Bifurcation diagram

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

Kuramoto-Sivashinsky dynamics on the center-unstable manifold

Abstract: This paper studies the dynamical behavior of solutions of the Kuramoto–Sivashinsky partial differential equation with periodic boundary conditions on a spatial interval $[ 0,h ]$. The length h is the bifurcation parameter and reduction is made to a two-(complex-)dimensional system on a local center-unstable manifold near the second bifurcation point $h_2 $ from the trivial solution. The resulting $O( 2 )$-equivariant system displays all the behavior found in high precision simulations of the partial differential equation near this bifurcation point. In particular, bifurcation sequences to stable traveling waves, unstable modulated traveling waves, and attracting heteroclinic cycles are reproduced qualitatively and quantitatively within $1\%$ in the parameter range $h_2 \pm 20\% $. A clear understanding of the global dynamical behavior in this region is thus obtained.

Bifurcation diagram and pattern formation of phase slip centers in superconducting wires driven with electric currents.

Abstract: We provide here new insights into the classical problem of a one-dimensional superconducting wire exposed to an applied electric current using the time-dependent Ginzburg-Landau model. The most striking feature of this system is the well-known appearance of oscillatory solutions exhibiting phase slip centers (PSC's) where the order parameter vanishes. Retaining temperature and applied current as parameters, we present a simple yet definitive explanation of the mechanism within this nonlinear model that leads to the PSC phenomenon and we establish where in parameter space these oscillatory solutions can be found. One of the most interesting features of the analysis is the evident collision of real eigenvalues of the associated PT-symmetric linearization, leading as it does to the emergence of complex elements of the spectrum.

Hopf Bifurcation Analysis: A Frequency Domain Approach

Abstract: The Hopf bifurcation theorem continuation of bifurcation curves on the parameter plane degenerate bifurcations in the space of system parameters high-order Hopf bifurcation formulas Hopf bifurcation in nonlinear systems with time delays birth of multiple limit cycles appendix.

Additive Noise Destroys a Pitchfork Bifurcation

Abstract: In the deterministic pitchfork bifurcation the dynamical behavior of the system changes as the parameter crosses the bifurcation point. The stable fixed point loses its stability. Two new stable fixed points appear. The respective domains of attraction of those two fixed points split the state space into two macroscopically distinct regions. It is shown here that this bifurcation of the dynamical behavior disappears as soon as additive white noise of arbitrarily small intensity is incorporated the model. The dynamical behavior of the disturbed system remains the same for all parameter values. In particular, the system has a (random) global attractor, and this attractor is a one-point set for all parameter values. For any parameter value all solutions converge to each other almost surely (uniformly in bounded sets). No splitting of the state space into distinct regions occurs, not even into random ones. This holds regardless of the intensity of the disturbance.