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Relevant bibliographies by topics / Hopf bifurcation

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Academic literature on the topic 'Hopf bifurcation'

Author: Grafiati

Published: 4 June 2021

Last updated: 15 March 2025

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Journal articles on the topic "Hopf bifurcation"

1

Xu, Chaoqun, and Sanling Yuan. "Spatial Periodic Solutions in a Delayed Diffusive Predator–Prey Model with Herd Behavior." International Journal of Bifurcation and Chaos 25, no. 11 (October 2015): 1550155. http://dx.doi.org/10.1142/s0218127415501552.

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A delayed diffusive predator–prey model with herd behavior subject to Neumann boundary conditions is studied both theoretically and numerically. Applying Hopf bifurcation analysis, we obtain the critical conditions under which the model generates spatially nonhomogeneous bifurcating periodic solutions. It is shown that the spatially homogeneous Hopf bifurcations always exist and that the spatially nonhomogeneous Hopf bifurcations will arise when the diffusion coefficients are suitably small. The explicit formulae for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by employing the normal form theory and center manifold theorems for partial functional differential equations.

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SONG, YONGLI, JUNJIE WEI, and MAOAN HAN. "LOCAL AND GLOBAL HOPF BIFURCATION IN A DELAYED HEMATOPOIESIS MODEL." International Journal of Bifurcation and Chaos 14, no. 11 (November 2004): 3909–19. http://dx.doi.org/10.1142/s0218127404011697.

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In this paper, we consider the following nonlinear differential equation [Formula: see text] We first consider the existence of local Hopf bifurcations, and then derive the explicit formulas which determine the stability, direction and other properties of bifurcating periodic solutions, using the normal form theory and center manifold reduction. Further, particular attention is focused on the existence of the global Hopf bifurcation. By using the global Hopf bifurcation theory due to Wu [1998], we show that the local Hopf bifurcation of (1) implies the global Hopf bifurcation after the second critical value of the delay τ. Finally, numerical simulation results are given to support the theoretical predictions.

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Yan, Xiang-Ping, and Wan-Tong Li. "Global existence of periodic solutions in a simplified four-neuron BAM neural network model with multiple delays." Discrete Dynamics in Nature and Society 2006 (2006): 1–18. http://dx.doi.org/10.1155/ddns/2006/57254.

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We consider a simplified bidirectional associated memory (BAM) neural network model with four neurons and multiple time delays. The global existence of periodic solutions bifurcating from Hopf bifurcations is investigated by applying the global Hopf bifurcation theorem due to Wu and Bendixson's criterion for high-dimensional ordinary differential equations due to Li and Muldowney. It is shown that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of the sum of two delays. Numerical simulations supporting the theoretical analysis are also included.

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Xu, Changjin, Maoxin Liao, and Xiaofei He. "Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays." International Journal of Applied Mathematics and Computer Science 21, no. 1 (March 1, 2011): 97–107. http://dx.doi.org/10.2478/v10006-011-0007-0.

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Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays In this paper, a two-species Lotka-Volterra predator-prey model with two delays is considered. By analyzing the associated characteristic transcendental equation, the linear stability of the positive equilibrium is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and direction of Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using normal form theory and center manifold theory. Some numerical simulations for supporting the theoretical results are also included.

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Zhai, Yanhui, Ying Xiong, Xiaona Ma, and Haiyun Bai. "Global Hopf Bifurcation Analysis for an Avian Influenza Virus Propagation Model with Nonlinear Incidence Rate and Delay." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/242410.

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The paper investigated an avian influenza virus propagation model with nonlinear incidence rate and delay based on SIR epidemic model. We regard delay as bifurcating parameter to study the dynamical behaviors. At first, local asymptotical stability and existence of Hopf bifurcation are studied; Hopf bifurcation occurs when time delay passes through a sequence of critical values. An explicit algorithm for determining the direction of the Hopf bifurcations and stability of the bifurcation periodic solutions is derived by applying the normal form theory and center manifold theorem. What is more, the global existence of periodic solutions is established by using a global Hopf bifurcation result.

