Academic literature on the topic 'Diagrammes de bifurcation'
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Journal articles on the topic "Diagrammes de bifurcation"
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WANG, HUAILEI, and HAIYAN HU. "BIFURCATION ANALYSIS OF A DELAYED DYNAMIC SYSTEM VIA METHOD OF MULTIPLE SCALES AND SHOOTING TECHNIQUE." International Journal of Bifurcation and Chaos 15, no. 02 (February 2005): 425–50. http://dx.doi.org/10.1142/s0218127405012326.
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This paper presents a detailed study on the bifurcation of a controlled Duffing oscillator with a time delay involved in the feedback loop. The first objective is to determine the bifurcating periodic motions and to obtain the global diagrams of local bifurcations of periodic motions with respect to time delay. In order to determine the bifurcation point, an analysis on the stability switches of the trivial equilibrium is first performed for all possible parametric combinations. Then, by means of the method of multiple scales, an analysis on the local bifurcation of periodic motions is given. The static bifurcation diagrams on the amplitude-delay plane exhibit two kinds of local bifurcations of periodic motions, namely the saddle-node bifurcation and the pitchfork bifurcation, indicating a sudden emergence of two periodic motions with different stability and a Hopf bifurcation, respectively, in the sense of dynamic bifurcation. The second objective is to develop a shooting technique to locate both stable and unstable periodic motions of autonomous delay differential equations such that the periodic motions and their stability predicted using the method of multiple scales could be verified. The efficacy of the shooting scheme is well illustrated by some examples via phase trajectory and time history. It is shown that periodic motions located by the shooting method agree very well with those predicted on the bifurcation diagrams. Finally, the paper presents some interesting features of this simple, but dynamics-rich system.
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WANG, HUAILEI, HAIYAN HU, and ZAIHUA WANG. "GLOBAL DYNAMICS OF A DUFFING OSCILLATOR WITH DELAYED DISPLACEMENT FEEDBACK." International Journal of Bifurcation and Chaos 14, no. 08 (August 2004): 2753–75. http://dx.doi.org/10.1142/s0218127404010990.
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This paper presents a systematic study on the dynamics of a controlled Duffing oscillator with delayed displacement feedback, especially on the local bifurcations of periodic motions with respect to the time delay. The study begins with the analysis of the stability switches of the trivial equilibrium of the system with various parametric combinations and gives the critical values of time delay, where the trivial equilibrium may change its stability. It shows that as the time delay increases from zero to the positive infinity, the trivial equilibrium undergoes a different number of stability switches for different parametric combinations, and becomes unstable at last for all parametric combinations. Then, the method of multiple scales and the numerical computation method are jointly used to obtain a global diagram of local bifurcations of periodic motions with respect to the time delay for each type of parametric combinations. The diagrams indicate two kinds of local bifurcations. One is the saddle-node bifurcation and the other is the pitchfork bifurcation, of which the former means the sudden emerging of two periodic motions with different stability and the latter implies the Hopf bifurcation in the sense of dynamic bifurcation. A novel feature, referred to as the property of "periodicity in delay", is observed in the global diagrams of local bifurcations and used to justify the validity of infinite number of bifurcating branches in the bifurcation diagrams. The stability of the periodic motions is discussed not only from the high-order approximation of the asymptotic solution, but also from viewpoint of basin of attraction, which gives a good explanation for coexisting periodic motions and quasi-periodic motions, as well as an overall idea of basin of attraction. Afterwards, a conventional Poincaré section technique is used to reveal the abundant dynamic structures of a tori bifurcation sequence, which shows that the system will repeat similar quasi-periodic motions several times, with an increase of time delay, enroute to a chaotic motion. Finally, a new Poincaré section technique is proposed as a comparison with the conventional one, and the results show that the dynamical structures on the two kinds of Poincaré sections are topologically symmetric in rotation.
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XUEJUN, GAO. "BIFURCATION BEHAVIORS OF THE TWO-STATE VARIABLE FRICTION LAW OF A ROCK MASS SYSTEM." International Journal of Bifurcation and Chaos 23, no. 11 (November 2013): 1350184. http://dx.doi.org/10.1142/s0218127413501848.
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Based on the stability and bifurcation theory of dynamical systems, the bifurcation behaviors and chaotic motions of the two-state variable friction law of a rock mass system are investigated by the bifurcation diagrams based on the continuation method and the Poincaré maps. The stick-slip of the rock mass is formulated as an initial values problem for an autonomous system of three coupled nonlinear ordinary differential equations (ODEs) of first order. The results of linear stability analysis indicate that there is an equilibrium position in the rock mass system. Furthermore, numerical results of nonlinear analysis indicate that the equilibrium position loses its stability from a sup-critical Hopf bifurcation point, and then the bifurcating periodic motion evolves into chaotic motion through a series of period-doubling bifurcations with the decreasing of the control parameter. The stick-slip and chaotic motions evolve into infinity in the end with some unstable periodic motions.
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Fan, Li, and Sanyi Tang. "Global Bifurcation Analysis of a Population Model with Stage Structure and Beverton–Holt Saturation Function." International Journal of Bifurcation and Chaos 25, no. 12 (November 2015): 1550170. http://dx.doi.org/10.1142/s0218127415501709.
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In the present paper, we perform a complete bifurcation analysis of a two-stage population model, in which the per capita birth rate and stage transition rate from juveniles to adults are density dependent and take the general Beverton–Holt functions. Our study reveals a rich bifurcation structure including codimension-one bifurcations such as saddle-node, Hopf, homoclinic bifurcations, and codimension-two bifurcations such as Bogdanov–Takens (BT), Bautin bifurcations, etc. Moreover, by employing the polynomial analysis and approximation techniques, the existences of equilibria, Hopf and BT bifurcations as well as the formulas for calculating their bifurcation sets have been provided. Finally, the complete bifurcation diagrams and associate phase portraits are obtained not only analytically but also confirmed and extended numerically.
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Eskandari, Zohreh, Javad Alidousti, and Reza Khoshsiar Ghaziani. "Codimension-One and -Two Bifurcations of a Three-Dimensional Discrete Game Model." International Journal of Bifurcation and Chaos 31, no. 02 (February 2021): 2150023. http://dx.doi.org/10.1142/s0218127421500231.
