The brief time-reversibility of the local Lyapunov exponents for a small chaotic Hamiltonian system

• We consider the local Lyapunov spectrum for a four-dimensional Hamilton system.
• Its stable periodic motion can be reversed for long times.
• In the chaotic motion, time reversal occurs only for a short time.
• Perturbations will change this short unstable case into a different stable case.
• These observations might relate chaos to the Second Law of Thermodynamics.

We consider the local (instantaneous) Lyapunov spectrum for a four-dimensional Hamiltonian system. Its stable periodic motion can be reversed for long times. Its unstable chaotic motion, with two symmetric pairs of exponents, cannot. In the latter case reversal occurs for more than a thousand fourth-order Runge–Kutta time steps, followed by a transition to a new set of paired Lyapunov exponents, unrelated to those seen in the forward time direction. The relation of the observed chaotic dynamics to the Second Law of Thermodynamics is discussed.