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

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.

💡 Research Summary

The paper investigates the dynamics of two coupled nonlinear oscillators that are driven by a periodic external force. By performing an exact separation of the internal motion from the overall motion, the authors derive a single fourth‑order nonlinear differential equation that governs the internal degree of freedom. This equation retains all nonlinear terms and therefore provides a more faithful representation of the system than the usual second‑order approximations commonly employed in the literature.

To obtain steady‑state periodic solutions of the fourth‑order equation, the authors apply the Krylov‑Bogoliubov‑Mitropolsky (KBM) averaging method. Assuming a weak nonlinearity, weak damping, and a slowly varying amplitude and phase, they substitute a trial solution of the form (x(t)=A(t)\cos(\Omega t+\phi(t))) into the equation, average over the fast oscillation, and derive evolution equations for the amplitude (A) and phase (\phi). Setting the time derivatives of (A) and (\phi) to zero yields an algebraic relationship between the forcing frequency (\Omega) and the steady‑state amplitude (A). This relationship is an algebraic curve in the ((\Omega, A)) plane, which can possess multiple branches, self‑intersections, and cusp points.

The central focus of the study is the qualitative transformation—or “metamorphosis”—of these amplitude‑frequency curves when control parameters (forcing amplitude, damping coefficient, nonlinear stiffness, etc.) are varied near singular points of the curve. Singular points include fold (turning) points where two branches coalesce, cusp points where a branch changes direction, and bifurcation points where a new branch emerges. Near such points, infinitesimal changes in parameters can cause a dramatic re‑shaping of the curve: a previously stable solution may disappear, a new stable branch may appear, or the system may jump between widely separated amplitudes for the same forcing frequency.

Through systematic parameter sweeps and numerical continuation, the authors map out regions of the parameter space where these metamorphoses occur. They demonstrate that when the forcing amplitude exceeds a critical value, a fold point can annihilate, giving rise to a closed loop in the amplitude‑frequency diagram. This loop is associated with hysteresis: as the forcing frequency is swept up and down, the system follows different branches, leading to a jump phenomenon observable in experiments. In regimes of low damping and strong nonlinearity, cusp‑like structures appear, indicating the possibility of abrupt transitions and even chaotic responses.

The paper’s findings have practical implications for the design and control of nonlinear resonant devices such as MEMS resonators, nonlinear optical cavities, and metamaterial structures. By identifying the parameter regimes where singular points and associated metamorphoses occur, engineers can either avoid undesirable jumps and instability or deliberately exploit multistability for switching applications.

In conclusion, the authors provide a rigorous derivation of a fourth‑order nonlinear equation that captures the full internal dynamics of two coupled driven oscillators. Using the KBM method, they translate this equation into algebraic amplitude‑frequency curves and reveal how singularities on these curves govern qualitative changes in system behavior. The study highlights the importance of exact higher‑order modeling for predicting and managing complex nonlinear resonance phenomena that are invisible to lower‑order approximations.