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In this paper, bifurcation analysis of a three-dimensional discrete game model is provided. Possible codimension-one (codim-1) and codimension-two (codim-2) bifurcations of this model and its iterations are investigated under variation of one and two parameters, respectively. For each bifurcation, normal form coefficients are calculated through reduction of the system to the associated center manifold. The bifurcations detected in this paper include transcritical, fold, flip (period-doubling), Neimark–Sacker, period-doubling Neimark–Sacker, resonance 1:2, resonance 1:3, resonance 1:4 and fold-flip bifurcations. Moreover, we depict bifurcation diagrams corresponding to each bifurcation with the aid of numerical continuation method. These bifurcation curves not only confirm our analytical results, but also reveal a richer dynamics of the model especially in the higher iterations.
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SALAS, F., F. GORDILLO, J. ARACIL, and R. REGINATTO. "CODIMENSION-TWO BIFURCATIONS IN INDIRECT FIELD ORIENTED CONTROL OF INDUCTION MOTOR DRIVES." International Journal of Bifurcation and Chaos 18, no. 03 (March 2008): 779–92. http://dx.doi.org/10.1142/s0218127408020641.
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This paper provides further results on bifurcation analysis of indirect field oriented control of induction motors. Previous results presented on this subject [Bazanella & Reginatto, 2000, 2001; Gordillo et al., 2002] are summarized and extended by means of a codimension-two bifurcation analysis. It is shown that codimension-two bifurcation phenomena, such as a Bogdanov–Takens and zero–Hopf bifurcations, occur in IFOC as a result of parameter mismatch and certain setting of the proportional-integral speed controller. Conditions for the existence of such bifurcations are derived analytically, as long as possible, and bifurcation diagrams are presented with the help of simulation and numerical bifurcation analysis.
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WEI, HSIU-CHUAN. "NUMERICAL REVISIT TO A CLASS OF ONE-PREDATOR, TWO-PREY MODELS." International Journal of Bifurcation and Chaos 20, no. 08 (August 2010): 2521–36. http://dx.doi.org/10.1142/s0218127410027143.
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Some observations are made on a class of one-predator, two-prey models via numerical analysis. The simulations are performed with the aid of an adaptive grid method for constructing bifurcation diagrams and cell-to-cell mapping for global analysis. A two-dimensional bifurcation diagram is constructed to show that regions of coexistence of all three species, which imply the balance of competitive and predatory forces, are surrounded by regions of extinction of one or two species. Two or three coexisting attractors which may have a chaotic member are found in some regions of the bifurcation diagram. Their separatrices are computed to show the domains of attraction. The bifurcation diagram also contains codimension-two bifurcation points including Bogdanov–Takens, Gavrilov–Guckenheimer, and Bautin bifurcations. The dynamics in the vicinity of these codimension-two bifurcation points are discussed. Some global bifurcations including homoclinic and heteroclinic bifurcations are investigated. They can account for the disappearance of chaotic attractors and limit cycles. Bifurcations of limit cycles such as transcritical and saddle-node bifurcations are also studied in this work. Finally, some relevant calculations of Lyapunov exponents and power spectra are included to support the chaotic properties.
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ROCŞOREANU, CARMEN, NICOLAIE GIURGIŢEANU, and ADELINA GEORGESCU. "CONNECTIONS BETWEEN SADDLES FOR THE FITZHUGH–NAGUMO SYSTEM." International Journal of Bifurcation and Chaos 11, no. 02 (February 2001): 533–40. http://dx.doi.org/10.1142/s0218127401002213.
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By studying the two-dimensional FitzHugh–Nagumo (F–N) dynamical system, points of Bogdanov–Takens bifurcation were detected (Sec. 1). Two of the curves of homoclinic bifurcation emerging from these points intersect each other at a point of double breaking saddle connection bifurcation (Sec. 2). Numerical investigations of the bifurcation curves emerging from this point, in the parameter plane, allowed us to find other types of codimension-one and -two bifurcations concerning the connections between saddles and saddle-nodes, referred to as saddle-node–saddle connection bifurcation and saddle-node–saddle with separatrix connection bifurcation, respectively. The local bifurcation diagrams corresponding to these bifurcations are presented in Sec. 3. An analogy between the bifurcation corresponding to the point of double homoclinic bifurcation and the point of double breaking saddle connection bifurcation is also presented in Sec. 3.
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MOIOLA, JORGE LUIS. "ON THE COMPUTATION OF LOCAL BIFURCATION DIAGRAMS NEAR DEGENERATE HOPF BIFURCATIONS OF CERTAIN TYPES." International Journal of Bifurcation and Chaos 03, no. 05 (October 1993): 1103–22. http://dx.doi.org/10.1142/s0218127493000921.
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The computation of local bifurcation diagrams near degenerate Hopf bifurcations of certain types using feedback system theory and harmonic balance techniques is presented. This approach also provides the analytical expressions for the defining and the nondegeneracy conditions in the so-called frequency domain counterpart. A classical graphical method is easily adapted to carry on the continuation of the oscillatory branches to depict the local bifurcation diagrams. Moreover, several higher-order harmonic balance approximations are implemented to compare the accuracy of the computed solutions. The results are presented using local bifurcation diagrams, phase portrait plots and period diagrams, with similar ones obtained by using AUTO.
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AGLIARI, ANNA, GIAN-ITALO BISCHI, ROBERTO DIECI, and LAURA GARDINI. "GLOBAL BIFURCATIONS OF CLOSED INVARIANT CURVES IN TWO-DIMENSIONAL MAPS: A COMPUTER ASSISTED STUDY." International Journal of Bifurcation and Chaos 15, no. 04 (April 2005): 1285–328. http://dx.doi.org/10.1142/s0218127405012685.
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In this paper we describe some sequences of global bifurcations involving attracting and repelling closed invariant curves of two-dimensional maps that have a fixed point which may lose stability both via a supercritical Neimark bifurcation and a supercritical pitchfork or flip bifurcation. These bifurcations, characterized by the creation of heteroclinic and homoclinic connections or homoclinic tangles, are first described through qualitative phase diagrams and then by several numerical examples. Similar bifurcation phenomena can also be observed when the parameters in a two-dimensional parameter plane cross through many overlapping Arnold's tongues.
Dissertations / Theses on the topic "Diagrammes de bifurcation"
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Dutour, Sikirić Mathieu. "Bifurcation vers l'état d'Abrikosov et diagramme de phase." Paris 11, 1999. http://www.theses.fr/1999PA112313.
