Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

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

Global orbit of a complicated nonlinear system with the global dynamic frequency method: 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Journal of Low Frequency Noise, Vibration and Active Control" SAGE

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

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


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.

Loading next page...
 
/lp/sage/global-orbit-of-a-complicated-nonlinear-system-with-the-global-dynamic-8XH5sp9B1h

References (35)

Publisher
SAGE
Copyright
Copyright © 2022 by SAGE Publications Ltd unless otherwise noted. Manuscript content on this site is licensed under Creative Commons Licenses
ISSN
0263-0923
eISSN
2048-4046
DOI
10.1177/1461348420919193
Publisher site
See Article on Publisher Site

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.

Journal

"Journal of Low Frequency Noise, Vibration and Active Control"SAGE

Published: Nov 4, 2020

Keywords: Homoclinic bifurcation; hyperbolic function; chaos

There are no references for this article.