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Zhang, Huayong, Ju Kang, Tousheng Huang, Xuebing Cong, Shengnan Ma, and Hai Huang. "Hopf Bifurcation, Hopf-Hopf Bifurcation, and Period-Doubling Bifurcation in a Four-Species Food Web." Mathematical Problems in Engineering 2018 (September 27, 2018): 1–21. http://dx.doi.org/10.1155/2018/8394651.

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Complex dynamics of a four-species food web with two preys, one middle predator, and one top predator are investigated. Via the method of Jacobian matrix, the stability of coexisting equilibrium for all populations is determined. Based on this equilibrium, three bifurcations, i.e., Hopf bifurcation, Hopf-Hopf bifurcation, and period-doubling bifurcation, are analyzed by center manifold theorem, bifurcation theorem, and numerical simulations. We reveal that, influenced by the three bifurcations, the food web can exhibit very complex dynamical behaviors, including limit cycles, quasiperiodic behaviors, chaotic attractors, route to chaos, period-doubling cascade in orbits of period 2, 4, and 8 and period 3, 6, and 12, periodic windows, intermittent period, and chaos crisis. However, the complex dynamics may disappear with the extinction of one of the four populations, which may also lead to collapse of the food web. It suggests that the dynamical complexity and food web stability are determined by the food web structure and existing populations.

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Cai, Yongli, Zhanji Gui, Xuebing Zhang, Hongbo Shi, and Weiming Wang. "Bifurcations and Pattern Formation in a Predator–Prey Model." International Journal of Bifurcation and Chaos 28, no. 11 (October 2018): 1850140. http://dx.doi.org/10.1142/s0218127418501407.

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In this paper, we investigate the spatiotemporal dynamics of a Leslie–Gower predator–prey model incorporating a prey refuge subject to the Neumann boundary conditions. We mainly consider Hopf bifurcation and steady-state bifurcation which bifurcate from the constant positive steady-state of the model. In the case of Hopf bifurcation, by the center manifold theory and the normal form method, we establish the bifurcation direction and stability of bifurcating periodic solutions; in the case of steady-state bifurcation, by the local and global bifurcation theories, we prove the existence of the steady-state bifurcation, and find that there are two typical bifurcations, Turing bifurcation and Turing–Hopf bifurcation. Via numerical simulations, we find that the model exhibits not only stationary Turing pattern induced by diffusion which is dependent on space and independent of time, but also temporal periodic pattern induced by Hopf bifurcation which is dependent on time and independent of space, and spatiotemporal pattern induced by Turing–Hopf bifurcation which is dependent on both time and space. These results may enrich the pattern formation in the predator–prey model.

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Xu, Changjin. "Bifurcation Analysis for a Predator-Prey Model with Time Delay and Delay-Dependent Parameters." Abstract and Applied Analysis 2012 (2012): 1–20. http://dx.doi.org/10.1155/2012/264870.

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A class of stage-structured predator-prey model with time delay and delay-dependent parameters is considered. Its linear stability is investigated and Hopf bifurcation is demonstrated. Using normal form theory and center manifold theory, some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained. Finally, numerical simulations are performed to verify the analytical results.

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Niu, Ben, Yuxiao Guo, and Yanfei Du. "Hopf Bifurcation Induced by Delay Effect in a Diffusive Tumor-Immune System." International Journal of Bifurcation and Chaos 28, no. 11 (October 2018): 1850136. http://dx.doi.org/10.1142/s0218127418501365.

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Tumor-immune interaction plays an important role in the tumor treatment. We analyze the stability of steady states in a diffusive tumor-immune model with response and proliferation delay [Formula: see text] of immune system where the immune cell has a probability [Formula: see text] in killing tumor cells. We find increasing time delay [Formula: see text] destabilizes the positive steady state and induces Hopf bifurcations. The criticality of Hopf bifurcation is investigated by deriving normal forms on the center manifold, then the direction of bifurcation and stability of bifurcating periodic solutions are determined. Using a group of parameters to simulate the system, stable periodic solutions are found near the Hopf bifurcation. The effect of killing probability [Formula: see text] on Hopf bifurcation values is also discussed.

