Discrete breathers in a mechanical metamaterial

We consider a previously experimentally realized discrete model that describes a mechanical metamaterial consisting of a chain of pairs of rigid units connected by flexible hinges. Upon analyzing the linear band structure of the model, we identify parameter regimes in which this system may possess discrete breather solutions with frequencies inside the gap between optical and acoustic dispersion bands. We compute numerically exact solutions of this type for several different parameter regimes and investigate their properties and stability. Our findings demonstrate that upon appropriate parameter tuning within experimentally tractable ranges, the system exhibits a plethora of discrete breathers, with multiple branches of solutions that feature period-doubling and symmetry-breaking bifurcations, in addition to other mechanisms of stability change such as saddle-center and Hamiltonian Hopf bifurcations. The relevant stability analysis is corroborated by direct numerical computations examining the dynamical properties of the system and paving the way for potential further experimental exploration of this rich nonlinear dynamical lattice setting.

Figure 1

Top panel: Discrete chain of cross-shaped rigid units. Bottom panel: Kinematic variables and parameters. Adapted from Fig. 6 in Supplemental Material for Ref. [14].

Figure 2

Optical branch of the dispersion relation for (a) ${\varphi }_0<{\varphi }_0^{″}$ and (b) ${\varphi }_0\ge {\varphi }_0^{″}$. The corresponding values of ${\varphi }_0$ are indicated in each panel; see also the discussion in the text. Inset in panel (a) zooms in on the $k$ values near $\pi$. Here $\alpha =1.8$, $K_s=0.02$, $K_{\theta }=1.5\times {10}^{-4}$.

Figure 3

Optical (blue) and acoustic (red) branches for (a) ${\varphi }_0=26\pi /180\approx 0.4538$, $\alpha =1.8$, $K_s=0.02$, $K_{\theta }=1.5\times {10}^{-4}$; (b) ${\varphi }_0=8\pi /180\approx 0.1396$, $\alpha =5$, $K_s=0.02$, $K_{\theta }=0.01$; (c) ${\varphi }_0=10\pi /180\approx 0.1745$, $\alpha =5$, $K_s=0.02$, $K_{\theta }=0.01$. The dashed horizontal lines indicate the maximum ${\omega }_{-}\left(\pi \right)$ of the acoustic branch and ${\omega }_{+}\left(k_{\max }\right)/2$, half of the maximum of the optical branch. When the optical branch is above ${\omega }_{-}\left(\pi \right)$, condition (12) holds, and when it is above ${\omega }_{+}\left(k_{\text{max}}\right)/2$, condition (13) holds.

Figure 4

(a) $G$ defined in Eq. (12) as a function of ${\varphi }_0$. The horizontal line is $G=0$, and the two vertical lines indicate ${\varphi }_0={\varphi }_0^{*}=0.1400$ and ${\varphi }_0={\varphi }_0^{″}=0.5888$. (b) $S$ defined in Eq. (13) as a function of ${\varphi }_0$. The horizontal line is $S=0$ and the two vertical lines indicate ${\varphi }_0={\varphi }_0^{**}=0.2811$ and ${\varphi }_0={\varphi }_0^{″}=0.5888$. Here $\alpha =1.8$, $K_s=0.02$, $K_{\theta }=1.5\times {10}^{-4}$.

Figure 5

(a) $G$ defined in Eq. (12) as a function of ${\varphi }_0$. The horizontal line is $G=0$ and the two vertical lines indicate ${\varphi }_0={\varphi }_0^{*}=0.0992$ and ${\varphi }_0={\varphi }_0^{″}=0.1588$. (b) $S$ defined in Eq. (13) as a function of ${\varphi }_0$. The horizontal line is $S=0$ and the two vertical lines indicate ${\varphi }_0={\varphi }_0^{″}=0.1588$ and ${\varphi }_0={\varphi }_0^{***}=0.3906$. Here $\alpha =5$, $K_s=0.02$, $K_{\theta }=0.01$.

