Computation of periodic solution bifurcations in ODEs using bordered systems

We consider numerical methods for the computation and continuation of the three generic secondary periodic solution bifurcations in autonomous ODEs, namely the fold, the period-doubling (or flip) bifurcation, and the torus (or Neimark–Sacker) bifurcation. In the fold and flip cases we append one scalar equation to the standard periodic BVP that defines the periodic solution; in the torus case four scalar equations are appended. Evaluation of these scalar equations and their derivatives requires the solution of linear BVPs, whose sparsity structure (after discretization) is identical to that of the linearization of the periodic BVP. Therefore the calculations can be done using existing numerical linear algebra techniques, such as those implemented in the software AUTO and COLSYS

On the Takens-Bogdanov Bifurcation in the Chua’s Equation

The analysis of the Takens-Bogdanov bifurcation of the equilibrium at the origin in the Chua’s equation with a cubic nonlinearity is carried out. The local analysis provides, in first approximation, different bifurcation sets, where the presence of several dynamical behaviours (including periodic, homoclinic and heteroclinic orbits) is predicted. The local results are used as a guide to apply the adequate numerical methods to obtain a global understanding of the bifurcation sets. The study of the normal form of the Takens-Bogdanov bifurcation shows the presence of a degenerate (codimension-three) situation, which is analyzed in both homoclinic and heteroclinic cases

Bifurcation and stability for Nonlinear Schroedinger equations with double well potential in the semiclassical limit

We consider the stationary solutions for a class of Schroedinger equations with a symmetric double-well potential and a nonlinear perturbation. Here, in the semiclassical limit we prove that the reduction to a finite-mode approximation give the stationary solutions, up to an exponentially small term, and that symmetry-breaking bifurcation occurs at a given value for the strength of the nonlinear term. The kind of bifurcation picture only depends on the non-linearity power. We then discuss the stability/instability properties of each branch of the stationary solutions. Finally, we consider an explicit one-dimensional toy model where the double well potential is given by means of a couple of attractive Dirac's delta pointwise interactions.Comment: 46 pages, 4 figure

Semi-global analysis of periodic and quasi-periodic k:1 and k:2 resonances

The present paper investigates a family of nonlinear oscillators at Hopf bifurcation, driven by a small quasi-periodic forcing. In particular, we are interested in the situation that at bifurcation and for vanishing forcing strength, the driving frequency and the normal frequency are in k:1 or k:2 resonance. For small but nonvanishing forcing strength, a semi-global normal form system is found by averaging and applying a van der Pol transformation. The bifurcation diagram is organised by a codimension 3 singularity of nilpotent-elliptic type. A fairly complete analysis of local bifurcations is given; moreover, all the nonlocal bifurcation curves predicted by Dumortier et al. (1991) are found numerically.