2012 Volume 3 Issue 4 Pages 508-520
We have encountered serious difficulties when we analyze bifurcation phenomena in a high-dimensional oscillator by Lyapunov analysis. Our model uses an eight-dimensional piecewise-linear oscillator containing a hysteresis element. The numerical results show that the solutions do not arrive at a stationary state quite often even if we use piecewise-linear exact solutions and remove more than 100,000 transient iteration of the Poincaré map. Furthermore, we calculate the Lyapunov exponents by averaging 1,000,000 iteration of the Jacobian matrix of the Poincaré map after removing transient 500,000 iteration. The values of the Lyapunov exponents themselves could be trustworthy because piecewise-linear exact solutions have been employed. However, it is very difficult to classify the solutions from the value of the calculated Lyapunov exponents even if the attractor is not chaotic. This is because the bifurcation structure of a higher-dimensional torus is extremely complicated. In some cases, it is very hard to settle how many iteration is necessary to calculate reasonable Lyapunov exponents, and to distinguish whether Lyapunov exponents are exactly zero or not from the values of the numerically calculated Lyapunov exponents. This is a major concern of this study. These complex situations are spotlighted by observing attractors and drawing the graph of the Lyapunov exponents.