Construction of chaotic regions

Abstract

Non-linear oscillators can exhibit non-periodic chaotic response under periodic excitation. A point in the parametric space is said to be within the chaotic regions if the response under periodic excitation is chaotic. The objective is to construct the boundaries of the qualitatively different solution regions of an oscillator against its parameters. Chaos results when all possible periodic solutions are unstable but remain bounded. It is commonly associated with the accelerated bifurcation of higher order subharmonics. Periodic solutions are found by the incremental harmonic balance method. The stability of the solution is checked by the Floquet theory to determine the bifurcation points and the region boundaries. The boundaries are extended in an incremental manner. By means of the Newtonian algorithm, accurate solutions along the transition boundaries are traced without difficulties. The Duffing equation is taken as an example and the result is compared with that of Ueda [1].