Journal articles: '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.

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Ji, J. C., and Terry Brown. "Periodic and Chaotic Motion of a Time-Delayed Nonlinear System Under Two Coexisting Families of Additive Resonances." International Journal of Bifurcation and Chaos 27, no. 05 (May 2017): 1750066. http://dx.doi.org/10.1142/s0218127417500663.

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A time-delayed quadratic nonlinear mechanical system can exhibit two coexisting stable bifurcating solutions (SBSs) after two-to-one resonant Hopf bifurcations occur in the corresponding autonomous time-delayed system. One SBS is of small-amplitude and has the Hopf bifurcation frequencies (HBFs), while the other is of large-amplitude and contains the shifted Hopf bifurcation frequencies (the shifted HBFs). When the forcing frequency is tuned to be the sum of two HBFs or the sum of two shifted HBFs, two families of additive resonances can be induced in the forced response. The forced response under the additive resonance related to the HBFs can demonstrate periodic, quasi-periodic and chaotic motion. On the contrary, the forced response under the additive resonance associated with the shifted HBFs may exhibit period-three periodic motion and quasi-periodic motion. Bifurcation diagrams, time trajectories, frequency spectra, phase portraits and Poincaré sections are presented to show periodic, quasi-periodic, and chaotic motion of the time-delayed nonlinear system under the two families of additive resonances.

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REVEL, GUSTAVO, DIEGO M. ALONSO, and JORGE L. MOIOLA. "A GALLERY OF OSCILLATIONS IN A RESONANT ELECTRIC CIRCUIT: HOPF-HOPF AND FOLD-FLIP INTERACTIONS." International Journal of Bifurcation and Chaos 18, no. 02 (February 2008): 481–94. http://dx.doi.org/10.1142/s0218127408020409.

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In this work, the dynamics of a coupled electric circuit is studied. Several bifurcation diagrams associated with the truncated normal form of the Hopf-Hopf bifurcation are presented. The bifurcation curves are obtained by numerical continuation methods. The existence of quasi-periodic solutions with two (2D torus) and three (3D torus) frequency components is shown. These, in certain way, are close (or have a tendency to end up) to chaotic motion. Furthermore, two fold-flip bifurcations are detected in the vicinity of the Hopf-Hopf bifurcation, and are classified correspondingly. The analysis is completed with time simulations, the continuation of several limit cycle bifurcations and the indication of resonance points.

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ALGABA, A., M. MERINO, and A. J. RODRÍGUEZ-LUIS. "TAKENS–BOGDANOV BIFURCATIONS OF PERIODIC ORBITS AND ARNOLD'S TONGUES IN A THREE-DIMENSIONAL ELECTRONIC MODEL." International Journal of Bifurcation and Chaos 11, no. 02 (February 2001): 513–31. http://dx.doi.org/10.1142/s0218127401002286.

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In this paper we study Arnold's tongues in a ℤ2-symmetric electronic circuit. They appear in a rich bifurcation scenario organized by a degenerate codimension-three Hopf–pitchfork bifurcation. On the one hand, we describe the transition open-to-closed of the resonance zones, finding two different types of Takens–Bogdanov bifurcations (quadratic and cubic homoclinic-type) of periodic orbits. The existence of cascades of the cubic Takens–Bogdanov bifurcations is also pointed out. On the other hand, we study the dynamics inside the tongues showing different Poincaré sections. Several bifurcation diagrams show the presence of cusps of periodic orbits and homoclinic bifurcations. We show the relation that exists between two codimension-two bifurcations of equilibria, Takens–Bogdanov and Hopf–pitchfork, via homoclinic connections, period-doubling and quasiperiodic motions.

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KOSTOVA, TANYA, RENUKA RAVINDRAN, and MARIA SCHONBEK. "FITZHUGH–NAGUMO REVISITED: TYPES OF BIFURCATIONS, PERIODICAL FORCING AND STABILITY REGIONS BY A LYAPUNOV FUNCTIONAL." International Journal of Bifurcation and Chaos 14, no. 03 (March 2004): 913–25. http://dx.doi.org/10.1142/s0218127404009685.

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We study several aspects of FitzHugh–Nagumo's (FH–N) equations without diffusion. Some global stability results as well as the boundedness of solutions are derived by using a suitably defined Lyapunov functional. We show the existence of both supercritical and subcritical Hopf bifurcations. We demonstrate that the number of all bifurcation diagrams is 8 but that the possible sequential occurrences of bifurcation events is much richer. We present a numerical study of an example exhibiting a series of various bifurcations, including subcritical Hopf bifurcations, homoclinic bifurcations and saddle-node bifurcations of equilibria and of periodic solutions. Finally, we study periodically forced FH–N equations. We prove that phase-locking occurs independently of the magnitude of the periodic forcing.

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Roulet, Javier, and Gabriel B. Mindlin. "A Diagrammatic Representation of Phase Portraits and Bifurcation Diagrams of Two-Dimensional Dynamical Systems." International Journal of Bifurcation and Chaos 27, no. 13 (December 15, 2017): 1730045. http://dx.doi.org/10.1142/s0218127417300452.

