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: This paper will show that the addition of a slower Hebbian learning mechanism onto the Hopfield networks makes the resulting global dynamics to drive the network into a stable oscillatory regime, through a succession of intermittent and quasiperiodic regimes.
69 citations
TL;DR: In this paper, a new fractional-order hyperchaotic system based on the Lorenz system is presented, and the chaotic behaviors are validated by the positive Lyapunov exponents.
Abstract: In this paper, a new fractional-order hyperchaotic system based on the Lorenz system is presented. The chaotic behaviors are validated by the positive Lyapunov exponents. Furthermore, the fractional Hopf bifurcation is investigated. It is found that the system admits Hopf bifurcations with varying fractional order and parameters, respectively. Under different bifurcation parameters, some conditions ensuring the Hopf bifurcations are proposed. Numerical simulations are given to illustrate and verify the results.
69 citations
TL;DR: In this paper, the bifurcation and dynamical behaviour of the system of N globally coupled identical phase oscillators introduced by Hansel, Mato and Meunier, in the cases N = 3 and N = 4, were studied.
69 citations
TL;DR: The normal form of this singularity is calculated explicitly and both local and global bifurcations of the unfolding are analyzed by exploring a close relationship between the derived normal form and the truncated amplitude system for the fold-Hopf b ifurcation of ODEs.
Abstract: The fold-flip bifurcation occurs if a map has a fixed point with multipliers +1 and -1 simultaneously. In this paper the normal form of this singularity is calculated explicitly. Both local and global bifurcations of the unfolding are analyzed by exploring a close relationship between the derived normal form and the truncated amplitude system for the fold-Hopf bifurcation of ODEs. Two examples are presented, the generalized Henon map and an extension of the Lorenz-84 model. In the latter example the first-, second- and third-order derivatives of the Poincare map are computed using variational equations to find the normal form coefficients.
69 citations
TL;DR: In this article, the authors studied the codimension two and three Hopf bifurcations and the pertinent Lyapunov stability coefficients in the Watt governor system.
Abstract: This paper pursues the study carried out by the authors in Stability and Hopf bifurcation in the Watt governor system [14], focusing on the codimension one Hopf bifurcations in the centrifugal Watt governor differential system, as presented in Pontryagin's book Ordinary Differential Equations, [13]. Here are studied the codimension two and three Hopf bifurcations and the pertinent Lyapunov stability coefficients and bifurcation diagrams, illustrating the number, types and positions of bifurcating small amplitude periodic orbits, are determined. As a consequence it is found a region in the space of parameters where an attracting periodic orbit coexists with an attracting equilibrium.
69 citations