Nonreciprocal transition waves in active lattices

Active systems, such as contraction of muscle fibers and action potential propagation in neural networks, exhibit efficient directional transport of information and energy. Bringing these biological principles into engineered systems can overcome the limitations of passive metamaterials in tunability after fabrication, spontaneous reset, and the realization of dynamical nonreciprocity. However, a unified and transparent analytical framework for reversible nonreciprocal wave propagation governed by coupled fast-slow dynamics remains absent. We develop a mechanical lattice composed of active bistable elements incorporating a displacement-driven slow recovery variable to emulate biologically inspired recovery processes and capture the evolution of active stiffness. Discrete and continuum models are derived to describe the lattice dynamics. Building on continuum descriptions, we employ singular perturbation and matched asymptotic expansions, complemented by machine learning, to enable analytical characterization. Then we define a metric of nonreciprocal energy transport and map the key parameters to quantify how system parameters jointly control wave speed and propagation direction. Nonlinear results reveal that forward and backward transition waves propagate at markedly different steady speeds, demonstrating pronounced nonreciprocity. The key mechanism arises from the coupling between fast excitation and slow recovery, which produces a refractory effect that resets the energy landscape and therefore breaks time reversal symmetry in a geometrically symmetric structure. During the evolution of active stiffness, an intermediate phase emerges and co-propagates with the mechanical wave and selectively suppresses returning waves under appropriate timing conditions. Consequently, reversible and tunable nonreciprocal transition waves are achieved. We obtain bidirectional wave solutions, along with analytical expressions for wave speed, width and energy flux as well as their conditions of validity. Theoretical predictions agree closely with numerical simulations and demonstrate that the nonreciprocity depends on the recovery rate in a nonlinear manner. This study provides a general theoretical framework and practical design guidelines for bioinspired intelligent metamaterials that enable reversible directional energy routing, signal isolation, and mechanical logic, and it also informs the design of field controlled phase change materials and multiphysics devices.