Volume-preserving normal forms of Hopf-zero singularity

TL;DR

This paper develops a method for computing volume-preserving normal forms of Hopf-zero singularities, identifying unique generators of first integrals, and applying these to complex dynamical systems like R"ossler and Kuramoto--Sivashinsky equations.

Contribution

It introduces a Lie algebra framework for volume-preserving normal forms and computes infinite level parametric normal forms for non-degenerate perturbations.

Findings

Unique generator of first integrals for systems with quadratic parts

Explicit formulas for infinite level normal forms

Application to modified R"ossler and Kuramoto--Sivashinsky equations

Abstract

A practical method is described for computing the unique generator of the algebra of first integrals associated with a large class of Hopf-zero singularity. The set of all volume-preserving classical normal forms of this singularity is introduced via a Lie algebra description. This is a maximal vector space of classical normal forms with first integral; this is whence our approach works. Systems with a non-zero condition on their quadratic parts are considered. The algebra of all first integrals for any such system has a unique (modulo scalar multiplication) generator. The infinite level volume-preserving parametric normal forms of any non-degenerate perturbation within the Lie algebra of any such system is computed, where it can have rich dynamics. The associated unique generator of the algebra of first integrals are derived. The symmetry group of the infinite level normal forms are also discussed. Some necessary formulas are derived and applied to appropriately modified R\"{o}ssler and generalized Kuramoto--Sivashinsky equations to demonstrate the applicability of our theoretical results. An approach (introduced by Iooss and Lombardi) is applied to find an optimal truncation for the first level normal forms of these examples with exponentially small remainders. The numerically suggested radius of convergence (for the first integral) associated with a hypernormalization step is discussed for the truncated first level normal forms of the examples. This is achieved by an efficient implementation of the results using Maple.