Restoring the reciprocity invariance in nonlinear systems with broken mirror symmetry

Abstract

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.

Figures (7)

• Figure 1: Phase nonreciprocity in the coupled nonlinear system. Panel (a1) shows the output norms for the forward configuration, blue curve ($N^F$), and the backward configuration, orange curve ($N^B$), as a function of the forcing frequency, $\omega_f$. The insets (a2) and (a3) show magnifications of the respective colored regions in panel (a1) near the points where the curves $N^F$ and $N^B$ intersect. The temporal traces for the forward output displacement $x^F_2$, blue curve, and the backward output displacement $x^B_1$, orange curve, are shown in panels (b) and (c) at the values of $\omega_f$ (indicated in each panel) where $N^F=N^B$ in panels (a2) and (a3). In all panels, solid lines represent stable responses, and dashed lines represent unstable responses. Other parameters which are not indicated in the figure are $(\zeta,k_c,k_N,P)=(0.05, 5, 1,0.5)$.
• Figure 2: Evolution of the locus of phase nonreciprocity ($N^F=N^B$) as a function of the symmetry-breaking parameter $r$ for the second (out-of-phase) mode. The zeros of $R$ and $\Delta\phi$ coincide near $r=6.70$. Other parameters in Eq. (\ref{['eq:EOM']}) which are not indicated in the figure are $(\mu, \zeta,k_c,k_N,P)=(1.5, 0.05, 5, 1,0.5)$.
• Figure 3: Reciprocal dynamics in the system with broken mirror symmetry. Panel (a1) and (b1) show the output norms for the forward configuration, blue curve ($N^F$), and the backward configuration, orange curve ($N^B$), as a function of the forcing frequency, $\omega_f$, for $\mu=1.5$ and two values of $r$. The insets (a2) and (b2) show magnifications of the respective colored regions in panels (a1) and (a2), respectively, near the intersection points where reciprocity is achieved ($R\approx0$). The reciprocal response for the forward output displacement $x^F_2$, blue curve, and the backward output displacement $x^B_1$, orange curve, are shown in panels (a3) and (b3). In all panels, solid curves represent stable response and dashed curves represent unstable response; $R=3.287\cdot10^{-4}$ in (a3) and $R=6.172\cdot10^{-8}$ in (b3). Other parameters which are not indicated in the figure are $(\zeta,k_c,k_N,P)=(0.05, 5, 1,0.5)$.
• Figure 4: Loci of approximate reciprocity $R=\epsilon$ in the $(\mu,r)$-parameter plane. Purple lines corresponds to the locus of reciprocity near the first primary resonance, where darker shades represents lower values of $\epsilon$. The grey line corresponds to the locus of reciprocity near the second primary resonance. Solid lines represent stable solutions and dashed lines represent unstable solutions. Other parameters which are not indicated in the figure are $(\zeta,k_c,k_N,P)=(0.05, 5, 1,0.5)$.
• Figure 5: Reciprocal dynamics in the system with broken mirror symmetry. Panel (a1) and (b1) show the output norms for the forward configuration, blue curve ($N^F$), and the backward configuration, orange curve ($N^B$), as a function of the forcing frequency, $\omega_f$, for two different sets of values for $\mu$ and $\alpha$. The insets (a2) and (b2) show magnifications of the respective colored regions in panels (a1) and (a2), respectively, near the intersection points where reciprocity is achieved ($R\approx0$). The reciprocal response for the forward output displacement $x^F_2$, blue curve, and the backward output displacement $x^B_1$, orange curve, are shown in panels (a3) and (b3). In all panels, solid curves represent stable response and dashed curves represent unstable response; solutions in panels (a3) and (b3) are stable. Other parameters which are not indicated in the figure are $(r,\zeta,k_c,k_N,P)=(1,0.05, 5, 1,0.5)$.