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GUO, SHANGJIANG, and YUAN YUAN. "PATTERN FORMATION IN A RING NETWORK WITH DELAY." Mathematical Models and Methods in Applied Sciences 19, no. 10 (October 2009): 1797–852. http://dx.doi.org/10.1142/s0218202509004005.

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We consider a ring network of three identical neurons with delayed feedback. Regarding the coupling coefficients as bifurcation parameters, we obtain codimension one bifurcation (including a Fold bifurcation and Hopf bifurcation) and codimension two bifurcations (including Fold–Fold bifurcations, Fold–Hopf bifurcations and Hopf–Hopf bifurcations). We also give concrete formulas for the normal form coefficients derived via the center manifold reduction that provide detailed information about the bifurcation and stability of various bifurcated solutions. In particular, we obtain stable or unstable equilibria, periodic solutions, and quasi-periodic solutions.

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Dissertations / Theses on the topic "Hopf bifurcation"

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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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Books on the topic "Hopf bifurcation"

1

van der Meer, Jan-Cees. The Hamiltonian Hopf Bifurcation. Berlin, Heidelberg: Springer Berlin Heidelberg, 1985. http://dx.doi.org/10.1007/bfb0080357.

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Goodrich, John W. Hopf bifurcation in the driven cavity. [Washington, D.C.]: NASA, 1989.

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E, Gustafson Karl, Halasi Kadosa, United States. National Aeronautics and Space Administration., and Lewis Research Center. Institute for Computational Mechanics in Propulsion., eds. Hopf bifurcation in the driven cavity. [Washington, D.C.]: NASA, 1989.

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G, Chen, ed. Hopf bifurcation analysis: A frequency domain approach. Singapore: World Scientific, 1996.

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Dellnitz, Michael. Hopf-Verzweigung in Systemen mit Symmetrie und deren numerische Behandlung. Ammersbek bei Hamburg: Verlag an der Lottbek, 1989.

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Fiedler, Bernold. Global bifurcation of periodic solutions with symmetry. Berlin: Springer-Verlag, 1988.

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Magal, Pierre. Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models. Providence, R.I: American Mathematical Society, 2009.

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1963-, Ruan Shigui, ed. Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models. Providence, R.I: American Mathematical Society, 2009.

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Oral, Zeki Okan. Hopf bifurcations in path control of marine vehicles. Monterey, Calif: Naval Postgraduate School, 1993.

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Jan Cornelis van der Meer. Hamiltonian Hopf Bifurcation. Springer London, Limited, 2006.

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Book chapters on the topic "Hopf bifurcation"

1

Smith, Hal. "Hopf Bifurcation." In An Introduction to Delay Differential Equations with Applications to the Life Sciences, 87–118. New York, NY: Springer New York, 2010. http://dx.doi.org/10.1007/978-1-4419-7646-8_6.

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Diekmann, Odo, Sjoerd M. Verduyn Lunel, Stephan A. van Gils, and Hanns-Otto Walther. "Hopf bifurcation." In Applied Mathematical Sciences, 287–301. New York, NY: Springer New York, 1995. http://dx.doi.org/10.1007/978-1-4612-4206-2_11.

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Sun, Xiaojuan. "Hopf Bifurcation." In Encyclopedia of Systems Biology, 903. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4419-9863-7_531.

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Chen, Yushu, and Andrew Y. T. Leung. "Hopf Bifurcation." In Bifurcation and Chaos in Engineering, 176–229. London: Springer London, 1998. http://dx.doi.org/10.1007/978-1-4471-1575-5_6.

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Misbah, Chaouqi. "Hopf Bifurcation." In Complex Dynamics and Morphogenesis, 115–38. Dordrecht: Springer Netherlands, 2016. http://dx.doi.org/10.1007/978-94-024-1020-4_5.

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Liebscher, Stefan. "Zero-Hopf Bifurcation." In Bifurcation without Parameters, 103–8. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-10777-6_11.

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Liebscher, Stefan. "Double-Hopf Bifurcation." In Bifurcation without Parameters, 109–13. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-10777-6_12.

