Bifurcations And Chaos In Duffing Equation

In this paper, we discuss the dynamics of the During equation with even-odd nonlinear restoring force, one external forcing and a phase shift. At present,there are less attention to the equation with even-odd nonlinear-restoring force.By using local bifurcation theory, second-order averaging methods, Melnikovtheory and chaotic theories in dynamical systems, the conditions of existencesfor primary resonance, second-order subharmonic, third-order subharmonic, m-order subharmonic and chaos are given. Numerical simulations including bifur-cation diagram, bifurcation surfaces, phase portraits, not only show the con-sistence with the theoretical analysis, but also exhibit the new dynamical be-haviors. We show the onset of chaos, chaos suddenly converging to period 1orbit, chaos suddenly disappearing, cascades of inverse period-doubling bifurca-tion,priod-doubling bifurcation,symmetry period-doubling bifurcations of priod-3orbit,symmetry-breaking of period orbits,interleaving occurence of chaotic behav-iors and period-1 orbit,a great abundance of periodic windows in transient chaoticregions with interior crises and boundary crisis,varied attractors.The paper consists of two chapters. Chapter 1 is the preparation knowledge.A brief review of local bifurcation theory, second-order averaging methods andMelnikov theory is presented.In chapter 2, the Duffing equation with even-odd nonlinear restoring force,one external forcing and a phase shift is discussed, We give the conditions ofexistences and bifurcations for primary resonance, second-order subharmonic,third-order subharmoinc, m-order subharmonic, chaos, and the results of thenumerical simulation.