Journal articles: 'Bifurcation and Chaos theory'

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Yang, Ting. "Multistability and Hidden Attractors in a Three-Dimensional Chaotic System." International Journal of Bifurcation and Chaos 30, no. 06 (May 2020): 2050087. http://dx.doi.org/10.1142/s021812742050087x.

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This paper proposes a novel three-dimensional autonomous chaotic system. Interestingly, when the system has infinitely many stable equilibria, it is found that the system also has infinitely many hidden chaotic attractors. We show that the period-doubling bifurcations are the routes to chaos. Moreover, the Hopf bifurcations at all equilibria are investigated and it is also found that all the Hopf bifurcations simultaneously occur. Furthermore, the approximate expressions and stabilities of bifurcating limit cycles are obtained by using normal form theory and bifurcation theory.

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PENG, MINGSHU, and YUAN YUAN. "STABILITY, SYMMETRY-BREAKING BIFURCATIONS AND CHAOS IN DISCRETE DELAYED MODELS." International Journal of Bifurcation and Chaos 18, no. 05 (May 2008): 1477–501. http://dx.doi.org/10.1142/s0218127408021117.

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In this paper, we use the standard bifurcation theory to study rich dynamics of time-delayed coupling discrete oscillators. Equivariant bifurcations including equivariant Neimark–Sacker bifurcation, equivariant pitchfork bifurcation and equivariant periodic doubling bifurcation are analyzed in detail. In the application, we consider a ring of identical discrete delayed Ikeda oscillators. Multiple oscillation patterns, such as multiple stable equilibria, stable limit cycles, stable invariant tori and multiple chaotic attractors, are shown.

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Chen, Qin, and Jianguo Gao. "Delay Feedback Control of the Lorenz-Like System." Mathematical Problems in Engineering 2018 (June 28, 2018): 1–13. http://dx.doi.org/10.1155/2018/1459272.

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We choose the delay as a variable parameter and investigate the Lorentz-like system with delayed feedback by using Hopf bifurcation theory and functional differential equations. The local stability of the positive equilibrium and the existence of Hopf bifurcations are obtained. After that the direction of Hopf bifurcation and stability of periodic solutions bifurcating from equilibrium is determined by using the normal form theory and center manifold theorem. In the end, some numerical simulations are employed to validate the theoretical analysis. The results show that the purpose of controlling chaos can be achieved by adjusting appropriate feedback effect strength and delay parameters. The applied delay feedback control method in this paper is general and can be applied to other nonlinear chaotic systems.

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Liu, Yongjian, Xiezhen Huang, and Jincun Zheng. "Chaos and bifurcation in the controlled chaotic system." Open Mathematics 16, no. 1 (November 3, 2018): 1255–65. http://dx.doi.org/10.1515/math-2018-0105.

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AbstractIn this paper, chaos and bifurcation are explored for the controlled chaotic system, which is put forward based on the hybrid strategy in an unusual chaotic system. Behavior of the controlled system with variable parameter is researched in detain. Moreover, the normal form theory is used to analyze the direction and stability of bifurcating periodic solution.

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XU, XU, HAIYAN HU, and HUAILEI WANG. "DYNAMICS OF A TWO-DIMENSIONAL DELAYED SMALL-WORLD NETWORK UNDER DELAYED FEEDBACK CONTROL." International Journal of Bifurcation and Chaos 16, no. 11 (November 2006): 3257–73. http://dx.doi.org/10.1142/s021812740601677x.

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This paper presents a detailed analysis on the dynamics of a two-dimensional delayed small-world network under delayed state feedback control. On the basis of stability switch criteria, the equilibrium is studied, and the stability conditions are determined. This study shows that with properly chosen delay and gain in the delayed feedback path, the controlled small-world delayed network may have stable equilibrium, or periodic solutions resulting from the Hopf bifurcation, or the multistability solutions via three types of codimension two bifurcations. Moreover, the direction of Hopf bifurcation and stability of the bifurcation periodic solutions are determined by using the normal form theory and center manifold theorem. In addition, the study shows that the controlled model exhibits period-doubling bifurcations which lead eventually to chaos; and the chaos can also directly occur via the bifurcations from the quasi-periodic solutions. The results show that the delayed feedback is an effective approach in order to generate or annihilate complex behaviors in practical applications.

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LI, CHUNGUANG, GUANRONG CHEN, XIAOFENG LIAO, and JUEBANG YU. "HOPF BIFURCATION AND CHAOS IN TABU LEARNING NEURON MODELS." International Journal of Bifurcation and Chaos 15, no. 08 (August 2005): 2633–42. http://dx.doi.org/10.1142/s0218127405013575.

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In this paper, we consider the nonlinear dynamical behaviors of some tabu learning neuron models. We first consider a tabu learning single neuron model. By choosing the memory decay rate as a bifurcation parameter, we prove that Hopf bifurcation occurs in the neuron. The stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation are determined by applying the normal form theory. We give a numerical example to verify the theoretical analysis. Then, we demonstrate the chaotic behavior in such a neuron with sinusoidal external input, via computer simulations. Finally, we study the chaotic behaviors in tabu learning two-neuron models, with linear and quadratic proximity functions respectively.

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HARRISON, MARY ANN, and YING-CHENG LAI. "BIFURCATION TO HIGH-DIMENSIONAL CHAOS." International Journal of Bifurcation and Chaos 10, no. 06 (June 2000): 1471–83. http://dx.doi.org/10.1142/s0218127400000967.

