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Geometry and stability of dynamical systems.

Raffaele Punzi¹, Mattias N R Wohlfarth

¹Zentrum für Mathematische Physik und II. Institut für Theoretische Physik, Universität Hamburg, 22761 Hamburg, Germany. raffaele.punzi@desy.de

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|June 13, 2009

Summary

This study redefines stability for dynamical systems using geometry. It proposes new definitions for local and global stability, offering intrinsic measures for Lagrangian systems.

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Area of Science:

Dynamical Systems Theory
Differential Geometry
Mathematical Physics

Background:

Stability analysis is crucial for understanding dynamical systems.
Existing definitions of stability often rely on external geometric structures.
Intrinsic geometric properties of systems are key to robust stability definitions.

Purpose of the Study:

To provide a geometric perspective on the global and local stability of dynamical systems.
To develop intrinsic definitions of stability for general and Lagrangian systems.
To address limitations in current stability analysis methods.

Main Methods:

Geometric analysis of dynamical systems.
Definition of stability based on seminorms and linear connections.
Application to second-order and Lagrangian systems.
Analysis of Maupertuis-Jacobi geodesics.

Main Results:

Global Lyapunov stability requires seminorms on perturbation bundles.
A novel definition of local stability is proposed using linear connections.
Lagrangian systems possess intrinsic global and local stability notions.
These intrinsic definitions overcome limitations of geodesic-based analyses.

Conclusions:

Geometric structures are essential for unambiguous stability definitions.
Intrinsic stability measures for Lagrangian systems offer significant advantages.
The proposed framework enhances the understanding of dynamical system stability.