Symmetry of periodic solutions of five coupled oscillators in a ring

1996 ◽  
Vol 79 (5) ◽  
pp. 60-70
Author(s):  
Yuuji Katsuta ◽  
Hiroshi Kawakami
Keyword(s):  
Periodic Solutions ◽  
Coupled Oscillators

Coupled Oscillatory Systems with 𝔻4 Symmetry and Application to van der Pol Oscillators

2016 ◽  
Vol 26 (08) ◽  
pp. 1650141 ◽  
Author(s):  
Adrian C. Murza ◽  
Pei Yu
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Coupled Oscillators ◽  
Invariant Manifolds ◽  
Van Der Pol ◽  
Normal Form Theory ◽  
Oscillatory Systems ◽  
Weakly Coupled ◽  
Form Theory ◽  
Van Der Pol Oscillators

In this paper, we study the dynamics of autonomous ODE systems with [Formula: see text] symmetry. First, we consider eight weakly-coupled oscillators and establish the condition for the existence of stable heteroclinic cycles in most generic [Formula: see text]-equivariant systems. Then, we analyze the action of [Formula: see text] on [Formula: see text] and study the pattern of periodic solutions arising from Hopf bifurcation. We identify the type of periodic solutions associated with the pairs [Formula: see text] of spatiotemporal or spatial symmetries, and prove their existence by using the [Formula: see text] Theorem due to Hopf bifurcation and the [Formula: see text] symmetry. In particular, we give a rigorous proof for the existence of a fourth branch of periodic solutions in [Formula: see text]-equivariant systems. Further, we apply our theory to study a concrete case: two coupled van der Pol oscillators with [Formula: see text] symmetry. We use normal form theory to analyze the periodic solutions arising from Hopf bifurcation. Among the families of the periodic solutions, we pay particular attention to the phase-locked oscillations, each of them being embedded in one of the invariant manifolds, and identify the in-phase, completely synchronized motions. We derive their explicit expressions and analyze their stability in terms of the parameters.

scholarly journals A NEW TREATMENT FOR PERIODIC SOLUTIONS AND COUPLED OSCILLATORS

2011 ◽  
Vol 65 (2) ◽  
pp. 197-217 ◽  
Author(s):  
Shin-Ichiro EI ◽  
Kunishige OHGANE
Keyword(s):  
Periodic Solutions ◽  
Coupled Oscillators ◽  
New Treatment

Irreversible Passive Energy Transfer in the Damped Response of Coupled Oscillators With Essential Nonlinearity

2005 ◽  
Author(s):  
G. Kerschen ◽  
Y. S. Lee ◽  
A. F. Vakakis ◽  
D. M. McFarland ◽  
L. A. Bergman
Keyword(s):  
Energy Transfer ◽  
Periodic Solutions ◽  
Coupled Oscillators ◽  
Topological Structure ◽  
Seismic Isolation ◽  
Degrees Of Freedom ◽  
Linear Oscillator ◽  
Resonance Capture ◽  
Low Frequencies ◽  
Flutter Suppression

We study, numerically and analytically the dynamics of passive energy transfer from a damped linear oscillator to an essentially nonlinear end attachment. This transfer is caused either by fundamental or subharmonic resonance capture, and in some cases is initiated by nonlinear beat phenomena. It is shown that, due to the essential nonlinearity, the end attachment is capable of passively absorbing broadband energy both at high and low frequencies, acting, in essence, as a passive broadband boundary controller. Complicated transitions in the damped dynamics can be interpreted based on the topological structure and bifurcations of the periodic solutions of the underlying undamped system. Moreover, complex resonance capture cascades are numerically encountered when we increase the number of degrees of freedom of the system. The ungrounded, essentially nonlinear end attachment discussed in this work can find application in numerous practical settings, including vibration and shock isolation of structures, seismic isolation, flutter suppression and packaging.

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