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We treat the problem of characterizing in a systematic way the qualitative features of two-dimensional dynamical systems. To that end, we construct a representation of the topological features of phase portraits by means of diagrams that discard their quantitative information. All codimension 1 bifurcations are naturally embodied in the possible ways of transitioning smoothly between diagrams. We introduce a representation of bifurcation curves in parameter space that guides the proposition of bifurcation diagrams compatible with partial information about the system.

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WEI, HSIU-CHUAN. "THE DYNAMICS OF THE LUO–RUDY MODEL." International Journal of Bifurcation and Chaos 20, no. 12 (December 2010): 4055–66. http://dx.doi.org/10.1142/s0218127410028185.

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Computer simulation of mathematical models of mammalian ventricular myocytes has become a very promising tool for understanding the underlying mechanisms of cardiac arrhythmias and may provide useful predictions of treatment in order to prevent fatal arrhythmias. In this paper, we employ the Luo–Rudy (LR) model of membrane action potential of the mammalian ventricular cell. Simulation is performed with the aid of a direct method for locating the equilibria and by an adaptive grid method for constructing bifurcation diagrams. Two-parameter bifurcation diagrams are constructed to show the parameter sets in which (a) spontaneous oscillations exist, (b) the equilibrium corresponding to the resting potential is stable, or (c) the equilibrium with the most depolarized potential is stable. Multiple attractors are detected near the boundary of these parameter sets. The generation of normal and abnormal pacemaking activities is elucidated via bifurcation analysis. Spontaneous oscillations appear and disappear via saddle-node homoclinic bifurcations, Hopf bifurcations, homoclinic bifurcations, or fold bifurcations of limit cycles. Limit cycles may become chaotic via period-adding cascades as the extracellular potassium concentration [K]o decreases. Pacemaking activities may also be generated when the time-independent potassium current is blocked.

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ORRELL, DAVID, and LEONARD A. SMITH. "VISUALIZING BIFURCATIONS IN HIGH DIMENSIONAL SYSTEMS: THE SPECTRAL BIFURCATION DIAGRAM." International Journal of Bifurcation and Chaos 13, no. 10 (October 2003): 3015–27. http://dx.doi.org/10.1142/s0218127403008387.

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This paper presents methods to visualize bifurcations in flows of nonlinear dynamical systems, using the Lorenz '96 systems as examples. Three techniques are considered; the first two, density and max/min diagrams, are analagous to the bifurcation diagrams used for maps, which indicate how the system's behavior changes with a control parameter. However the diagrams are generally harder to interpret than the corresponding diagrams of maps, due to the continuous nature of the flow. The third technique takes an alternative approach: by calculating the power spectrum at each value of the control parameter, a plot is produced which clearly shows the changes between periodic, quasi-periodic, and chaotic states, and reveals structure not shown by the other methods.

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Zhang, Yingying, and Yicang Zhou. "The Bifurcation of Two Invariant Closed Curves in a Discrete Model." Discrete Dynamics in Nature and Society 2018 (May 30, 2018): 1–12. http://dx.doi.org/10.1155/2018/1613709.

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A discrete population model integrated using the forward Euler method is investigated. The qualitative bifurcation analysis indicates that the model exhibits rich dynamical behaviors including the existence of the equilibrium state, the flip bifurcation, the Neimark-Sacker bifurcation, and two invariant closed curves. The conditions for existence of these bifurcations are derived by using the center manifold and bifurcation theory. Numerical simulations and bifurcation diagrams exhibit the complex dynamical behaviors, especially the occurrence of two invariant closed curves.

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Zhang, Huayong, Shengnan Ma, Tousheng Huang, Xuebing Cong, Zichun Gao, and Feifan Zhang. "Complex Dynamics on the Routes to Chaos in a Discrete Predator-Prey System with Crowley-Martin Type Functional Response." Discrete Dynamics in Nature and Society 2018 (2018): 1–18. http://dx.doi.org/10.1155/2018/2386954.

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We present in this paper an investigation on a discrete predator-prey system with Crowley-Martin type functional response to know its complex dynamics on the routes to chaos which are induced by bifurcations. Via application of the center manifold theorem and bifurcation theorems, occurrence conditions for flip bifurcation and Neimark-Sacker bifurcation are determined, respectively. Numerical simulations are performed, on the one hand, verifying the theoretical results and, on the other hand, revealing new interesting dynamical behaviors of the discrete predator-prey system, including period-doubling cascades, period-2, period-3, period-4, period-5, period-6, period-7, period-8, period-9, period-11, period-13, period-15, period-16, period-20, period-22, period-24, period-30, and period-34 orbits, invariant cycles, chaotic attractors, sub-flip bifurcation, sub-(inverse) Neimark-Sacker bifurcation, chaotic interior crisis, chaotic band, sudden disappearance of chaotic dynamics and abrupt emergence of chaos, and intermittent periodic behaviors. Moreover, three-dimensional bifurcation diagrams are utilized to study the transition between flip bifurcation and Neimark-Sacker bifurcation, and a critical case between the two bifurcations is found. This critical bifurcation case is a combination of flip bifurcation and Neimark-Sacker bifurcation, showing the nonlinear characteristics of both bifurcations.

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De Paula, Aline S., Marcelo A. Savi, Vahid Vaziri, Ekaterina Pavlovskaia, and Marian Wiercigroch. "Experimental bifurcation control of a parametric pendulum." Journal of Vibration and Control 23, no. 14 (November 1, 2015): 2256–68. http://dx.doi.org/10.1177/1077546315613237.

