Speed limits on classical chaos

Abstract

Uncertainty in the initial conditions of dynamical systems can cause exponentially fast divergence of trajectories, a signature of deterministic chaos. Here, we derive a classical uncertainty relation that sets a speed limit on the rates of local observables underlying this behavior. For systems with a time-invariant stability matrix, this general speed limit simplifies to classical analogues of the Mandelstam-Tamm versions of the time-energy uncertainty relation. This classical bound derives from our definition of Fisher information in terms of Lyapunov vectors on tangent space, analogous to the quantum Fisher information defined in terms of wavevectors on Hilbert space. This information measures fluctuations in local stability of the state space and sets a lower bound on the time of classical, dynamical systems to evolve between two distinguishable states. The bounds it sets apply to systems that are open or closed, conservative or dissipative, actively driven or passively evolving, and directly connect the geometries of phase space and information.