Exact nonlinear fourth-order equation for two coupled nonlinear oscillators: metamorphoses of resonance curves

TL;DR

This paper derives an exact nonlinear fourth-order equation for two coupled oscillators and analyzes how resonance curves change near singular points, revealing qualitative shifts in dynamics.

Contribution

It introduces an exact fourth-order equation for coupled oscillators and studies metamorphoses of resonance curves using algebraic and bifurcation analysis.

Findings

Resonance curves undergo qualitative metamorphoses near singular points.

Amplitude profiles are algebraic curves analyzed via the Krylov-Bogoliubov-Mitropolsky method.

Dynamics exhibit qualitative changes in the vicinity of singular points.

Abstract

We study dynamics of two coupled periodically driven oscillators. The internal motion is separated off exactly to yield a nonlinear fourth-order equation describing inner dynamics. Periodic steady-state solutions of the fourth-order equation are determined within the Krylov-Bogoliubov-Mitropolsky approach - we compute the amplitude profiles, which from mathematical point of view are algebraic curves. In the present paper we investigate metamorphoses of amplitude profiles induced by changes of control parameters near singular points of these curves. It follows that dynamics changes qualitatively in the neighbourhood of a singular point.