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The aim of the study is to maintain the desired period-1 rotation of the parametric pendulum over a wide range of the excitation parameters. Here the Time-Delayed Feedback control method is employed to suppress those bifurcations, which lead to loss of stability of the desired rotational motion. First, the nonlinear dynamic analysis is carried out numerically for the system without control. Specifically, bifurcation diagrams and basins of attractions are computed showing co-existence of oscillatory and rotary attractors. Then numerical bifurcation diagrams are experimentally validated for a typical set of the system parameters giving undesired bifurcations. Finally, the control has been implemented and investigated both numerically and experimentally showing a good qualitative agreement.

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CHAY, TERESA REE, YIN SHUI FAN, and YOUNG SEEK LEE. "BURSTING, SPIKING, CHAOS, FRACTALS, AND UNIVERSALITY IN BIOLOGICAL RHYTHMS." International Journal of Bifurcation and Chaos 05, no. 03 (June 1995): 595–635. http://dx.doi.org/10.1142/s0218127495000491.

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Biological systems offer many interesting examples of oscillations, chaos, and bifurcations. Oscillations in biology arise because most cellular processes contain feedbacks that are appropriate for generating rhythms. These rhythms are essential for regulating cellular function. In this tutorial review, we treat two interesting nonlinear dynamic processes in biology that give rise to bursting, spiking, chaos, and fractals: endogenous electrical activity of excitable cells and Ca2+ releases from the Ca2+ stores in nonexcitable cells induced by hormones and neurotransmitters. We will first show that each of these complex processes can be described by a simple, yet elegant, mathematical model. We then show how to utilize bifurcation analyses to gain a deeper insight into the mechanisms involved in the neuronal and cellular oscillations. With the bifurcating diagrams, we explain how spiking can be transformed to bursting via a complex type of dynamic structure when the key parameter in the model varies. Understanding how this parameter would affect the bifurcation structure is important in predicting and controlling abnormal biological rhythms. Although we describe two very different dynamic processes in biological rhythms, we will show that there is universality in their bifurcation structures.

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Grebogi, Celso. "Linear Scaling Laws in Bifurcations of Scalar Maps." Zeitschrift für Naturforschung A 49, no. 12 (December 1, 1994): 1207–11. http://dx.doi.org/10.1515/zna-1994-1216.

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Abstract A global scaling property for bifurcation diagrams of periodic orbits of smooth scalar maps with both one and two dimensional parameter spaces is examined. It is argued that for both parameter spaces bifurcations within a periodic window of a given scalar map are well approximated by a linear transformation of the bifurcation diagram of a canonical map.

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Nelson, M. I., and H. S. Sidhu. "Bifurcation phenomena for an oxidation reaction in a continuously stirred tank reactor. II Diabatic operation." ANZIAM Journal 45, no. 3 (January 2004): 303–26. http://dx.doi.org/10.1017/s1446181100013389.

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AbstractWe extend an investigation into the bifurcation phenomena exhibited by an oxidation reaction in an adiabatic reactor to the case of a diabatic reactor. The primary bifurcation parameter is the fuel fraction; the inflow pressure and inflow temperature are the secondary bifurcation parameters. The inclusion of heat loss in the model does not change the static steady-state bifurcation diagram; the organising centre is a pitchfork singularity for both the adiabatic and diabatic reactors. However, unlike the adiabatic reactor, Hopf bifurcations may occur in the diabatic reactor. We construct the degenerate Hopf bifurcation curve by determining the double-Hopf locus. When the steady-state and degenerate Hopf bifurcation diagrams are combined it is found that there are 23 generic steady-state diagrams over the parameter region of interest. The implications of these structures from the perspective of flammability in the CSTR are discussed.

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Chen, Qiaoling, Zhidong Teng, Junli Liu, and Feng Wang. "Codimension-2 Bifurcation Analysis and Control of a Discrete Mosquito Model with a Proportional Release Rate of Sterile Mosquitoes." Complexity 2020 (August 17, 2020): 1–18. http://dx.doi.org/10.1155/2020/3075487.

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This paper concerns a discrete wild and sterile mosquito model with a proportional release rate of sterile mosquitoes. It is shown that the discrete model undergoes codimension-2 bifurcations with 1 : 2, 1 : 3, and 1 : 4 strong resonances by applying the bifurcation theory. Some numerical simulations, including codimension-2 bifurcation diagrams, maximum Lyapunov exponents diagrams, and phase portraits, are also presented to illustrate the validity of theoretical results and display the complex dynamical behaviors. Moreover, two control strategies are applied to the model.

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Ito, Daisuke, Tetsushi Ueta, Takuji Kousaka, and Kazuyuki Aihara. "Bifurcation Analysis of the Nagumo–Sato Model and Its Coupled Systems." International Journal of Bifurcation and Chaos 26, no. 03 (March 2016): 1630006. http://dx.doi.org/10.1142/s0218127416300068.

