitória em sistemas elétricos de potência, uma caracterização completa da fronteira da região de estabilidade foi desenvolvida. Esta caracterização foi desenvolvida sob a suposição de que o sistema dinâmico é bem conhecido e que os parâmetros de seu modelo são constantes. Na prática, variações de parâmetros ocorrem e bifurcações desta podem ocorrer. Nesta tese, desenvolveremos uma caracterização completa da fronteira da região de estabilidade de sistemas dinâmicos autônomos não lineares admitindo a existência de pontos de equilíbrio não hiperbólicos do tipo Hopf na fronteira da região de estabilidade. Sob certas condições de transversalidade, apresentaremos uma caracterização completa da fronteira da região de estabilidade admitindo tanto a presença de pontos de equilíbrio não hiperbólicos do tipo Hopf como também a existência de órbitas periódicas na fronteira. Ofereceremos também uma caracterização da fronteira da região de estabilidade fraca do ponto de equilíbrio não hiperbólico Hopf supercrítico do tipo zero e uma caracterização topológica da sua região de atração. Além disso, exibiremos resultados relativos ao comportamento da região de estabilidade de um ponto de equilíbrio assintoticamente estável e da sua fronteira na vizinhança do valor crítico de bifurcação do tipo Hopf.Asymptotically stable equilibrium points of nonlinear dynamical systems are generally not globally stable. In most cases, there is a subset of initial conditions, called stability region (or attraction area), in which trajectories tend to the equilibrium point when time approaches innity. Due to the importance of stability regions in applications, and mainly motivated by the problem of transient stability analysis in electric power systems, a complete characterization of the boundary of the stability region was developed. This characterization was developed under the assumption that the dynamic system is well known and the parameters of its model are constant. In practice, parameter variations happen and bifurcations may occur. In this thesis, we will develop a complete characterization of the boundary of the stability region of autonomous nonlinear dynamical systems admitting the existence of non-hyperbolic equilibrium points of the type Hopf on the boundary of the stability region. Under certain transversality conditions, we present a complete characterization of the boundary of the stability region admitting the presence of both non-hyperbolic equilibrium points of the type Hopf and periodic orbits on the boundary. Also a complete characterization of the boundary of the region of weak stability of a supercritical Hopf non-hyperbolic equilibrium point of the type zero and a topological characterization of its region of attraction is developed. Furthermore, the behavior of the stability region of an asymptotically stable equilibrium point and its boundary in the neighborhood of a critical value of bifurcation of the type Hopf is studied.
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Adimy, Mostafa. "Perturbation par dualité, interprétation par la théorie des semi-groupes intégrés : application à l'étude du problème de bifurcation de Hopf dans le cadre des équations à retard." Pau, 1991. http://www.theses.fr/1990PAUUA001.
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Bekkal, Brikci Fadia. "Modélisation du cycle cellulaire et couplage avec la dynamique de population cellulaire." Paris 6, 2005. http://www.theses.fr/2005PA066042.
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Fritz, Guillaume. "Etude des phénomènes de crissement pour les freins automobiles : Modélisation non-linéaire et conception robuste." Ecully, Ecole centrale de Lyon, 2007. http://bibli.ec-lyon.fr/exl-doc/TH_T2091_gfritz.pdf.
