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

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.

Figures (14)

• Figure 1: The "trimer" model and its normal modes. Each colored disc represents an identical mass. Each grounding spring (red) has a time-modulated linear component and a nonlinear component. The modulation phase of each grounding spring increases by $2\pi/3$ moving one position clockwise around the trimer (blue $\to$ green $\to$ orange). The coupling springs (gold) add a linear restoring force in the vertical direction. The positions and blue arrows represent the displacements and velocities of the masses from snapshots of the three normal modes of the linear system without modulation. The dots indicate the times and positions of the motion that is shown in the diagrams.
• Figure 2: Floquet spectra of the trimer at various parameter values. The colored lines are the absolute values of the Floquet eigenvalues of the system at the value of the parameter shown on the horizontal axis. When an eigenvalue exceeds $|\rho|=1$ (black lines), amplification occurs. The vertical lines indicate values where the fixed point formula Eq. \ref{['eq:trimerfp']} predicts amplification will no longer occur. For the plots where $\epsilon$, $\delta$, and $\gamma$ are varied, the other parameters are held fixed; the fixed values are $k = 1$, $\epsilon = 0.01$, $\delta = 0.1$, and $\gamma = 2\omega = 4$. For the plot with varied $k$, $g$ is also varied to maintain the relationship $\gamma = 2\omega = 2\sqrt{1+3k}$.
• Figure 3: Numerical solution of the full equations of motion. The parameters are $\epsilon = 0.06$, $\delta = 0.6$, $k = 1$, $\omega = \sqrt{1+3k} = 2$, $\gamma = 2\omega = 4$. The system is initialized in its unique amplified Floquet eigenstate, normalized to have magnitude 0.1. The red line shows the amplitude obtained by solving the reduced averaged equations \ref{['eq:realreducedeom']}. Blue boxes indicate the extents of the insets below. Top: The full trace from $t = 0$ to $t = 250$. The purple line is an exponential approximation whose growth rate is obtained from an averaging analysis of Eq. \ref{['eq:unaveraged']} in section \ref{['sec:avgdyntimescales']}; the growth rate obtained from this process also matches the perturbative calculation of the amplifying Floquet multiplier for Eq. \ref{['eq:unaveraged']} in landau1960mechanics. The time $t_{nl}$ marks the transition to nonlinear behavior, also described in Section \ref{['sec:avgdyntimescales']}. Blue inset: The initial exponential amplification. Green inset: The steady state showing chiral motion (each coordinate oscillating out-of-phase by $2\pi/3$). Note that the two insets show the length of time, so the frequencies in each interval can be compared.
• Figure 4: (a): Phase portraits of the reduced averaged equations computed from the conserved quantity. The initial conditions for the amplitude index the different curves; the phase is initialized at the fixed point value. (b): Numerical solutions to the full equations of motion compared to the amplitude found by solving the reduced averaged equation. The colors of the amplitudes correspond to the colors of the phase curves in the left figure. The vertical range shown in the plots at right is $-1.1\leq x_j\leq 1.1$. As the system moves deeper into the nonlinear regime (higher amplitudes), the averaging approximation should break down; this may explain the deviation between the averaged and exact trajectories at the top of the right figure. The parameter values are $\epsilon = 0$, $\delta = 0.6$, $k = 1$, $\gamma = 2\omega = 2\sqrt{1+3k} = 4$.
• Figure 5: Comparison of the integral expression for the trimer period (Equations \ref{['eq:periodclosed']} and \ref{['eq:periodopen']}, solid lines) with periods extracted from numerical integration of the dynamical equations. Each curve corresponds to a different set of system parameters, and each position on the horizontal axis represents the initial value (alternatively, a different level curve of the conserved quantity). Solid curves show the period predicted by the integral expression and dots show the numerically-computed period. The parameters are $k = 1, \delta = 0.1$ (purple); $k = 2, \delta = 0.1$ (brown); $k=1,\delta=0.6$ (pink); $k=2,\delta=0.6$ (gray). All traces have zero damping (i.e., $\epsilon = 0$). See Appendix \ref{['app:methods']} for a detailed description of the methods used to generate this figure.