Absolute Stability of Chaotic Asynchronous Multi-Interactions Schemes for Solving ODE

Multi Interaction Systems, used in the context of Virtual Reality, are dedicated to real-time interactive simulations. They open the way to the in virtuo experimentation, especially useful in the domain of biochemical kinetics. To this purpose, chaotic and asynchronous scheduling of autonomous processes is based upon desynchronization of phenomena involved in the system. It permits interactivity, especially the capability to add or remove phenomena in the course of a simulation. It provides methods of resolution of ordinary differential systems and partial derivative equations. Proofs of convergence for these methods have been established, but the problem of absolute stability, although it is crucial when considering multiscale or stiff problems, has not yet been treated. The aim of this article is to present absolute stability conditions for chaotic and asynchronous schemes. We give criteria so as to predict instability thresholds, and study in details the significant example of a damped spring-mass system. Our results, which make use of random matrices products theory, stress the point that the desynchronization of phenomena, and a random scheduling of their activations, can lead to instability.

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