Equilibria and bifurcations in contact dynamics

TL;DR

This paper systematically analyzes equilibria and bifurcations in contact vector fields, revealing their structure, types, and conditions for various bifurcations, with implications for higher-dimensional systems.

Contribution

It provides a detailed classification of equilibria and bifurcations in contact dynamics, including new insights into eigenvalue structures and bifurcation types in different dimensions.

Findings

Equilibria occur at singular or tangent points of the Hamiltonian zero-level set.

Eigenvalues at equilibria include a principal coefficient and quadruplets, with specific relations.

Type I and Type II codimension 1 equilibria lead to saddle-node bifurcations, with special features in higher dimensions.

Abstract

We provide a systematic study of equilibria of contact vector fields and the bifurcations that occur generically in 1-parameter families, and express the conclusions in terms of the Hamiltonian functions that generate the vector fields. Equilibria occur at points where the zero-level set of the Hamiltonian function is either singular or is tangent to the contact structure. The eigenvalues at an equilibrium have an interesting structure: there is always one particular real eigenvalue of any equilibrium, related to the contact structure, that we call the principal coefficient, while the other eigenvalues arise in quadruplets, similar to the symplectic case except they are translated by a real number equal to half the principal coefficient. There are two types of codimension 1 equilibria, named Type I, arising where the zero-set of the Hamiltonian is singular, and Type II where it is not, but there is a degeneracy related again to the principal coefficient and the contact of the zero level-set of the Hamiltonian with the contact structure. Both give rise generically to saddle-node bifurcations. Some special features include: (i) for Type II singularities, Hopf bifurcations cannot occur in dimension 3, but they may in dimension 5 or more; (ii) for Type I singularities, a fold-Hopf bifurcation can occur with codimension 1 in any dimension, and (iii) again for Type I, and in dimension at least 5, a fold-multi-Hopf bifurcation (where several pairs of eigenvalues pass through the imaginary axis simultaneously together with one through the origin) may also occur with codimension 1.