Controlling effect of geometrically defined local structural changes on chaotic Hamiltonian systems.

Area of Science:

• * Mathematical Physics
• * Dynamical Systems Theory
• * Chaos Theory

Background:

• * Characterizing chaotic conservative Hamiltonian systems is crucial for understanding complex dynamics.
• * Existing methods like Lyapunov criteria have limitations in local instability assessment.
• * Geometric approaches using Riemannian metrics offer novel perspectives on system dynamics.

Purpose of the Study:

• * To extend the characterization of chaotic Hamiltonian systems using a conformal metric.
• * To derive new, energy-dependent criteria for system instability.
• * To demonstrate a method for controlling chaotic systems by locally modifying potentials.

Main Methods:

• * Definition of a conformal metric tensor derived from the Hamiltonian structure.
• * Analysis of geodesic equations and their relation to Hamilton equations via tangent space mapping.
• * Calculation of dynamical curvature from the geodesic deviation to establish instability criteria.

Main Results:

• * Geodesic equations successfully reproduce Hamilton equations for potential models.
• * A novel, energy-dependent dynamical curvature criterion for instability was developed.
• * This criterion allows for local modification of potentials to achieve stable motion.

Conclusions:

• * The developed dynamical curvature criterion provides an alternative to Lyapunov criteria for instability.
• * The method offers a minimal, local approach to controlling chaotic Hamiltonian systems.
• * This geometric framework facilitates targeted stabilization of unstable dynamics.