Controlling Effect of Geometrically Defined Local Structural Changes on Chaotic Hamiltonian Systems

TL;DR

This paper introduces a local geometric criterion based on curvature to control chaos in Hamiltonian systems by modifying the potential, offering a minimal and effective method for stabilizing chaotic dynamics.

Contribution

It extends geometric analysis of Hamiltonian chaos to potential models using conformal metrics and develops a local curvature-based control method for chaos suppression.

Findings

Curvature-based criteria identify unstable regions in Hamiltonian systems.

Local potential modifications can stabilize chaotic trajectories.

The method provides a minimal intervention approach for chaos control.

Abstract

An effective characterization of chaotic conservative Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor derived from the structure of the Hamiltonian has been extended to a wide class of potential models of standard form through definition of a conformal metric. The geodesic equations reproduce the Hamilton equations of the original potential model through an inverse map in the tangent space. The second covariant derivative of the geodesic deviation in this space generates a dynamical curvature, resulting in (energy dependent) criteria for unstable behavior different from the usual Lyapunov criteria. We show here that this criterion can be constructively used to modify locally the potential of a chaotic Hamiltonian model in such a way that stable motion is achieved. Since our criterion for instability is local in coordinate space, these results provide a new and minimal method for achieving control of a chaotic system.