  Circumventing the reciprocity invariance has posed an interesting challenge
in the design of modern devices for wave engineering. In passive devices,
operating the device in the nonlinear response regime is a common means for
realizing nonreciprocity. Because mirror-symmetric systems are trivially
reciprocal, breaking the mirror symmetry is a necessary requirement for
nonreciprocal dynamics to exist in nonlinear systems. However, the response of
an asymmetric nonlinear system is not necessarily nonreciprocal. In this work,
we report on the existence of stable, steady-state nonlinear reciprocal
dynamics in coupled asymmetric systems subject to external harmonic excitation.
We restore reciprocity in the asymmetric system by tuning two symmetry-breaking
parameters simultaneously. We identify response regimes in the vicinity of the
primary resonances of the system where the steady-state left-to-right
transmission characteristics are identical to the right-to-left characteristics
in terms of frequency, amplitude and phase. We interpret these regimes of
reciprocal dynamics in the context of phase nonreciprocity, wherein incident
waves undergo a nonreciprocal phase shift depending on their direction of
travel. We hope these findings help design devices with new functionalities for
controlling and steering of elastic waves.

Applied Physics;Dynamical Systems