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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, a general model of nonlinear systems with distributed delays is derived from Chen's system with the weak kernel, and the local stability is analyzed by using the Routh-Hurwitz criterion, where the direction and the stability of the bifurcating periodic solutions are determined using the normal form theory and the center manifold theorem.
Abstract: In this paper, a general model of nonlinear systems with distributed delays is studied. Chen’s system can be derived from this model with the weak kernel. After the local stability is analyzed by using the Routh–Hurwitz criterion, Hopf bifurcation is studied, where the direction and the stability of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Some numerical simulations for justifying the theoretical analysis are also presented. Chaotic behavior of Chen’s system with the strong kernel is also found through numerical simulation, in which some waveform diagrams, phase portraits, and bifurcation plots are presented and analyzed.
39 citations
TL;DR: In this article, a numerical method which uses only rather classical tools of Numerical Analysis such as Newton's method and routines for ordinary differential equations, unstable periodic solutions of differential-difference equations can be computed.
Abstract: By employing a numerical method which uses only rather classical tools of Numerical Analysis such as Newton's method and routines for ordinary differential equations, unstable periodic solutions of differential-difference equations can be computed. The method is applied to determine bifurcation diagrams with backward bifurcation.
39 citations
TL;DR: This work extends previous work on ideal replicators to include the square root rate and other possible nonlinearities, which it couple with an enzymatic sink, and obtains exact general relations for the Poincare-Adronov-Hopf and Turing bifurcations.
Abstract: Chemical self-replication of oligonucleotides and helical peptides exhibits the so-called square root rate law. Based on this rate we extend our previous work on ideal replicators to include the square root rate and other possible nonlinearities, which we couple with an enzymatic sink. For this generalized model, we consider the role of cross diffusion in pattern formation, and we obtain exact general relations for the Poincare-Adronov-Hopf and Turing bifurcations, and our generalized results include the Higgins, Autocatalator, and Templator models as specific cases.
39 citations
TL;DR: Haag et al. as mentioned in this paper studied the bicomplex formulation of the time-dependent variational principle and found that in fact there are three tangent bifurcations for very small gain-loss contributions which coalesce in a cusp-bifurcation.
Abstract: A Bose-Einstein condensate in a double-well potential features stationary solutions even for attractive contact interaction as long as the particle number and therefore the interaction strength do not exceed a certain limit. Introducing balanced gain and loss into such a system drastically changes the bifurcation scenario at which these states are created. Instead of two tangent bifurcations at which the symmetric and antisymmetric states emerge, one tangent bifurcation between two formerly independent branches arises [D. Haag et al., Phys. Rev. A 89, 023601 (2014)]. We study this transition in detail using a bicomplex formulation of the time-dependent variational principle and find that in fact there are three tangent bifurcations for very small gain-loss contributions which coalesce in a cusp bifurcation.
39 citations
TL;DR: In this article, a class of discrete-time systems with two neurons is considered and its linear stability is investigated and Neimark-Sacker bifurcation is demonstrated by analyzing the corresponding characteristic equation, in particular, the explicit formula for determining the direction of NEI and the stability of periodic solution is obtained by using the normal form method and the center manifold theory for discrete time system developed by Kuznetsov.
39 citations