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High-dimensional chaos has been an area of growing recent investigation. The questions of how dynamical systems become high-dimensionally chaotic with multiple positive Lyapunov exponents, and what the characteristic features associated with the transition are, remain less investigated. In this paper, we present one possible route to high-dimensional chaos. By this route, a subsystem becomes chaotic with one positive Lyapunov exponent via one of the known routes to low-dimensional chaos, after which the complementary subsystem becomes chaotic, leading to additional positive Lyapunov exponents for the whole system. A characteristic feature of this route is that the additional Lyapunov exponents pass through zero smoothly. As a consequence, the fractal dimension of the chaotic attractor changes continuously through the transition, in contrast to the transition to low-dimensional chaos at which the fractal dimension changes abruptly. We present a heuristic theory and numerical examples to illustrate this route to high-dimensional chaos.

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Abbasi, Muhammad Aqib, and Qamar Din. "Under the influence of crowding effects: Stability, bifurcation and chaos control for a discrete-time predator–prey model." International Journal of Biomathematics 12, no. 04 (May 2019): 1950044. http://dx.doi.org/10.1142/s179352451950044x.

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The interaction between predators and preys exhibits more complicated behavior under the influence of crowding effects. By taking into account the crowding effects, the qualitative behavior of a prey–predator model is investigated. Particularly, we examine the boundedness as well as existence and uniqueness of positive steady-state and stability analysis of the unique positive steady-state. Moreover, it is also proved that the system undergoes Hopf bifurcation and flip bifurcation with the help of bifurcation theory. Moreover, a chaos control technique is proposed for controlling chaos under the influence of bifurcations. Finally, numerical simulations are provided to illustrate the theoretical results. These results of numerical simulations demonstrate chaotic long-term behavior over a broad range of parameters. The presence of chaotic behavior in the model is confirmed by computing maximum Lyapunov exponents.

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Chen, Qiaoling, Zhidong Teng, and Zengyun Hu. "Bifurcation and control for a discrete-time prey–predator model with Holling-IV functional response." International Journal of Applied Mathematics and Computer Science 23, no. 2 (June 1, 2013): 247–61. http://dx.doi.org/10.2478/amcs-2013-0019.

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The dynamics of a discrete-time predator-prey model with Holling-IV functional response are investigated. It is shown that the model undergoes a flip bifurcation, a Hopf bifurcation and a saddle-node bifurcation by using the center manifold theorem and bifurcation theory. Numerical simulations not only exhibit our results with the theoretical analysis, but also show the complex dynamical behaviors, such as the period-3, 6, 9, 12, 20, 63, 70, 112 orbits, a cascade of period-doubling bifurcations in period-2, 4, 8, 16, quasi-periodic orbits, an attracting invariant circle, an inverse period-doubling bifurcation from the period-32 orbit leading to chaos and a boundary crisis, a sudden onset of chaos and a sudden disappearance of the chaotic dynamics, attracting chaotic sets and non-attracting sets. We also observe that when the prey is in chaotic dynamics the predator can tend to extinction or to a stable equilibrium. Specifically, we stabilize the chaotic orbits at an unstable fixed point by using OGY chaotic control.

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Rotshtein, Alexander, Denys Katielnikov, and Ludmila Pustylnik. "Reliability Modeling and Optimization Using Fuzzy Logic and Chaos Theory." International Journal of Quality, Statistics, and Reliability 2012 (October 23, 2012): 1–9. http://dx.doi.org/10.1155/2012/847416.

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Fuzzy sets membership functions integrated with logistic map as the chaos generator were used to create reliability bifurcations diagrams of the system with redundancy of the components. This paper shows that increasing in the number of redundant components results in a postponement of the moment of the first bifurcation which is considered as most contributing to the loss of the reliability. The increasing of redundancy also provides the shrinkage of the oscillation orbit of the level of the system’s membership to reliable state. The paper includes the problem statement of redundancy optimization under conditions of chaotic behavior of influencing parameters and genetic algorithm of this problem solving. The paper shows the possibility of chaos-tolerant systems design with the required level of reliability.

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11

Barnett, William A., Apostolos Serletis, and Demitre Serletis. "NONLINEAR AND COMPLEX DYNAMICS IN ECONOMICS." Macroeconomic Dynamics 19, no. 8 (November 7, 2014): 1749–79. http://dx.doi.org/10.1017/s1365100514000091.

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This paper is an up-to-date survey of the state of the art in dynamical systems theory relevant to high levels of dynamical complexity, characterizing chaos and near-chaos, as commonly found in the physical sciences. The paper also surveys applications in economics and finance. This survey does not include bifurcation analyses at lower levels of dynamical complexity, such as Hopf and transcritical bifurcations, which arise closer to the stable region of the parameter space. We discuss the geometric approach (based on the theory of differential/difference equations) to dynamical systems and make the basic notions of complexity, chaos, and other related concepts precise, having in mind their (actual or potential) applications to economically motivated questions. We also introduce specific applications in microeconomics, macroeconomics, and finance and discuss the policy relevance of chaos.

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Din, Qamar, and Muhammad Asad Iqbal. "Bifurcation Analysis and Chaos Control for a Discrete-Time Enzyme Model." Zeitschrift für Naturforschung A 74, no. 1 (December 19, 2018): 1–14. http://dx.doi.org/10.1515/zna-2018-0254.

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AbstractBasically enzymes are biological catalysts that increase the speed of a chemical reaction without undergoing any permanent chemical change. With the application of Euler’s forward scheme, a discrete-time enzyme model is presented. Further investigation related to its qualitative behaviour revealed that discrete-time model shows rich dynamics as compared to its continuous counterpart. It is investigated that discrete-time model has a unique trivial equilibrium point. The local asymptotic behaviour of equilibrium is discussed for discrete-time enzyme model. Furthermore, with the help of the bifurcation theory and centre manifold theorem, explicit parametric conditions for directions and existence of flip and Hopf bifurcations are investigated. Moreover, two existing chaos control methods, that is, Ott, Grebogi and Yorke feedback control and hybrid control strategy, are implemented. In particular, a novel chaos control technique, based on state feedback control is introduced for controlling chaos under the influence of flip and Hopf bifurcations in discrete-time enzyme model. Numerical simulations are provided to illustrate theoretical discussion and effectiveness of newly introduced chaos control method.