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Cette thèse s’articule autour de deux thèmes principaux : la compréhension fine et l’analyse robuste du crissement de frein. Afin de prédire le crissement, une méthodologie linéaire basée sur un modèle éléments finis représentatif d’un frein réel est proposée dans un premier temps. Ce modèle permet de mettre en évidence le concept de coalescence de modes et de le généraliser afin de considérer le crissement comme un phénomène multiparamétrique de couplage de modes. Une étude originale de l’effet de l’amortissement sur la stabilité du système est également réalisée. Dans un second temps, une méthodologie non-linéaire incluant un calcul statique non-linéaire, une linéarisation du problème et une analyse aux valeurs propres complexes est adoptée. L’influence de la position d’équilibre statique sur le comportement dynamique du système est expliquée. Des courbes de coalescence présentant des discontinuités sont mises en évidence. Ces discontinuités dues aux changements de position d’équilibre statique expliquent la nature fugace du crissement lorsque le système est faiblement chargé. Pour concevoir au mieux un système de freinage, il est important de prendre en compte la problématique du crissement au plus tôt dans le processus de conception. Un modèle de conception est proposé et est capable de représenter à moindre coût le comportement nominal du système ainsi que sa variabilité aux paramètres. Ce modèle permet de hiérarchiser les paramètres dimensionnants en fonction de leur impact sur le crissement. Afin d’évaluer les performances du système en prenant en compte la variabilité aux conditions d’utilisation et d’environnement, une démarche nommée matrix test numérique est proposée et permet de balayer toutes les conditions dans lesquelles le système est susceptible d’être utilisé. Une démarche de conception robuste, qui s’appuie sur les matrix tests numériques, est enfin proposée afin de trouver une solution plus performante et plus robuste que la solution initiale vis à vis des différentes dispersions. Cette approche aboutit à une solution qui présente un risque de crissement quatre fois inférieur à la solution initialeThis study deals with advanced understanding and robust analysis of brake squeal. First, a linear methodology based on the finite element model of an actual brake corner is proposed. This model points out the mode lock-in mechanism, which may be generalized to consider squeal as a multiparametric mode coupling phenomenon. An original study dealing with the effect of damping on the system stability has been conducted. Then, a non-linear methodology including a non-linear static step, a linearization process and a complex eigenvalue analysis is applied. The effect of static position on dynamic behaviour is explained and coalescence curves featuring discontinuities are shown. These discontinuities, which are induced by changes in static position, account for the fugitive nature of squeal when the brake is weakly loaded. It is important to consider squeal upstream in the development process to improve braking systems. A cost effective design model, which is able to reproduce with accuracy the nominal behaviour of the system and its variability to parameters, has been proposed. Thanks to this model, major parameters are ranked with respect to their effet on squeal. In order to assess the system performance considering operative and environment parameters, a numerical matrix test methodology has been proposed and sweeps all the conditions the brake may face. A robust design approach based on numerical matrix tests has been undertaken to find a more competitive and more robust solution
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Lemée, Thomas. "Shear-flow instabilities in closed flow." Thesis, Paris 11, 2013. http://www.theses.fr/2013PA112038.
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Cette étude se concentre sur la compréhension de la physique des instabilités dans différents écoulements de cisaillement, particulièrement la cavité entraînée et la cavité thermocapillaire, où l'écoulement d'un fluide incompressible est assuré soit par le mouvement d’une ou plusieurs parois, soit par des contraintes d’origine thermique.Un code spectral a été validé sur le cas très étudié de la cavité entrainée par une paroi mobile. Il est démontré dans ce cas que l'écoulement transit d'un régime stationnaire à un instationnaire au-delà d'une valeur critique du nombre de Reynolds. Ce travail est le premier à donner une interprétation physique de l'évolution non monotonique du nombre de Reynolds critique en fonction du facteur d'aspect. Lorsque le fluide est entraîné par deux parois mobiles, la cavité entraînée possède un plan de symétrie particulièrement sensible. Des solutions asymétriques peuvent être observés en plus de la solution symétrique au-dessus d'une certaine valeur du nombre de Reynolds. La transition oscillatoire entre la solution symétrique et les solutions asymétriques est expliquée physiquement par les forces en compétition. Dans le cas asymétrique, l'évolution de la topologie permet à l'écoulement de rester stationnaire avec l'augmentation du nombre de Reynolds. Lorsque l'équilibre est perdu une instabilité se manifeste par l'apparition d'un régime oscillatoire dans l'écoulement asymétrique.Dans une cavité thermocapillaire rectangulaire avec une surface libre, Smith et Davis prévoient deux types d'instabilités convectives thermiques: des rouleaux longitudinaux stationnaires et des ondes