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 singular Hopf bifurcation from a basic steady state to these relaxation oscillations was studied. But the authors focused on the symmetry of the harmonic oscillations near the Hopf point and did not consider the reverse transition from stable steady states to periodic solutions.
Abstract: Relaxation oscillations characterized by two quite different time scales are described by mathematical models of the form $x_t = f(x,y,\lambda ,\varepsilon )$ and $y_t = \varepsilon g(x,y,\lambda ,\varepsilon )$ where $\varepsilon \ll 1$ and $\lambda $ is the control parameter. In this paper, we study the singular Hopf bifurcation from a basic steady state to these relaxation oscillations. Our bifurcation analysis shows how the harmonic oscillations near the bifurcation point progressively change to become pulsed, triangular oscillations.In the second part of the paper, we present a numerical study of the FitzHugh–Nagumo equations for nerve conduction. We first observe that the numerical results are in good agreement with the analytical predictions. We then consider the switching from a stable steady state to a stable periodic solution, or the reverse transition. Our purpose is to explain the annihilation experiments described in the nerve conduction literature.
180 citations
TL;DR: It is shown that locally asymptotic stability of a positive equilibrium point does not imply global stability for this class of predator-prey models.
Abstract: We describe the bifurcation diagram of limit cycles that appear in the first realistic quadrant of the predator-prey model proposed by R. M. May [ Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, 1974]. In particular, we give a qualitative description of the bifurcation curve when two limit cycles collapse on a semistable limit cycle and disappear. Moreover, we show that locally asymptotic stability of a positive equilibrium point does not imply global stability for this class of predator-prey models.
179 citations
01 Jan 1995
TL;DR: In this paper, the main modes of behavior of a food chain model, composed of logistic prey and Holling type II predator and superpredator, are discussed through bifurcation analysis, alternating between a normal form approach and numerical continuation.
Abstract: The main modes of behavior of a food chain model, composed of logistic prey and Holling type II predator and superpredator, are discussed in this paper. The study is carried out through bifurcation analysis, alternating between a normal form approach and numerical continuation. The two-parameter bifurcation diagram of the model contains Hopf, fold and transcritical bifurcation curves of equilibria as well as flip, fold and transcritical bifurcation curves of limit cycles. The appearance of chaos in the model is proved to be related to a Hopf bifurcation and a degenerate homoclinic bifurcation in the prey-predator subsystem. The boundary of the chaotic region is shown to have a very peculiar structure.
179 citations
TL;DR: The main modes of behavior of a food chain model composed of logistic prey and Holling type II predator and superpredator are discussed in this paper and the appearance of chaos in the model is proved to be related to a Hopf bifurcation and a degenerate homoclinic b ifurcation in the prey-predator subsystem.
Abstract: The main modes of behavior of a food chain model composed of logistic prey and Holling type II predator and superpredator are discussed in this paper. The study is carried out through bifurcation analysis, alternating between a normal form approach and numerical continuation. The two-parameter bifurcation diagram of the model contains Hopf, fold, and transcritical bifurcation curves of equilibria as well as flip, fold, and transcritical bifurcation curves of limit cycles. The appearance of chaos in the model is proved to be related to a Hopf bifurcation and a degenerate homoclinic bifurcation in the prey-predator subsystem. The boundary of the chaotic region is shown to have a very peculiar structure.
178 citations
TL;DR: In this article, a one-dimensional iterative chaotic map with infinite collapses within a symmetrical region was proposed, and the stability of fixed points and that around the singular point were analyzed.
Abstract: A one-dimensional iterative chaotic map with infinite collapses within symmetrical region [-1, O)/spl cup/(O, +1] is proposed. The stability of fixed points and that around the singular point are analyzed. Higher Lyapunov exponents of proposed map show stronger chaotic characteristics than some iterative and continuous chaotic models usually used. There exist inverse bifurcation phenomena and special main periodic windows at certain positions shown in the bifurcation diagram, which can explain the generation mechanism of chaos. The chaotic model with good properties can be generated if choosing the parameter of the map properly. Stronger inner pseudorandom characteristics can also be observed through /spl chi//sup 2/ test on the supposition of even distribution. This chaotic model may have many advantages in practical use.
178 citations