Bifurcation diagram

About: Bifurcation diagram is a research topic. Over the lifetime, 8379 publications have been published within this topic receiving 139664 citations.

Bifurcation Analysis of a Predator-Prey System Involving Group Defence

Abstract: A class of ODEs of generalized Gause type modeling predator-prey interaction is considered. The prey are assumed to exhibit a phenomenon called group defence, that is, predation is decreased or even eliminated due to the ability of the prey to defend or disguise themselves as their numbers increase.Using the carrying capacity of the environment as the bifurcation parameter, it is shown that the model undergoes a sequence of bifurcations that includes a homoclinic bifurcation as well as a Hopf bifurcation. Conditions (that hold even in the case of no group defence) that ensure a subcritical Hopf bifurcation and also the spontaneous appearance of a semistable periodic orbit that splits into a pair (one stable and one unstable) of periodic orbits are given.Ecological ramifications are considered. Unlike the classical model, sufficient enrichment of the environment combined with group defence leads to extinction of the predator (deterministically) for almost all initial conditions, providing strong support fo...

Chaos and Hopf bifurcation of a finance system

Abstract: The complex dynamical behavior of a finance system is investigated in this paper The Ruelle–Takens route to chaos and strange nonchaotic attractors (SNA) are found through numerical simulations Then the system with time-delayed feedback is considered and the stability and Hopf bifurcation of the controlled system are investigated This research has important theoretical and practical meanings

Grazing, skipping and sliding: Analysis of the non-smooth dynamics of the DC/DC buck converter

Abstract: This paper provides an analytical insight into the observed nonlinear behaviour of a simple widely used power electronic circuit (the buck converter) and draws parallels with a wider class of piecewise-smooth systems. After introducing the buck converter model and background, the most fascinating features of its dynamical behaviour are reviewed. So-called grazing and sliding solutions are discussed and their role in determining many of the buck converter's dynamical oddities is demonstrated. In particular, a local map is studied which explains how grazing bifurcations cause sharp turning points in the bifurcation diagram of periodic orbits. Moreover, these orbits are shown to accumulate onto a sliding trajectory through a `spiralling' impact adding scenario. The structure of such a diagram is derived analytically and is shown to be closely related to the analysis of homoclinic bifurcations. The results are shown to match perfectly with numerical simulations. The sudden jump to large-scale chaos and the fingered structure of the resulting attractor are also explained.

Dynamical Systems V: Bifurcation Theory and Catastrophe Theory

Abstract: Nonlinear Dynamical Systems and Carleman LinearizationProbability Theory IIIMethods of Qualitative Theory in Nonlinear DynamicsMathematics of Complexity and Dynamical SystemsFundamentals of Dynamical Systems and Bifurcation TheoryElements of Applied Bifurcation TheoryGeneral Topology IIIQuasi-Periodic Motions in Families of Dynamical SystemsDynamical Systems VDynamical Systems in NeuroscienceDynamical Systems VPartial Differential Equations VIINormal Forms, Bifurcations and Finiteness Problems in Differential EquationsNonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector FieldsBifurcation Theory and Methods of Dynamical SystemsPartial Differential Equations IXElements of Applied Bifurcation TheoryCommutative Harmonic Analysis IICatastrophe TheorySmooth Dynamical SystemsHandbook of Dynamical SystemsBifurcation Theory of Functional Differential EquationsFunctional Analysis IDynamical Systems VPartial Differential Equations IIGlobal Bifurcation Theory and Hilbert’s Sixteenth ProblemNonlocal BifurcationsAlgebraic Geometry IIDynamical Systems IXPartial Differential Equations VIIINormal Forms and Bifurcation of Planar Vector FieldsLocal and Semi-Local Bifurcations in Hamiltonian Dynamical SystemsDynamical Systems VIIIAlgebraic Geometry IVElements of Applied Bifurcation TheoryCommutative Harmonic Analysis IIINumerical Bifurcation Analysis of MapsGlobal Analysis of Dynamical SystemsProbability Theory IIIBifurcations and Chaos in Piecewise-smooth Dynamical Systems

Dynamic analysis, circuit realization, control design and image encryption application of an extended Lü system with coexisting attractors

Abstract: This paper introduces an extended Lu system with coexisting attractors. The number and stability of equilibria are determined. The coexisting attractors of the system are displayed by the bifurcation diagrams, Lyapunov exponent spectrum, phase portraits. It is shown that the system has a pair of strange attractors, a pair of limit cycles, a pair of point attractors for different initial conditions. The circuit implementation of the chaotic attractor and coexisting attractors of the system are presented. The control problem of the system is studied as well. A controller is designed to stabilize the system to the origin and realize the switching between two chaotic attractors based on the passive control method. Moreover, a chaotic image encryption algorithm is proposed according to the system. The performance of the algorithm is numerically analyzed.