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, an asymptotic method was used to show that secondary bifurcation occurs in a model biochemical reaction studied as a simplified example of morphogenesis, and the secondary branches were presented.
Abstract: We use an asymptotic method to show that secondary bifurcation occurs in a model biochemical reaction studied as a simplified example of morphogenesis. The secondary bifurcation points are computed, and asymptotic expansions for the secondary branches are presented.
41 citations
TL;DR: In this article, the complex dynamical behaviors of a three-dimensional continuous autonomous system which is described as x ˙ = ax − yz, y ˘ = − by + xz, z ˚ = − cz + x 2.
41 citations
TL;DR: In this paper, the amplitude map for codimension-2 bifurcations of fixed points of dissipative diffeomorphisms with a pair of complex eigenvalues together with either an eigenvalue − 1 or another such a pair.
Abstract: We study codimension-2 bifurcations of fixed points of dissipative diffeomorphisms with a pair of complex eigenvalues together with either an eigenvalue − 1 or another such a pair. In the previous studies only cubic normal forms were considered. However, in some cases the unfolding requires higher-order terms and these are investigated here. We (re)derive the normal forms and reduce them to a single amplitude map. This map is similar to the amplitude system for the double-Hopf bifurcation of vector fields. We show how the critical normal form coefficients determine the general bifurcation picture for this amplitude map. Representative nonsymmetric perturbations of the normal forms are studied numerically. Our case studies show a detailed picture near various bifurcation curves, which is richer than known theoretical predictions. For arbitrary maps with these bifurcations we give explicit formulas for critical normal form coefficients on center manifolds. We apply them to an example from robotics where we ...
41 citations
TL;DR: In this article, the authors present experimental evidence of coexisting periodic attractors in a semiconductor laser subject to external optical injection and derive a third-order differential equation for the phase of the laser field.
Abstract: We present experimental evidence of coexisting periodic attractors in a semiconductor laser subject to external optical injection. The coexisting attractors appear after the semiconductor laser has undergone a Hopf bifurcation from the locked steady state. We consider the single-mode rate equations and derive a third-order differential equation for the phase of the laser field. We then analyse the bifurcation diagram of the time-periodic states in terms of the frequency detuning and the injection rate and show the existence of multiple periodic attractors.
41 citations
TL;DR: In this paper, the authors consider planar piecewise smooth differential systems with a discontinuity line, and characterize the critical crossing cycle bifurcation, also termed as homoclinic connection to a fold.
41 citations