Resonant Bands of a Mathieu-Duffing Oscillator With a Twin-Well Potential

2002 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Parametric Excitation ◽  
Duffing Oscillator ◽  
Resonance Interaction ◽  
Energy Approach ◽  
Potential Wells ◽  
Spectrum Method ◽  
Numerical Predictions ◽  
Left And Right ◽  
Energy Increment

The (M:1)-resonant bands in the left and right potential wells are skew-symmetric, and the (2M:1)-resonant bands of the large orbit motion are symmetric. The analytical conditions for the onset and destruction of a resonant band are developed through the incremental energy approach. The numerical predictions of such the onset and destruction are also completed by the energy increment spectrum method. The sub-resonance interaction occurs for strong excitations, which needs to be further investigated. These results are applicable to the small orbit and large orbit motions of post-buckled structure under a parametric excitation.

Resonant Layers in Nonlinear Dynamics

1998 ◽  
Vol 65 (3) ◽  
pp. 727-736 ◽  
Author(s):  
R. P. S. Han ◽  
A. C. J. Luo
Keyword(s):  
Nonlinear Dynamics ◽  
Renormalization Group ◽  
Numerical Simulations ◽  
Duffing Oscillator ◽  
Energy Approach ◽  
New Method ◽  
Mapping Technique ◽  
Standard Mapping ◽  
Left And Right ◽  
Group Technique

A new method based on an incremental energy approach and the standard mapping technique is proposed for the study of resonant layers in nonlinear dynamics. To demonstrate the procedure, the method is applied to four types of Duffing oscillators. The appearance, disappearance and accumulated disappearance strengths of the resonant layers for each type of oscillator are derived. A quantitative check of the appearance strength is performed by computing its value using three independent methods: Chirikov overlap criterion, renormalization group technique, and numerical simulations. It is also observed that for the case of the twin-well Duffing oscillator, its perturbed left and right wells are asymmetric.

RESONANT LAYERS IN A PARAMETRICALLY EXCITED PENDULUM

2002 ◽  
Vol 12 (02) ◽  
pp. 409-419 ◽  
Author(s):  
ALBERT C. J. LUO
Keyword(s):  
Hamiltonian Systems ◽  
Analytical Method ◽  
Numerical Approach ◽  
Mapping Method ◽  
Numerical Results ◽  
Numerical Prediction ◽  
Parametrically Excited ◽  
Spectrum Method ◽  
Poincare Mapping ◽  
Energy Increment

The energy increment spectrum method is developed for the numerical prediction of a specific primary resonant layer, and the width of the resonant layer can be estimated through the energy increment spectrum. This numerical approach is applied to investigate the (2M:1)-librational and (M:1)-rotational, resonant layers in a parametrically excited pendulum, and the corresponding analytical conditions for such resonant layers are developed. The numerical approach predicts the appearance and disappearance of resonant layers in nonlinear Hamiltonian systems rather than the conventional Poincaré mapping method. Illustrations of the analytical and numerical results for the appearance and disappearance of the resonant layers are given. The width of the resonant layers in the paremetric pendulum is computed. The analytical method should be further improved through renormalization.

Investigations of Stochastic Layers in Nonlinear Dynamics

1999 ◽  
Vol 122 (1) ◽  
pp. 36-41 ◽  
Author(s):  
Albert C. J. Luo ◽  
Ray P. S. Han
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Investigation ◽  
Duffing Oscillator ◽  
Minimum Energy ◽  
Energy Approach ◽  
Energy Spectra ◽  
Poincaré Mapping ◽  
Periodically Driven ◽  
Poincare Mapping ◽  
Stochastic Layer

The onset of a new resonance in the stochastic layer is predicted numerically through the maximum and minimum energy spectra when the energy jump in the spectra occurs. The incremental energy approach among all the established, analytic approaches gives the best prediction of the onset of resonance in the stochastic layer compared to numerical investigation. The stochastic layers in the periodically-driven pendulum are discussed as another example. Illustrations of stochastic layers in the twin-well Duffing oscillator and the periodically-driven pendulum are given through the Poincare´ mapping sections. [S0739-3717(00)00701-7]

scholarly journals CHARACTERISTICS OF STOCHASTIC RESONANCE IN ASYMMETRIC DUFFING OSCILLATOR

2011 ◽  
Vol 21 (09) ◽  
pp. 2729-2739 ◽  
Author(s):  
S. ARATHI ◽  
S. RAJASEKAR ◽  
J. KURTHS
Keyword(s):  
Stochastic Resonance ◽  
Residence Time ◽  
Noise Intensity ◽  
Signal To Noise Ratio ◽  
Duffing Oscillator ◽  
Symmetric Case ◽  
At Resonance ◽  
Left And Right ◽  
Double Well Potential ◽  
Range Of Values

We study the characteristics of stochastic resonance (SR) in the Duffing oscillator with three types of asymmetries in its double-well potential. The asymmetries controlled by a parameter α are introduced in the potential by varying (i) the depth of the left-well alone, (ii) the location of the minimum of the left-well alone and (iii) both depth and location of the minimum of the left-well alone. The characteristics of SR in the asymmetric cases are different from the symmetric case (α = 1). We find that asymmetry has a strong influence on the optimum noise intensity at which signal-to-noise ratio (SNR) is maximum, mean residence time at resonance and the probability distribution of residence time in the left- and right-wells. For a range of values of α, α ≠ 1, SNR is found to be relatively higher than for α = 1.

