Trigonometric polynomials methods to simulate oscillating chaotic systems

Numerical simulations are an important tool in the understanding of chaotic dynamic systems. Limitations inherent to the simulations may lead to erroneous interpretations of the behavior of these systems. Among these limitations are superstability (suppression of the chaos of chaotic systems), computational chaos (induction of chaos in non-chaotic systems) and apparent contradictions between theoretical predictions and observed responses in the simulations. These issues are present whether the solver is fixed-step or variable-step. If we are to trust the output of numerical approaches when solving dynamic systems, it is of utmost importance to know whether any simulations are yielding the correct answers, or whether such answers have been affected by errors. Such errors are either inherent to the numerical approach or due to rounding. In both cases, they may alter the outcome to the point of not representing the true dynamics of the system. Simulations ought to reflect the system’s true behavior. The outcomes of the simulations must be coherent with the simulated system. If one wants to avoid numerical approaches that may induce computational chaos, superstability or theoretical discrepancies, it is necessary to choose the right numerical method. This research investigates the potential selection of a numerical method capitalizing on known characteristics of the dynamic system. The use of special methods that tap on some known features of the chaotic dynamic system at hand has received little attention thus far. In this work, we propose the use of trigonometric polynomials methods (TPM’s) derived by Gautschi [1]. Gautschi’s method leverages on the oscillatory response inherent to chaotic systems. We hypothesize that using Gautschi’s method should result in simulations that adhere more accurately to the real solution. This is in contrast to the use of traditional methods commonly applied for these kinds of simulations. We give evidence that the adequate choice of an special method tapping on prior knowledge should outperformed an otherwise default choice. Empirical evidence is given in the form of simulations over a set of chaotic systems. As additional advantages, superstability is avoided and choosing of the integration step is straight-forward as it is based on the system frequency.