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Large-amplitude nonlinear normal modes of the discrete sine lattices
Valeri V Smirnov1, Leonid I Manevitch1
1Institute of Chemical Physics, RAS, 4 Kosygin Street, Moscow 119991, Russia.
Physical Review. E
|March 17, 2017
Summary
We analytically describe large-amplitude oscillations in finite pendulum chains, revealing complex behaviors beyond simplified models. This work offers insights into nonlinear dynamics and polymer analogs, crucial for understanding complex physical systems.
Area of Science:
- Nonlinear Dynamics
- Condensed Matter Physics
- Statistical Mechanics
Background:
- Harmonically coupled pendulums are a fundamental model with broad physics applications.
- Previous studies primarily focused on infinite chains, limiting understanding of finite systems.
- Finite discrete chains serve as analytical analogs for coarse-grained polymer models in molecular dynamics.
Purpose of the Study:
- To develop an analytical description for large-amplitude stationary oscillations in finite discrete pendulum systems.
- To investigate nonlinear normal modes and their dispersion relations without amplitude restrictions (near π).
- To analyze the limitations of the sine-Gordon equation approximation for finite chains.
Main Methods:
- Analytical description of stationary oscillations.
- Derivation of dispersion relations for nonlinear normal modes at arbitrary amplitudes.
- Analysis of mode interactions, stability, and localization phenomena.
- Comparison with numerical simulations of finite-length chains.
Main Results:
- Dispersion relations for nonlinear normal modes are derived for arbitrary amplitudes.
- The long-wavelength (sine-Gordon) approximation is shown to be inadequate for large amplitudes (around π/2).
- Complex zone structures arise at large amplitudes due to multiple resonances between modes.
- Shorter wavelength modes can exhibit lower frequencies than longer wavelength modes.
- Resonant interactions lead to oscillation localization, with explicit instability and localization thresholds determined.
Conclusions:
- The analytical approach provides accurate predictions for finite pendulum chains, surpassing the sine-Gordon approximation.
- Complex nonlinear phenomena, including mode resonances and oscillation localization, are characteristic of finite systems.
- This research enhances the understanding of nonlinear normal modes and their stability in discrete systems with potential applications in polymer physics.
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