The largest Lyapunov exponent as a tool for detecting relative changes in the particle positions

Dynamics of the dissipatively driven Frenkel-Kontorova model with asymmetric deformable substrate potential is examined by analyzing response function, the largest Lyapunov exponent and Poincar\'{e} sections for two neighboring particles. The obtained results show that the largest Lyapunov exponent, besides being used for investigating integral quantities, can be used for detecting microchanges in chain configuration. Slight changes in relative positions of the particles are registered through jumps of the largest Lyapunov exponent in the pinning regime. The occurrence of such jumps is highly dependent on type of commensurate structure and deformation of substrate potential. The obtained results also show that the minimal force required to initiate collective motion of the chain is not dependent on the number of Lyapunov exponent jumps in the pinning regime.Comment: 10 pages, 8 figure

Analysis of vibrational resonance in an oscillator with exponential mass variation

This study investigated vibrational resonance (VR) in a Duffing-type oscillator with position-dependent mass (PDM) distribution defined by spatially varying exponential function. The role of two PDM parameters, the fixed rest mass and nonlinear strength on observed resonances was investigated from the analytical and numerical computation of response amplitude, which is a measure of the amplification of a low-frequency (LF) signal through the introduction and modulation of a high-frequency (HF) signal in a weakly driven nonlinear system. The method of direct separation of motion was used to analytically compute the response amplitude, while the numerically computed response amplitude was obtained from the Fourier spectrum of the output signal. Single resonance peaks with good agreement between the numerically and the analytically computed responses were observed for the traditional HF-induced VR and the PDM-induced resonances. The results demonstrated that spatial mass perturbation can play the roles of HF signals typically used in traditional VR setups. The results of this investigation corroborate earlier reports that stated PDM parameters can complement the HF signal to control the observed resonance peaks. However, the exponentially varying PDM parameters did not initiate double or multiple resonances as reported for other mass distributions such as the regular mass function and the doubly-singular mass function. This study communicates that the nature of the PDM distribution actually determines the possibility of generating new peaks from observed resonances

Modulation instability gain and localized waves by modified Frenkel-Kontorova model of higher order nonlinearity

In this paper, modulation instability and nonlinear supratransmission are investigated in a one-dimensional chain of atoms using cubic-quartic nonlinearity coefficients. As a result, we establish the discrete nonlinear evolution equation by using the multi-scale scheme. To calculate the modulation instability gain, we use the linearizing scheme. Particular attention is given to the impact of the higher nonlinear term on the modulation instability. Following that, full numerical integration was performed to identify modulated wave patterns, as well as the appearance of a rogue wave. Through the nonlinear supratransmission phenomenon, one end of the discrete model is driven into the forbidden bandgap. As a result, for driving amplitudes above the supratransmission threshold, the solitonic bright soliton and modulated wave patterns are satisfied. An important behavior is observed in the transient range of time of propagation when the bright solitonic wave turns into a chaotic solitonic wave. These results corroborate our analytical investigations on the modulation instability and show that the one-dimensional chain of atoms is a fruitful medium to generate long-lived modulated waves

A study of poststenotic shear layer instabilities

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