Nonintegrability of dynamical systems near degenerate equilibria

We prove that general three- or four-dimensional systems %of differential equations are real-analytically nonintegrable near degenerate equilibria in the Bogoyavlenskij sense under additional weak conditions when the Jacobian matrices have a zero and pair of purely imaginary eigenvalues or two incommensurate pairs of purely imaginary eigenvalues at the equilibria. For this purpose, we reduce their integrability to that of the corresponding Poincare-Dulac normal forms and further to that of simple planar systems, and use a novel approach for proving the analytic nonintegrability of planar systems. Our result also implies that general three- and four-dimensional systems exhibiting fold-Hopf and double-Hopf codimension-two bifurcations, respectively, are real-analytically nonintegrable under the weak conditions. To demonstrate these results, we give two examples for the Rossler system and coupled van der Pol oscillators.Comment: 19 page

Periodic perturbations of codimension-two bifurcations with a double zero eigenvalue in dynamical systems with symmetry

We study bifurcation behavior in periodic perturbations of two-dimensional symmetric systems exhibiting codimension-two bifurcations with a double eigenvalue when the frequencies of the perturbation terms are small. We transform the periodically perturbed system to a simpler one which is a periodic perturbation of the normal form for codimension-two bifurcations with a double zero eigenvalue and symmetry, and apply the subharmonic and homoclinic Melnikov methods to analyze bifurcations occurring in the system. In particular, we show that there exist transverse homoclinic or heteroclinic orbits, which yield chaotic dynamics, in wide parameter regions. These results can be applied to three or higher-dimensional systems and even to infinite-dimensional systems with the assistance of center manifold reduction and the invariant manifold theory. We illustrate our theory for a pendulum subjected to position and velocity feedback control when the desired position is periodic in time. We also give numerical computations by the computer tool AUTO to demonstrate the theoretical results.Comment: 40 pages, 21 figure

Heteroclinic connections between triple collisions and relative periodic orbits in the isosceles three-body problem

We study the isosceles three-body problem and show that there exist infinitely many families of relative periodic orbits converging to heteroclinic cycles between equilibria on the collision manifold in Devaney's blown-up coordinates. Towards this end, we prove that two types of heteroclinic orbits exist in much wider parameter ranges than previously detected, using self-validating interval arithmetic calculations, and we appeal to the previous results on heteroclinic orbits. Moreover, we give numerical computations for heteroclinic and relative periodic orbits to demonstrate our theoretical results. The numerical results also indicate that the two types of heteroclinic orbits and families of relative periodic orbits exist in wider parameter regions than detected in the theory and that some of them are related to Euler orbits