Figure 6

(a) Energy $H$ as a function of frequency $\omega$. The two insets show the strain variable defined in Eq. (4) and the angle variable at points $A$, $B$, and $C$ along the solution curve. (b) $H\left(\omega \right)$ for the single-period (blue curve) and double-period (red and green curves) solution branches at twice their frequency. The bifurcation points are marked by $d$ and $e$. The two insets show the strain and angle variables at points $D$ and $E$ along the double-period solution curves. (c) Maximum modulus $\left|\mu \right|$ of Floquet multipliers versus frequency $\omega$ along the single-period (blue) and double-period (red and green) solution branches. The insets show the corresponding Floquet multipliers near the unit circle. While the double-period solution along the red curve coincides with the single period solution (blue curve) at the bifurcation point $e$, the Floquet multipliers for the double-period solution are squares of those for the single-period one, resulting in the gap between the blue and red curves. (d) Upper panel: Largest modulus $\left|\mu \right|$ of the real Floquet multipliers as a function of frequency $\omega$ along the blue single-period and green double-period solution curves near the bifurcation point $d$. Lower panel: Second-largest modulus $\left|\mu \right|$ of the real Floquet multipliers as a function of $\omega$ along the blue single-period and red double-period solution curves near the bifurcation point $e$. Note that these real Floquet multipliers are negative for the blue curve and positive for the red and green curves. Here and in the remainder of this section we have $\alpha =1.8$, $K_s=0.02$, $K_{\theta }=1.5\times {10}^{-4}$, $N=200$, and ${\varphi }_0=26\pi /180$.

Figure 7

(a) Enlarged view of the green double-period solution curve. The dashed vertical lines indicate the local minimum (left) and maximum (right) of $H\left(\omega \right)$ along the upper branch of the multivalued curve. Points $x_1,\cdots ,x_9$ correspond to the Floquet multiplier panels shown in (b). The inset shows an enlarged view near the turning point. Point $x_5$, which is very close to $x_6$, is marked by a larger red circle. (b) Floquet multipliers near $\mu =1$ for points marked in (a). The arrows indicate how the Floquet multipliers evolve as the branch is transversed.

Figure 8

(a) Maximum modulus $\left|\mu \right|$ of Floquet multipliers versus frequency $\omega$ along the blue branch prior to the bifurcation. The vertical line indicates the frequency $\omega =1.229$, at which the top optical and bottom acoustic arcs shown in Fig. 9 below first intersect (see text for details). Here $\left|\mu \right|>1$ corresponds to oscillatory instabilities, as shown in the insets, where the red curve is part of the unit circle. (b) Floquet multipliers $\mu$ at the start of the continuation ($\omega =1.57$).

Figure 9

Numerically computed Floquet multipliers (dark blue crosses) and arcs of Floquet multipliers in Eq. (18) corresponding to top optical (${\omega }_{+}\left(k\right)$, red arc) and bottom acoustic ($-{\omega }_{-}\left(k\right)$, light blue arc) dispersion bands at (a) $\omega =1.57$, (b) $\omega =1.4$, (c) $\omega =1.201$.

Figure 10

The strain and angle variables near the right end of the chain for the phantom breather with frequency $\omega =0.9972$ (solid red) and the regular (localized) discrete breather with frequency $\omega =1.02$ (dashed blue).

Figure 11

The amplitude spectrum $P\left(\omega \right)$ obtained using the FFT for different values of the prescribed breather frequency $\tilde{\omega }$: (a) $\tilde{\omega }=1.1$, (b) $\tilde{\omega }=1.02$. The left and right shaded strips in each of the bottom panels indicate the acoustic and optical dispersion bands, respectively. The dashed vertical lines indicate $\tilde{\omega }$ and $2\tilde{\omega }$ in both panels and $\tilde{\omega }/2$ in panel (b). It is clear that the frequencies associated with the breather do not resonate with the linear spectral bands in the cases shown.

Figure 12

Initial guess for (a) displacement $u_n={\varepsilon }_{u}tanh\left(\delta (n-N/2)\right)$; (b) angle ${\theta }_n$ obtained from the $\pi$-mode eigenvector (see text for details). Here ${\varepsilon }_u=0.05$ and $\delta =0.15$.

Figure 13

Energy $H$ of the computed DB solutions as a function frequency $\omega$. Blue, red, and green curves are branches of solutions that have even symmetry, while the asymmetric solution curves are shown in black. The legend in the figure holds for all four curves, and the inequalities refer to the maximum modulus of the Floquet multiplier. The parts of the curves where there are only oscillatory instabilities with the maximum modulus of the Floquet multipliers exceeding 1.009 are indicated by thin dotted segments, while solid segments indicate the portions where there are no exponential instabilities, and the maximum modulus of the Floquet multipliers is below 1.009. Here and in the remainder of this subsection, we have $\alpha =5$, $K_s=0.02$, $K_{\theta }=0.01$, $N=200$, and ${\varphi }_0=10\pi /180$.

