Controllability near a homoclinic bifurcation

TL;DR

This paper investigates how control-affine systems near a homoclinic bifurcation behave in terms of controllability, revealing how control sets evolve with parameters and identifying conditions for the existence of periodic solutions close to homoclinic orbits.

Contribution

It introduces a new parameter based on a split function that characterizes the behavior of control sets near a homoclinic bifurcation in low-dimensional systems.

Findings

Control sets depend on a bifurcation parameter and control range size.

A split function determines the control set behavior near the bifurcation.

Existence of periodic solutions close to homoclinic orbits under control.

Abstract

Controllability properties are studied for control-affine systems depending on a parameter and with constrained control values. The uncontrolled systems in dimension two and three are subject to a homoclinic bifurcation. This generates two families of control sets depending on a parameter in the involved vector fields and the size of the control range. A new parameter given by a split function for the homoclinic bifurcation determines the behavior of these control sets. It is also shown that there are parameter regions where the uncontrolled equation has no periodic orbits, while the controlled systems have periodic solutions arbitrarily close to the homoclinic orbit