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Misbah, Chaouqi. "Bifurcation de Hopf." In Dynamiques complexes et morphogenèse, 109–28. Paris: Springer Paris, 2011. http://dx.doi.org/10.1007/978-2-8178-0194-0_5.

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Golubitsky, Martin, and David G. Schaeffer. "The Hopf Bifurcation." In Applied Mathematical Sciences, 337–96. New York, NY: Springer New York, 1985. http://dx.doi.org/10.1007/978-1-4612-5034-0_8.

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Maruyama, Toru. "Hopf Bifurcation Theorem." In Monographs in Mathematical Economics, 301–45. Singapore: Springer Singapore, 2018. http://dx.doi.org/10.1007/978-981-13-2730-8_11.

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Conference papers on the topic "Hopf bifurcation"

1

Lin, Guojian, Balakumar Balachandran, and Eyad H. Abed. "Bifurcation Behavior of a Supercavitating Vehicle." In ASME 2006 International Mechanical Engineering Congress and Exposition. ASMEDC, 2006. http://dx.doi.org/10.1115/imece2006-14052.

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In this effort, a numerical study of the bifurcation behavior of a supercavitating vehicle is conducted. The nonsmoothness of this system is due to the planing force acting on the vehicle. With a focus on dive-plane dynamics, bifurcations with respect to a quasi-static variation of the cavitation number are studied. The system is found to exhibit rich and complex dynamics including nonsmooth bifurcations such as the grazing bifurcation and smooth bifurcations such as Hopf bifurcations, cyclic-fold bifurcations, and period-doubling bifurcations, chaotic attractors, transient chaotic motions, and crises. The tailslap phenomenon of the supercavitating vehicle is identified as a consequence of the Hopf bifurcation followed by a grazing event. It is shown that the occurrence of these bifurcations can be delayed or triggered earlier by using dynamic linear feedback control aided by washout filters.

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Ikeda, Takashi. "Bifurcation Phenomena Caused by Two Nonlinear Dynamic Absorbers." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-34714.

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The characteristics of two nonlinear vibration absorbers simultaneously attached to structures under harmonic excitation are investigated. The frequency response curves are theoretically determined using van der Pol’s method. It is found from the theoretical analysis that pitchfork bifurcations may appear on a part of the response curves which are stable in a system with one nonlinear dynamic absorber. Three steady-state solutions with different amplitudes appear just after the pitchfork bifurcation. After that, Hopf bifurcations may occur depending on the values of the system parameters, and amplitude- and phase-modulated motion including a chaotic vibration appears after the Hopf bifurcation. Lyapunov exponents are numerically calculated to prove the occurrence of a chaotic vibration. In addition, it is also found that only Hopf bifurcations, not pitchfork bifurcations, can occur even when the linear and nonlinear dynamic absorbers are combined.

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Rand, Richard, Albert Barcilon, and Tina Morrison. "Parametric Resonance of Hopf Bifurcation." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84016.

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We investigate the dynamics of a system consisting of a simple, harmonic oscillator with small nonlinearity, small damping and small parametric forcing in the neighborhood of 2:1 resonance. We assume that the unforced system exhibits the birth of a stable limit cycle as the damping changes sign from positive to negative (a supercritical Hopf bifurcation). Using perturbation methods and numerical integration, we investigate the changes which occur in long-time behavior as the damping parameter is varied. We show that for large positive damping, the origin is stable, whereas for large negative damping a quasiperiodic behavior occurs. These two steady states are connected by a complicated series of bifurcations which occur as the damping is varied.

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Cadou, Jean-Marc, Yann Guevel, and Gregory Girault. "Stability Analysis of 2D Flows by Numerical Tools Based on the Asymptotic Numerical Method: Application to the One and Two Sides Lid Driven Cavity." In ASME 2012 11th Biennial Conference on Engineering Systems Design and Analysis. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/esda2012-82446.