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13

Guan, Junbiao. "Bifurcation Analysis and Chaos Control in Genesio System with Delayed Feedback." ISRN Mathematical Physics 2012 (February 8, 2012): 1–12. http://dx.doi.org/10.5402/2012/843962.

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We investigate the local Hopf bifurcation in Genesio system with delayed feedback control. We choose the delay as the parameter, and the occurrence of local Hopf bifurcations are verified. By using the normal form theory and the center manifold theorem, we obtain the explicit formulae for determining the stability and direction of bifurcated periodic solutions. Numerical simulations indicate that delayed feedback control plays an effective role in control of chaos.

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INABA, NAOHIKO, MUNEHISA SEKIKAWA, TETSURO ENDO, and TAKASHI TSUBOUCHI. "REVEALING THE TRICK OF TAMING CHAOS BY WEAK HARMONIC PERTURBATIONS." International Journal of Bifurcation and Chaos 13, no. 10 (October 2003): 2905–15. http://dx.doi.org/10.1142/s0218127403008272.

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Taming chaos by weak harmonic perturbations has been a hot topic in recent years. In this paper, the authors investigate a scenario for the mechanism of taming chaos via bifurcation theory, and assert that this phenomenon is caused by a slight shift of the saddle node bifurcation curves.

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JIANG, XUEWEI, DI YUAN, and YI XIAO. "CHAOTIC DYNAMICS OF A FIVE-DIMENSIONAL NONLINEAR NETWORK." International Journal of Modern Physics C 18, no. 03 (March 2007): 335–42. http://dx.doi.org/10.1142/s012918310700956x.

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The dynamics of a five-dimensional nonlinear network based on the theory of Chinese traditional medicine is studied by the Lyapunov exponent spectrum, Poincaré, power spectrum and bifurcation diagrams. The result shows that this system has complex dynamical behaviors, such as chaotic ones. It also shows that the system evolves into chaos through a series of period-doubling bifurcations.

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Chin, Char-Ming, and A. H. Nayfeh. "Bifurcation and Chaos in Externally Excited Circular Cylindrical Shells." Journal of Applied Mechanics 63, no. 3 (September 1, 1996): 565–74. http://dx.doi.org/10.1115/1.2823335.

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The nonlinear response of an infinitely long cylindrical shell to a primary excitation of one of its two orthogonal flexural modes is investigated. The method of multiple scales is used to derive four ordinary differential equations describing the amplitudes and phases of the two orthogonal modes by (a) attacking a two-mode discretization of the governing partial differential equations and (b) directly attacking the partial differential equations. The two-mode discretization results in erroneous solutions because it does not account for the effects of the quadratic nonlinearities. The resulting two sets of modulation equations are used to study the equilibrium and dynamic solutions and their stability and hence show the different bifurcations. The response could be a single-mode solution or a two-mode solution. The equilibrium solutions of the two orthogonal third flexural modes undergo a Hopf bifurcation. A combination of a shooting technique and Floquet theory is used to calculate limit cycles and their stability. The numerical results indicate the existence of a sequence of period-doubling bifurcations that culminates in chaos, multiple attractors, explosive bifurcations, and crises.

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Al-Basyouni, K. S., and A. Q. Khan. "Discrete-Time Predator-Prey Model with Bifurcations and Chaos." Mathematical Problems in Engineering 2020 (November 12, 2020): 1–14. http://dx.doi.org/10.1155/2020/8845926.

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In this paper, local dynamics, bifurcations and chaos control in a discrete-time predator-prey model have been explored in ℝ + 2 . It is proved that the model has a trivial fixed point for all parametric values and the unique positive fixed point under definite parametric conditions. By the existing linear stability theory, we studied the topological classifications at fixed points. It is explored that at trivial fixed point model does not undergo the flip bifurcation, but flip bifurcation occurs at the unique positive fixed point, and no other bifurcations occur at this point. Numerical simulations are performed not only to demonstrate obtained theoretical results but also to tell the complex behaviors in orbits of period-4, period-6, period-8, period-12, period-17, and period-18. We have computed the Maximum Lyapunov exponents as well as fractal dimension numerically to demonstrate the appearance of chaotic behaviors in the considered model. Further feedback control method is employed to stabilize chaos existing in the model. Finally, existence of periodic points at fixed points for the model is also explored.

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Yang, Jihua, Erli Zhang, and Mei Liu. "Bifurcation Analysis and Chaos Control in a Modified Finance System with Delayed Feedback." International Journal of Bifurcation and Chaos 26, no. 06 (June 15, 2016): 1650105. http://dx.doi.org/10.1142/s0218127416501054.

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We investigate the effect of delayed feedback on the finance system, which describes the time variation of the interest rate, for establishing the fiscal policy. By local stability analysis, we theoretically prove the existences of Hopf bifurcation and Hopf-zero bifurcation. By using the normal form method and center manifold theory, we determine the stability and direction of a bifurcating periodic solution. Finally, we give some numerical solutions, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable equilibrium or periodic orbit.

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Chang, Shun-Chang. "Controlling Chaos through Period-Doubling Bifurcations in Attitude Dynamics for Power Systems." Mathematical Problems in Engineering 2020 (December 4, 2020): 1–10. http://dx.doi.org/10.1155/2020/8853459.

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This paper addresses the complex nonlinear dynamics involved in controlling chaos in power systems using bifurcation diagrams, time responses, phase portraits, Poincaré maps, and frequency spectra. Our results revealed that nonlinearities in power systems produce period-doubling bifurcations, which can lead to chaotic motion. Analysis based on the Lyapunov exponent and Lyapunov dimension was used to identify the onset of chaotic behavior. We also developed a continuous feedback control method based on synchronization characteristics for suppressing of chaotic oscillations. The results of our simulation support the feasibility of using the proposed method. The robustness of parametric perturbations on a power system with synchronization control was analyzed using bifurcation diagrams and Lyapunov stability theory.

