Higher order nonlinear response in random resistor networks: numerical studies for arbitrary nonlinearity

1995 ◽  
Vol 99 (4) ◽  
pp. 599-603 ◽  
Author(s):  
G. M. Zhang
Keyword(s):  
Nonlinear Response ◽  
Higher Order ◽  
Numerical Studies ◽  
Resistor Networks

Waveform Invariance in Nonlinear Periodic Systems Using Higher-Order Multiple Scales

2016 ◽  
Author(s):  
Matthew Fronk ◽  
Michael J. Leamy
Keyword(s):  
Perturbation Analysis ◽  
Multiple Scales ◽  
Initial Conditions ◽  
Higher Order ◽  
Linear Quadratic ◽  
Numerical Studies ◽  
Specific Order ◽  
Spectral Variance ◽  
Force Displacement

This paper carries-out a higher-order, multiple scales perturbation analysis on nonlinear monoatomic and diatomic chains with the intent of predicting invariant waveforms. The chains incorporate linear, quadratic, and cubic force-displacement relationships, and linear dampers. Multi-harmonic results for 1st and 2nd order expansions are reported in closed form, while results for the 3rd order are computed numerically on a case-by-case basis, thus avoiding difficulties associated with large symbolic expressions. Dimensionless parameters are introduced which characterize the amplitude-dependent nonlinear nature of a given chain. Interpretation of the perturbation solutions suggests that the nonlinear chains support certain waveforms which propagate invariantly; i.e., the spectral content does not change significantly over time and space. Numerical simulations confirm this finding using initial conditions corresponding to a specific order of the perturbation solution, and subsequent FFT’s of the response track the growth (or decay) of spatial harmonic content. A variance parameter computes mean fluctuation of the harmonics about their initial values. For a variety of parameter sets, the numerical studies confirm that spectral variance reduces when waves receive 2nd order initial conditions as compared to 1st order ones. Furthermore, chains given 3rd order initial conditions exhibit smaller variance when compared to those given 1st and 2nd order ones. The studies’ results suggest that introducing higher-order multiple scales perturbation analysis captures long-term, non-localized invariant waves (or cnoidal waves), which have the potential for propagating coherent information over long distances.

Spectral Analyses of Nonlinear Interactions

1995 ◽  
Author(s):  
Balakumar Balachandran ◽  
Khalil A. Khan
Keyword(s):  
Coupled Oscillators ◽  
Higher Order ◽  
Nonlinear Interactions ◽  
Phase Coupling ◽  
Spectral Analyses ◽  
Weakly Nonlinear ◽  
Numerical Studies ◽  
Analytical Approximations ◽  
Higher Order Spectra

Abstract Signals pertaining to motions of nonlinearly coupled oscillators are studied using higher-order spectral analyses. The analyses is used to understand the role of phase coupling in nonlinear interactions between two or more Fourier components. For certain motions of weakly nonlinear systems, analytical approximations are obtained for relevant higher-order spectra and coherence functions. Numerical studies are conducted to verify analytical predictions and to illustrate the usefulness of spectral analyses for different cases.

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