A generalized harmonic function perturbation method for determining limit cycles and homoclinic orbits of Helmholtz—Duffing oscillator

2013 ◽  
Vol 332 (21) ◽  
pp. 5508-5522 ◽  
Author(s):  
Zhenbo Li ◽  
Jiashi Tang ◽  
Ping Cai
Keyword(s):  
Harmonic Function ◽  
Perturbation Method ◽  
Limit Cycles ◽  
Duffing Oscillator ◽  
Homoclinic Orbits ◽  
Generalized Harmonic Function

scholarly journals Homoclinic orbits and Lie rotated vector fields

2000 ◽  
Vol 24 (3) ◽  
pp. 187-192
Author(s):  
Jie Wang ◽  
Chen Chen
Keyword(s):  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Vector Fields ◽  
Homoclinic Orbits ◽  
Singular Points ◽  
Definition Of

Based on the definition of Lie rotated vector fields in the plane, this paper gives the property of homoclinic orbit as parameter is changed and the singular points are fixed on Lie rotated vector fields. It gives the conditions of yielding limit cycles as well.

scholarly journals Multiple Bifurcations of a Cylindrical Dynamical System

2016 ◽  
Vol 46 (1) ◽  
pp. 33-52
Author(s):  
Ning Han ◽  
Qingjie Cao
Keyword(s):  
Dynamical System ◽  
Limit Cycles ◽  
Homoclinic Orbits ◽  
Complex Dynamic ◽  
Pendulum System ◽  
Sd Oscillator ◽  
Bistable State ◽  
Different Types ◽  
Coupling Dynamics ◽  
Dynamic Bifurcation

Abstract This paper focuses on multiple bifurcations of a cylindrical dynamical system, which is evolved from a rotating pendulum with SD oscillator. The rotating pendulum system exhibits the coupling dynamics property of the bistable state and conventional pendulum with the ho- moclinic orbits of the first and second type. A double Andronov-Hopf bifurcation, two saddle-node bifurcations of periodic orbits and a pair of homoclinic bifurcations are detected by using analytical analysis and nu- merical calculation. It is found that the homoclinic orbits of the second type can bifurcate into a pair of rotational limit cycles, coexisting with the oscillating limit cycle. Additionally, the results obtained herein, are helpful to explore different types of limit cycles and the complex dynamic bifurcation of cylindrical dynamical system.

Stability and Perturbations of Homoclinic Loops in a Class of Piecewise Smooth Systems

2015 ◽  
Vol 25 (09) ◽  
pp. 1550114 ◽  
Author(s):  
Shuang Chen ◽  
Zhengdong Du
Keyword(s):  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Piecewise Linear ◽  
Small Neighborhood ◽  
Homoclinic Orbits ◽  
Piecewise Smooth ◽  
Piecewise Smooth Systems ◽  
Planar System ◽  
The Stability ◽  
First Time

Like for smooth systems, a typical method to produce multiple limit cycles for a given piecewise smooth planar system is via homoclinic bifurcation. Previous works only focused on limit cycles that bifurcate from homoclinic orbits of piecewise-linear systems. In this paper, we consider for the first time the same problem for a class of general nonlinear piecewise smooth systems. By introducing the Dulac map in a small neighborhood of the hyperbolic saddle, we obtain the approximation of the Poincaré map for the nonsmooth homoclinic orbit. Then, we give conditions for the stability of the homoclinic orbit and conditions under which one or two limit cycles bifurcate from it. As an example, we construct a nonlinear piecewise smooth system with two limit cycles that bifurcate from a homoclinic orbit.

On the limit cycles, period-doubling, and quasi-periodic solutions of the forced Van der Pol-Duffing oscillator

2017 ◽  
Vol 78 (4) ◽  
pp. 1217-1231 ◽  
Author(s):  
Jifeng Cui ◽  
Wenyu Zhang ◽  
Zeng Liu ◽  
Jianglong Sun
Keyword(s):  
Periodic Solutions ◽  
Limit Cycles ◽  
Duffing Oscillator ◽  
Period Doubling ◽  
Van Der Pol

EFFECT OF FRACTIONAL DERIVATIVE PROPERTIES ON THE PERIODIC SOLUTION OF THE NONLINEAR OSCILLATIONS

Fractals ◽  
2020 ◽  
Vol 28 (07) ◽  
pp. 2050095 ◽  
Author(s):  
YUSRY O. EL-DIB ◽  
NASSER S. ELGAZERY
Keyword(s):  
Periodic Solution ◽  
Fractional Derivative ◽  
Perturbation Method ◽  
Nonlinear Oscillations ◽  
Duffing Oscillator ◽  
Solution Process ◽  
Auxiliary Parameter ◽  
Periodic Force ◽  
Definition Of ◽  
Perturbed Equation

A periodic solution of the time-fractional nonlinear oscillator is derived based on the Riemann–Liouville definition of the fractional derivative. In this approach, the particular integral to the fractional perturbed equation is found out. An enhanced perturbation method is developed to study the forced nonlinear Duffing oscillator. The modified homotopy equation with two expanded parameters and an additional auxiliary parameter is applied in this proposal. The basic idea of the enhanced method is to apply the annihilator operator to construct a simplified equation freeness of the periodic force. This method makes the solution process for the forced problem much simpler. The resulting equation is valid for studying all types of possible resonance states. The outcome shows that this alteration method overcomes all shortcomings of the perturbation method and leads to obtain a periodic solution.

scholarly journals Novel Hyperbolic Homoclinic Solutions of the Helmholtz-Duffing Oscillators

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
Yang-Yang Chen ◽  
Shu-Hui Chen ◽  
Wei-Wei Wang
Keyword(s):  
Perturbation Method ◽  
Duffing Oscillator ◽  
Homoclinic Solution ◽  
Homoclinic Solutions ◽  
The Self ◽  
Hyperbolic Function

The exact and explicit homoclinic solution of the undamped Helmholtz-Duffing oscillator is derived by a presented hyperbolic function balance procedure. The homoclinic solution of the self-excited Helmholtz-Duffing oscillator can also be obtained by an extended hyperbolic perturbation method. The application of the present homoclinic solutions to the chaos prediction of the nonautonomous Helmholtz-Duffing oscillator is performed. Effectiveness and advantage of the present solutions are shown by comparisons.

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