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The Nagumo–Sato model is a simple mathematical expression of a single neuron, and it is categorized as a discrete-time hybrid dynamical system. To compute bifurcation sets in such a discrete-time hybrid dynamical system accurately, conditions for periodic solutions and bifurcations are formulated herewith as a boundary value problem, and Newton’s method is implemented to solve that problem. As the results of the analysis, the following properties are obtained: border-collision bifurcations play a dominant role in dynamical behavior of the model; chaotic regions are distinguished by tangent bifurcations; and multistable attractors are observed in its coupled system. We demonstrate several bifurcation diagrams and corresponding topological properties of periodic solutions.

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Moon, Jae-Sung, and Mark W. Spong. "Classification of periodic and chaotic passive limit cycles for a compass-gait biped with gait asymmetries." Robotica 29, no. 7 (March 23, 2011): 967–74. http://dx.doi.org/10.1017/s0263574711000178.

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In this paper we study the problem of passive walking for a compass-gait biped with gait asymmetries. In particular, we identify and classify bifurcations leading to chaos caused by the gait asymmetries because of unequal leg masses. We present bifurcation diagrams showing step period versus the ratio of leg masses at various walking slopes. The cell mapping method is used to find stable limit cycles as the parameters are varied. It is found that a variety of bifurcation diagrams can be grouped into six stages that consist of three expanding and three contracting stages. The analysis of each stage shows that marginally stable limit cycles exhibit period-doubling, period-remerging, and saddle-node bifurcations. We also show qualitative changes regarding chaos, i.e., generation and extinction of chaos follow cyclic patterns in passive dynamic walking.

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Li, Xiaodong, Weipeng Zhang, Fengjie Geng, and Jicai Huang. "The Twisting Bifurcations of Double Homoclinic Loops with Resonant Eigenvalues." Abstract and Applied Analysis 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/152518.

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The twisting bifurcations of double homoclinic loops with resonant eigenvalues are investigated in four-dimensional systems. The coexistence or noncoexistence of large 1-homoclinic orbit and large 1-periodic orbit near double homoclinic loops is given. The existence or nonexistence of saddle-node bifurcation surfaces is obtained. Finally, the complete bifurcation diagrams and bifurcation curves are also given under different cases. Moreover, the methods adopted in this paper can be extended to a higher dimensional system.

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Chen, K. T., K. F. Yarn, H. Y. Chen, C. C. Tsai, W. J. Luo, and C. N. Chen. "Aspect Ratio Effect on Laminar Flow Bifurcations in a Curved Rectangular Tube Driven by Pressure Gradients." Journal of Mechanics 33, no. 6 (October 30, 2017): 831–40. http://dx.doi.org/10.1017/jmech.2017.93.

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AbstractThis study investigated the flow bifurcations of flows driven by a pressure gradient in a rectangular curved tube. When fluid flows within a curved tube, due to the centrifugal effect, secondary vortices can be induced in the cross section of the tube. The secondary flow states are dependent on the magnitude of the pressure gradient (q) and the aspect ratio (γ). In this study, the continuation method was applied to investigate the flow bifurcations in a curved tube with increasing pressure gradient (1 < q < 6000) and aspect ratio (0.9 < γ < 1.4).The bifurcation diagrams are composed of solution branches, which are linked by limiting points or bifurcation points. The flow states in a solution branch belong to the same group. The ranges of the flow states and the relationship between the states can also be derived from the bifurcation diagrams. In this study, two types of bifurcation were found, one in the range of 0.9 < γ < 1.17, and another in the range of 1.18 < γ < 1.4. The ranges of stable flow solutions and the distributions of limit and bifurcation points in both pressure gradient and aspect ratio are derived in this study.

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Zhang, Tiansi, and Dianli Zhao. "Bifurcation of an Orbit Homoclinic to a Hyperbolic Saddle of a Vector Field inR4." Discrete Dynamics in Nature and Society 2015 (2015): 1–6. http://dx.doi.org/10.1155/2015/571838.

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We perform a bifurcation analysis of an orbit homoclinic to a hyperbolic saddle of a vector field inR4. We give an expression of the gap between returning points in a transverse section by renormalizing system, through which we find the existence of homoclinic-doubling bifurcation in the case1+α>β>ν. Meanwhile, after reparametrizing the parameter, a periodic-doubling bifurcation appears and may be close to a saddle-node bifurcation, if the parameter is varied. These scenarios correspond to the occurrence of chaos. Based on our analysis, bifurcation diagrams of these bifurcations are depicted.

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KHIBNIK, ALEXANDER I., DIRK ROOSE, and LEON O. CHUA. "ON PERIODIC ORBITS AND HOMOCLINIC BIFURCATIONS IN CHUA’S CIRCUIT WITH A SMOOTH NONLINEARITY." International Journal of Bifurcation and Chaos 03, no. 02 (April 1993): 363–84. http://dx.doi.org/10.1142/s021812749300026x.

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We present the bifurcation analysis of Chua’s circuit equations with a smooth nonlinearity, described by a cubic polynomial. Our study focuses on phenomena that can be observed directly in the numerical simulation of the model, and on phenomena which are revealed by a more elaborate analysis based on continuation techniques and bifurcation theory. We emphasize how a combination of these approaches actually works in practice. We compare the dynamics of Chua’s circuit equations with piecewise-linear and with smooth nonlinearity. The dynamics of these two variants are similar, but we also present some differences. We conjecture that this similarity is due to the central role of homoclinicity in this model. We describe different ways in which the type of a homoclinic bifurcation influences the behavior of branches of periodic orbits. We present an overview of codimension 1 bifurcation diagrams for principal periodic orbits near homoclinicity for three-dimensional systems, both in the generic case and in the case of odd symmetry. Most of these diagrams actually occurs in the model. We found several homoclinic bifurcations of codimension 2, related to the so called resonant conditions. We study one of these bifurcations, a double neutral saddle loop.