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LIU, J. T., X. D. YANG, and L. Q. CHEN. "BIFURCATIONS AND CHAOS OF AN AXIALLY MOVING PLATE UNDER EXTERNAL AND PARAMETRIC EXCITATIONS." International Journal of Structural Stability and Dynamics 12, no. 04 (July 2012): 1250023. http://dx.doi.org/10.1142/s021945541250023x.

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The chaos and bifurcations in transverse motion of an axially moving thin plate under external and parametric excitations are studied herein. The geometric nonlinearity is introduced by using the von Karman large deflection theory. The coupled partial differential equations of transverse deflection and stress are truncated into a set of ordinary differential equations. By using the Poincaré map and the largest Lyapunov exponent, the dynamical behaviors including chaos are identified based on numerical solutions of the ordinary differential equations. The bifurcation diagrams are presented for different parameters, such as axially moving velocity, damping, external and parametric excitation amplitudes. The chaos is detected in both cases of external and parametric excitations. The interesting relevance between onset of chaos with the corresponding linear instability range are indicated in the external and parametric responses.

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MAISTRENKO, YURI L., OLEKSANDR V. POPOVYCH, and PETER A. TASS. "CHAOTIC ATTRACTOR IN THE KURAMOTO MODEL." International Journal of Bifurcation and Chaos 15, no. 11 (November 2005): 3457–66. http://dx.doi.org/10.1142/s0218127405014155.

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The Kuramoto model of globally coupled phase oscillators is an essentially nonlinear dynamical system with a rich dynamics including synchronization and chaos. We study the Kuramoto model from the standpoint of bifurcation and chaos theory of low-dimensional dynamical systems. We find a chaotic attractor in the four-dimensional Kuramoto model and study its origin. The torus destruction scenario is one of the major mechanisms by which chaos arises. L. P. Shilnikov has made decisive contributions to its discovery. We show also that in the Kuramoto model the transition to chaos is in accordance with the torus destruction scenario. We present the general bifurcation diagram containing phase chaos, Cherry flow as well as periodic and quasiperiodic dynamics.

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Zhou, Liangqiang, Ziman Zhao, and Fangqi Chen. "Stability and Hopf bifurcation analysis of a new four-dimensional hyper-chaotic system." Modern Physics Letters B 34, no. 29 (July 24, 2020): 2050327. http://dx.doi.org/10.1142/s0217984920503273.

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With both analytical and numerical methods, local dynamic behaviors including stability and Hopf bifurcation of a new four-dimensional hyper-chaotic system are studied in this paper. All the equilibrium points and their stability conditions are obtained with the Routh–Hurwitz criterion. It is shown that there may exist one, two, or three equilibrium points for different system parameters. Via Hopf bifurcation theory, parameter conditions leading to Hopf bifurcation is presented. With the aid of center manifold and the first Lyapunov coefficient, it is also presented that the Hopf bifurcation is supercritical for some certain parameters. Finally, numerical simulations are given to confirm the analytical results and demonstrate the chaotic attractors of this system. It is also shown that the system may evolve chaotic motions through periodic bifurcations or intermittence chaos while the system parameters vary.

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Zhao, Huitao, Yaowei Sun, and Zhen Wang. "Control of Hopf Bifurcation and Chaos in a Delayed Lotka-Volterra Predator-Prey System with Time-Delayed Feedbacks." Abstract and Applied Analysis 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/104156.

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A delayed Lotka-Volterra predator-prey system with time delayed feedback is studied by using the theory of functional differential equation and Hassard’s method. By choosing appropriate control parameter, we investigate the existence of Hopf bifurcation. An explicit algorithm is given to determine the directions and stabilities of the bifurcating periodic solutions. We find that these control laws can be applied to control Hopf bifurcation and chaotic attractor. Finally, some numerical simulations are given to illustrate the effectiveness of the results found.

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Baydemir, Pinar, Huseyin Merdan, Esra Karaoglu, and Gokce Sucu. "Complex Dynamics of a Discrete-Time Prey–Predator System with Leslie Type: Stability, Bifurcation Analyses and Chaos." International Journal of Bifurcation and Chaos 30, no. 10 (August 2020): 2050149. http://dx.doi.org/10.1142/s0218127420501497.

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Dynamic behavior of a discrete-time prey–predator system with Leslie type is analyzed. The discrete mathematical model was obtained by applying the forward Euler scheme to its continuous-time counterpart. First, the local stability conditions of equilibrium point of this system are determined. Then, the conditions of existence for flip bifurcation and Neimark–Sacker bifurcation arising from this positive equilibrium point are investigated. More specifically, by choosing integral step size as a bifurcation parameter, these bifurcations are driven via center manifold theorem and normal form theory. Finally, numerical simulations are performed to support and extend the theoretical results. Analytical results show that an integral step size has a significant role on the dynamics of a discrete system. Numerical simulations support that enlarging the integral step size causes chaotic behavior.

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Talla, F. Calvin, Robert Tchitnga, P. H. Louodop Fotso, Romanic Kengne, Bonaventure Nana, and Anaclet Fomethe. "Unexpected Behaviors in a Single Mesh Josephson Junction Based Self-Reproducing Autonomous System." International Journal of Bifurcation and Chaos 30, no. 07 (June 15, 2020): 2050097. http://dx.doi.org/10.1142/s0218127420500972.

