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

Learn More →

Complex dynamics in physical pendulum equation with suspension axis vibrations

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 for Ω = nω + ɛν, n = 1 − 4, where ν is not rational to ω. 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 parameters α, δ, f 0 and Ω; 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Complex dynamics in physical pendulum equation with suspension axis vibrations

Loading next page...
 
/lp/springer-journals/complex-dynamics-in-physical-pendulum-equation-with-suspension-axis-3j9DNlkhYV

References (46)

Publisher
Springer Journals
Copyright
Copyright © 2010 by Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer Berlin Heidelberg
Subject
Mathematics; Theoretical, Mathematical and Computational Physics; Math Applications in Computer Science; Applications of Mathematics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-008-8276-6
Publisher site
See Article on Publisher Site

Abstract

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 for Ω = nω + ɛν, n = 1 − 4, where ν is not rational to ω. 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 parameters α, δ, f 0 and Ω; 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.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Dec 12, 2009

There are no references for this article.