Multi-core and many-core SPMD parallel algorithms for construction of basins of attraction

Abstract

Construction of basins of attraction, used for the analysis of nonlinear dynamical systems which present multistability, are computationaly very expensive. Because of the long runtime needed, in many cases, the construction of basins does not have any practical use. Numerical time integration is currently the bottleneck of algorithms used for the construction of such basins. The integrations related to each set of initial conditions are independent of each other. The assignment of each integration to a separate thread seems very attractive, and parallel algorithms which use this approach to construct the basins are presented here. Two versions are considered, one for multi-core and another for many-core architectures, both based on a SPMD approach. The algorithm is tested on three systems, the classic nonlinear Duffing system, a non-ideal system exhibiting the Sommerfeld effect and an immunodynamic system. The results for all examples demonstrate the versatility of the proposed parallel algorithm, showing that the multi-core parallel algorithm using MPI has nearly an ideal speedup and efficiency.

Stability of a nonideally excited Duffing oscillator

“…which is exactly the same as (40) except for a larger driving torque at resonance. The corresponding torquespeed curves near resonance are represented in Fig.…”

“…uncertainties on the values of the system parameters) [1]. In addition, some effort should be devoted to determining or estimating the basins of attraction of the different attractors present in the system [20,40]. This would help to quantify the probability of resonant capture and passage through resonance.…”

“…Fig 6. Torque-speed curves near resonance for parameter values(40) stable-since we have d < 0-while the one on the right branch (point B) is unstable. These fixed points can be readily computed by introducing parameter values (40) into expressions (26):…”