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In the literature, existing Josephson junction based oscillators are mostly driven by external sources. Knowing the different limits of the external driven systems, we propose in this work a new autonomous one that exhibits the unusual and striking multiple phenomena among which coexist the multiple hidden attractors in self-reproducing process under the effect of initial conditions. The eight-term autonomous chaotic system has a single nonlinearity of sinusoidal type acting on only one of the state variables. A priori, the simplicity of the system does not predict the richness of its dynamics. We also find that a limit cycle attractor widens to a parameter controlling coexisting multiple-scroll attractors through the splitting and the inverse splitting of periods. Multiple types of bifurcations are found including period-doubling and period-splitting (antimonotonicity) sequences to chaos, crisis and Hopf type bifurcation. To the best of our knowledge, some of these interesting phenomena have not yet been reported in similar class of autonomous Josephson junction based circuits. Moreover, analytical investigations based on the Hopf theory analysis lead to the expressions that determine the direction of appearance of the Hopf bifurcation, confirming the existence and determining the stability of bifurcating periodic solutions. To observe this latter bifurcation and to illustrate the theoretical analysis, numerical simulations are performed. Chaos can be easily controlled by the frequency of the linear oscillator, the superconducting junction current, as well as the gain of the amplifier or circuit component values. The circuit and Field Programmable Gate Arrays (FPGA)-based implementation of the system are presented as well.

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Din, Qamar, A. A. Elsadany, and Samia Ibrahim. "Bifurcation Analysis and Chaos Control in a Second-Order Rational Difference Equation." International Journal of Nonlinear Sciences and Numerical Simulation 19, no. 1 (February 23, 2018): 53–68. http://dx.doi.org/10.1515/ijnsns-2017-0077.

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AbstractThis work is related to dynamics of a second-order rational difference equation. We investigate the parametric conditions for local asymptotic stability of equilibria. Center manifold theorem and bifurcation theory are implemented to discuss the parametric conditions for existence and direction of period-doubling bifurcation and pitchfork bifurcation at trivial equilibrium point. Moreover, the parametric conditions for existence and direction of Neimark–Sacker bifurcation at positive steady state are investigated with the help of bifurcation theory. The chaos control in the system is discussed through implementation of OGY feedback control method. In particular, we stabilize the chaotic orbits at an unstable fixed point by using OGY chaotic control. Finally, numerical simulations are provided to illustrate theoretical results. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the system.

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Din, Qamar. "Stability, bifurcation analysis and chaos control for a predator-prey system." Journal of Vibration and Control 25, no. 3 (August 12, 2018): 612–26. http://dx.doi.org/10.1177/1077546318790871.

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We study qualitative behavior of a modified prey–predator model by introducing density-dependent per capita growth rates and a Holling type II functional response. Positivity of solutions, boundedness and local asymptotic stability of equilibria were investigated for continuous type of the prey–predator system. In order to discuss the rich dynamics of the proposed model, a piecewise constant argument was implemented to obtain a discrete counterpart of the continuous system. Moreover, in the case of a discrete-time prey–predator model, the boundedness of solutions and local asymptotic stability of equilibria were investigated. With the help of the center manifold theorem and bifurcation theory, we investigated whether a discrete-time model undergoes period-doubling and Neimark–Sacker bifurcation at its positive steady-state. Finally, two novel generalized hybrid feedback control methods are presented for chaos control under the influence of period-doubling and Neimark–Sacker bifurcations. In order to illustrate the effectiveness of the proposed control strategies, numerical simulations are presented.

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Rojas, Raul. "Chaos als neues naturwissenschafliches Paradigma." PROKLA. Zeitschrift für kritische Sozialwissenschaft 22, no. 88 (September 1, 1992): 374–87. http://dx.doi.org/10.32387/prokla.v22i88.1060.

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Tue author describes the theory of chaos as a scientific revolution which reaches beyond the microphysical indeterminism of Quantum Mechanics by abolishing determinism also on the macroscopical level. With a simple non-linear system the notions bifurcation, strange attractor, universality of chaos, self-organization and criticality are explained and illustrated through examples.

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Zhang, Hai Long, Ning Zhang, Fu Hong Min, and En Rong Wang. "Analysis on Chaotic Vibrations of the Magneto-Rheological Suspension System." Applied Mechanics and Materials 826 (February 2016): 28–34. http://dx.doi.org/10.4028/www.scientific.net/amm.826.28.

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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 nonlinear systems stability theory. With the bifurcation diagram and corresponding Lyapunov exponent spectrum diagrams detected through numerical calculation, we can observe the complex dynamical behaviors and oscillating mechanism of alternating periodic oscillations, quasi-periodic oscillations and chaotic oscillations with different profiles of road excitation, as well as the dynamical evolution to chaos by period-doubling bifurcations, saddle-node bifurcations and reverse period-doubling bifurcations.

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Paidoussis, M. P., G. X. Li, and R. H. Rand. "Chaotic Motions of a Constrained Pipe Conveying Fluid: Comparison Between Simulation, Analysis, and Experiment." Journal of Applied Mechanics 58, no. 2 (June 1, 1991): 559–65. http://dx.doi.org/10.1115/1.2897220.

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A refined analytical model is presented for the dynamics of a cantilevered pipe conveying fluid and constrained by motion limiting restraints. Calculations with the discretized form of this model with a progressively increasing number of degrees of freedom, N, show that convergence is achieved with N = 4 or 5, which agrees with previously performed fractal dimension calculations of experimental data. Theory shows that, beyond the Hopf bifurcation, as the flow is increased, a pitchfork bifurcation is followed by a cascade of period doubling bifurcations leading to chaos, which is in qualitative agreement with observation. The numerically computed theoretical critical flow velocities are in excellent quantitative agreement (5–10 percent) with experimental values for the thresholds of the Hopf and period doubling bifurcations and for the onset of chaos. An approximation for the critical flow velocity for the loss of stability of the post-Hopf limit cycle is also obtained by using center manifold concepts and normal form techniques for a simplified version of the analytical model; it is found that the values obtained in this manner are approximately within 10 percent of those computed numerically.

