On the reliability of numerical studies of stochasticity I: Existence of time averages

Abstract

The stochastic properties of classical dynamical systems are often studied by means of numerical computations of orbits up to very large times, so that the accumulation of numerical errors would appear to destroy the reliability of the computations. We discuss this problem on the basis of a theorem of Anosov and Bowen which implies that, if the errors at each step are small enough, for Anosov systems the computations of time averages are reliable even for infinite times. We test numerically from this point of view three classical examples

Noise-enhanced spontaneous chaos in semiconductor superlattices at room temperature

“…[5,15,27] at 300 K using a standard stochastic Euler method. To calculate the largest Lyapunov exponent (LLE), we have simultaneously integrated all perturbed and unperturbed trajectories and used the Benettin et al algorithm [30]. LLE calculations with the Gao et al algorithm [12,31] give the same results.…”

Numerical integration of variational equations

“…An N D Hamiltonian system has 2N (possibly nondistinct) LCEs, which are ordered as χ 1 ≥ χ 2 ≥ · · · ≥ χ 2N . In [18] a theorem was formulated, which led directly to the development of a numerical technique for the computation of all LCEs, based on the time evolution of many deviation vectors, kept linearly independent through a Gram-Schmidt orthonormalization procedure. The theoretical framework, as well as the corresponding numerical method for the computation of all LCEs (usually called the 'standard method' ), were given in [16,17].…”

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“…Non-equilibrium extensions were shown to exhibit symmetry about a point other than zero [6,7], leading to the discovery that these extensions also contain hidden Hamiltonian structure [8,9]. Also known for more than twenty years is the algorithm for numerical computation of Lyapunov exponents due to Benettin and others [10,11]. Later, a constraint method was introduced [12].…”