Bifurcation diagram

About: Bifurcation diagram is a research topic. Over the lifetime, 8379 publications have been published within this topic receiving 139664 citations.

Stochastic bifurcation in noise-driven lasers and Hopf oscillators.

Abstract: This paper considers nonlinear dynamics in an ensemble of uncoupled lasers, each being a limit-cycle oscillator, which are driven by the same external white Gaussian noise. As the external-noise strength increases, there is an onset of synchronization and then subsequent loss of synchrony. Local analysis of the laser equations shows that synchronization becomes unstable via stochastic bifurcation to chaos, defined as a passing of the largest Lyapunov exponent through zero. The locus of this bifurcation is calculated in the three-dimensional parameter space defined by the Hopf parameter, amount of amplitude-phase coupling, and external-noise strength. Numerical comparison between the laser system and the normal form of Hopf bifurcation uncovers a square-root law for this stochastic bifurcation as well as strong enhancement in noise-induced chaos due to the laser's relaxation oscillation.

Railway vehicle chaos and asymmetric hunting

Abstract: SUMMARY In this paper we present the results of a refined investigation of the dynamical behaviour of Cooperrider's complex bogie. The earlier results were presented in [4] and [7]. It was discovered, that one of the solution branches in [4] and [5] was one of an asymmetric, periodic oscillation - albeit with a very small offset, but it indicates, that the asymmetric oscillation is the generic mode at speeds much lower than has hitherto been found. The bifurcation diagram has been completed, a new type of bifurcation discovered and the other asymmetric branch determined. Furthermore we discovered chaotic motion of the bogie at much lower speeds than reported in [5] and [8][, and we present the result here. Finally we present a new solution branch, which represents an unstable, symmetric oscillation. It has the interesting property, that it turns stable in a small speed range for very high speeds. It has a smaller amplitude than the coexisting chaos. Such behaviour is not uncommon in dynamical systems, see...

Introduction to Dynamic Bifurcation.

Abstract: : Dynamic bifurcation theory in differential equations is concerned with the changes that occur in the structure of the limit sets of solutions as parameters in the vector field are varied. For example, if the vector field is the gradient of a function with a finite number of critical points, then the omega-limit set of each orbit is an equilibrium point. Thus, one must be concerned with how the number of equilibrium points changes with the parameters (this is usually called static bifurcation theory), how the stability properties of the equilibrium points change and the manner in which the equilibrium points are connected to each other by orbits. If the vector field is not the gradient of a function, then other types of limiting motions can occur; for example, periodic orbits, invariant tori, homoclinic and heteroclinic orbits. The purpose of these notes is to give an introduction to the methods used in determining how these more complicated limit sets change as parameters vary. (Author)

Stabilizing unstable periodic orbits in the Lorenz equations using time-delayed feedback control.

Abstract: For many years it was believed that an unstable periodic orbit with an odd number of real Floquet multipliers greater than unity cannot be stabilized by the time-delayed feedback control mechanism of Pyragas. A recent paper by Fiedler et al. Phys. Rev. Lett. 98, 114101 (2007) uses the normal form of a subcritical Hopf bifurcation to give a counterexample to this theorem. Using the Lorenz equations as an example, we demonstrate that the stabilization mechanism identified by Fiedler et al. for the Hopf normal form can also apply to unstable periodic orbits created by subcritical Hopf bifurcations in higher-dimensional dynamical systems. Our analysis focuses on a particular codimension-two bifurcation that captures the stabilization mechanism in the Hopf normal form example, and we show that the same codimension-two bifurcation is present in the Lorenz equations with appropriately chosen Pyragas-type time-delayed feedback. This example suggests a possible strategy for choosing the feedback gain matrix in Pyragas control of unstable periodic orbits that arise from a subcritical Hopf bifurcation of a stable equilibrium. In particular, our choice of feedback gain matrix is informed by the Fiedler et al. example, and it works over a broad range of parameters, despite the fact that a center-manifold reduction of the higher-dimensional problem does not lead to their model problem.

Bifurcation analysis of a Morris---Lecar neuron model

Abstract: In this paper, we investigate the dynamical behaviors of a Morris---Lecar neuron model. By using bifurcation methods and numerical simulations, we examine the global structure of bifurcations of the model. Results are summarized in various two-parameter bifurcation diagrams with the stimulating current as the abscissa and the other parameter as the ordinate. We also give the one-parameter bifurcation diagrams and pay much attention to the emergence of periodic solutions and bistability. Different membrane excitability is obtained by bifurcation analysis and frequency-current curves. The alteration of the membrane properties of the Morris---Lecar neurons is discussed.