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Pellegrini, L., C. Tablino Possio, and G. Biardi. "An Example of How Nonlinear Dynamics Tools Can be Successfully Applied to A Chemical System." Fractals 05, no. 03 (September 1997): 531–47. http://dx.doi.org/10.1142/s0218348x97000425.

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The nonlinear dynamic behavior of a Proportional-Integral controlled Continuously Stirred Tank Reactor (CSTR) is analyzed in depth progressing from chaos characterization, through the high codimension bifurcation theory, up to the application of Controlling Chaos techniques. All these tools can be successfully applied to recognize, to avoid and to use chaos in practical applications, so that the nonlinear dynamic theory turns out to be an indispensable science to constrain dynamic systems to work in the most suitable operative conditions.

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Feng, Guo, Ding Yin, and Li Jiacheng. "Neimark–Sacker Bifurcation and Controlling Chaos in a Three-Species Food Chain Model through the OGY Method." Discrete Dynamics in Nature and Society 2021 (June 25, 2021): 1–13. http://dx.doi.org/10.1155/2021/6316235.

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The dynamics behavior of a discrete-time three-species food chain model is investigated. By using bifurcation theory, it is shown that the equilibrium point of the system loses its stability, and the system undergoes Neimark–Sacker bifurcation, which leads to chaos as the parameter changes. The chaotic motion is controlled on the stable periodic period-1 orbit using the implementation of the hybrid control strategy. The factor affecting the control time of chaos is also studied. Numerical simulations are consistent with the theoretical analysis. The results of this research prove that the chaos control method can be extended to the higher-dimensional biological model and can be realized.

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Arafa, Ayman A., Yong Xu, and Gamal M. Mahmoud. "Chaos Suppression via Integrative Time Delay Control." International Journal of Bifurcation and Chaos 30, no. 14 (November 2020): 2050208. http://dx.doi.org/10.1142/s0218127420502089.

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A general strategy for suppressing chaos in chaotic Burke–Shaw system using integrative time delay (ITD) control is proposed, as an example. The idea of ITD is that the feedback is integrated over a time interval. Physically, the chaotic system responds to the average information it receives from the feedback. The main feature of integrative is that the stability of the chaotic system occurs over a wider range of the space parameters. Controlling chaotic systems with ITD has not been discussed before as far as we know. Stability and the existence of Hopf bifurcation are studied which demonstrate that the switch stability occurs at critical values of the time delay. Employing the normal form theory and center manifold argument, an explicit formula is derived to determine the stability and the direction of the bifurcating periodic solutions. Numerically, the bifurcation diagram and the eigenvalues of the corresponding characteristic equations are computed to supply a clear interpretation for suppressing chaos via ITD. Furthermore, ITD method is compared with the time delayed feedback (TDF) control numerically. This comparison shows that the stability area with ITD is larger than TDF which demonstrates the feasibility and effectiveness of the ITD. Other examples of chaotic systems can be similarly investigated.

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Yang, Ting. "Dynamical Analysis on a Finance System with Nonconstant Elasticity of Demand." International Journal of Bifurcation and Chaos 30, no. 10 (August 2020): 2050148. http://dx.doi.org/10.1142/s0218127420501485.

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This paper investigates a finance system with nonconstant elasticity of demand. First, under some conditions, the system has invariant algebraic surfaces and the analytic expressions of the surfaces are given. Furthermore, when the two surfaces coincide and become one surface, the dynamics on the surface are analyzed and a globally stable equilibrium is found. Second, by using the normal form theory, the Hopf bifurcation is studied and the approximate expression and stability of the bifurcating periodic orbit are obtained. Third, the chaotic behaviors are investigated and the route to chaos is period-doubling bifurcations. Moreover, it is found that the system has coexisting attractors, including periodic attractor and periodic attractor, chaotic attractor and chaotic attractor. With the change of parameter, the two chaotic attractors coincide and then a symmetrical chaotic attractor arises.

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XIAO, MIN, and JINDE CAO. "BIFURCATION ANALYSIS AND CHAOS CONTROL FOR LÜ SYSTEM WITH DELAYED FEEDBACK." International Journal of Bifurcation and Chaos 17, no. 12 (December 2007): 4309–22. http://dx.doi.org/10.1142/s0218127407019974.

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Time-delayed feedback has been introduced as a powerful tool to control unstable periodic orbits or control unstable steady states. In the present paper, regarding the delay as a parameter, we investigate the effect of delay on the dynamics of Lü system with delayed feedback. After the effect of delay on the steady states is analyzed, Hopf bifurcation is studied, where the direction, stability and other properties of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Finally, we provide several numerical simulations, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a stable periodic orbit.

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Getchell, Morgan. "Chaos Theory and Emergent Behavior in the West Virginia Water Crisis." Journal of International Crisis and Risk Communication Research 1, no. 2 (October 15, 2018): 173–200. http://dx.doi.org/10.30658/jicrcr.1.2.1.

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Chaos theory holds that systems act in unpredictable, nonlinear ways and that their behavior can only be observed, never predicted. This is an informative model for an organization in crisis. The West Virginia water contamination crisis, which began on January 9, 2014, fits the criteria of a system in chaos. This study employs a close case study method to examine this case through the lens of chaos theory and its tenets: sensitivity to initial conditions, bifurcation, fractals, strange attractors, and self-organization. In particular, close attention is paid to emergent organizations and how their embodiment of strange attractor values spurred the self-organization process for this chaotic system.

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Wang, Jingyue, Haotian Wang, and Tie Wang. "External Periodic Force Control of a Single-Degree-of-Freedom Vibroimpact System." Journal of Control Science and Engineering 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/570137.

