Harmonics And Bifurcations Of Harmonics And Chaos In Duffing Equation

In this paper , we discuss the dynamics of the Duffing equation with fifth nonlinear restoring force, one external forcing and a phase shift. At present, there are less attention to the effect of the phase shift on dynamics.By using local bifurcation theory, second-order averaging methods, Mel-nikov theory and chaotic theories in dynamical systems, the conditions of existences for primary resonance, second-order subharmonic, third-order subhar-monic, m-order subharmonic and chaos are given. Numerical simulations including bifurcation diagram, bifurcation surfaces, phase portraits, not only show the consistence with the theoretical analysis, but also exhibit the new dynamical behaviors. We show the onset of chaos, chaos suddenly converging to period 1 orbit, chaos suddenly disappearing, one-band and double-band chaos, period-doubling bifurcations from period 1, 2, and 3 orbits, chaotic attractor with complex period-windows (period-2, 3 and 5), interior crisis, boundary crisis and period-3 bubble.The paper consists of two chapters. Chapter 1 is the preparation knowledge. A brief review of local bifurcation theory, second-order averaging methods and Melnikov theory is presented.In chapter 2, the Duffing equation with fifth nonlinear restoring force, one external forcing and a phase shift is discussed, We give the conditions of existences and bifurcations for primary resonance, second-order subharmonic, third-order subharmoinc, m-order subharmonic, chaos, and the results of the numerical simulation.