Choice of Time Step in the Simulation of Continuous Time Dynamical Systems

Authors: İnanç BİROL, Avadis HACINLIYAN

Abstract: In the simulation of a dynamcal system, if the system at hand possesses a chaotic nature, the choice of time step is crucial. Characterizing a chaotic system, the precission of measurements play a major role for such systems that almost any imprecission in the observables shows a tendency of exponentially propogating in time. Ironically, to reveal this property, one has to simulate the system, as it is yet the only available method to compute the Lyapunov exponents of a system (the quantities which carry the fingerprint of chaos or non-chaos). Continuous time simulation algorithms to compute the Lyapunov exponents augments a given system by equations that govern the time evolution of a set of unit basis vectors about a fiducial trajectory. One such method is Wiesel algorithm. In this work, it is shown that the Wiesel algorithm represents a system which exactly maintains an orthonormal basis for the tangent flow. The choice of simulation time steps for chaotic systems is considered and suggested bounds are verified for the Lorenz and R\"ossler systems.

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