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 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.

THE EFFECT OF NONLINEAR DAMPING ON THE UNIVERSAL ESCAPE OSCILLATOR

1999 ◽  
Vol 09 (04) ◽  
pp. 735-744 ◽  
Author(s):  
MIGUEL A. F. SANJUÁN
Keyword(s):  
Energy Dissipation ◽  
Period Doubling ◽  
Nonlinear Damping ◽  
Basins Of Attraction ◽  
Period Doubling Bifurcation ◽  
Damping Coefficient ◽  
Nonlinear Dissipation ◽  
Fractal Basin Boundaries ◽  
Basin Boundaries

This paper analyzes the role of nonlinear dissipation on the universal escape oscillator. Nonlinear damping terms proportional to the power of the velocity are assumed and an investigation on its effects on the dynamics of the oscillator, such as the threshold of period-doubling bifurcation, fractal basin boundaries and how the basins of attraction are destroyed, is carried out. The results suggest that increasing the power of the nonlinear damping, has similar effects as of decreasing the damping coefficient for a linearly damped case, showing the very importance of the level or amount of energy dissipation.

ANALYTICAL ESTIMATES OF THE EFFECT OF NONLINEAR DAMPING IN SOME NONLINEAR OSCILLATORS

2000 ◽  
Vol 10 (09) ◽  
pp. 2257-2267 ◽  
Author(s):  
JOSÉ L. TRUEBA ◽  
JOAQUÍN RAMS ◽  
MIGUEL A. F. SANJUÁN
Keyword(s):  
Nonlinear Oscillator ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Nonlinear Damping ◽  
Damping Coefficient ◽  
Critical Parameters ◽  
Nonlinear Dissipation ◽  
Simple Pendulum ◽  
Melnikov Theory ◽  
Damped Systems

This paper reports on the effect of nonlinear damping on certain nonlinear oscillators, where analytical estimates provided by the Melnikov theory are obtained. We assume general nonlinear damping terms proportional to the power of velocity. General and useful expressions for the nonlinearly damped Duffing oscillator and for the nonlinearly damped simple pendulum are computed. They provide the critical parameters in terms of the damping coefficient and damping exponent, that is, the power of the velocity, for which complicated behavior is expected. We also consider generalized nonlinear damped systems, which may contain several nonlinear damping terms. Using the idea of Melnikov equivalence, we show that the effect of nonlinear dissipation can be equivalent to a linearly damped nonlinear oscillator with a modified damping coefficient.

Effect of nonlinear dissipation on the basin boundaries of a driven two-well Rayleigh–Duffing oscillator

2009 ◽  
Vol 39 (3) ◽  
pp. 1092-1099 ◽  
Author(s):  
M. Siewe Siewe ◽  
Hongjun Cao ◽  
Miguel A.F. Sanjuán
Keyword(s):  
Duffing Oscillator ◽  
Nonlinear Dissipation ◽  
Basin Boundaries

scholarly journals Melnikov Chaos in a Modified Rayleigh–Duffing Oscillator with ϕ6 Potential

2016 ◽  
Vol 26 (05) ◽  
pp. 1650085 ◽  
Author(s):  
C. H. Miwadinou ◽  
A. V. Monwanou ◽  
L. A. Hinvi ◽  
A. A. Koukpemedji ◽  
C. Ainamon ◽  
...  
Keyword(s):  
Chaotic Behavior ◽  
Duffing Oscillator ◽  
Basin Of Attraction ◽  
Melnikov Method ◽  
Nonlinear Damping ◽  
External Excitation ◽  
Melnikov Criterion ◽  
Single Well ◽  
Quadratic Damping ◽  
The Basin Of Attraction

The chaotic behavior of the modified Rayleigh–Duffing oscillator with [Formula: see text] potential and external excitation is investigated both analytically and numerically. The so-called oscillator models, for example, ship rolling motions. The single well and triple well potential cases are considered. Melnikov method is applied and the conditions for the existence of homoclinic and heteroclinic chaos are obtained. The effects of nonlinear damping on roll motion of ships are analyzed in detail. As it is known, nonlinear roll damping is a very important parameter in estimating ship responses. It is noted that the pure and unpure quadratic damping parameters affect the Melnikov criterion in the heteroclinic and homoclinic cases respectively while the pure cubic parameter affects the amplitude in both cases. The predictions have been tested with numerical simulations based on the basin of attraction. It is pointed out that certain quadratic damping effects are contrary to cubic damping effect.

scholarly journals Tuning nonlinear damping in graphene nanoresonators by parametric–direct internal resonance

