Weakly Coupled Distributed Calculation of Lyapunov Exponents for Non-Linear Dynamical Systems

Abstract

Numerical estimation of Lyapunov exponents in non-linear dynamical systems results in a very high computational cost. This is due to the large-scale computational cost of several Runge-Kutta problems that need to be calculated. In this work we introduce a parallel implementation based on MPI (Message Passing Interface) for the calculation of the Lyapunov exponents for a multidimensional dynamical system, considering a weakly coupled algorithm. Since we work on an academic high-latency cluster interconnected with a gigabit switch, the design has to be oriented to reduce the number of messages required. With the design introduced in this work, the computing time is drastically reduced, and the obtained performance leads to close to optimal speed-up ratios. The implemented parallelisation allows us to carry out many experiments for the calculation of several Lyapunov exponents with a low-cost cluster. The numerical experiments showed a high scalability, which we showed with up to 68 cores.

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“…In the context of a bounded system, the presence of a positive LE is an indicator of chaos in nonlinear dynamical systems exhibiting seemingly random and unpredictable behavior. Therefore, the computation of the maximal LE is as a valuable tool for detecting a chaotic behavior [42].…”