Phase space volume preserving dynamics for deterministic dynamical systems

Area of Science:

• * Statistical mechanics
• * Nonlinear dynamics
• * Classical chaos

Background:

• * Linearized dynamics typically cause chaotic systems to exhibit unphysical phase space volume collapse.
• * This collapse arises from exponential alignment of tangent vectors, irrespective of true volume compressibility.
• * Existing models struggle to preserve volume in non-Hamiltonian and dissipative systems.

Purpose of the Study:

• * To propose an alternative linearized dynamics that preserves phase space volume.
• * To develop a generalized Liouville equation applicable to non-Hamiltonian systems.
• * To provide an invariant measure for dissipative dynamics and an evolution equation for the density matrix.

Main Methods:

• * Rectifying the generalized Liouville equation using an operator derived from the anti-symmetric part of the stability matrix.
• * Defining a time-evolution operator that generates orthogonal transformations, preserving volume elements.
• * Analyzing tangent space dynamics using complete basis vectors without re-orthogonalization.

Main Results:

• * Phase space volume invariance is achieved even for non-Hamiltonian and dissipative chaotic systems.
• * An evolution equation analogous to the quantum Liouville-von Neumann equation is derived.
• * The method allows computation of Lyapunov exponent spectra and Gibbs entropy flow rates for various chaotic models.

Conclusions:

• * The proposed dynamics offers a physically realistic description of chaotic systems by preserving phase space volume.
• * This approach provides a robust framework for analyzing dissipative, transient, and driven chaotic dynamics.
• * The method is numerically convenient and applicable to diverse classical systems.