Bifurcation diagram

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

Bifurcation structure of the driven van der pol oscillator

Abstract: The bifurcation set in the three-dimensional parameter space of the periodically driven van der Pol oscillator has been investigated by continuation of local bifurcation curves. The two regions in which the driving frequency ω is greater or less than the limit cycle frequency ω0 of the nondriven oscillator are considered separately. For the case ω > ω0, the subharmonic region, the extent and location of the largest Arnol'd tongues are shown, as well as the period-doubling cascades and chaotic attractors that appear within most of them. Special attention is paid to the pattern of the bifurcation curves in the transitional region between low and large dampings that is difficult to approach analytically. In the case ω < ω0, the ultraharmonic region, a recurrent pattern of the bifurcation curves is found for small values of the damping d. At medium damping the structure of the bifurcation curves becomes involved. Period-doubling sequences and chaotic attractors occur.

Peakons and their bifurcation in a generalized camassa–holm equation

Abstract: Camassa and Holm [1993] recently derived a new dispersive shallow water equation known as the Camassa–Holm equation. They showed that it also has solitary wave solutions which have a discontinuous first derivative at the wave peak and thus are called "peakons". In this paper, from the mathematical point of view, we study the peakons and their bifurcation of the following generalized Camassa–Holm equation \[ u_t+2ku_x-u_{xxt}+au^mu_x = 2u_xu_{xx} + uu_{xxx} \] with a>0, k∈ℝ, m∈ℕ and the integral constants taken as zero. Using the bifurcation method of the phase plane, we first give the phase portrait bifurcation, then give the integral expressions of peakons through the bifurcation curves and the phase portraits, and finally obtain the peakon bifurcation parameter value and the number of peakons. For m=1, 2, 3, we give the explicit expressions for the peakons. It seems that the bifurcation method of phase planes is good for the study of peakons in nonlinear integrable equations.

Symmetry-Breaking Bifurcation in the Nonlinear Schrödinger Equation with Symmetric Potentials

Abstract: We consider the focusing (attractive) nonlinear Schrodinger (NLS) equation with an external, symmetric potential which vanishes at infinity and supports a linear bound state. We prove that the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non-degenerate local maxima at zero. Under a generic assumption we show that the bifurcation is either a subcritical or supercritical pitchfork. In the particular case of double-well potentials with large separation, the power of nonlinearity determines the subcritical or supercritical character of the bifurcation. The results are obtained from a careful analysis of the spectral properties of the ground states at both small and large values for the corresponding eigenvalue parameter.

Degenerate Hopf bifurcation formulas and Hilbert's 16th problem

Abstract: This paper presents explicit formulas for the solution of degenerate Hopf bifurcation problems for general systems of differential equations of dimension $n \geqq 2$ , with smooth vector fields. The main new result is the general solution of the problem for a weak focus of order 3. For bifurcation problems with a distinguished parameter, we present the formulas for the defining conditions of all cases with codimension $ \leqq 3$. The formulas have been applied to Hilbert’s 16th problem, yielding a new proof of Bautin’s theorem, and correcting an error in Bautin’s formula for the third focal value. The approach used is the Lyapunov–Schmidt method. A review of five other approaches is given, along with literature references and comparisons to the present work.