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Complex Dynamics in Physical Pendulum Equation with Suspension Axis Vibrations


  The physical pendulum equation with suspension axis vibrations is investigated.By using Melnikov s method,we prove the conditions for the existence of chaos under periodic perturbations.By using second-order averaging method and Melinikov s method,we give the conditions for the existence of chaos in an averaged system under quasi-periodic perturbations forΩ=nω+ευ,n=1-4,whereυis not rational toω.We are not able to prove the existence of chaos for n=5-15,but show the chaotic behavior for n=5 by numerical simulation.By numerical simulation we check on our theoretical analysis and further exhibit the complex dynamical behavior,including the bifurcation and reverse bifurcation from period-one to period-two orbits;the onset of chaos,the entire chaotic region without periodic windows,chaotic regions with complex periodic windows or with complex quasi-periodic windows;chaotic behaviors suddenly disappearing,or converting to period-one orbit which means that the system can be stabilized to periodic motion by adjusting bifurcation parametersα,δ,fo andΩ;and the onset of invariant torus or quasi-periodic behaviors,the entire invariant torus region or quasi-periodic region without periodic window,quasi-periodic behaviors or invariant torus behaviors suddenly disappearing or converting to periodic orbit;and the jumping behaviors which including from periodone orbit to anther period-one orbit,from quasi-periodic set to another quasi-periodic set;and the interleaving occurrence of chaotic behaviors and invariant torus behaviors or quasi-periodic behaviors;and the interior crisis; and the symmetry breaking of period-one orbit;and the different nice chaotic attractors.However,we haven t find the cascades of period-doubling bifurcations under the quasi-periodic perturbations and show the differences of dynamical behaviors and technics of research between the periodic perturbations and quasi-periodic perturbations.……