Generating the periodic solutions for forcing van der Pol oscillators by the Iteration Perturbation method

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.

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Synchronization or cluster synchronization in coupled Van der Pol oscillators networks with different topological types

Physica Scripta ◽

10.1088/1402-4896/ac46f3 ◽

2021 ◽

Author(s):

Shuai Wang ◽

Yong Li

Keyword(s):

Periodic Solutions ◽

Phase Synchronization ◽

Van Der Pol Oscillator ◽

Cluster Synchronization ◽

Critical Conditions ◽

Complete Synchronization ◽

Van Der Pol ◽

Special Groups ◽

Different Types ◽

Van Der Pol Oscillators

Abstract In this paper, we try to discuss the mechanism of synchronization or cluster synchronization in the coupled Van der Pol oscillator networks with different topology types by using the theory of rotating periodic solutions. The synchronous solutions here are transformed into rotating periodic solutions of some dynamical systems. By analyzing the bifurcation of rotating periodic solutions, the critical conditions of synchronous solutions are given in three different networks. We use the rotating periodic matrix in the rotating periodic theory to judge various types of synchronization phenomena, such as complete synchronization, anti-phase synchronization, periodic synchronization, or cluster synchronization. All rotating periodic matrices which satisfy the exchange invariance of multiple oscillators form special groups in these networks. By using the conjugate classes of these groups, we obtain various possible synchronization solutions in the three networks. In particular, we find symmetry has different effects on synchronization in different networks. The network with better symmetry has more elements in the corresponding group, which may have more types of synchronous solutions. However, different types of symmetry may get the same type of synchronous solutions or different types of synchronous solutions, depending on whether their corresponding rotating periodic matrices are similar.

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Periodic solutions of multi-degree-of-freedom strongly nonlinear coupled van der Pol oscillators by homotopy analysis method

Acta Mechanica ◽

10.1007/s00707-010-0405-7 ◽

2010 ◽

Vol 217 (3-4) ◽

pp. 269-285 ◽

Cited By ~ 6

Author(s):

W. Zhang ◽

Y. H. Qian ◽

M. H. Yao ◽

S. K. Lai

Keyword(s):

Periodic Solutions ◽

Homotopy Analysis Method ◽

Degree Of Freedom ◽

Analysis Method ◽

Van Der Pol ◽

Strongly Nonlinear ◽

Homotopy Analysis ◽

Van Der Pol Oscillators

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Analysis of Nonlinear Systems by Truncated Point Mappings

Applied Mechanics Reviews ◽

10.1115/1.3152412 ◽

1989 ◽

Vol 42 (11S) ◽

pp. S83-S92 ◽

Cited By ~ 1

Author(s):

Ramesh S. Guttalu ◽

Henryk Flashner

Keyword(s):

Nonlinear Systems ◽

Periodic Solutions ◽

Parametric Analysis ◽

Analytical Techniques ◽

Mapping Method ◽

Order Of Approximation ◽

Van Der Pol ◽

Point Mapping ◽

Mathieu’S Equation ◽

Van Der Pol Oscillators

This paper summarizes results obtained by the authors regarding the utility of truncated point mappings which have been recently published in a series of papers. The method described here is applicable to the analysis of multidimensional, multiparameter, periodic nonlinear systems by means of truncated point mappings. Based on multinomial truncation, an explicit analytical expression is determined for the point mapping in terms of the states and parameters of the system to any order of approximation. By combining this approach with analytical techniques, such as the perturbation method employed here, we obtain a powerful tool for finding periodic solutions and for analyzing their stability. The versatility of truncated point mapping method is demonstrated by applying it to study the limit cycles of van der Pol and coupled van der Pol oscillators, the periodic solutions of the forced Duffing’s equation and for a parametric analysis of periodic solutions of Mathieu’s equation.

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Periodic Solutions for Coupled Van Der Pol Oscillators of Three-Degree-of-Freedom Solved by Homotopy Analysis Method

2010 International Conference on Computing, Control and Industrial Engineering ◽

10.1109/ccie.2010.54 ◽

2010 ◽

Author(s):

Youhua Qian

Keyword(s):

Periodic Solutions ◽

Homotopy Analysis Method ◽

Degree Of Freedom ◽

Analysis Method ◽

Van Der Pol ◽

Homotopy Analysis ◽

Van Der Pol Oscillators ◽

Three Degree Of Freedom

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Time Delay Control for Two van der Pol Oscillators

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4002390 ◽

2010 ◽

Vol 6 (1) ◽

Cited By ~ 2

Author(s):

Attilio Maccari

Keyword(s):

Time Delay ◽

Vibration Control ◽

Periodic Solutions ◽

Original System ◽

Excitation Amplitude ◽

Van Der Pol ◽

Flow Equations ◽

Technical Requirements ◽

Stable Periodic ◽

Van Der Pol Oscillators

A method for time delay vibration control of the principal and fundamental resonances of two nonlinearly coupled van der Pol oscillators is investigated Using the asymptotic perturbation method, four slow-flow equations on the amplitude and phase of the oscillators are obtained. Their fixed points correspond to a two-period quasi-periodic phase-locked motion for the original system. In the system without control, stable periodic solutions (if any) exist only for fixed values of amplitude and phase and depend on the system parameters and excitation amplitude. In many cases, the amplitudes of these solutions do not correspond to the technical requirements. On the contrary, it is demonstrated that, if vibration control terms are added, stable two-period quasi-periodic solutions with arbitrarily chosen amplitudes can be accomplished. Therefore, an effective vibration control is possible if appropriate time delay and feedback gains are chosen.

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