Nonlinear skin breathing modes in one-dimensional nonreciprocal mechanical lattices

TL;DR

The paper addresses how nonreciprocity and onsite nonlinearity enable time-periodic, boundary-localized skin modes in a 1D lattice. It employs multiple-scale analysis and numerical pseudo-arclength continuation to construct nonlinear skin breathing modes emerging from linear skin modes for arbitrary nonreciprocity strength. It derives a nonlinear frequency shift dependent on amplitude, nonlinearity type, lattice size, and nonreciprocity, and shows that some modes form end breathers when harmonics enter spectral gaps; Floquet analysis reveals regimes of linear stability and instability with skin-localized perturbations. The results extend the nonlinear skin effect from stationary to time-periodic excitations, offering a route to engineer breathing modes in nonreciprocal mechanical metamaterials.

Abstract

We investigate the interplay of nonreciprocity and nonlinearity in a one-dimensional nonlinear Klein-Gordon chain of classical oscillators coupled by asymmetric springs, akin to a mechanical analogue of the Hatano-Nelson model with onsite nonlinearity. Using multiple-scale analysis, we show that families of nonlinear skin breathing modes -- time-periodic, boundary-localized oscillations -- emerge from their linear counterparts at any nonreciprocal strength. We derive an explicit nonlinear frequency shift for these families of nonlinear breathing modes, showing its dependence on amplitude, nonlinearity type, lattice size, and nonreciprocity, and we predict the emergence of genuine skin end breathers at the boundary once their oscillation frequency and higher harmonics enter the spectral gaps of the linear spectrum. Numerical pseudo-arclength continuation confirms full families of solutions for both hardening and softening nonlinearities. Furthermore, the Floquet analysis shows that these modes can be either linearly stable or unstable, with Floquet eigenvectors exhibiting skin localization inherited from the asymmetric couplings. Our results extend the nonlinear non-Hermitian skin effect from stationary modes to intrinsically time-periodic excitations, providing a pathway to engineer and control breathing modes in nonreciprocal mechanical metamaterials.

Figures (7)

