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Abstract
Global orbits connect the saddle points in an infinite period through the homoclinic and heteroclinic types of manifolds. Different from the periodic movement analysis, it requires special strategies to obtain expression of the orbit and detect the associated profound dynamic behaviors, such as chaos. In this paper, a global dynamic frequency method is applied to detect the homoclinic and heteroclinic bifurcation of the complicated nonlinear systems. The so-called dynamic frequency refers to the newly introduced frequency that varies with time t, unlike the usual static variable. This new method obtains the critical bifurcation value as well as the analytic expression of the orbit by using a standard five-step hyperbolic function-balancing procedure, which represents the influence of the higher harmonic terms on the global orbit and leads to a significant reduction of calculation workload. Moreover, a new homoclinic manifold analysis maps the periodic excitation onto the target global manifold that transfers the chaos discussion of non-autonomous systems into the orbit computation of the general autonomous system. That strategy unifies the global bifurcation analysis into a standard orbit approximation procedure. The numerical simulation results are shown to compare with the predictions.
Highlights
Nonlinear oscillations have attracted the attention of many researchers in the fields of physics, applied mathematics, and engineering fields
Effective methods have been developed to obtain the periodic solutions of the nonlinear oscillators,[1] such as the variational iteration method,[2,3] the homotopy perturbation method,[4,5] the variational approach,[6,7] and the Hamiltonian approach.[8,9]
Zhang et al.[15] computed the undetermined fundamental frequency of periodic solution, so that the homoclinic bifurcation appeared in terms of the vanishing frequency
Summary
Nonlinear oscillations have attracted the attention of many researchers in the fields of physics, applied mathematics, and engineering fields. The main differences between the global dynamic frequency and other analytic approaches are: (1) perform the discussion through energy-based equation and not the original differential equation; (2) balance limited hyperbolic terms to find the orbit expressions up to any order approximation without the usual termination, while the increase in the computation error from the HB method is attributed to the truncation of the higher-order nonlinear terms. Those points will be demonstrated by applying the proposed approaches to more specific cases to show the efficiency of the method.
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