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This work deals with the complete stability analysis in the case of two dimensional fluid flows searching for steady and Hopf bifurcations. The stability analysis starts with the computation of steady bifurcations. It is realized by first considering the monitoring of an indicator which is a scalar function. The indicator is computed via a perturbation method: the Asymptotic Numerical Method. Steady bifurcation point corresponds to the zero of this indicator. From this singular point, all the steady bifurcated branches are computed by using the perturbation method. Then the stability analysis is pursued with the computation of Hopf bifurcations by a hybrid method. This latter consists in coupling a continuation method with a direct Newton solver. The continuation method allows the bifurcation indicator to be determined using alternating reduced order and full size steps resolution. The quantities coming from the bifurcation indicator are used as initial approximations for the direct method iterations. Then an augmented system whose solutions are Hopf bifurcation is solved by a direct method of Newton kind. The examples of the one side and two sides lid driven cavities show the reliability and the efficiency of the proposed numerical tools.

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Wang, Yuefang, Lefeng Lu¨, and Yingxi Liu. "On Multiple Hopf Bifurcations of Airflow Excited Vibration of a Translating String." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-34451.

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This paper presents the stability and bifurcation of transverse motion of translating strings excited by a steady wind flowfield. The stability of the equilibrium configuration is presented for loss of stability and generation of limit cycles via the Hopf bifurcation. It is demonstrated that there are single, double and quadruple Hopf bifurcations in the parametric space that lead to the limit cycle motion. The method of Incremental Harmonic Balance is used to solve the limit cycle response of which the stability is determined by computation of the Floquet multipliers. For the forced vibration, it is pointed out that the periodic and quasi-periodic motions exist as parameters are changed. The quench frequency and the Neimark-Sacker (NS) bifurcation and flip bifurcation are obtained. The continuity software MATCONT is adopted and the Resonance 1:1, 1:3 and 1:4 as well as NS to NS bifurcations are presented. The bifurcation behavior reveals the complexity of the string’s motion response induce by aerodynamic excitations.

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Sterpu, Mihaela. "Bifurcation in coupled Hopf oscillators." In MATHEMATICAL ANALYSIS AND APPLICATIONS: International Conference on Mathematical Analysis and Applications. AIP, 2006. http://dx.doi.org/10.1063/1.2205043.

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Bajaj, Anil K., Joseph M. Johnson, and Seo Il Chang. "Amplitude Dynamics of an Autoparametric Two Degree-of-Freedom System." In ASME 1991 Design Technical Conferences. American Society of Mechanical Engineers, 1991. http://dx.doi.org/10.1115/detc1991-0326.

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Abstract Forced, weakly nonlinear oscillations of a two degree-of-freedom autoparametric system are studied for resonant excitations. The method of averaging is used to obtain first order approximations to the response of the system. In the subharmonic case of internal and external resonance, where the external excitation is in the neighborhood of the higher natural frequency, a complete bifurcation analysis of the averaged equations is undertaken. The “locked pendulum” mode of response bifurcates to coupled-mode motion for some excitation frequencies and forcing amplitudes. The coupled-mode response can undergo Hopf bifurcation to limit cycle motions, though it requires mistuning from the exact internal resonance condition. The Hopf bifurcation sets are constructed and dynamic steady solutions of the amplitude or averaged equations are investigated using software packages AUTO and KAOS. It is shown that both super- and sub-critical Hopf bifurcations arise and the limit cycles quickly undergo period-doubling bifurcations to chaos. These imply chaotic amplitude modulated motions for the system.

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Yu, Pei. "A General Formula for Controlling Hopf Bifurcation." In ASME 2002 International Mechanical Engineering Congress and Exposition. ASMEDC, 2002. http://dx.doi.org/10.1115/imece2002-32397.

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An explicit general formula is proposed for controlling Hopf bifurcation using state feedback. This method can be used to either delay (or even eliminate) an existing Hopf bifurcation or change a subcritical Hopf bifurcation to supercritical. The Lorenz system is used to illustrate the application of the formula.

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Marquez, Richard. "Hopf bifurcation in TCP/adaptive RED." In 2007 46th IEEE Conference on Decision and Control. IEEE, 2007. http://dx.doi.org/10.1109/cdc.2007.4434898.

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Verduzco, F., and J. Alvarez. "Hopf bifurcation control for affine systems." In Proceedings of the 2004 American Control Conference. IEEE, 2004. http://dx.doi.org/10.23919/acc.2004.1386712.

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