Nonlinear stabilization of parametrically amplified chiral modes

Linear models simplify complex dynamics, providing a tractable basis for understanding and designing useful responses in coupled systems. When driven, such systems can harbor unstable growing modes that eventually push the response into the nonlinear regime. Since even small nonlinearities can dramatically alter system behavior, the applicability of linear models to designing response in driven coupled systems is not guaranteed. In this work, we show that nonlinearity can stabilize a coupled, parametrically amplified mechanical system while retaining desirable features of the linearized dynamics. Specifically, we analyze a coupled oscillator ring with a parametric modulation designed to generate an amplified linear mode that breaks chiral symmetry. We find that nonlinearity limits the exponential divergence of the amplified mode, guiding the system to a steady state that inherits the chirality of the linear motion. The existence and amplitude of the steady-state motion are predicted using averaging theory. Our work demonstrates that Floquet theory and averaging theory can be combined to predict and design stable symmetry-breaking responses in periodically driven nonlinear systems and provides an example of a driven nonlinear driven system that avoids thermalization indefinitely.