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Nous etudions dans cette these la fonctionnelle de ginzburg-landau dans r 3 sur des couples de fonctions (, $$a) qui verifient des conditions de periodicite de jauge en x 3 et selon un reseau discret de (x 2, x 3). Nous montrons que le probleme variationnel est equivalent au probleme de la minimisation d'une autre fonctionnelle sur un tore. Dans le cadre de la demonstration, un fibre vectoriel non trivial apparait. On se limite alors pour la suite a une quantification de 1. On montre ensuite que la fonctionnelle admet un minimum sur l'espace fonctionnel h 1 qui verifie un systeme d'equations aux derivees partielles appele systeme de ginzburg-landau. Le minimum est c par l'ellipticite du systeme d'equations de ginzburg-landau. On montre qu'il y a une bifurcation du couple (0, 0) pour le champ critique h e x t = k ou k est un parametre caracteristique du systeme. On etudie alors la stabilite de la solution bifurquee. On etudie la dependance de l'energie minimale a l'egard de la geometrie du tore. Enfin nous decrivons toutes les solutions du systeme d'equations de ginzburg-landau dans la limite k tend vers l'infini. Dans le dernier chapitre nous donnons pour notre modele la structure du diagramme des phases en precisant quelle regions sont normales, supraconductrices pure, mixte.
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Assemat, Pauline. "Dynamique non-linéaire des écoulements confinés : application à l'instabilité de Marangoni-Bénard et aux écoulements entre surfaces texturées." Toulouse 3, 2008. http://thesesups.ups-tlse.fr/1225/.
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Le travail porte sur deux problématiques scientifiques : la formation de structures convectives induites par l'instabilité de Marangoni-Bénard et les propriétés de transport des écoulements entre surfaces texturées. Bien que physiquement distincts, ces deux systèmes présentent les points communs d'être assujettis à de fortes contraintes spatiales. Il sont analysés par le biais de la théorie des bifurcations. L'étude de la convection de Marangoni-Bénard a été menée dans des géométries cylindriques à section transverse circulaire et faiblement elliptique. La comparaison des deux situations dans le régime non-linéaire a été menée par l'étude des changements induits sur les diagrammes de bifurcation eux mêmes interprétés par la théorie des bifurcations en présence de symétries. Nous avons ensuite mené l'étude de cette instabilité en présence de mélanges fluides binaires sujets à l'effet Soret et dans des couches fluides bidimensionnelles. Ce travail a révélé la formation de structures convectives spatialement localisées appelées convectons dont nous avons révélé la formation sur un fond d'ondes de plus faible amplitude. Enfin, nous avons étudié les propriétés de transport des écoulements entre surfaces texturées. Le système étudié est confiné transversalement à la direction de l'écoulement ce qui place cette étude dans le contexte de la microfluidique et de l'élaboration de micro-mélangeurs passifs. La simulation numérique et l'analyse des propriétés de transport de traceurs passifs est menée sur les équations issues d'un développement asymptotique faiblement inertiel dans un canal formé d'une succession périodique de cellules texturéesThe work focuses on two different physical situations: the convective structures resulting from the Marangoni-Bénard instability and the flow between patterned surfaces. The two systems are spatially constrained and are analysed using dynamical systems theories. Marangoni-Bénard convection has been studied in cylindrical geometries with either a circular or a weakly elliptical cross-section. The comparison of the two situations is carried out in the non-linear regime and the corresponding bifurcation diagrams are analysed using bifurcation theory with symmetries. Two-dimensional Marangoni convection in binary mixtures with Soret effect has also been studied in large periodic domains. The results show the formation of steady convective structures localized in space called convectons and the onset of stable convectons embedded in a background of small amplitude standing waves. Finally, the transport properties of flows in between patterned surfaces under weak inertia influence is studied. The flow is induced by a constant applied pressure gradient and the velocity field is calculated using an extension of the lubrication approximation taking into account the first order inertial corrections. Trajectories of tracers are obtained by integrating numerically the quasi-analytic velocity field. The transport properties are analysed by the study of Poincaré sections and their invariants
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Terrien, Soizic. "Instruments de la famille des flûtes : analyse des transitions entre régimes." Thesis, Aix-Marseille, 2014. http://www.theses.fr/2014AIXM4756.
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La diversité des régimes des instruments de la famille des flûtes a été mise en évidence à de nombreuses reprises : régimes statiques, périodiques, ou non périodiques. Cependant, de nombreux aspects de la dynamique de ces instruments demeurent mal compris. Pour les musiciens comme pour les facteurs d'instruments, les transitions entre régimes revêtent une importance particulière : d'une part elles correspondent à des changements de notes, et d'autre part la production d'un régime donné est conditionnée par les paramètres de facture (liés à la fabrication de l'instrument), et de contrôle (ajustés en permanence par l'instrumentiste). On s'attache dans ce document à caractériser les transitions entre régimes dans les flûtes, en lien avec des problématiques de facture et de jeu. Différentes approches sont mises en place. Des approches expérimentales d'une part, avec des mesures sur musicien et sur bouche artificielle. Par ailleurs, un modèle physique de l'instrument - un système dynamique à retard de type neutre - est étudié, par intégration temporelle d'une part, mais également par collocation orthogonale et continuation, donnant ainsi accès aux diagrammes de bifurcations.Croiser les résultats de ces différentes approches permet de mieux appréhender différents phénomènes : hystérésis associée aux changements de régime, ou mécanisme d'apparition des régimes non périodiques. L'influence de paramètres de facture et de contrôle est également étudiée : le rôle majeur de la géométrie interne du canal des flûtes à bec est mis en évidence, et l'influence de la dynamique de la pression dans la bouche du musicien sur les seuils de changement de régimes est caractériséeVarious studies have highlighted the diversity of regimes in flute-like instruments : static, periodic or non periodic regimes. However, some aspects of their dynamics remain poorly understood. Both for flute players and makers, transitions between regimes are particularly important : on the one hand, they correspond to a change of the note played, and on the other hand, production of a given regime is determined by parameters related to making and to playing of the instrument. In this document, we are interested in caracteristics of regime change in flute-like instruments, in relation with making and playing issues.Different approches are considered. First, experimental methods, with measurement on both musician and an artificial mouth. On the other hand, a physical model of the instrument - a system of delay differential equations of neutral type - is studied, through time-domain integration, and using orthogonal collocation coupled to numerical continuation. This last approach provides access to bifurcation diagrams.Considering results of these different methods, it becomes possible to better understand different experimental phenomena, such as regime change and associated hysteresis, or production mechanisms of non periodic regimes. Influence of different parameters is further studied : the crucial importance of the channel geometry in recorders is highlighted, and the influence of the mouth pressure dynamics on regime change thresholds is analysed
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Rihana, Sandy. "Modélisation de l'activité électrique utérine." Compiègne, 2008. http://www.theses.fr/2008COMP1742.