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Sen, Moitri, and Malay Banerjee. "Rich Global Dynamics in a Prey–Predator Model with Allee Effect and Density Dependent Death Rate of Predator." International Journal of Bifurcation and Chaos 25, no. 03 (March 2015): 1530007. http://dx.doi.org/10.1142/s0218127415300074.

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In this work we have considered a prey–predator model with strong Allee effect in the prey growth function, Holling type-II functional response and density dependent death rate for predators. It presents a comprehensive study of the complete global dynamics for the considered system. Especially to see the effect of the density dependent death rate of predator on the system behavior, we have presented the two parametric bifurcation diagrams taking it as one of the bifurcation parameters. In course of that we have explored all possible local and global bifurcations that the system could undergo, namely the existence of transcritical bifurcation, saddle node bifurcation, cusp bifurcation, Hopf-bifurcation, Bogdanov–Takens bifurcation and Bautin bifurcation respectively.

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SIETTOS, CONSTANTINOS I., IOANNIS G. KEVREKIDIS, and DIMITRIOS MAROUDAS. "COARSE BIFURCATION DIAGRAMS VIA MICROSCOPIC SIMULATORS: A STATE-FEEDBACK CONTROL-BASED APPROACH." International Journal of Bifurcation and Chaos 14, no. 01 (January 2004): 207–20. http://dx.doi.org/10.1142/s0218127404009193.

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We present and illustrate a feedback control-based framework that enables microscopic/stochastic simulators to trace their "coarse" bifurcation diagrams, characterizing the dependence of their expected dynamical behavior on parameters. The framework combines the so-called "coarse time stepper" and arc-length continuation ideas from numerical bifurcation theory with linear dynamic feedback control. An augmented dynamical system is formulated, in which the bifurcation parameter evolution is linked with the microscopic simulation dynamics through feedback laws. The augmentation stably steers the system along both stable and unstable portions of the open-loop bifurcation diagram. The framework is illustrated using kinetic Monte Carlo simulations of simple surface reaction schemes that exhibit both coarse regular turning points and coarse Hopf bifurcations.

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Miino, Yuu, and Tetsushi Ueta. "Influence of Bifurcation Structures Revealed by Refinement of a Nonlinear Conductance in JosephsonJunction Element." Complexity 2018 (December 2, 2018): 1–10. http://dx.doi.org/10.1155/2018/8931525.

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We conduct a bifurcation analysis of a single-junction superconducting quantum interferometer with an external flux. We approximate the current-voltage characteristics of the conductance in the equivalent circuit of the JJ by using two types of functions: a linear function and a piecewise linear (PWL) function. We describe a method to compute the local stability of the solution orbit and to solve the bifurcation problem. As a result, we reveal the bifurcation structure of the systems in a two-dimensional parameter plane. By making a comparison between the linear and PWL cases, we find a difference in the shapes of their bifurcation sets in the two-dimensional parameter plane even though there are no differences in the one-dimensional bifurcation diagrams or the trajectories. As for the influence of piecewise linearization, we discovered that grazing bifurcations terminate the calculation of the local bifurcations, because they drastically change the stability of the periodic orbit.

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KOLOKOLOV, YURY, and ANNA MONOVSKAYA. "ESTIMATING THE UNCERTAINTY OF THE BEHAVIOR OF A PWM POWER CONVERTER BY ANALYZING A SET OF EXPERIMENTAL BIFURCATION DIAGRAMS." International Journal of Bifurcation and Chaos 23, no. 04 (April 2013): 1350063. http://dx.doi.org/10.1142/s0218127413500636.

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One of the main faults of a PWM power converter is linked to losing the operating process stability because of bifurcations. A bifurcation diagram contains information on the evolution of the behavior of a PWM power converter that could be theoretically used to prevent the bifurcations. Surprisingly, applying the bifurcation analysis is not yet typical in engineering practice. One of the reasons seems to be in the fundamental properties of the PWM power converter dynamics caused by the unavoidable uncertainty of its behavior near a bifurcation point. We propose a new approach to estimating this uncertainty. By analyzing a set of experimental bifurcation diagrams, our approach allows to determine both the location of the uncertainty zone and the quantitative regularities of the behavior within this zone. The proposed approach can be applied to design and maintenance, including fault diagnosis, and also to scientific research of the nonlinear dynamics regularities. Our results are experimentally verified by using the "PWM DC drive" setup.

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Armbruster, D., and G. Dangelmayr. "Coupled stationary bifurcations in non-flux boundary value problems." Mathematical Proceedings of the Cambridge Philosophical Society 101, no. 1 (January 1987): 167–92. http://dx.doi.org/10.1017/s0305004100066500.

