Nonintegrability of dynamical systems near degenerate equilibria

TL;DR

This paper proves that most three- and four-dimensional dynamical systems near certain degenerate equilibria are analytically nonintegrable, using a novel reduction approach and examples like Rossler and van der Pol systems.

Contribution

It establishes the nonintegrability of complex systems near degenerate equilibria by reducing the problem to planar systems and introducing a new proof method.

Findings

Most systems near degenerate equilibria are nonintegrable.

The method applies to systems with specific eigenvalue configurations.

Examples include Rossler system and coupled van der Pol oscillators.

Abstract

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.