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Durant ces dernières décennies, l'activité électrique utérine origine des contractions menant à l'accouchement constitue une étude de recherche primordiale pour la prévention et pour la détection des accouchements prématurés. La modélisation mathématique et la simulation informatique sont devenues des outils indispensables pour la compréhension de différents phénomènes électrophysiologiques afin de prédire, et d'agir en cas d'anomalie. Sachant que le contrôle de l'excitabilité utérine s'avère avoir des conséquences thérapeutiques importantes, nous avons choisi de débuter le modèle à l'échelle cellulaire. L'analyse dynamique de ce modèle a permis de montrer l'efficacité de certains traitements tocolytiques tels que les bloqueurs des canaux calciques et les ouvreurs des canaux potassiques. Le contrôle de la contractilité utérine ne se limite pas au niveau cellulaire mais s'étend aussi au niveau tissulaire. Nous avons démontré comment un modèle de propagation biophysique permet de reproduire le couplage électrique réduit entre les cellules en début de grossesse et le couplage fort et synchronisé à l'approche du terme. Cette propagation a permis d'estimer un électromyogramme utérin de surface. Ce travail de thèse, quoique innovant et intéressant reste dans une première étape préliminaire. Il en porte en lui de futurs axes de recherches et de développement pluridisciplinaires prometteurs, dans l'objectif de fournir un modèle numérique de l'activité électrique utérine, contribuant à la compréhension de phénomènes physiologiques et à la prédiction d'accouchement prématuréIt is hypothesized that uterine electrical activity is efficiently correlated to the uterine contractions appearance. Once, forceful contractions appear, delivery is near. Therefore, the understanding of the genesis and of the propagation of the uterine electrical activity may provide an efficient tool to diagnosis preterm labour. Moreover, the control of uterine excitability seems to have important therapeutic consequences in controlling preterm labour. Modelling the electrical activity in uterine tissue is an important step for the understanding of physiological uterine contractile mechanisms. It would permit to reconstruct the uterine EMG. This work presents an electrophysiological model of the uterine cell that incorporates ion channel models at the cell level. The dynamical analysis of the uterine cell model allows a better apprehension of the main physiological effects on the cell's reponse. The cellular electrical activity will be integrated in a two dimension model, represented by the reaction diffusion equations, and will serve to the spatio-temporel integration at the uterine level for EMG reconstruction. This model validates some key physiological hypotheses considering uterine excitability and propagation
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Mayol, Serra Catalina. "Dinàmica no lineal de sistemes làsers: potencials de Lyapunov i diagrames de bifurcacions." Doctoral thesis, Universitat de les Illes Balears, 2002. http://hdl.handle.net/10803/9430.
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En aquest treball s'ha estudiat la dinàmica dels làsers de classe A i de classe B en termes del potencial de Lyapunov. En el cas que s'injecti un senyal al làser o es modulin alguns dels paràmetres, apareix un comportament moltmés complex i s'estudia el conjunt de bifurcacions.1) Als làsers de classe A, la dinàmica determinista s'ha interpretat com el moviment damunt el potencial de Lyapunov. En la dinàmica estocàstica s'obté un flux sostingut per renou per a la fase del camp elèctric.
2) Per als làsers de classe A amb senyal injectat, s'ha descrit el conjunt de bifurcacions complet i s'ha determinat el conjunt d'amplituds i freqüències en el quals el làser respon
ajustant la seva freqüència a la del camp extern.
3) S'ha obtingut un potencial de Lyapunov pels làsers de classe B, només vàlid en el cas determinista, que inclou els termes de saturació de guany i d'emissió espontània.
4) S'ha realitzat un estudi del conjunt de bifurcacions parcial al voltant del règim tipus II de la singularitat Hopf--sella--node en un làser de classe B amb senyal injectat.
5) S'han identificat les respostes òptimes pels làsers de semiconductor sotmesos a modulació periòdica externa. S'han obtingut les corbes que donen la resposta màxima per cada tipus de resonància en el pla definit per l'amplitud relativa de modulació i la freqüència de modulació.
In this work we have studied the dynamics of both class A and class B lasers in terms of Lyapunov potentials. In the case of an injected signal or when some laser parameters are modulated, and more complex behaviour is expected, the bifurcation set is studied. The main results are the following:
1) For class A lasers, the deterministic dynamics has been interpreted as a movement on the potential landscape. In the stochastic dynamics we have found a noise sustained flow for the phase of the electric field.
2) For class A lasers with an injected signal, we have been able to describe the whole bifurcation set of this system and to determine the set of amplitudes frequencies for which the laser responds adjusting its frequency to that of the external field.
3) In the case of class B lasers, we have obtained a Lyapunov potential only valid in the deterministic case, including spontaneous emission and gain saturation terms. The fixed point corresponding to the laser in the on state has been interpreted as a minimum in this potential. Relaxation to this minimum is reached through damped oscillations.
4) We have performed a study of the partial bifurcation set around the type II regime of the Hopf-saddle-node singularity in a class B laser with injected signal.
5) We have identified the optimal responses of a semiconductor laser subjected to an external periodic modulation. The lines that give a maximum response for each type of resonance are obtained in the plane defined by the relative amplitude modulation and frequency modulation.
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Bouktir, Yasser. "Étude des phénomènes d'instabilités, bifurcation et endommagement en mise en forme des matériaux." Thesis, Université de Lorraine, 2018. http://www.theses.fr/2018LORR0217/document.