hydrothermales instationnaires. L'apparition de ses instabilités a été mis en évidence à plusieurs reprises expérimentalement et numériquement. Alors que les applications impliquent souvent plus d'une surface libre, il semble qu'il y ait peu de connaissances sur l'écoulement thermocapillaire entraînée avec deux surfaces libres. Un film liquide libre soumis à des contraintes thermocapillaires possède un plan de symétrie particulier comme dans le cas de la cavité entrainée par deux parois mobiles. Une étude de stabilité linéaire avec deux profils de vitesse pour le film liquide libre est présentée avec différents nombres de Prandtl. Au-delà d'un nombre de Marangoni critique, il est découvert que ces états de base sont sensibles à quatre types d'instabilités convectives thermiques qui peuvent conserver ou briser la symétrie du système. Les mécanismes qui permettent de prédire ces instabilités sont également découverts et interpréter en fonction de la valeur du nombre de Prandtl du fluide. La comparaison avec les travaux de Smith et Davis est faite. Une simulation numérique directe permet de valider les résultats obtenus avec l'étude de stabilité de linéaireThis study focuses on the understanding of the physics of different instabilities in driven cavities, specifically the lid-driven cavity and the thermocapillarity driven cavity where flow in an incompressible fluid is driven either due to one or many moving walls or due to surface stresses that appear from surface tension gradients caused by thermal gradients. A spectral code is benchmarked on the well-studied case of the lid-cavity driven by one moving wall. In this case, It is shown that the flow transit form a steady regime to unsteady regime beyond a critical value of the Reynolds number. This work is the first to give a physical interpretation of the non-monotonic evolution of the critical Reynolds number versus the size of the cavity. When the fluid is driven by two facing walls moving in the same direction, the cavity possesses a plane of symmetry particularly sensitive. Thus, asymmetrical solutions can be observed in addition to the symmetrical solution above a certain value of the Reynolds number. The oscillatory transition between the symmetric solution and asymmetric solutions is explained physically by the forces in competition. In the asymmetric case, the change of the topology allows the flow to remain steady with increasing the Reynolds number. When the equilibrium is lost, an instability manifests by the appearance of an oscillatory regime in the asymmetric flow. In a rectangular cavity thermocapillary with a free surface, Smith and Davis found two types of thermal convective instabilities: steady longitudinal rolls and unsteady hydrothermal waves. The appearance of its instability has been highlighted repeatedly experimentally and numerically. While applications often involve more than a free surface, it seems that there is little knowledge about the thermocapillary driven flow with two free surfaces. A free liquid film possesses a particular plane of symmetry as in the case of the two-sided lid-driven cavity. A linear stability analysis for the free liquid film with two velocity profiles is presented with various Prandtl numbers. Beyond a critical Marangoni number, it is observed that these basic states are sensitive to four types of thermal convective instabilities, which can keep or break the symmetry of the system. Mechanisms that predict these instabilities are discovered and interpreted according to the value of the Prandtl number of the fluid. Comparison with the work of Smith and Davis is made. A direct numerical simulation is done to validate the results obtained with the linear stability analysis
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Caton, François. "Écoulement de Taylor-Couette stratifié : régimes, bifurcations et transport." Université Joseph Fourier (Grenoble ; 1971-2015), 1998. http://www.theses.fr/1998GRE10071.
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Cette these est consacree a l'etude experimentale et theorique de l'ecoulement de taylor-couette en presence d'une stratification axiale stable. Une analyse de stabilite lineaire predit le caractere oscillatoire du seuil d'instabilite. Les visualisations montrent que le premier regime instable est un regime d'ondes stationnaires axisymetriques. Un modele theorique base sur l'existence d'ondes internes confinees est en bon accord avec les mesures des fluctuations temporelles de densite dans ce regime. Le second regime est un regime de vortex non-axisymetriques dont la taille verticale est diminuee par la stratification, les vortex etant presents par paires a une meme hauteur et tournant a la vitesse moyenne dans l'entrefer. La transition entre le regime des ondes et celui de vortex presente de l'hysteresis. Un diagramme des bifurcations du systeme pour les deux premiers regimes est alors propose. Dans le regime suivant, nous avons note la coexistence de structures stationnaires et de defauts. Enfin nous nous sommes interesses a un regime presentant des cellules ressemblant a des vortex de taylor, mais couples par paires. Ces deux regimes presentent des spectres faiblement turbulents. Enfin, une etude du transport d'un traceur passif dans les differents regimes a ete realisee, mettant en evidence l'existence de regimes de diffusion anormale au temps courts. Au temps longs, une diffusion normale est obtenue, le coefficient de diffusion effectif augmentant de facon quadratique avec le taux de rotation du cylindre interieur.