Extensions of the phase space linearization (PSL) technique for non-linear oscillators

Abstract

This paper is devoted to further exploration of the PSL technique developed earlier in its companion paper. To start with, it is shown that the method is not only applicable for obtaining one-periodic orbits, but can be made use of for obtaining every conceivable orbit, including subharmonic or periodic-doubled orbits, quasi-periodic orbits and even chaotic orbits. This is numerically illustrated by constructing various orbits of Ueda's, Duffing-Holmes and van der Pol's oscillators. Next, the separatrix, that separates the basins of attraction of the stable limit cycles of Duffing-Holmes' oscillator, is constructed using the PSL procedure. A possibility of predicting the chaotic diffusion of trajectories based on a heuristic argument of a near-tangency of stable and unstable limit cycles of Duffing-Holmes' oscillator is also discussed. Finally, the PSL scheme is made use of to compute various characteristic quantities such as Fourier spectra, Liapunov characteristic exponents and probability density functions. Many new results are presented to establish the versatility of the PSL method.