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L’objectif de ce sujet de thèse est de prédire l’apparition des instabilités plastiques (striction diffuse et striction localisée) dans les matériaux métalliques. Ces matériaux sont décrits par des modèles de comportement élasto-plastique couplés à l’endommagement. L'approche de Lemaitre, reliant l'endommagement à la déformation plastique équivalente et au taux de restitution de la densité d'énergie élastique, est adoptée. Parmi les différents critères et indicateurs qui sont considérés pour la prédiction des instabilités matériau, la théorie de bifurcation et les critères de type force maximum sont tout particulièrement analysés et comparés. Un objectif important de cette étude consiste à déterminer les mécanismes déstabilisants clés associés à cette modélisation du comportement, ainsi que l’impact des différents aspects physiques et des paramètres matériau sur l’apparition de la striction. Les développements résultants sont appliqués à une sélection représentative de matériaux métalliques afin prédire leurs limites de formabilités. Cette approche combinant des lois de comportement et critères de striction peut être utilisée comme outil théorique et numérique d’aide à la conception de nouveaux matériaux à ductilité amélioréeThe aim of the present work is to predict the occurrence of plastic instabilities (diffuse and localized necking) in thin sheet metals. The prediction of these plastic instabilities is undertaken using an elastic–plastic model coupled with ductile damage, which is then combined with various plastic instability criteria theory. The bifurcation-based criteria and the maximum force criterion used in this work are formulated within a general three-dimensional modeling framework, and then applied for the particular case of plane-stress conditions for sheet metals. Some theoretical relationships or links between the different investigated necking criteria are established, which allows a hierarchical classification in terms of their conservative character in predicting critical necking strains. The resulting numerical tool is implemented into the finite element code ABAQUS/Standard to predict forming limit diagrams, in both situations of a fully three-dimensional formulation and a plane-stress framework. This approach, that combines constitutive equations to necking criteria, serves as a useful tool in the design of new materials with improved ductility
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Risler, Thomas. "Comportement critique d'oscillateurs couples ; Groupe de renormalisation et classe d'universalite." Phd thesis, Université Pierre et Marie Curie - Paris VI, 2003. http://tel.archives-ouvertes.fr/tel-00004449.
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Les etonnantes performances de l'organe auditif des mammiferes sontnotamment dues aux proprietes generiques des oscillateurs critiques
couples qui constituent le systeme. Cette these presente une etude
des proprietes critiques generiques des
systemes spatialement etendus d'oscillateurs stochastiques couples,
operant dans le voisinage d'une instabilite oscillante homogene ou
bifurcation de Hopf. Dans ce contexte, cette bifurcation constitue un
point critique dynamique hors equilibre, exhibant des proprietes
universelles qui sont canoniquement decrites par l'equation
Ginzburg-Landau complexe en presence de bruit. La formulation du probleme
en termes d'une theorie statistique dynamique des champs non hamiltonienne
nous permet d'etudier le comportement critique du systeme a l'aide des
techniques de la renormalisation dynamique perturbative.
Dans un cas particulier, une analogie exacte avec le modele O(2) dynamique
nous permet d'ecrire une relation generalisee de la relation
fluctuation-dissipation et de deduire le comportement critique du systeme
directement a partir des etudes anterieures. Dans le cas general,
nous etablissons la structure du groupe de renormalisation de la theorie
dans un espace de dimension
4-epsilon, en lui adaptant les schemas de renormalisation de Wilson et
de Callan-Symanzik. La presence d'une frequence caracteristique dans le
systeme - la frequence des oscillations spontanees a la transition -
impose d'associer aux transformations de renormalisation un changement de
referentiel oscillant dependant de l'echelle. Nous effectuons le
calcul a l'ordre de deux boucles en theorie des perturbations, et montrons
que la classe d'universalite du modele est decrite par le point fixe du
modele dynamique dissipatif
O(2) dans un referentiel oscillant bien choisi. Ainsi, bien que la
dynamique soit hautement hors equilibre et brise les relations de bilan
detaille, une relation fluctuation-dissipation generalisee est
asymptotiquement restauree a la transition. Cette relation prevoit
l'existence de fortes contraintes sur les principales observables
experimentales : la fonction de correlation a deux points et la fonction
de reponse lineaire a un stimulus sinusoidal.
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Basak, Gancheva Inna. "Explicit integration of some integrable systems of classical mechanics." Doctoral thesis, Universitat Politècnica de Catalunya, 2011. http://hdl.handle.net/10803/125114.
Full textAbstract:
The main objective of the thesis is the analytical and geometrical study of several integrable finite-dimentional dynamical systems of classical mechanics,
which are closely related, namely:
- the classical generalization of the Euler top: the Zhukovski-Volterra (ZV) system describing the free motion of a gyrostat, i.e., a rigid body carrying
a symmetric rotator whose axis is fixed in the body;
- the Steklov-Lyapunov integrable case of the Kirchhoff equations describing the motion of a rigid body in an ideal incompressible liquid;
- a nontrivial integrable generalization of the Steklov-Lyapunov system found by V.Rubanovskii: it describes the motion of a gyrostat in an ideal
fluid in presence of a non-zero circulation.
In our study we obtained explicit solution of the Zhukovski-Volterra ([2] and the Steklov-Lyapunov systems in terms of sigma- or theta-functions,
and performed a bifurcation analysis of these systems, as well as of the Rubanovskii generalization.
One should note that the solution of the ZV system was first given by V. Volterra, who, however, presented only its structure, but not the explicit formulas.
The thesis gives a new alternative solution of this system by using an algebraic parametrization of the angular momentum. This allowed us to find
poles and zeros of angular momentum in an algebraic way. The parametrization was also used to find an explicit solution for the Euler precession angle,
and, as a consequence, to solve the Poisson equations describing the motion of the gyrostat in space.
Similarly, by giving a geometric interpretation of the separating variables, and using the Weierstrass root functions, we reconstructed the thetafunctional
solution of the Steklov-Lyapunov systems, which was first given by F. Kötter in 1899 without a derivation ([3]).
In the study of bifurcations and singularities of the ZV system we used its bi-Hamiltonian structure ([1]. According the new method, the solution is
critical, if there exist a parameter of corresponding family of Poisson brackets, for wich the rang of the brackets with this parameter drops. Applying
new technics, based on the property of the system of being bi-Hamiltonian, we construct the bifurcation diagram of the ZV system. We also find the
equilibrium points of the system, check the non-degeneracy condition for such points in the sense of the singularity theory of Hamiltonian systems,
determine the types of equilibria points, and verify whether they are stable or not. We also describe the topological type of common levels of the first
integrals of the ZV system. Similar problems have been discussed in many papers, but the goal of our work is to study the system and demonstrate the
above techniques. It is a remarkable fact that using the bi-Hamiltonian property makes it possible to answer all the above questions practically without
any difficult computations.