Chaotic Motions in Resonant Separatrix Bands of a Parametrically Excited Pendulum

2000 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Numerical Results ◽  
Numerical Prediction ◽  
Chaotic Motions ◽  
Parametrically Excited ◽  
Spectrum Method ◽  
Energy Increment

Abstract The analytical conditions for the presence of primary (2M:1)-librational and (M:1)-rotational, resonant bands in a parametrically excited pendulum are obtained. The energy increment spectrum method is also developed for the numerical prediction of a specific primary resonant band. Illustrations of the analytical and numerical results for the appearance and destruction of the resonant bands are given for a comparison.

ESTIMATION OF SYMMETRY BREAKING AND ESCAPE BY OBSERVATION OF MANIFOLD TANGENCIES

1995 ◽  
Vol 05 (03) ◽  
pp. 883-890 ◽  
Author(s):  
M.J. CLIFFORD ◽  
S.R. BISHOP
Keyword(s):  
Symmetry Breaking ◽  
Parametric Excitation ◽  
Potential Well ◽  
Potential Wells ◽  
Symmetric Potential ◽  
Initial Change ◽  
Symmetry Breaking Bifurcation

A heteroclinic tangency marking the initial change in secondary Birkhoff signature is identified as a good indicator of the chaotic escape from a symmetric potential well under parametric excitation as it occurs shortly after the symmetry breaking bifurcation which precedes the final event. This is compared to the criterion for asymmetric potential wells, and the lack of any change in primary Birkhoff signature highlights the differences in the two cases.

GLOBAL TANGENCY AND TRANSVERSALITY OF PERIODIC FLOWS AND CHAOS IN A PERIODICALLY FORCED, DAMPED DUFFING OSCILLATOR

2008 ◽  
Vol 18 (01) ◽  
pp. 1-49 ◽  
Author(s):  
ALBERT C. J. LUO
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
First Integral ◽  
Poincaré Mapping ◽  
Nonlinear Dynamical ◽  
Periodic Flows ◽  
Poincare Mapping ◽  
Integral Quantity ◽  
Energy Increment

This paper presents how to apply a newly developed general theory for the global transversality and tangency of flows in n-dimensional nonlinear dynamical systems to a 2-D nonlinear dynamical system (i.e. a periodically forced, damped Duffing oscillator). The global tangency and transversality of the periodic and chaotic motions to the separatrix for such a nonlinear system are discussed to help us understand the complexity of chaos in nonlinear dynamical systems. This paper presents the concept that the global transversality and tangency to the separatrix are independent of the Melnikov function (or the energy increment). Chaos in nonlinear dynamical systems makes the exact energy increment quantity to be chaotic no matter if the nonlinear dynamical systems have separatrices or not. The simple zero of the Melnikov function cannot be used to simply determine the existence of chaos in nonlinear dynamical systems. Through this paper, the expectation is that, from now on, one can use the alternative aspect to look into the complexity of chaos in nonlinear dynamical systems. Therefore, in this paper, the analytical conditions for global transversality and tangency of 2-D nonlinear dynamical systems are presented. The first integral quantity increment (i.e. the energy increment) for a certain time interval is achieved for periodic flows and chaos in the 2-D nonlinear dynamical systems. Under the perturbation assumptions and convergent conditions, the Melnikov function is recovered from the first integral quantity increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The corresponding analytical conditions for the global transversality and tangency to the separatrix are obtained and verified by numerical simulations. The switching planes and the corresponding local and global mappings are defined on the separatrix. The mapping structures are developed for local and global periodic flows passing through the separatrix. The mapping structures of global chaos in the damped Duffing oscillator are also discussed. Bifurcation scenarios of the damped Duffing oscillator are presented through the traditional Poincaré mapping section and the switching planes. The first integral quantity increment (i.e. L-function) is presented to observe the periodicity of flows. In addition, the global tangency of periodic flows in such an oscillator is measured by the G-function and G(1)-function, and is verified by numerical simulations. The first integral quantity increment of periodic flows is zero for their complete periodic cycles. Numerical simulations of chaos in such a Duffing oscillator are carried out through the Poincaré mapping sections. The conservative energy distribution, G-function and L-function along the displacement of Poincaré mapping points are presented to observe the complexity of chaos. The first integral quantity increment (i.e. L-function) of chaotic flows at the Poincaré mapping points is nonzero and chaotic. The switching planes of chaos are presented on the separatrix for a better understanding of the global transversality to the separatrix. The switching point distribution on the separatrix is presented and the switching G-function on the separatrix is given to show the global transversality of chaos on the separatrix. The analytical conditions are obtained from the new theory rather than the Melnikov method. The new conditions for the global transversality and tangency are more accurate and independent of the small parameters.

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