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 model most often used by ecologists to describe interactions between predator and prey populations is analyzed with reference to the case of periodically varying parameters, and a complete bifurcation diagram for periodic solutions of period one and two is obtained by means of a continuation technique.
Abstract: The model most often used by ecologists to describe interactions between predator and prey populations is analyzed in this paper with reference to the case of periodically varying parameters. A complete bifurcation diagram for periodic solutions of period one and two is obtained by means of a continuation technique. The results perfectly agree with the local theory of periodically forced Hopf bifurcation. The two classical routes to chaos, i.e., cascade of period doublings and torus destruction, are numerically detected.
94 citations
TL;DR: In this article, an SIR model with a standard incidence rate and a nonlinear recovery rate was established to consider the impact of available resource of the public health system especially the number of hospital beds.
94 citations
TL;DR: In this paper, the Yamada model for self-pulsations in (semiconductor) lasers with saturable absorber is considered and a complete picture of all possible dynamics is presented in terms of two-dimensional bifurcation diagrams, in which they find a Bogdanov-takens bifurycation as an organizing center.
94 citations
TL;DR: The Holling--Tanner model for predator-prey systems has two Hopf bifurcation points for certain parameters and the dependence of the environmental parameters on the underlying bIfurcation structure is uncovered using two-timing.
Abstract: The Holling--Tanner model for predator-prey systems has two Hopf bifurcation points for certain parameters. The dependence of the environmental parameters on the underlying bifurcation structure is uncovered using two-timing. Emphasis is on how the bifurcation diagram changes as the Hopf bifurcation points separate. Two degenerate cases require a modification of conventional two-timing. When the two Hopf bifurcation points nearly coalesce, the two stable periodic solution branches are shown to be connected. As a ratio of linear growth rates varies, the Hopf bifurcation points separate further and one limit cycle becomes unstable. This situation can correspond to an outbreak in populations. The modified two-timing analysis analytically captures the unstable and stable limit cycles of the new branch.
94 citations
TL;DR: In this article, a chaotic fractional-order modified hybrid optical system is presented, which is further investigated by means of Poincare mapping, parameter phase portraits, and the largest Lyapunov exponents.
Abstract: In this paper, a chaotic fractional-order modified hybrid optical system is presented. Some basic dynamical properties are further investigated by means of Poincare mapping, parameter phase portraits, and the largest Lyapunov exponents. Fractional Hopf bifurcation conditions are proposed; it is found that Hopf bifurcation occurs on the proposed system when the fractional-order varies and passes a sequence of critical values. The chaotic motion is validated by the positive Lyapunov exponent. Finally, some numerical simulations are also carried out to illustrate our results.
94 citations