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.
Highlights
IntroductionThe study of non-linear differential equations began with the rise of differential equations themselves, formally, the birth of the modern field of non-linear dynamical systems began in 1962 when
The study of non-linear differential equations began with the rise of differential equations themselves, formally, the birth of the modern field of non-linear dynamical systems began in 1962 whenEdward Lorenz, a MIT meteorologist, computationally simulated a set of differential equations for fluid convection in the atmosphere
As Lyapunov exponents give a measure of the separation of closely adjacent solutions to the set of differential equations when the system has evolved into a steady state, their numerical calculation has always led to high processing times
Summary
The study of non-linear differential equations began with the rise of differential equations themselves, formally, the birth of the modern field of non-linear dynamical systems began in 1962 when. Edward Lorenz, a MIT meteorologist, computationally simulated a set of differential equations for fluid convection in the atmosphere In such simulations, he noticed a complicated behaviour that seemed to depend sensitively on initial conditions, and here he found the “Lorenz” strange attractor. As Lyapunov exponents give a measure of the separation of closely adjacent solutions (in initial conditions) to the set of differential equations when the system has evolved into a steady state (after a very long time), their numerical calculation has always led to high processing times. In order to fully exploit the richness of the non-linear dynamical behaviour embodied in the set of ODEs, the full Lyapunov spectrum is required This way, the calculation of Lyapunov exponents is a high-computational-load problem that is benefited by parallelisation techniques.
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