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Multiple resonance in coupled Duffing oscillators and nonlinear normal modes.

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This study investigates resonance in coupled Duffing oscillators. Researchers found hysteretical unstable regions and nonlinear normal modes in a two-oscillator system, revealing complex dynamics under specific driving conditions.

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Area of Science:

  • Nonlinear Dynamics
  • Oscillatory Systems
  • Coupled Oscillator Physics

Background:

  • Duffing oscillators are fundamental models in nonlinear dynamics.
  • Understanding resonance phenomena in coupled systems is crucial for various physical applications.
  • Previous studies often focused on single oscillators or simpler coupling schemes.

Purpose of the Study:

  • To analytically and numerically investigate resonance in a chain of N linearly coupled damped Duffing oscillators.
  • To develop a general recursion scheme for describing resonance curves in multi-oscillator systems.
  • To analyze the specific case of N=2 oscillators under high driving amplitude and stiffness.

Main Methods:

  • Analytical calculation of fixed points of the dynamical system.
  • Numerical simulations using a fourth-order multivariate Runge-Kutta method.
  • Development of a general recursion scheme relating oscillator amplitudes and driving frequency.

Main Results:

  • A general recursion scheme was established to describe resonance curves for N coupled Duffing oscillators.
  • Hysteretical unstable regions were identified in the resonance curves for a two-oscillator system (N=2) under specific conditions.
  • Nonlinear normal modes, manifesting as quasiperiodic oscillations, were observed in unstable driving frequency regimes.

Conclusions:

  • The study provides a comprehensive framework for analyzing resonance in coupled Duffing oscillator chains.
  • The findings highlight the existence of complex nonlinear behaviors, including hysteresis and quasiperiodicity, in these systems.
  • The identified nonlinear normal modes offer insights into the energy transfer and stability of coupled oscillatory systems.