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Ata Keşkekler ◽  
Oriel Shoshani ◽  
Martin Lee ◽  
Herre S. J. van der Zant ◽  
Peter G. Steeneken ◽  
...  
Keyword(s):  
Parametric Resonance ◽  
Internal Resonance ◽  
Microscopic Theory ◽  
Fold Increase ◽  
Nonlinear Damping ◽  
Supporting Evidence ◽  
Nonlinear Dissipation ◽  
Modal Interactions ◽  
Solid State Physics ◽  
Internal Resonances

AbstractMechanical sources of nonlinear damping play a central role in modern physics, from solid-state physics to thermodynamics. The microscopic theory of mechanical dissipation suggests that nonlinear damping of a resonant mode can be strongly enhanced when it is coupled to a vibration mode that is close to twice its resonance frequency. To date, no experimental evidence of this enhancement has been realized. In this letter, we experimentally show that nanoresonators driven into parametric-direct internal resonance provide supporting evidence for the microscopic theory of nonlinear dissipation. By regulating the drive level, we tune the parametric resonance of a graphene nanodrum over a range of 40–70 MHz to reach successive two-to-one internal resonances, leading to a nearly two-fold increase of the nonlinear damping. Our study opens up a route towards utilizing modal interactions and parametric resonance to realize resonators with engineered nonlinear dissipation over wide frequency range.

scholarly journals Analysis of the self-similar solutions of a generalized Burger's equation with nonlinear damping

2001 ◽  
Vol 7 (3) ◽  
pp. 253-282 ◽  
Author(s):  
Ch. Srinivasa Rao ◽  
P. L. Sachdev ◽  
Mythily Ramaswamy
Keyword(s):  
Initial Conditions ◽  
Nonlinear Damping ◽  
Nonlinear Ordinary Differential Equation ◽  
The Self ◽  
Similar Reduction ◽  
Generalized Burgers Equation ◽  
Different Types ◽  
Self Similar ◽  
Parameter Ranges ◽  
Similar Solutions

The nonlinear ordinary differential equation resulting from the self-similar reduction of a generalized Burgers equation with nonlinear damping is studied in some detail. Assuming initial conditions at the origin we observe a wide variety of solutions – (positive) single hump, unbounded or those with a finite zero. The existence and nonexistence of positive bounded solutions with different types of decay (exponential or algebraic) to zero at infinity for specific parameter ranges are proved.

A study on the role of the initial conditions and the nonlinear dissipation in the non-Hermitian effective Hamiltonian approach

Optik ◽  
2018 ◽  
Vol 174 ◽  
pp. 114-120 ◽  
Author(s):  
Santiago Echeverri-Arteaga ◽  
Herbert Vinck-Posada ◽  
Edgar A. Gómez
Keyword(s):  
Initial Conditions ◽  
Effective Hamiltonian ◽  
Hamiltonian Approach ◽  
Nonlinear Dissipation

Stability and Bifurcation Analysis of Periodic Motions in a Periodically Forced Duffing Oscillator

2011 ◽  
Author(s):  
Albert C. J. Luo ◽  
Jianzhe Huang
Keyword(s):  
Approximate Solution ◽  
Bifurcation Analysis ◽  
Duffing Oscillator ◽  
Approximate Solutions ◽  
Nonlinear Damping ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Stability And Bifurcation Analysis

In this paper, the analytical, approximate solutions of period-1 motions in the nonlinear damping, periodically forced, Duffing oscillator is obtained. The corresponding stability and bifurcation analysis of the HB2 approximate solution of period-1 motions in the forced Duffing oscillator is carried out. Numerical illustrations of period-1 motions are presented.

Optimal Control of Homoclinic Bifurcation in a Periodically Driven Helmholtz Oscillator

2001 ◽  
Author(s):  
Stefano Lenci ◽  
Giuseppe Rega
Keyword(s):  
Excitation Amplitude ◽  
Harmonic Excitation ◽  
Homoclinic Bifurcation ◽  
Critical Threshold ◽  
Control Procedure ◽  
Practical Case ◽  
Reference Case ◽  
Stable And Unstable Manifolds ◽  
Periodically Driven ◽  
Theoretical Predictions

Abstract The problem of avoiding the homoclinic bifurcation of the hilltop saddle of the Helmholtz equation by a shrewd choice of the shape of the external excitation is considered. The distance between the perturbed manifolds is computed by means of the Melnikov’s method, and its dependence on the shape of the excitation is emphasized. Successively, it is shown how it is possible to determine a theoretical optimal excitation which maximizes the distance between stable and unstable manifolds for a fixed excitation amplitude or, equivalently, which maximizes the critical amplitude for homoclinic bifurcation. The practical case of a finite number of subharmonics is considered in detail. The corresponding optimal problems are solved numerically and the related optimal excitations are given. It is shown that when the number of subharmonics increases, the critical threshold for homoclinic bifurcation tends to double with respect to the reference case of harmonic excitation. Some numerical simulations are finally performed to verify the theoretical predictions and the effectiveness of the control procedure.

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