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 article, the role of the function x|x| as a chaos generator in non-autonomous systems is investigated, and it is rigorously proven via the Melnikov function method that this particular quadratic function induces Smale horseshoes to the Duffing-like system.
Abstract: This paper investigates the role of the function x|x| as a chaos generator in nonautonomous systems. A Duffing-like nonautonomous oscillator is used for illustration. It is rigorously proven via the Melnikov function method that this particular quadratic function induces Smale horseshoes to the Duffing-like system. Moreover, its physical meaning as an energy function is demonstrated, which provides a critical value for the emergence of chaos. Simulations with bifurcation analysis are given for better understanding of the underlying dynamics.
82 citations
TL;DR: In this paper, a time-delayed feedback controller is proposed in order to control chaos and Hopf bifurcation in a fractional-order memristor-based chaotic system with time delay.
Abstract: In this paper, a time-delayed feedback controller is proposed in order to control chaos and Hopf bifurcation in a fractional-order memristor-based chaotic system with time delay. The associated characteristic equation is established by regarding the time delay as a bifurcation parameter. A set of conditions which ensure the existence of the Hopf bifurcation are gained by analyzing the corresponding characteristic equation. Then, we discuss the influence of feedback gain on the critical value of fractional order and time delay in the controlled system. Theoretical analysis shows that the controller is effective in delaying the Hopf bifurcation critical value via decreasing the feedback gain. Finally, some numerical simulations are presented to prove the validity of our theoretical analysis and confirm that the time-delayed feedback controller is valid in controlling chaos and Hopf bifurcation in the fractional-order memristor-based system.
82 citations
TL;DR: The aim of this paper is the study of the long-term behavior of population communities described by piecewise smooth models (known as Filippov systems) by proposing a relatively simple method, called the puzzle method, to construct the complete bifurcation diagram step-by-step.
81 citations
TL;DR: A general bifurcation in the synchronized dynamics of time-delay-coupled nonlinear oscillators is presented, being observed in regimes of oscillator death as well as in periodic, quasiperiodic, and chaotic dynamics.
Abstract: We present a general bifurcation in the synchronized dynamics of time-delay-coupled nonlinear oscillators. The relative phase between the oscillators jumps from zero to $\ensuremath{\pi}$ as a function of the coupling; this phase-flip bifurcation is accompanied by a discontinuous change in the frequency of the synchronized oscillators. This phenomenon is of broad relevance, being observed in regimes of oscillator death as well as in periodic, quasiperiodic, and chaotic dynamics. Time-delay coupling is necessary for the phase-flip bifurcation. We illustrate the phenomenon, and present analytical results for paradigmatic nonlinear systems. Possible applications are discussed.
81 citations
TL;DR: In this paper, the authors investigated nonlinear properties of a stratospheric vacillation model in the light of bifurcation theory, and obtained three branches of the steady solutions by use of Powell's hybrid method and the pseudo-arclength continuation method.
Abstract: Nonlinear properties of a stratospheric vacillation model are investigated numerically in the light of bifurcation theory. The model is exactly the same as that used by Holton and Mass, which describes the wave-zonal flow interaction in a β-channel under a nonconservative constraint with zonal-flow forcing and wave dissipation. A set of 81 nonlinear ordinary differential equations with variables depending on time is obtained by a severe truncation and vertical differencing. All of the external parameters are fixed in time. The amplitude of the wave forcing or the intensity of zonal wind forcing at the bottom boundary is changed as a bifurcation parameter. Three branches of the steady solutions are obtained by use of Powell's hybrid method and the pseudo-arclength continuation method. Linear stability of these solution branches is investigated by solving an eigenvalue problem in the linearized system. In some range of the bifurcation parameter, there exists a multiplicity of stable steady solution...
81 citations