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AbstractCoupled stationary bifurcations in nonlinear operator equations for functions, which are defined on a real interval with non-flux boundary conditions at the ends, are analysed in the framework of imperfect bifurcation theory. The bifurcation equations resulting from a Lyapunov–Schmidt reduction possess a natural structure which can be obtained by taking real parts of a diagonal action in ℂ2 of the symmetry group 0(2). A complete unfolding theory is developed and bifurcation equations are classified up to codimension two. Structurally stable bifurcation diagrams are given and their dependence on the wave numbers of the unstable modes is clarified.

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FUJIMOTO, KEN'ICHI, and TETSUYA YOSHINAGA. "BIFURCATIONS IN AN ITERATIVE OPTIMIZATION PROCESS FOR INTENSITY MODULATED RADIATION THERAPY." International Journal of Bifurcation and Chaos 19, no. 03 (March 2009): 1087–95. http://dx.doi.org/10.1142/s0218127409023512.

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We investigate bifurcations in an iterative method for optimizing an objective function derived from the intensity modulated radiation therapy (IMRT). Through the bifurcation analysis of discrete dynamical systems for small IMRT plans, we obtained bifurcation diagrams showing suitable parameter values for the convergence of fixed points corresponding to global or local minima of the objectives, and we found that rich nonlinear phenomena including a chaotic state can occur. We also illustrate that a similar bifurcation phenomenon occurs in a large IMRT system. The results on bifurcation structure are useful for suggesting the parameter design of IMRT plans with normal operation.

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BUZZI, CLAUDIO AGUINALDO, TIAGO DE CARVALHO, and MARCO ANTONIO TEIXEIRA. "ON THREE-PARAMETER FAMILIES OF FILIPPOV SYSTEMS — THE FOLD–SADDLE SINGULARITY." International Journal of Bifurcation and Chaos 22, no. 12 (December 2012): 1250291. http://dx.doi.org/10.1142/s0218127412502914.

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This paper presents results concerning bifurcations of 2D piecewise-smooth vector fields. In particular, the generic unfoldings of codimension-three fold–saddle singularities of Filippov systems, where a boundary-saddle and a fold coincide, are considered and the bifurcation diagrams exhibited.

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SRINIVASAN, K., K. THAMILMARAN, and A. VENKATESAN. "CLASSIFICATION OF BIFURCATIONS AND CHAOS IN CHUA'S CIRCUIT WITH EFFECT OF DIFFERENT PERIODIC FORCES." International Journal of Bifurcation and Chaos 19, no. 06 (June 2009): 1951–73. http://dx.doi.org/10.1142/s0218127409023846.

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We study the effect of different periodic excitations like sine, square, triangle and sawtooth waves on Chua's circuit and show that the circuit can undergo distinctly modified bifurcation structure, generation of new periodic regimes, induction of crises and so on. In particular, we point out that under the influence of different periodic excitations, a rich variety of bifurcation phenomena, including the familiar period-doubling sequence, intermittent route to chaos and period-adding sequences, reverse bifurcations, remerging chaotic band attractors, a large number of coexisting periodic attractors exist in the system. The analysis is carried out numerically using phase portraits, two-parameter phase diagrams in the forcing amplitude-frequency plane and one-parameter bifurcation diagrams. The chaotic dynamics of this circuit is also realized experimentally.

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Li, Songtao, Qunhong Li, and Zhongchuan Meng. "Dynamic Behaviors of a Two-Degree-of-Freedom Impact Oscillator with Two-Sided Constraints." Shock and Vibration 2021 (April 1, 2021): 1–14. http://dx.doi.org/10.1155/2021/8854115.

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The dynamic model of a vibroimpact system subjected to harmonic excitation with symmetric elastic constraints is investigated with analytical and numerical methods. The codimension-one bifurcation diagrams with respect to frequency of the excitation are obtained by means of the continuation technique, and the different types of bifurcations are detected, such as grazing bifurcation, saddle-node bifurcation, and period-doubling bifurcation, which predicts the complexity of the system considered. Based on the grazing phenomenon obtained, the zero-time-discontinuity mapping is extended from the single constraint system presented in the literature to the two-sided elastic constraint system discussed in this paper. The Poincare mapping of double grazing periodic motion is derived, and this compound mapping is applied to obtain the existence conditions of codimension-two grazing bifurcation point of the system. According to the deduced theoretical result, the grazing curve and the codimension-two grazing bifurcation points are validated by numerical simulation. Finally, various types of periodic-impact motions near the codimension-two grazing bifurcation point are illustrated through the unfolding diagram and phase diagrams.

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XU, YANCONG, DEMING ZHU, and FENGJIE GENG. "CODIMENSION 3 HETEROCLINIC BIFURCATIONS WITH ORBIT AND INCLINATION FLIPS IN REVERSIBLE SYSTEMS." International Journal of Bifurcation and Chaos 18, no. 12 (December 2008): 3689–701. http://dx.doi.org/10.1142/s0218127408022652.

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Heteroclinic bifurcations with orbit-flips and inclination-flips are investigated in a four-dimensional reversible system by using the method originally established in [Zhu, 1998; Zhu & Xia, 1998]. The existence and coexistence of R-symmetric homoclinic orbit and R-symmetric heteroclinic orbit, R-symmetric homoclinic orbit and R-symmetric periodic orbit are obtained. The double R-symmetric homoclinic bifurcation is found, and the continuum of R-symmetric periodic orbits accumulating into a homoclinic orbit is also demonstrated. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation diagrams are drawn.

