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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: A new no-equilibrium chaotic system that is constructed by adding a tiny perturbation to a simple chaotic flow having a line equilibrium is introduced.
Abstract: This paper introduces a new no-equilibrium chaotic system that is constructed by adding a tiny perturbation to a simple chaotic flow having a line equilibrium. The dynamics of the proposed system are investigated through Lyapunov exponents, bifurcation diagram, Poincare map and period-doubling route to chaos. A circuit realization is also represented. Moreover, two other new chaotic systems without equilibria are also proposed by applying the presented methodology.
167 citations
TL;DR: From the propagation of localized infinitesimal disturbances in DDS, it is shown the existence of convective type instabilities and a widely studied class of DDS is mapped onto an evolution rule for a spatial system with drift and diffusion.
Abstract: The interpretation of delayed dynamical systems (DDS) in terms of a suitable spatiotemporal dynamics is put on a rigorous ground by deriving amplitude equations in the vicinity of a Hopf bifurcation. We show that comoving Lyapunov exponents can be defined and computed in a DDS. From the propagation of localized infinitesimal disturbances in DDS, we show the existence of convective type instabilities. Moreover, a widely studied class of DDS is mapped onto an evolution rule for a spatial system with drift and diffusion.
166 citations
TL;DR: In this paper, a symmetry breaking front bifurcation is studied in a wide class of reaction-diffusion systems and the effects it has on pattern formation and pattern dynamics.
Abstract: Domain patterns in reaction-diffusion systems often contain two spatial scales; a long scale determined by a typical domain size, and a short scale pertaining to front structures separating different domains. Such patterns naturally develop in bistable and excitable systems, but may also appear far beyond Hopf and Turing bifurcations. The global behaviour of domain patterns strongly depends on the fronts' inner structures. In this paper we study a symmetry breaking front bifurcation expected to occur in a wide class of reaction-diffusion systems, and the effects it has on pattern formation and pattern dynamics. We extend previous works on this type of front bifurcation and clarify the relations among them. We show that the appearance of front multiplicity beyond the bifurcation point allows the formation of persistent patterns rather than transient ones. In a different parameter regime, we find that the front bifurcation outlines a transition from oscillating (or breathing) patterns to travelling ones. Near a boundary we find that fronts beyond the bifurcation can reflect, while those below it either bind to the boundary or disappear.
166 citations
TL;DR: In this paper, the structure of nonnegative steady-state solutions of a system of reaction-diffusion equations arising in ecology is investigated and ranges of parameters are found for which there exist nontrivial steady state solutions.
Abstract: In this paper the structure of the nonnegative steady-state solutions of a system of reaction-diffusion equations arising in ecology is investigated. The equations model a situation in which a predator species and a prey species inhabit the same region and the interaction terms are of Holling–Tanner type sothat the predator has finite appetite. Prey and predator birth-rates are treated as bifurcation parameters and the theorems of global bifurcation theory are adapted so that they apply easily to the system. Thus ranges of parameters are found for which there exist nontrivial steady-state solutions.
165 citations
TL;DR: In this article, the conditions of existence for flip bifurcation and Hopf Bifurcation were derived by using center manifold theorem and bifurlcation theory, and numerical simulation results not only show the consistence with the theoretical analysis but also display the new and interesting dynamical behaviors, including period-3, 5, 6, 7, 8,8, 9, 10, 12, 18, 20, 22, 30,30, 39-orbits in different chaotic regions.
Abstract: The discrete-time predator–prey system obtained by Euler method is investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory. And numerical simulation results not only show the consistence with the theoretical analysis but also display the new and interesting dynamical behaviors, including period-3, 5, 6, 7, 8, 9, 10, 12, 18, 20, 22, 30, 39-orbits in different chaotic regions, attracting invariant circle, period-doubling bifurcation from period-10 leading to chaos, inverse period-doubling bifurcation from period-5 leading to chaos, interior crisis and boundary crisis, intermittency mechanic, onset of chaos suddenly and sudden disappearance of the chaotic dynamics, attracting chaotic set, and non-attracting chaotic set. In particular, we observe that when the prey is in chaotic dynamic, the predator can tend to extinction or to a stable equilibrium. The computations of Lyapunov exponents confirm the dynamical behaviors. The analysis and results in this paper are interesting in mathematics and biology.
165 citations