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A single-degree-of-freedom mechanical model of vibro-impact system is established. Bifurcation and chaos in the system are revealed with the time history diagram, phase trajectory map, and Poincaré map. According to the bifurcation and chaos of the actual vibro-impact system, the paper puts forward external periodic force control strategy. The method of controlling chaos by external periodic force feedback controller is developed to guide chaotic motions towards regular motions. The stability of the control system is also analyzed especially by theory. By selecting appropriate feedback coefficients, the unstable periodic orbits of the original chaotic orbit can be stabilized to the stable periodic orbits. The effectiveness of this control method is verified by numerical simulation.

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Ray, Arnob, and Dibakar Ghosh. "Another New Chaotic System: Bifurcation and Chaos Control." International Journal of Bifurcation and Chaos 30, no. 11 (September 15, 2020): 2050161. http://dx.doi.org/10.1142/s0218127420501618.

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We propose a new simple three-dimensional continuous autonomous model with two nonlinear terms and observe the dynamical behavior with respect to system parameters. This system changes the stability of fixed point via Hopf bifurcation and then undergoes a cascade of period-doubling route to chaos. We analytically derive the first Lyapunov coefficient to investigate the nature of Hopf bifurcation. We investigate well-separated regions for different kinds of attractors in the two-dimensional parameter space. Next, we introduce a timescale ratio parameter and calculate the slow manifold using geometric singular perturbation theory. Finally, the chaotic state annihilates by decreasing the value of the timescale ratio parameter.

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Grzywna, Zbigniew J., and Zuzanna Siwy. "Chaos in Ionic Transport Through Membranes." International Journal of Bifurcation and Chaos 07, no. 05 (May 1997): 1115–23. http://dx.doi.org/10.1142/s0218127497000911.

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The chaotic nature of ionic current through nuclear filters has been detected and analyzed by means of Liapunov exponents and bifurcation diagrams. The "crowd model" describing the nonlinear behavior of single channel ionic current, based on the family of logistic maps, has been formulated and applied to analysis of the patch clamp recordings. Some characteristic features of the system read out from the bifurcation diagrams of the two-parameter logistic map, and also from the extended Hodgkin–Huxley theory have been presented. The relation between the logistic difference schemes (and, at the same time, the class of logistic differential equations) and the logistic maps as well as some possibilities of using the crowd model to biological membranes have also been shown.

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Yang, Yujing, and Wenzhe Tang. "Research on a 3D Predator-Prey Evolutionary System in Real Estate Market." Complexity 2018 (2018): 1–13. http://dx.doi.org/10.1155/2018/6154940.

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This paper establishes a model on the upstream and downstream relationship among private enterprises, provincial and local officials, and the central government in the real estate market using the population ecology theory of mutual relations among individual species from the perspective of business ecosystem. A dynamic model is introduced and the complex dynamical behaviors of such a predator-prey model are investigated by means of numerical simulation. The local stability conditions and complex dynamics are investigated, and the existence of chaos is discussed in the sense of Marotto theorem; bifurcation diagrams, Lyapunov exponents, sensitivity analysis for initial values, and time history figure of the system are mapped out and discussed. This shows that there are two routes to complicated dynamics, one of which is the cascade of flip bifurcations resulting in periodic cycles (and chaos), and the other one is Neimark-Sacker bifurcation which produces attractive invariant closed curves. We arrive at conclusions that the phenomenon of chaos is harmful to private enterprises, and unstable behavior is often unfavorable. Thus, linear feedback control is applied to drive the model to a stable state when the system exhibits chaotic behaviors, achieving the goal of eliminating the negative effects to a large extent.

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Chen, Xueli, and Lishun Ren. "Bifurcation Analysis and Chaos Control in a Discrete-Time Parasite-Host Model." Discrete Dynamics in Nature and Society 2017 (2017): 1–17. http://dx.doi.org/10.1155/2017/9275474.

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A discrete-time parasite-host system with bifurcation is investigated in detail in this paper. The existence and stability of nonnegative fixed points are explored and the conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation are derived by using the center manifold theorem and bifurcation theory. And we also prove the chaos in the sense of Marotto. The numerical simulations not only illustrate the consistence with the theoretical analysis, but also exhibit other complex dynamical behaviors, such as bifurcation diagrams, Maximum Lyapunov exponents, and phase portraits. More specifically, when the integral step size is chosen as a bifurcation parameter, this paper presents the finding of period orbits, attracting invariant cycles and chaotic attractors of the discrete-time parasite-host system. Specifically, we have stabilized the chaotic orbits at an unstable fixed point by using the feedback control method.

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BOI, LUCIANO. "GEOMETRY OF DYNAMICAL SYSTEMS AND TOPOLOGICAL STABILITY: FROM BIFURCATION, CHAOS AND FRACTALS TO DYNAMICS IN NATURAL AND LIFE SCIENCES." International Journal of Bifurcation and Chaos 21, no. 03 (March 2011): 815–67. http://dx.doi.org/10.1142/s0218127411028842.

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The aim of this article is to review some basic concepts of the geometric theory of dynamical systems and stability. In this context, we also consider the related fundamental notions of broken symmetry, bifurcation and chaos. That of bifurcation is a very sophisticated mathematical concept, which displays a number of local and global behaviors of those spaces within which a large variety of natural forms unfold. The suited theoretical framework for understanding deeply the concept of bifurcation is the study of singularities of mappings, their topological structures and their classification into equivalence classes. Furthermore, we consider the fundamental role played by the phenomena of breaking symmetry and chaos in the evolution and organization of various natural and living systems. In the last part of the paper, we present some striking features and results of nonlinearity and stability in the framework of the geometrical theory of dynamical systems.

