Analysis of homoclinic bifurcation in Duffing oscillator under two-frequency excitation: Peculiarity of using Melnikov method in combination with averaging technique

Chaos Theory ◽  
2011 ◽  
Author(s):  
Vladimir Ryabov ◽  
Kenta Fukushima
Keyword(s):  
Duffing Oscillator ◽  
Melnikov Method ◽  
Homoclinic Bifurcation ◽  
Averaging Technique ◽  
Frequency Excitation

scholarly journals Effect of Nonlinear Dissipation on the Basin Boundaries of a Driven Two-Well Modified Rayleigh–Duffing Oscillator

2015 ◽  
Vol 25 (02) ◽  
pp. 1550024 ◽  
Author(s):  
C. H. Miwadinou ◽  
A. V. Monwanou ◽  
J. B. Chabi Orou
Keyword(s):  
Initial Conditions ◽  
Duffing Oscillator ◽  
Excitation Amplitude ◽  
Nonlinear Damping ◽  
Homoclinic Bifurcation ◽  
Nonlinear Dissipation ◽  
Transition To Chaos ◽  
Melnikov Criterion ◽  
Quadratic Nonlinearities ◽  
Basin Boundaries

This paper considers the effect of nonlinear dissipation on the basin boundaries of a driven two-well modified Rayleigh–Duffing oscillator where pure cubic, unpure cubic, pure quadratic and unpure quadratic nonlinearities are considered. By analyzing the potential, an analytic expression is found for the homoclinic orbit. The Melnikov criterion is used to examine a global homoclinic bifurcation and transition to chaos. Unpure quadratic parameter and parametric excitation amplitude effects are found on the critical Melnikov amplitude μ cr . Finally, the phase space of initial conditions is carefully examined in order to analyze the effect of the nonlinear damping, and particularly how the basin boundaries become fractalized.

scholarly journals Control of Bistability in a Delayed Duffing Oscillator

2012 ◽  
Vol 2012 ◽  
pp. 1-5 ◽  
Author(s):  
Mustapha Hamdi ◽  
Mohamed Belhaq
Keyword(s):  
Hopf Bifurcation ◽  
Multiple Scales ◽  
Duffing Oscillator ◽  
Stable Limit Cycle ◽  
Slow Flow ◽  
Stable Limit ◽  
Nontrivial Solutions ◽  
Slow Dynamic ◽  
High Frequency Excitation ◽  
Frequency Excitation

The effect of a high-frequency excitation on nontrivial solutions and bistability in a delayed Duffing oscillator with a delayed displacement feedback is investigated in this paper. We use the technique of direct partition of motion and the multiple scales method to obtain the slow dynamic of the system and its slow flow. The analysis of the slow flow provides approximations of the Hopf and secondary Hopf bifurcation curves. As a result, this study shows that increasing the delay gain, the system undergoes a secondary Hopf bifurcation. Further, it is indicated that as the frequency of the excitation is increased, the Hopf and secondary Hopf bifurcation curves overlap giving birth in the parameter space to small regions of bistability where a stable trivial steady state and a stable limit cycle coexist. Numerical simulations are carried out to validate the analytical finding.

Optimal Control of Homoclinic Bifurcation: Theoretical Treatment and Practical Reduction of Safe Basin Erosion in the Helmholtz Oscillator

2003 ◽  
Vol 9 (3-4) ◽  
pp. 281-315 ◽  
Author(s):  
Stefano Lenci ◽  
Giuseppe Rega
Keyword(s):  
Finite Number ◽  
Control Method ◽  
Mathematical Problem ◽  
Melnikov Method ◽  
Excitation Amplitude ◽  
Homoclinic Bifurcation ◽  
Theoretical Treatment ◽  
Stable And Unstable Manifolds ◽  
Unstable Manifolds ◽  
Basin Erosion

