Computational sensitivity of a high dimensional dynamical oscillator

The difficulty arises when we carry out Lyapunov analysis for a high dimensional oscillator. Our model is an eight-dimensional oscillator with a hysteresis element. This oscillator is piecewise-linear, and therefore, the explicit solution in each branch are obtained explicitly. We define the return map rigorously by using these explicit solutions. Numerical results show that we cannot often obtain a stationary solution even if we remove the transient 100,000 iterations of the return map. Furthermore, we encounter the following case: Lyapunov exponents are calculated by averaging 1,000,000 iterations of the Jacobian matrix of the return map to calculate the Lyapunov exponents. However, we cannot simply estimate and classify the solutions from the value of Lyapunov exponents in some cases even if the objective attractor is not chaotic, because the structure of oscillators with high dimensions are extremely complex.