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KUZNETSOV, YU A., S. MURATORI, and S. RINALDI. "BIFURCATIONS AND CHAOS IN A PERIODIC PREDATOR-PREY MODEL." International Journal of Bifurcation and Chaos 02, no. 01 (March 1992): 117–28. http://dx.doi.org/10.1142/s0218127492000112.

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The model most often used by ecologists to describe interactions between predator and prey populations is analyzed in this paper with reference to the case of periodically varying parameters. A complete bifurcation diagram for periodic solutions of period one and two is obtained by means of a continuation technique. The results perfectly agree with the local theory of periodically forced Hopf bifurcation. The two classical routes to chaos, i.e., cascade of period doublings and torus destruction, are numerically detected.

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Liu, Mei, Bo Sang, Ning Wang, and Irfan Ahmad. "Chaotic Dynamics by Some Quadratic Jerk Systems." Axioms 10, no. 3 (September 14, 2021): 227. http://dx.doi.org/10.3390/axioms10030227.

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This paper is about the dynamical evolution of a family of chaotic jerk systems, which have different attractors for varying values of parameter a. By using Hopf bifurcation analysis, bifurcation diagrams, Lyapunov exponents, and cross sections, both self-excited and hidden attractors are explored. The self-exited chaotic attractors are found via a supercritical Hopf bifurcation and period-doubling cascades to chaos. The hidden chaotic attractors (related to a subcritical Hopf bifurcation, and with a unique stable equilibrium) are also found via period-doubling cascades to chaos. A circuit implementation is presented for the hidden chaotic attractor. The methods used in this paper will help understand and predict the chaotic dynamics of quadratic jerk systems.

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Liu, Xia, Yanwei Liu, and Qiaoping Li. "Multiple Bifurcations and Chaos in a Discrete Prey-Predator System with Generalized Holling III Functional Response." Discrete Dynamics in Nature and Society 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/245421.

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A prey-predator system with the strong Allee effect and generalized Holling type III functional response is presented and discretized. It is shown that the combined influences of Allee effect and step size have an important effect on the dynamics of the system. The existences of Flip and Neimark-Sacker bifurcations and strange attractors and chaotic bands are investigated by using the center manifold theorem and bifurcation theory and some numerical methods.

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Li, Na, Wei Tan, and Huitao Zhao. "Hopf Bifurcation Analysis and Chaos Control of a Chaotic System withoutilnikov Orbits." Discrete Dynamics in Nature and Society 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/912798.

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This paper mainly investigates the dynamical behaviors of a chaotic system withoutilnikov orbits by the normal form theory. Both the stability of the equilibria and the existence of local Hopf bifurcation are proved in view of analyzing the associated characteristic equation. Meanwhile, the direction and the period of bifurcating periodic solutions are determined. Regarding the delay as a parameter, we discuss the effect of time delay on the dynamics of chaotic system with delayed feedback control. Finally, numerical simulations indicate that chaotic oscillation is converted into a steady state when the delay passes through a certain critical value.

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Elhadj, Zeraoulia. "An example of superstable quadratic mapping of the space." Facta universitatis - series: Electronics and Energetics 22, no. 3 (2009): 385–90. http://dx.doi.org/10.2298/fuee0903385e.

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It is shown rigorously in this paper that an elementary 3-D quadratic mapping is superstable, i.e. it is superstable for some ranges of its bifurcation parameters. Numerical results that confirm the theory are also given and discussed. These numerical results give a new route to chaos which we call: the superstable quasi-periodic route to chaos.

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Harb, Ahmad M., Issa Batarseh, Lamine M. Mili, and Mohamed A. Zohdy. "Bifurcation and Chaos Theory in Electrical Power Systems: Analysis and Control." Mathematical Problems in Engineering 2012 (2012): 1–2. http://dx.doi.org/10.1155/2012/573910.

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Chang, Hui, Zhen Wang, Yuxia Li, and Guanrong Chen. "Dynamic Analysis of a Bistable Bi-Local Active Memristor and Its Associated Oscillator System." International Journal of Bifurcation and Chaos 28, no. 08 (July 2018): 1850105. http://dx.doi.org/10.1142/s0218127418501055.

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This paper proposes a new type of memristor with two distinct stable pinched hysteresis loops and twin symmetrical local activity domains, named as a bistable bi-local active memristor. A detailed and comprehensive analysis of the memristor and its associated oscillator system is carried out to verify its dynamic behaviors based on nonlinear circuit theory and Hopf bifurcation theory. The local-activity domains and the edge-of-chaos domains of the memristor, which are both symmetric with respect to the origin, are confirmed by utilizing the mathematical cogent theory. Finally, the subcritical Hopf bifurcation phenomenon is identified in the subcritical Hopf bifurcation region of the memristor.

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Wu, Fang, and Junhai Ma. "The Chaos Dynamic of Multiproduct Cournot Duopoly Game with Managerial Delegation." Discrete Dynamics in Nature and Society 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/206961.

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Although oligopoly theory is generally concerned with the single-product firm, what is true in the real word is that most of the firms offer multiproducts rather than single products in order to obtain cost-saving advantages, cater for the diversity of consumer tastes, and provide a barrier to entry. We develop a dynamical multiproduct Cournot duopoly model in discrete time, where each firm has an owner who delegates the output decision to a manager. The principle of decision-making is bounded rational. And each firm has a nonlinear total cost function due to the multiproduct framework. The Cournot Nash equilibrium and the local stability are investigated. The tangential bifurcation and intermittent chaos are reported by numerical simulations. The results show that high output adjustment speed can lead to output fluctuations which are characterized by phases of low volatility with small output changes and phases of high volatility with large output changes. The intermittent route to chaos of Flip bifurcation and another intermittent route of Flip bifurcation which contains Hopf bifurcation can exist in the system. The study can improve our understanding of intermittent chaos frequently observed in oligopoly economy.

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Journal articles on the topic 'Bifurcation and Chaos theory'

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