Flat bands and compactons in mechanical lattices

Local configurational symmetry in lattice structures may give rise to stationary, compact solutions, even in the absence of disorder and nonlinearity. These compact solutions are related to the existence of flat dispersion curves (bands). Nonlinearity can destabilize such compactons. One common flat-band-generating system is the one-dimensional cross-stitch model, in which compactons were shown to exist for the photonic lattice with Kerr nonlinearity. The compactons exist there already in the linear regime and are not generally destructed by that nonlinearity. Smooth nonlinearity of this kind does not permit performing complete stability analysis for this chain. We consider a discrete mechanical system with flat dispersion bands, in which the nonlinearity exists due to impact constraints. In this case, one can use the concept of the saltation matrix for the analytic construction of the monodromy matrix. Besides, we consider a smooth nonlinear lattice with linearly connected massless boxes, each containing two symmetric anharmonic oscillators. In this model, the flat bands and discrete compactons also readily emerge. This system also permits performing comprehensive stability analysis, at least in the anticontinuum limit, due to the reduced number of degrees of freedom. In both systems, there exist two types of localization. The first one is the complete localization, and the second one is the more common exponential localization. The latter type is associated with discrete breathers (DBs). Two principal mechanisms for the loss of stability are revealed. The first one is the possible internal instability of the symmetric and/or antisymmetric solution in the individual unit cell of the chain. One can interpret this instability pattern as internal resonance between the compacton and the DB. The other mechanism is global instability related to resonance of the stationary solution with the propagation frequencies. Different instability mechanisms lead to different bifurcations at the stability threshold.

Figure 1

Sketch of the 1D horizontal “cross-stitch” chain. Lines: linear springs ($k$); double lines: shear springs ($k_{\text{shear}}$); (gray) anvils: rigid impactors; $\delta \ll L$.

Figure 2

Dispersion relations (frequencies given in units of $\sqrt{k/m}$), ${\gamma }_1=0,{\gamma }_2=1/2,{\gamma }_3=2,{\gamma }_4=3,\gamma \triangleq k_s/k$.

Figure 3

Stability bounds of the compacton mode for $N\gg 1$: numeric results in solid line (red online) and dash-dotted line (blue online) and analytic results in dashed black line. The labels “PF” and “NS” denote the pitchfork and Neimark-Sacker bifurcations, respectively.

Figure 4

Critical Floquet multiplier variation ($\gamma \gg 1$), compacton mode, $N=1$.

Figure 5

Typical $\gamma \gg 1$ full chain monodromy eigenvector, showing localization in compacton mode ($N=301$).

Figure 6

Breather mode stability bounds for $\gamma \ll 1,N=301$ (asymptotic estimate shown as dashed line).

Figure 7

Breather mode stability bounds for $\gamma \gg 1,N=301$ (asymptoties estimates shown as dashed lines).

Figure 8

Simulation results for the compacton — stable case: $N=9,k/k_{\text{shear}}=10,\omega /{\omega }_{\text{FB}}=2.2,{\dot{u}}_1\left(0\right)={10}^{-3}\sqrt{k/m}\mathrm{\Delta }$.

Figure 9

Simulation results for the compacton — “NS”-unstable case: $N=9,k/k_{\text{shear}}=10,\omega /{\omega }_{\text{FB}}=1.5,{\dot{u}}_1\left(0\right)={10}^{-3}\sqrt{k/m}\mathrm{\Delta }$.

Figure 10

Simulation results for the “DB” — stable case: $N=9,k/k_{\text{shear}}=\omega /{\omega }_{\text{FB}}=10,{\dot{v}}_1\left(0\right)={10}^{-3}\sqrt{k/m}\mathrm{\Delta }$.

Figure 11

Simulation results for the “DB” — “PF”-unstable case: $N=9,\frac{k}{k_{\text{shear}}}=10,\frac{\omega }{{\omega }_{\text{FB}}}=2.2,{\dot{v}}_1\left(0\right)={10}^{-3}\sqrt{k/m}\mathrm{\Delta }$.

Figure 12

Simulation results for (perturbed) compacton initial conditions — “PF”-unstable case (principal-site histories): $N=9,k/k_{\text{shear}}=10,\omega /{\omega }_{\text{FB}}=10,{\dot{u}}_1\left(0\right)={10}^{-3}\sqrt{k/m}\mathrm{\Delta }$.

Figure 13

Simulation results for (perturbed) compacton initial conditions — “PF”-unstable case (maximal-displacement-time profiles).

Figure 14

Sketch of a chain of massless boxes with internal oscillators (line across internal spring symbol denotes anharmonic addition to the potential).

Figure 15

Dispersion relations (flat band in solid), $\delta =k/k_1,\kappa =k/m$.

Figure 16

Schematic depiction of the elementary cell subjected to nonlinear analysis.

Figure 17

Linear stability bounds in solid line (red online) with asymptotic estimates in dash black and the two points chosen for numerical integration.

Figure 18

Unstable case — emergence of instability for the principal element in the chain (the compacton site) for short (top) and long (bottom) times, for $N=10,Y_0=0.8\sqrt{k/p},k/k_0=0.05,y_1\left(0\right)={10}^{-4}$.

Figure 19

Poincaré section: periodic energy exchange between modes (the parameters here correspond to global bifurcation for a single element, with the plot serving as a qualitative illustration for the case of a chain) for $\widehat{\epsilon }=0.0645,\widehat{\eta }=\frac{5}{2}$ (separatrices in red and blue online).

Figure 20

Unstable (compacton) case: emerging (from initial perturbation) breather-mode profile for $N=10,Y_0=0.8\sqrt{k/p},k/k_0=0.05,y_1\left(0\right)={10}^{-4}$.

Figure 21

Integration results — stable case (top to bottom): antisymmetric mode displacement history, symmetric mode displacement history, and symmetric (inset: antisymmetric)-mode maximal-displacement-time profile, for $N=10,Y_0=0.8\sqrt{\frac{k}{p}},\frac{k}{k_0}=0.5,y_1\left(0\right)={10}^{-4}$.

Figure 22

$v_1\left(\tau \right)$ — high-amplitude (unstable) case: $N=10,Y_0=10\sqrt{k/p},k/k_0=0.00005$.

Figure 23

Top to bottom: $y_1\left(\tau \right),y_n\left(t{|}_{{\parallel \mathbf{y}\parallel }_{\infty }^{max}}\right)$ — high-amplitude (unstable) case, $N=10,Y_0=10\sqrt{k/p},k/k_0=0.00005$.

Figure 24

$v_1\left(\tau \right)$ — high-amplitude (stable) case, $N=10,Y_0=10\sqrt{k/p},k/k_0=0.005$.

Figure 25

Top to bottom: $y_1\left(\tau \right),y_n\left(t{|}_{{\parallel \mathbf{y}\parallel }_{\infty }^{max}}\right)$ — high-amplitude (stable) case, $N=10,Y_0=10\sqrt{k/p},k/k_0=0.005$.

Figure 26

Poincaré sections in (normalized) symmetric-mode displacement and velocity, for energies just below and above the degenerate instability tongue (for $\widehat{\eta }=5/2$), specified by $\widehat{\epsilon }=0.048$ (left) and $\widehat{\epsilon }=0.049$ (right).

Figure 27

Poincaré sections, illustrating the effect of the degenerate instability tongue on the topology, for energies further above the critical line: KAM islands emanate from the compact mode as energy is increased ($\widehat{\eta }=5/2$ and $\widehat{\epsilon }=0.05$, left, and $\widehat{\epsilon }=0.0528$, right).