Topic
Bifurcation diagram
About: Bifurcation diagram is a research topic. Over the lifetime, 8379 publications have been published within this topic receiving 139664 citations.
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TL;DR: In this paper, the dynamics of a discrete-time predator-prey system is investigated in the closed first quadrant R + 2, and it is shown that the system undergoes flip bifurcation and Hopf bifurbation in the interior of R+2 by using center manifold theorem and bifurlcation theory.
Abstract: The dynamics of a discrete-time predator–prey system is investigated in the closed first quadrant R + 2 . It is shown that the system undergoes flip bifurcation and Hopf bifurcation in the interior of R + 2 by using center manifold theorem and bifurcation theory. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-5, 6, 9, 10, 14, 18, 20, 25 orbits, cascade of period-doubling bifurcation in period-2, 4, 8, quasi-periodic orbits and the chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors.
300 citations
TL;DR: In this paper, the authors apply the theory of singularities of differentiable mappings (SOMD) to study the effect of imperfections in a system subject to bifurcation.
Abstract: : This paper applies the theory of singularities of differentiable mappings - specifically the unfolding theorem - to study the effect of imperfections in a system subject to bifurcation. In a number of special cases we have classified (up to a suitable equivalence) all the possible perturbations of the bifurcation equations by a finite number of imperfection parameters. These cases include both bifurcation from a double eigenvalue and from a simple eigenvalue degenerate in the sense of Crandall-Rabinowitz.
299 citations
Book•
01 Nov 1985
TL;DR: In this article, the restricted problem of three bodies is considered and the Hamiltonian Hopf bifurcation is shown to preserve normal forms for energy-momentum maps.
Abstract: Preliminaries.- Normal forms for Hamiltonian functions.- Fibration preserving normal forms for energy-momentum maps.- The Hamiltonian Hopf bifurcation.- Nonintegrable systems at resonance.- The restricted problem of three bodies.
292 citations
Book•
02 Aug 2003
TL;DR: In this paper, the Lyapunov-Schmidt reduction for potential operators was used to reduce the hopf Bifurcation for Hamiltonian, Reversible, and Conservative systems.
Abstract: Introduction Appendix I Local Theory I1 The Implicit Function Theorem I2 The Method of Lyapunov-Schmidt I3 The Lyapunov-Schmidt Reduction for Potential Operators I4 An Implicit Function Theorem for One-Dimensional Kernels: Turning Points I5 Bifurcation with a One-Dimensional Kernel I6 Bifurcation Formulas (stationary case) I7 The Principle of Exchange of Stability (stationary case) I8 Hopf Bifurcation I9 Bifurcation Formulas for Hopf Bifurcation I10 A Lyapunov Center Theorem I11 Constrained Hopf Bifurcation for Hamiltonian, Reversible, and Conservative Systems I12 The Principle of Exchange of Stability for Hopf Bifurcation I13 Continuation of Periodic Solutions and Their Stability I14 Period Doubling Bifurcation and Exchange of Stability I15 Newton Polygon I16 Degenerate Bifurcation at a Simple Eigenvalue and Stability of Bifurcating Solutions I17 Degenerate Hopf Bifurcation and Floquet Exponents of Bifurcating Periodic Orbits I18 The Principle of Reduced Stability for Stationary and Periodic Solutions I19 Bifurcation with High-Dimensional Kernels, Multiparameter Bifurcation and Application of the Principle of Reduced Stability I20 Bifurcation from Infinity I21 Bifurcation with High-Dimensional Kernels for Potential Operators: Variational Methods I22 Notes and Remarks to Chapter I Appendix II Global Theory II1 The Brouwer Degree II2 The Leray Schauder Degree II3 Application of the Degree in Bifurcation Theory II4 Odd Crossing Numbers II5 A Degree for a Class of Proper Fredholm Operators and Global Bifurcation Theorems II6 A Global Implicit Function Theorem II7 Change of Morse Index and Local Bifurcation for Potential Operators II8 Notes and Remarks to Chapter II Appendix III Applications III1 The Fredholm Property of Elliptic Operators III2 Local Bifurcation for Elliptic Problems III3 Free Nonlinear Vibrations III4 Hopf Bifurcation for Parabolic Problems III5 Global Bifurcation and Continuation for Elliptic Problems III6 Preservation of Nodal Structure on Global Branches III7 Smoothness and Uniqueness of Global Positive Solution Branches III8 Notes and Remarks to Chapter III
278 citations
TL;DR: In this article, the authors investigated Hopf-like bifurcation phenomena and chaotic behavior in cellular neural networks and found that the chaotic attractor found here has properties similar to the famous double scroll attractor.
Abstract: Bifurcation phenomena and chaotic behavior in cellular neural networks are investigated. In a two-cell autonomous system, Hopf-like bifurcation has been found, at which the flow around the origin, an equilibrium point of the system, changes from asymptotically stable to periodic. As the parameter grows further, by reaching another bifurcation value, the generated limit cycle disappears and the network becomes convergent again. Chaos is also presented in a three-cell autonomous system. It is shown that the chaotic attractor found here has properties similar to the famous double scroll attractor. >
277 citations