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BRØNS, MORTEN, LARS KØLLGAARD VOIGT, and JENS NØRKÆR SØRENSEN. "Streamline topology of steady axisymmetric vortex breakdown in a cylinder with co- and counter-rotating end-covers." Journal of Fluid Mechanics 401 (December 25, 1999): 275–92. http://dx.doi.org/10.1017/s0022112099006588.

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Using a combination of bifurcation theory for two-dimensional dynamical systems and numerical simulations, we systematically determine the possible flow topologies of the steady vortex breakdown in axisymmetric flow in a cylindrical container with rotating end-covers. For fixed values of the ratio of the angular velocities of the covers in the range from −0.02 to 0.05, bifurcations of recirculating bubbles under variation of the aspect ratio of the cylinder and the Reynolds number are found. Bifurcation curves are determined by a simple fitting procedure of the data from the simulations. For the much studied case of zero rotation ratio (one fixed cover) a complete bifurcation diagram is constructed. Very good agreement with experimental results is obtained, and hitherto unresolved details are determined in the parameter region where up to three bubbles exist. For non-zero rotation ratios the bifurcation diagrams are found to change dramatically and give rise to other types of bifurcations.

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HUANG, JICAI, YIJUN GONG, and JING CHEN. "MULTIPLE BIFURCATIONS IN A PREDATOR–PREY SYSTEM OF HOLLING AND LESLIE TYPE WITH CONSTANT-YIELD PREY HARVESTING." International Journal of Bifurcation and Chaos 23, no. 10 (October 2013): 1350164. http://dx.doi.org/10.1142/s0218127413501642.

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The bifurcation analysis of a predator–prey system of Holling and Leslie type with constant-yield prey harvesting is carried out in this paper. It is shown that the model has a Bogdanov–Takens singularity (cusp case) of codimension at least 4 for some parameter values. Various kinds of bifurcations, such as saddle-node bifurcation, Hopf bifurcation, repelling and attracting Bogdanov–Takens bifurcations of codimensions 2 and 3, are also shown in the model as parameters vary. Hence, there are different parameter values for which the model has a limit cycle, a homoclinic loop, two limit cycles, or a limit cycle coexisting with a homoclinic loop. These results present far richer dynamics compared to the model with no harvesting. Numerical simulations, including the repelling and attracting Bogdanov–Takens bifurcation diagrams and corresponding phase portraits, and the existence of two limit cycles or an unstable limit cycle enclosing a stable multiple focus with multiplicity one, are also given to support the theoretical analysis.

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Farnoosh, Rahman, and Mahmood Parsamanesh. "Disease extinction and persistence in a discrete-time sis epidemic model with vaccination and varying population size." Filomat 31, no. 15 (2017): 4735–47. http://dx.doi.org/10.2298/fil1715735f.

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A discrete-time SIS epidemic model with vaccination is introduced and formulated by a system of difference equations. Some necessary and sufficient conditions for asymptotic stability of the equilibria are obtained. Furthermore, a sufficient condition is also presented. Next, bifurcations of the model including transcritical bifurcation, period-doubling bifurcation, and the Neimark-Sacker bifurcation are considered. In addition, these issues will be studied for the corresponding model with constant population size. Dynamics of the model are also studied and compared in detail with those found theoretically by using bifurcation diagrams, analysis of eigenvalues of the Jacobian matrix, Lyapunov exponents and solutions of the models in some examples.

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Zhang, Hailong, Enrong Wang, Fuhong Min, Ning Zhang, Chunyi Su, and Subhash Rakheja. "Nonlinear Dynamics Analysis of the Semiactive Suspension System with Magneto-Rheological Damper." Shock and Vibration 2015 (2015): 1–12. http://dx.doi.org/10.1155/2015/971731.

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This paper examines dynamical behavior of a nonlinear oscillator which models a quarter-car forced by the road profile. The magneto-rheological (MR) suspension system has been established, by employing the modified Bouc-Wen force-velocity (F-v) model of magneto-rheological damper (MRD). The possibility of chaotic motions in MR suspension is discovered by employing the method of nonlinear stability analysis. With the bifurcation diagrams and corresponding Lyapunov exponent (LE) spectrum diagrams detected through numerical calculation, we can observe the complex dynamical behaviors and oscillating mechanism of alternating periodic oscillations, quasiperiodic oscillations, and chaotic oscillations with different profiles of road excitation, as well as the dynamical evolutions to chaos through period-doubling bifurcations, saddle-node bifurcations, and reverse period-doubling bifurcations.

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Cao, Q. J., Y. W. Han, T. W. Liang, M. Wiercigroch, and S. Piskarev. "Multiple Buckling and Codimension-Three Bifurcation Phenomena of a Nonlinear Oscillator." International Journal of Bifurcation and Chaos 24, no. 01 (January 2014): 1430005. http://dx.doi.org/10.1142/s0218127414300055.

