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 existence of a tangent homoclinic orbit allows us to understand in a quantitative way a particular and regular sequence of cool flame-ignition oscillations observed in a thermokinetic model of hydrocarbon oxidation.
Abstract: Nonlinear autonomous dynamical systems with ahomoclinic tangency to a periodic orbit are investigated. We study the bifurcation sequences of the mixed-mode oscillations generated by the homoclinicity, which are shown to belong to two different types, depending on the nature of the Liapunov numbers of the basic periodic orbit. A detailed numerical analysis is carried out to show how the existence of a tangent homoclinic orbit allows us to understand in a quantitative way a particular and regular sequence of cool flame-ignition oscillations observed in a thermokinetic model of hydrocarbon oxidation. Chaotic cool flame oscillations are also observed in the same model. When the control parameter crosses a critical value, this chaotic set of trajectories becomes globally unstable and forms a Cantor-like hyperbolic repellor, and the ignition mechanism generates ahomoclinic tangency to the Cantor set of trajectories. The complex bifurcation diagram may be globally reconstructed from a one-dimensional dynamical system, thanks to the strong contractivity of thermokinetics. It is found that a symbolic dynamics with three symbols is necessary to classify the periodic windows of the complex bifurcation sequence observed numerically in this system.
76 citations
TL;DR: In this article, the global bifurcation diagram of a three-parameter family of cubic Lienard systems is derived, which is a universal character in that its bifuration diagram appears in many models from applications for which a combination of hysteretic and self-oscillatory behavior is essential.
Abstract: We derive the global bifurcation diagram of a three-parameter family of cubic Lienard systems. This family seems to have a universal character in that its bifurcation diagram (or parts of it) appears in many models from applications for which a combination of hysteretic and self-oscillatory behaviour is essential. The family emerges as a partial unfolding of a doubly degenerate Bogdanov-Takens point, that is, of the codimension-four singularity with nilpotent linear part and no quadratic terms in the normal form. We give a new presentation of a local four-parameter bifurcation diagram which is a candidate for the universal unfolding of this singularity.
76 citations
TL;DR: The present paper focuses on some global bifurcations, which precede the Neimark bIfurcation, and produce other attractors which coexist with the still attractive Cournot fixed point.
Abstract: An adaptive oligopoly model, where the demand function is isoelastic and the competitors operate under constant marginal costs, is considered. The Cournot equilibrium point then loses stability through a subcritical Neimark bifurcation. The present paper focuses some global bifurcations, which precede the Neimark bifurcation, and produce other attractors which coexist with the still attractive Cournot fixed point.
75 citations
TL;DR: It is shown by analytical and numerical methods that the dynamical scenario includes the coexistence of multiple stable periodic solutions and can be described in terms of the Eckhaus instability, which is well known in the context of spatially extended systems.
Abstract: The dynamical behavior of various physical and biological systems under the influence of delayed feedback or coupling can be modeled by including terms with delayed arguments in the equations of motion. In particular, the case of long delay may lead to complicated and high-dimensional dynamics. We investigate the effects of delay in systems that display an oscillatory instability (Hopf bifurcation) in the absence of delay. We show by analytical and numerical methods that the dynamical scenario includes the coexistence of multiple stable periodic solutions and can be described in terms of the Eckhaus instability, which is well known in the context of spatially extended systems.
75 citations
TL;DR: Nonlinear control scheme is proposed to the chaotic 3D dynamical system and the controlled system, depending on five parameters, can exhibit codimension one, two, and three Hopf bifurcations in a much larger parameter regain.
75 citations