Nonlinear stabilization of chiral modes in space-time modulated parametric oscillators

TL;DR

This paper demonstrates that chiral steady states in space-time modulated parametric oscillators persist under nonlinear effects, enabling robust nonreciprocal signal control in realistic systems.

Contribution

It introduces a nonlinear analysis showing the persistence of chiral states in coupled oscillators with cubic nonlinearity, supported by a reduced model and finite-element simulations.

Findings

Nonlinear chiral steady states exist with finite amplitudes.

The reduced model accurately predicts steady-state properties.

Simulations confirm relevance to elastic plate resonators.

Abstract

Phase control of parametric modulation in coupled oscillator networks enables sculpting of dynamical states with desired spatiotemporal symmetries. Symmetry-aware Floquet analysis successfully predicts such states in linear systems, but whether their symmetry properties persist under nonlinearity remains largely unexplored. Here, we establish the existence of nonlinear chiral steady states in a trio of coupled parametric oscillators with modulation phases chosen to selectively amplify a circulating mode in the linearized system. We find that a cubic nonlinearity arrests exponential growth of the amplified mode, producing a steady finite-amplitude motion that retains the expected chirality. By exploiting space-time symmetry, we reduce the dynamics to a single averaged equation that quantitatively predicts nonlinear trajectories, steady-state amplitudes, and characteristic time scales. The chiral steady states possess finite basins of attraction and are accessible from wide ranges of initial conditions and system parameters. Finite-element simulations of elastic plate resonators quantitatively reproduce these features, establishing the relevance of the reduced model to realistic continuum systems. Our results demonstrate that desirable properties of linear time-modulated systems, such as chirality and directional amplification, persist into strongly nonlinear regimes, opening pathways to robust nonreciprocal signal routing and amplification in parametrically driven platforms.