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In this paper, we investigate the global bifurcations and multiple bucklings of a nonlinear oscillator with a pair of strong irrational nonlinear restoring forces, proposed recently by Han et al. [2012]. The equilibrium stabilities of multiple snap-through buckling system under static loading are analyzed. It is found that complex bifurcations are exhibited of codimension-three with two parameters at the catastrophe point. The universal unfolding for the codimension-three bifurcation is also found to be equivalent to a nonlinear viscous damped system. The bifurcation diagrams and the corresponding codimension-three behaviors are obtained by employing subharmonic Melnikov functions for the existing singular closed orbits of homoclinic, tangent homoclinic, homo-heteroclinic and cuspidal heteroclinic, respectively.

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Li, Changzhi, Dhanagopal Ramachandran, Karthikeyan Rajagopal, Sajad Jafari, and Yongjian Liu. "Predicting Tipping Points in Chaotic Maps with Period-Doubling Bifurcations." Complexity 2021 (June 4, 2021): 1–10. http://dx.doi.org/10.1155/2021/9927607.

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In this paper, bifurcation points of two chaotic maps are studied: symmetric sine map and Gaussian map. Investigating the properties of these maps shows that they have a variety of dynamical solutions by changing the bifurcation parameter. Sine map has symmetry with respect to the origin, which causes multistability in its dynamics. The systems’ bifurcation diagrams show various dynamics and bifurcation points. Predicting bifurcation points of dynamical systems is vital. Any bifurcation can cause a huge wanted/unwanted change in the states of a system. Thus, their predictions are essential in order to be prepared for the changes. Here, the systems’ bifurcations are studied using three indicators of critical slowing down: modified autocorrelation method, modified variance method, and Lyapunov exponent. The results present the efficiency of these indicators in predicting bifurcation points.

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Mulugeta, Biruk Tafesse, Liping Yu, and Jingli Ren. "Bifurcation Analysis of a One-Prey and Two-Predators Model with Additional Food and Harvesting Subject to Toxicity." International Journal of Bifurcation and Chaos 31, no. 06 (May 2021): 2150089. http://dx.doi.org/10.1142/s0218127421500899.

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In this paper, a three-dimensional one-prey and two-predators model, with additional food and harvesting in the presence of toxicity is proposed. Additional food is being provided to one predator. The dynamics and bifurcations of the system are investigated using center manifold theorem, normal form theory and Sotomayor’s theorem. It is proved that the system undergoes transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, generalized Hopf bifurcation, Bogdanov–Takens bifurcation and cusp bifurcation with respect to different parameters. Bifurcation diagrams of the system with respect to toxic effect and harvesting effect are illustrated. The phase portraits and solution curves are also presented to verify the dynamic behavior. The results show that the combined effect of the factors has the power of transforming simple ecosystems into complex ecosystems.

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MANOEL, MIRIAM, and IAN STEWART. "DEGENERATE BIFURCATIONS WITH Z2⊕Z2-SYMMETRY." International Journal of Bifurcation and Chaos 09, no. 08 (August 1999): 1653–67. http://dx.doi.org/10.1142/s0218127499001140.

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Bifurcation problems with the symmetry group Z2⊕Z2 of the rectangle are common in applied science, for example, whenever a Euclidean invariant PDE is posed on a rectangular domain. In this work we derive normal forms for one-parameter bifurcations of steady states with symmetry of the group Z2⊕Z2. We study degeneracies of Z2⊕Z2-codimension 3 and modality 1. We also deduce persistent bifurcation diagrams when the system is subject to symmetry-preserving perturbations.

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Wu, Xiaoqin P. "Simple-Zero and Double-Zero Singularities of a Kaldor-Kalecki Model of Business Cycles with Delay." Discrete Dynamics in Nature and Society 2009 (2009): 1–29. http://dx.doi.org/10.1155/2009/923809.

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We study the Kaldor-Kalecki model of business cycles with delay in both the gross product and the capital stock. Simple-zero and double-zero singularities are investigated when bifurcation parameters change near certain critical values. By performing center manifold reduction, the normal forms on the center manifold are derived to obtain the bifurcation diagrams of the model such as Hopf, homoclinic and double limit cycle bifurcations. Some examples are given to confirm the theoretical results.

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De Dier, B., F. Walraven, R. Janssen, P. Van Rompay, and V. Hlavacek. "Bifurcation and Stability Analysis of a One-Dimensional Diffusion-Autocatalytic Reaction System." Zeitschrift für Naturforschung A 42, no. 9 (September 1, 1987): 994–1004. http://dx.doi.org/10.1515/zna-1987-0912.

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Results of a numerical analysis of a set of one-dimensional reaction -diffusion equations are presented. The basis of these equations is a model scheme of chemical reactions, involving auto-and cross-catalytic steps (“Brusselator”). The steady state problem is solved numerically, fully exploiting the properties of recently developed continuation codes. Bifurcation diagrams are constructed for zero flux boundary conditions. For a relatively large diffusivity of initial species the Brusselator displays a huge number of dissipative steady state structures. At low system lengths a mechanism of perturbed bifurcation may be percieved. Bifurcations coincide with turning points of asymmetric solution branches. Completely isolated solutions prove to exist as well. For the problem without limited diffusion of the initial species, a careful bifurcation analysis show s the existence of a number of higher order bifurcations. At some of these points asymmetric profiles emanate from other asymmetric structures. Bifurcation points and limit points do not necessarily coincide. Stability analysis shows that relatively few steady states are stable. Especially symmetric solutions are found to be stable.

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Journal articles on the topic 'Diagrammes de bifurcation'

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