The same method is applied to construct the bifurcation diagram for the Steklov-Lyapunov system, describe the zones of real motion, and analyze
stability of critical periodic solutions. Then the bifurcation analysis is extended to the Rubanovskii generalizaton. Here the main difficulty is that the number of different types of the bifurcation diagram is quite high, so we only describe general properties of the bifurcation curves, do stability analysis for closed trajectories, and equilibria.El objetivo principal de la tesis es el estudio analítico y geométrico de varios sistemas integrables dinámicos y finito-dimensionales de la mecánica clásica que están estrechamente vinculados, a saber: -La generalización clásica de Euler top: el sistema Zhukovski-Volterra (ZV) que describe el movimiento libre de un giróstato, es decir, un cuerpo rígido que lleva un rotor simétrico cuyo eje es fijo al cuerpo. - El caso del sistema integrable de Steklov-Lyapunov de las ecuaciones de Kirchhoff que describen el movimiento de un cuerpo rígido en un líquido incompresible ideal; - Una generalización no trivial del sistema integrable de Steklov-Lyapunov encontrado por V. Rubanovskii que describe el movimiento de un giróstato en un fluido ideal en presencia de una circulación distinta de cero. En nuestro estudio hemos obtenido una solución explícita de los sistemas de Zhukovski-Volterra [2] y de Steklov-Lyapunov en términos de funciones sigma- o theta y hemos realizado un análisis de la bifurcación de estos sistemas, así como de la generalización de Rubanovskii. Hay que señalar que la solución del sistema de ZV fue dado por primera vez por V. Volterra, que, sin embargo, presenta sólo su estructura, pero no las fórmulas explícitas. La tesis ofrece una nueva solución alternativa de este sistema mediante el uso de una parametrización algebraica del momento angular. Esto nos ha permitido encontrar polos y ceros del momento angular en forma algebraica. La parametrización también se utilizó para encontrar una solución explícita para el ángulo de precesión de Euler, y, en consecuencia, para resolver las ecuaciones de Poisson que describen el movimiento de un giróstato en el espacio. Del mismo modo, al dar una interpretación geométrica de las variables de separación, y utilizando las funciones de las raíces Weierstrass, hemos reconstruido la solución thetafunctional de los sistemas de Steklov-Lyapunov, que fue dado por primera vez por F. Kotter en 1899 sin una derivación ([3]). En el estudio de las bifurcaciones y las singularidades del sistema ZV hemos utilizado su estructura bi-Hamiltoniana ([1]). Según el nuevo método, la solución es crítica, si existe un parámetro de la familia correspondiente del paréntesis de Poisson, para que el rango de las paréntesis con este parámetro se disminuye. Aplicando las nuevas técnicas, basadas en la propiedad del sistema de ser bi-Hamiltoniana, construimos el diagrama de bifurcación del sistema ZV. También hemos encontrado los puntos de equilibrio del sistema, verificando la condición de no-degeneración de estos puntos, en el sentido de la teoría de singularidad de los sistemas hamiltonianos, determinando los tipos de puntos de equilibrio, y comprobando si son estables o no. También hemos descrito el tipo topológico de los niveles comunes de los primeros integrales del sistema de ZV. Problemas similares se han discutido en muchas obras, pero el objetivo de nuestro trabajo es estudiar el sistema y demostrar las técnicas anteriormente mencionadas. Es un hecho notable que el uso de la propiedad bi-Hamilton permite responder a todas las preguntas anteriores, prácticamente sin ningún cálculo difícil. El mismo método se aplica para construir el diagrama de bifurcación para el sistema de Steklov-Lyapunov, describir las zonas de movimiento real, y analizar la estabilidad de soluciones periódicas críticas. A continuación, el análisis de bifurcación se extiende a la generalización Rubanovskii. Aquí la principal dificultad consiste en que el número de diferentes tipos del diagrama de bifurcación es bastante alto, por lo que sólo se describen las propiedades generales de las curvas de bifurcación, y el análisis de estabilidad se hace para trayectorias cerradas, y equilibrios .
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Casonatto, Catiana. "Invariantes do tipo Vassiliev de aplicações estáveis de 3-variedade em \'R POT. 4\'." Universidade de São Paulo, 2011. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-18082011-090351/.
Full textAbstract:
Neste trabalho obtemos que o espaço dos invariantes locais do tipo Vassiliev de primeira ordem de aplicações estáveis de 3-variedade fechada orientada em \' R POT. 4\' é 4-dimensional. Damos uma interpretação geométrica para 2 dos 4 geradores deste espaço, a saber, \'I IND. Q\' o número de pontos quádruplos e \'I IND. C / P\' o número de pares de pontos do tipo crosscap/plano, da imagem de uma aplicação estável. Ao reduzir o espaço das aplicações para o das imersões esáaveis, obtemos que o espaço dos invariantes locais de imersões estáveis é 3-dimensional. Os invariantes que obtemos são: \'I IND. Q\' o número de pares de pontos quádruplos da imagem de uma imersão estável e dois índices de interseção \'I IND. I\'`+ e \'I IND. l\' introduzidos por V. Goryunov em [15]. Como início de um estudo que almejamos realizar sobre a geometria de uma m-variedade em \'R POT. m+1\' com singularidades, obtemos os tipos de contatos genéricos da suspensão do crosscap (única singularidade estavel de \'R POT. 3\' em \'R POT. 4\' ) com hiperplanos de \'R POT.4\'In this work we obtain that the space of first order local Vassiliev type invariants of stable maps of oriented 3-manifolds in \'R POT. 4\' is 4-dimensional. We give a geometric interpretation for two of the four generators of this space, namely, \'I IND. Q\' the number of quadruple points and \'I IND. C / P\' the number of pairs of points of crosscap/plane type, of the image of a stable map. In the case of stable immersions, we obtain that the space of local invariants of stable immersions is 3-dimensional. The invariants that we obtain are: \'I IND. Q\' the number of pairs of quadruple points of the image of a stable immersion and the positive and negative linking invariants \'I IND. I`+ and I\'I IND., l\' introduced by V. Goryunov in [15]. As a beging of a study that we want to realise about the geometry of a m-manifold in \'R POT. m+1\' with singularities, we obtain the generic contacts of the suspension of crosscap (the only stable singularity from \'R POT. 3\' to \'R POT. 4\') with hyperplanes of \'R POT. 4\'
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Balakireva, Irina. "Nonlinear dynamics of Kerr optical frequency combs." Thesis, Besançon, 2015. http://www.theses.fr/2015BESA2043.
