Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system

Zhang, Ling-Hao; Wang, Wei

doi:10.3934/dcds.2017024

Cited by 1 publication

(1 citation statement)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…27 took the series form manifolds to imitate the Shilnikov-type homoclinic orbit in a 3 D system and proposed a semi-analytical direct approach to construct the heteroclinic orbit. 28 For those non-autonomous cases, Mikhlin and Manucharyan 29 constructed the orbit expression of the autonomous system with the Padé approximation and found the critical value of chaos.…”

Section: Introductionmentioning

confidence: 99%

Global orbit of a complicated nonlinear system with the global dynamic frequency method

Wang

et al. 2020

Journal of Low Frequency Noise, Vibration and Active Control

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%