Phase space volume preserving dynamics for deterministic dynamical systems

Infinitesimal volumes stretch and contract as they coevolve with classical phase space trajectories according to linearized dynamics. Unless these tangent-space dynamics are modified, chaotic evolution causes the volume spanned by evolving tangent vectors to collapse. However, this collapse is unphysical and due to their exponential alignment along the most expanding direction, independent of the compressibility of the phase-space volume. Here, we propose an alternative linearized dynamics and rectify the generalized Liouville equation to preserve phase space volume, even for non-Hamiltonian systems. Within a classical density matrix theory, we define the time-evolution operator from the anti-symmetric part of the stability matrix so that phase space volume is time invariant. The operator generates orthogonal transformations without distorting volume elements, providing an invariant measure for dissipative dynamics and an evolution equation for the density matrix akin to the quantum mechanical Liouville–von Neumann equation. The compressibility of volume elements is determined by a nonorthogonal operator made from the symmetric part of the stability matrix. We analyze complete sets of basis vectors for the tangent space dynamics of chaotic systems, which may be dissipative, transient, or driven, without re-orthogonalization of tangent vectors. The linear harmonic oscillator, the Lorenz–Fetter model, and the Hénon–Heiles system demonstrate the computation of the instantaneous Lyapunov exponent spectrum and the local Gibbs entropy flow rate using these bases and show that it is numerically convenient.