Full textAbstract:
La présente thèse est consacrée à l’étude des peignes optiques de Kerr dans les résonateurs àmodes de galerie, au sein desquels la lumière peut être excitée par pompage externe. L’effet Kerrexistant dans ces résonateurs engendre des modes latéraux équidistants (dans le domaine spectral)de part et d’autre du mode excité, c’est à dire un peigne de fréquence. Cette thèse est diviséeen trois chapitres. Le premier est dédié à l’introduction de la génération de ces peignes et leurapplications. Le deuxième chapitre présente l’analyse de l’équation de Lugiato-Lefever, décrivantde manière analytique le système, et conduit à la construction de deux diagrammes de bifurcationpour les dispersions normale et anomale. Ils sont tracés en fonction des deux seuls paramètresexpérimentalement contrôlables une fois le résonateur fabriqué : la puissance du laser et sondécalage de fréquence. Ces diagrammes indiquent les plages de paramètres pour lesquels une,deux, ou trois solutions existent ainsi que leur stabilité. Les simulations numériques renseignentle type exact de solution associée à chaque aire (notamment les solitons brillants et sombres, lesbreathers, les peignes optiques de Kerr de premier et deuxième ordre, et un régime chaotique) ; cesdiagrammes indiquent donc les paramètres du laser à choisir afin de générer la solution souhaitée.Le troisième chapitre est dédié aux peignes de Kerr optique secondaires, lignes additionnelles dansle domaine spectral générées entre les lignes du peigne principal. Ils apparaissent en dispersionanormale, lorsque la quantité de photon pompe excède un seuil dit de second ordre, qui a étédéterminé numériquementThis thesis is dedicated to the study of the Kerr optical frequency combs in whispering gallery moderesonators, where the light can be excited by the extern pump. Due to the Kerr effect existing in theseresonators, the quasi-equidistant lines in the spectral domain are generated around the excited mode,that is the frequency comb. This thesis is devided in three chapters. The first one is dedicated to theintroduction of the Kerr comb generation and their applications.The second one presents the analysisof the Lugiato-Lefever equation used for the analytical study of the system, leading to the constructionof two bifurcation diagrams for the normal and anomalous dispersions. They are plotted for twoparameters, which can be controlled during experiments once the resonator has been fabricated,which are the pump power of the laser and its frequency detuning. These diagrams show the areas ofthe parameters for which one, two, or three solutions exist and their stability. The additional numericalsimulations show the exact type of the solution in each area (such as the bright and dark solitons,the breathers, the primary and secondary Kerr combs and chaotical regimes), finally these diagramsshow the parameters of the laser needed to be choosen for the generation of the desired solution.The third chapter is dedicated to the secondary Kerr combs, which are the additional lines generatedbetween the lines of the primary comb. They appear in the anomalous dispersion regime, when thequantity of the pump photons crosses the second-order threshold, which has been found numerically
Books on the topic "Diagrammes de bifurcation"
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A, Kuznet͡s︡ov I͡U︡, and Khibnik A. I, eds. Bifurkat͡s︡ionnye diagrammy dinamicheskikh sistem na ploskosti. Pushchino: Nauch. t͡s︡entr biologicheskikh issledovaniĭ AN SSSR v Pushchine, 1985.
Find full textBook chapters on the topic "Diagrammes de bifurcation"
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Nusse, Helena E., James A. Yorke, and Eric J. Kostelich. "Bifurcation Diagrams." In Dynamics: Numerical Explorations, 229–68. New York, NY: Springer US, 1994. http://dx.doi.org/10.1007/978-1-4684-0231-5_6.
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Nusse, Helena E., James A. Yorke, Brian R. Hunt, and Eric J. Kostelich. "Bifurcation Diagrams." In Dynamics: Numerical Explorations, 267–312. New York, NY: Springer New York, 1998. http://dx.doi.org/10.1007/978-1-4612-0219-6_6.
Full text
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Parker, Thomas S., and Leon O. Chua. "Bifurcation Diagrams." In Practical Numerical Algorithms for Chaotic Systems, 201–35. New York, NY: Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-3486-9_8.
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Ikeda, Kiyohiro, and Kazuo Murota. "Experimentally Observed Bifurcation Diagrams." In Imperfect Bifurcation in Structures and Materials, 122–49. New York, NY: Springer New York, 2002. http://dx.doi.org/10.1007/978-1-4757-3697-7_6.
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Ikeda, Kiyohiro, and Kazuo Murota. "Experimentally Observed Bifurcation Diagrams." In Imperfect Bifurcation in Structures and Materials, 125–48. New York, NY: Springer New York, 2010. http://dx.doi.org/10.1007/978-1-4419-7296-5_6.
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Ikeda, Kiyohiro, and Kazuo Murota. "Experimentally Observed Bifurcation Diagrams." In Imperfect Bifurcation in Structures and Materials, 141–64. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-21473-9_6.
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Morcillo, Daniel, Daniel Burbano, Fabiola Angulo, and Gerard Olivar. "Using Bifurcation Diagrams for Controlling Chaos." In Springer Proceedings in Mathematics & Statistics, 77–84. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-08266-0_6.
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Jarausch, Helmut, and Wolfgang Mackens. "Computing Bifurcation Diagrams for Large Nonlinear Variational Problems." In Large Scale Scientific Computing, 114–37. Boston, MA: Birkhäuser Boston, 1987. http://dx.doi.org/10.1007/978-1-4684-6754-3_8.
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Owen, Livia, and Eric Harjanto. "A Basic Manual for AUTO-07p in Computing Bifurcation Diagrams of a Predator-Prey Model." In Dynamical Systems, Bifurcation Analysis and Applications, 205–24. Singapore: Springer Singapore, 2019. http://dx.doi.org/10.1007/978-981-32-9832-3_11.
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Eilbeck, J. C., and J. E. Furter. "Understanding Steady-State Bifurcation Diagrams for a Model Reaction-Diffusion System." In Continuation and Bifurcations: Numerical Techniques and Applications, 25–41. Dordrecht: Springer Netherlands, 1990. http://dx.doi.org/10.1007/978-94-009-0659-4_3.
Full textConference papers on the topic "Diagrammes de bifurcation"
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Elgohary, Tarek A., and Tamás Kalmár-Nagy. "Nonlinear Analysis of a 2-DOF Piecewise Linear Aeroelastic System." In ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-70038.
Full textAbstract:
Aerodynamic forces for a 2-DOF aeroelastic system oscillating in pitch and plunge are modeled as a piecewise linear function. Equilibria of the piecewise linear model are obtained and their stability/bifurcations analyzed. Two of the main bifurcations are border collision and rapid/Hopf bifurcations. Continuation is used to generate the bifurcation diagrams of the system. Chaotic behavior following the intermittent route is also observed. To better understand the grazing phenomenon sets of initial conditions associated with the system behavior are defined and analyzed.
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Stiefs, Dirk, Thilo Gross, Ezio Venturino, and Ulrike Feudel. "Computing 3D Bifurcation Diagrams." In NUMERICAL ANALYSIS AND APPLIED MATHEMATICS: International Conference on Numerical Analysis and Applied Mathematics 2008. American Institute of Physics, 2008. http://dx.doi.org/10.1063/1.2991095.