• Figure 1: Schematic of a chain of classical oscillators interconnected with asymmetric elastic spring couplings. The relation between the force, $F$, and the spring's elongation, $\Delta y$ is given by $F_{\text{Right}\rightarrow \text{Left}} = a_{+}\Delta y$ (red) and $F_{\text{Left}\rightarrow \text{Right}} = a_{-}\Delta y$ (blue), where $a_{\pm} = 1 \mp \gamma$ are the elastic constants and $\gamma$ measures the nonreciprocal strength. In addition, the onsite restoring force is given by $F = \alpha y + g y^3$ (black), where $\alpha$ and $g$ are the onsite elastic and nonlinear spring coefficients, respectively, and $y$ denotes the oscillator displacement.
• Figure 2: Dependence of the numerically obtained linear frequency, $\omega_j$ against the wave number $j$ for the nonreciprocal KG chain of Fig. \ref{['fig:nonreciprocal_KG_chain']} with $N=24$, $\gamma =0.25$, $\alpha =1$, $g=0$, and fixed boundary conditions. The lower and upper frequency cutoffs are $\omega_{min} = (2 + \alpha - 2\sqrt{1-\gamma^2})^{1/2}$ and $\omega_{max} = (2 + \alpha + 2\sqrt{1-\gamma^2})^{1/2}$ respectively (see text for details). The bottom and top insets show the right eigenvectors, $u_{j,n}$ with wave numbers $j=1$ and $j=N$. (b) Procedure for the computation of a family of nonlinear skin breathing modes emerging from their linear counterparts.
• Figure 3: (a) Dependence of the nonlinear frequency $\Omega$ on the amplitude $\lVert \vec{y} \rVert^{2}$ for families of nonlinear skin breathing modes emerging from the linear skin modes with $j = 1$ (red), $j = 6$ (blue), $j = 10$ (cyan), $j = 14$ (yellow), $j=18$ (purple) and $j = 24$ (green), shown from the bottom to the top curves. The dashed portions of the curves indicate unstable modes, while the solid portions correspond to stable ones (this representation is qualitative). The parameters used in the computations are $N = 24$, $\alpha = 1$, $\gamma = 0.25$, and $g = +1$ (hardening). The numerical continuation is stopped when $\lVert \vec{y} \rVert^{2} = 3$, which roughly corresponds to weak and moderate nonlinear strengths. (b) Profile of a representative nonlinear skin breathing mode with $\lVert \vec{y}\rVert^{2} = 0.2$, belonging to the family with $j=24$, see star symbol. Note that its linear counterpart is shown in the upper inset of Fig. \ref{['fig:skin_modes_and_procedure']}(a). (c) Time evolution of the state in (b) used as the initial condition, clearly demonstrating an amplitude that oscillates in time. (d) Same as in (a), but for $g = -1$ (softening). (e) Same as in (b), but for the softening case shown in (d), see dot symbol. Note that its linear counterpart is depicted in the lower inset of Fig. \ref{['fig:skin_modes_and_procedure']}(a). (f) Same as in (c), but using the state in (e) as the initial condition.
• Figure 4: Dependence of the nonlinear frequency sensitivity factor $\mathcal{S} = \partial \widetilde{\Omega} / \partial S$ on the lattice size $N$, where $\widetilde{\Omega} = (\Omega - \omega)/\omega$ and $S = \lVert \vec{y} \rVert^{2}$. We compute $\mathcal{S}$ for fixed values $\gamma = 0.25$ (red squares), $\gamma = 0.4$ (blue dots), $\gamma = 0.55$ (cyan triangles), and $\gamma = 0.7$ (purple stars). The derivatives are evaluated using a two-point finite difference at $\lVert \vec{y}\rVert^{2} = 0.01$ and $0$NOTE0001. The dashed line is a guide to the eye showing $\mathcal{S}(N) \sim \exp(-2.5\,N)$ (see text for details). Inset: Dependence of $\mathcal{S}$ on $\gamma$ for three lattice sizes, $N = 24$, $N = 32$, and $N = 42$, shown as the blue, red, and cyan dotted symbols, respectively. We find that $\mathcal{S}(\gamma)\sim r^{-2}$, $r^{-2.5}$ and $r^{-3}$ for $N = 24$, $N = 32$, and $N = 42$ respectively. In all cases, the analytical predictions show as solid curves overlapping with the numerical results, see also Fig. \ref{['fig_app:freq_amplitude_theo_numerics']}.
• Figure 5: (a) Dependence of $\lvert \lambda \rvert$ against the amplitude $\lVert \vec{y}\rVert^{2}$ for the representative family emerging from the skin mode of $j=1$ with $N = 24$, $\alpha = 1$, $\gamma = 0.25$, and $g = -1$ [see also the red curve in Fig. \ref{['fig:nonlinear_continuation_results']}(d)]. Values of $\max \lvert \lambda \rvert \neq 1$ signal instability. (a)–(b) Floquet eigenvalues $\lambda$ in the complex plane for the representative nonlinear breathing modes with amplitudes $\lVert \vec{y}\rVert = 1$ [purple dashed horizontal line in (c)] and $\lVert \vec{y}\rVert^2 = 3$ [green dotted horizontal line in (c)]. Eigenvalues lying outside the unit circle (gray circle) along the real axis indicate real instabilities, while those outside the circle off the real axis correspond to complex instabilities. (d) Floquet eigenvectors $\lVert \vec{z}_l\rVert$ corresponding to the eigenvalues shown in (b). The vectors $\vec{z}_l$ are sorted in increasing order of $\lvert \lambda \rvert$. For clarity, each eigenvector $\lVert \vec{z}_l \rVert$ is normalized by $\max \lvert z_{l,n} \rvert$.