A control method of the homoclinic bifurcation is developed and applied to the nonlinear dynamics of the Helmholtz oscillator. The method consists of choosing the shape of external and/or parametric periodic excitations, which permits us to avoid, in an optimal manner, the transverse intersection of the stable and unstable manifolds of the hilltop saddle. The homoclinic bifurcation is detected by the Melnikov method, and its dependence on the shape of the excitation is shown. We successively investigate the mathematical problem of optimization, which consists of determining the theoretical optimal excitation that maximizes the distance between stable and unstable manifolds for fixed excitation amplitude or, equivalently, the critical amplitude for homoclinic bifurcation. The optimal excitations in the reduced case with a finite number of superharmonic corrections are first determined, and then the optimization problem with infinite superharmonics is investigated and solved under a constraint on the relevant amplitudes, which is necessary to guarantee the physical admissibility of the mathematical solution. The mixed case of a finite number of constrained superharmonics is also considered. Some numerical simulations are then performed aimed at verifying the Melnikov's theoretical predictions of the homoclinic bifurcations and showing how the optimal excitations are indeed able to separate stable and unstable manifolds. Finally, we numerically investigate in detail the effectiveness of the control method with respect to the basin erosion and escape phenomena, which are the most important and dangerous practical aspects of the Helmholtz oscillator.

Combination resonances in the response of the duffing oscillator to a three-frequency excitation

1998 ◽  
Vol 131 (3-4) ◽  
pp. 235-245 ◽  
Author(s):  
S. Yang ◽  
A. H. Nayfeh ◽  
D. T. Mook
Keyword(s):  
Duffing Oscillator ◽  
Frequency Excitation

scholarly journals The Focus Case of a Nonsmooth Rayleigh-Duffing Oscillator

2021 ◽  
Author(s):  
Zhaoxia Wang ◽  
Hebai Chen ◽  
Yilei Tang
Keyword(s):  
Limit Cycle ◽  
Duffing Oscillator ◽  
Pitchfork Bifurcation ◽  
Homoclinic Bifurcation ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Bifurcation Curve ◽  
Limit Cycle Bifurcation ◽  
Theoretical Results ◽  
Double Limit

Abstract In this paper, we study the global dynamics of a nonsmooth Rayleigh-Duffing equation x¨ + ax˙ + bx˙|x˙| + cx + dx3 = 0 for the case d > 0, i.e., the focus case. The global dynamics of this nonsmooth Rayleigh-Duffing oscillator for the case d < 0, i.e., the saddle case, has been studied completely in the companion volume [Int. J. Non-Linear Mech., 129 (2021) 103657]. The research for the focus case is more complex than the saddle case, such as the appearance of five limit cycles and the gluing bifurcation which means that two double limit cycle bifurcation curves and one homoclinic bifurcation curve are very adjacent occurs. We present bifurcation diagram, including one pitchfork bifurcation curve, two Hopf bifurcation curves, two double limit cycle bifurcation curves and one homoclinic bifurcation curve. Finally, numerical phase portraits illustrate our theoretical results.

scholarly journals Vibration of the Duffing Oscillator: Effect of Fractional Damping

2007 ◽  
Vol 14 (1) ◽  
pp. 29-36 ◽  
Author(s):  
Marek Borowiec ◽  
Grzegorz Litak ◽  
Arkadiusz Syta
Keyword(s):  
Lyapunov Exponent ◽  
Perturbation Methods ◽  
Duffing Oscillator ◽  
Homoclinic Bifurcation ◽  
Sufficient Condition ◽  
External Excitation ◽  
Fractional Damping ◽  
Duffing System ◽  
Transition To Chaos ◽  
Melnikov Criterion

We have applied the Melnikov criterion to examine a global homoclinic bifurcation and transition to chaos in a case of the Duffing system with nonlinear fractional damping and external excitation. Using perturbation methods we have found a critical forcing amplitude above which the system may behave chaotically. The results have been verified by numerical simulations using standard nonlinear tools as Poincare maps and a Lyapunov exponent. Above the critical Melnikov amplitude μ_c, which a sufficient condition of a global homoclinic bifurcation, we have observed the region with a transient chaotic motion.

Strong chaotification and robust chaos in the Duffing oscillator induced by two-frequency excitation

2021 ◽  
Vol 103 (2) ◽  
pp. 1955-1967
Author(s):  
André Gusso ◽  
Sebastian Ujevic ◽  
Ricardo L. Viana
Keyword(s):  
Duffing Oscillator ◽  
Frequency Excitation
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