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Wang, Dong-Mei, Wei Zhang, Mu-Rong Li, and Qian Wang. "Application of HDQM for the Analysis of Two-Dimensional Nonlinear Dynamics of Axially Accelerating Beam." In ASME 2014 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/detc2014-34139.
Full textAbstract:
In this paper, the two-dimensional nonlinear dynamics are investigated for the nonplanar nonlinear vibrations of an axially accelerating moving viscoelastic beam with in-plane and out-of-plane vibrations by the harmonic differential quadrature method (HDQM). The coupled nonlinear partial differential equations for the two-dimensional nonplanar nonlinear vibrations are discretized in space and time domains using HDQM and Runge-Kutta-Fehlberg methods respectively. Based on the numerical solutions, the nonlinear dynamical behaviors such as bifurcations and chaotic motions of the nonlinear system are investigated by using of the phase portrait and the bifurcation diagrams. The bifurcation diagrams for the in-plane and out-of-plane displacements via the mean axial velocity and the amplitude of velocity fluctuation are respectively presented while other parameters are fixed.
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Lelkes, János, and Tamás Kalmár-Nagy. "Effect of Structural Nonlinearity on a Piecewise Linear Aeroelastic System." In ASME 2019 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/detc2019-98208.
Full textAbstract:
Abstract Aeroelasticity is the study of the interaction of aerodynamic, elastic and inertia forces. When flexible structures, such as an airfoil, undergo wind excitation, divergence or flutter instability may arise. We study the dynamics of a two-degree-of-freedom (pitch and plunge) aeroelastic system with cubic structural nonlinearities. The aerodynamic forces are modeled as a piecewise linear function of the effective angle of attack. Stability and bifurcations of equilibria are analyzed. The effect of the structural nonlinearity is investigated. We find border collision, rapid, Hopf, saddle-node and pitchfork bifurcations. Bifurcation diagrams of the system were calculated utilizing MatCont and Mathematica.
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Wang, Yuefang, Yong Li, Yong Zhang, and Xiaoyan Wang. "Nonlinear Vibration and Bifurcation Analysis for Rotor-Seal-Bearing Systems of Centrifugal Compressors." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-34549.
Full textAbstract:
This paper presents the nonlinear coupling vibration and bifurcation of a high-speed centrifugal compressor with a labyrinth seal and two journal bearings. The rotor system is modeled as a Jeffcott rotor. The Muszynska’s model is used to express the seal force with multiple parameters. For the journal bearings, the model proposed by Zhang is adopted to express the excitation of unsteady oil-film force. The Runge-Kutta method is used to determine the vibration responses at the disc center and the two bearings. With parameters of rotation speed and pressure difference of the seal, bifurcation diagrams are presented to demonstrate the complexity in the rotor motions. Multiple periodic bifurcations are pointed out using two seal pressure differences. The intricate bifurcation behavior shows inherent interactions between forces of oil-film and seal, which reflect much more complicated rotor dynamics than the one with either of the excitations alone.
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Corea-Araujo, J. A., F. Gonzalez-Molina, J. A. Martinez, J. A. Barrado-Rodrigo, and L. Guasch-Pesquer. "Ferroresonance analysis using 3D bifurcation diagrams." In 2013 IEEE Power & Energy Society General Meeting. IEEE, 2013. http://dx.doi.org/10.1109/pesmg.2013.6672183.
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Bejar, Jose Ortiz, Juan J. Flores Romero, and Garibaldi Pineda Garcia. "Qualitative simulation over two-parameter bifurcation diagrams." In 2014 IEEE International Autumn Meeting on Power, Electronics and Computing (ROPEC). IEEE, 2014. http://dx.doi.org/10.1109/ropec.2014.7036332.
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Vörös, Illés, and Dénes Takács. "Bifurcation Analysis of a Lane Keeping Controller With Feedback Delay." In ASME 2020 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/detc2020-22387.
Full textAbstract:
Abstract The aim of this study is to highlight nonlinear behaviors and periodic orbits of the single-track vehicle model with a delayed feedback controller. Two widely used tire models, namely a linear tire characteristic and Pacejka’s Magic Formula are considered. Linearly stable domains of parameters such as the vehicle speed and the control gains are determined. Periodic solutions originating from Hopf bifurcation points are followed using numerical continuation and the results obtained with the two different tire models are compared. It is shown that neglecting the saturation of the tire lateral forces at total sliding might change the sense of certain Hopf bifurcations from subcritical to supercritical. The results are verified by numerical simulations. The resulting bifurcation diagrams aim to quantify the degree of robustness of these controllers with regards to the initial conditions at various parameter ranges in order to assure stable and safe operation.
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Oestreich, M., N. Hinrichs, K. Popp, and C. J. Budd. "Analytical and Experimental Investigation of an Impact Oscillator." In ASME 1997 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1997. http://dx.doi.org/10.1115/detc97/vib-3907.
Full textAbstract:
Abstract In the present paper a single-degree-of-freedom, periodically forced impact oscillator is investigated by numerics and experiments. At first the different types of motion and bifurcation diagrams are determined from experiments and compared to numerical results on the basis of the identified impact model. Also the nonsmooth third order system shows a rich bifurcational behaviour. On the basis of the identified nonsmooth model further analytical and numerical analysis can be applied. The Lyapunov exponents and the winding number are evaluated for the non-smooth system.
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Mustafa, G., and A. Ertas. "Nonlinear Interaction of a Parametrically-Excited Coupled Column-Pendulum Oscillator." In ASME 1993 Design Technical Conferences. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/detc1993-0134.
Full textAbstract:
Abstract A new vibration absorbing device is introduced for large flexible structures. The bifurcation diagrams obtained for the averaged system, indicate that the system loses stability via two distinct routes. One leading to a saddle-node bifurcation, normally associated with the jump phenomena. The second instability is due to the Hopf bifurcation, that results in amplitude modulated motion of the oscillator. A parameter range has been identified where these bifurcations coalesce. This phenomenon is a strong indicator of existence of homoclinic orbits. In addition to the regular solution branches, that bifurcate from the zero solution, the system also possesses isolated solutions (the so-called “isolas”) that form isolated loops bounded away from zero. As the forcing amplitude is varied, the isolas appear, disappear or coalesce with the regular solution branches. The response curves indicate that the column amplitude shows saturation. The pendulum acts as a vibration absorber over a range of frequency where the column response is saturated. However, there is also a frequency range over which a reverse flow of energy occurs, where the pendulum shows reduced amplitude at the cost of large amplitudes of the column.