Figure 14

(a) Energy $H$ as a function of frequency $\omega$ along the blue symmetric solution branch. The insets provide a enlarged view of the turning points. (b) Strain and angle variables for the solutions at points $A$, $B$, and $C$ in (a). (c) $H\left(\omega \right)$ along the red symmetric solution branch. The inset showing Floquet multipliers illustrates the emergence of an exponential instability. A pair of complex Floquet multipliers (blue crosses) associated with a solution before the transition collides to form two positive real multipliers (red crosses) associated with the solution after the collision. The corresponding symmetric multipliers inside the unit circle are not shown. (d) Strain and angle variables for the solutions at points $A$, $B$, and $C$ in (c). (e) Left panel: The unstable asymmetric branch connecting the red and blue symmetric branches. Right panel: Maximum real Floquet multiplier as a function of energy for the three branches. All solution profiles are shown at the time instances of maximal amplitude.

Figure 15

(a) Energy $H$ as a function of frequency $\omega$ along the blue and green symmetric solution branches and bifurcating black branches of asymmetric solutions, with $a$, $b$, $c$, and $d$ marking the bifurcation points. The insets show the solutions of the asymmetric and symmetric branches at points $A$, $B$, and $C$. (b) The stability exchange between the symmetric (blue) and the asymmetric (black) branch. Both the associated portion of the bifurcation diagram and the dominant multiplier of each branch associated with the instability growth rate are shown. (c) Energy $H$ versus frequency $\omega$ for the green symmetric solution branch with $d$ and $e$ marking the bifurcation points (see Fig. 16 for the asymmetric branch bifurcating from $e$). (d) The enlarged view of the region inside the rectangle in (c). The insets show the transition from exponential to oscillatory instabilities and vice versa that take place over the green symmetric curve. The red and blue crosses indicate Floquet multipliers $\mu$ outside the unit circle that correspond to solutions before and after the transition point, respectively.

Figure 16

(a) Energy $H$ as a function of frequency $\omega$ along the green symmetric solution branch and a branch of asymmetric solutions (different from the ones discussed earlier) bifurcating at point $e$. The insets include profiles of the angle variable at points $A$, $B$, and $C$. (b) The insets associated with points $D$ and $E$ show the emergence of a new pair of real Floquet multipliers. In both cases, there is an additional pair of real multipliers, which is not shown due to its larger magnitude. The inset associated with point $F$ illustrates the collision of two pairs of real multipliers to form two complex pairs. The red and blue crosses indicate Floquet multipliers outside the unit circle that correspond to solutions before and after the transition point, respectively.

Figure 17

(a) Energy $H$ as a function of frequency $\omega$ along an asymmetric solution branch that exists near the $k=\pi$ edge of the optical branch. The insets include profiles of the angle variable at points $A$, $B$, $C$, and $D$. (b) The same branch, with the inset showing the collision of two pairs of complex Floquet multipliers to form two real pairs. Blue and red crosses show the pairs outside the unit circle that correspond to solutions before and after the transition, respectively.

Figure 18

(a) Energy $H$ as a function of frequency $\omega$ along the red and blue symmetric solution curves. Points $A\text{–}L$ indicate the perturbed unstable solutions, while points $A^{*}-L^{*}$ mark the corresponding final states. The inset zooms in on the region including the end points. (b) Space-time plot of the displacement $u_n\left(t\right)$ for the solution corresponding to point $E$. Here $\epsilon ={10}^{-5}$ is the strength of the perturbation and $\mu =1.3596$ is the largest real Floquet multiplier. (c) Space-time plot of the angle ${\theta }_n\left(t\right)$. (d) Enlarged view of (c). Both (c) and (d) are shown in a logarithmic plot to facilitate the visualization of the small scales involving dispersive wave radiation as a result of the instability.

Figure 19

Energy $H$ of the computed DB solutions as a function frequency $\omega$. Blue and red curves bifurcate from the optical band at $k=0$, while the green curve is associated with the acoustic $\pi$ mode. All of the branches shown contain solutions with even symmetry. Thin dashed portions of the curves indicate the presence of the real multiplier pairs $(1/\mu ,\mu )$ with $\mu >1$. Along the thick dashed segments, there are also real multipliers $(1/\mu ,\mu )$ with $\mu <-1$. The parts of the curves where there are only oscillatory instabilities with the maximum modulus of the Floquet multipliers exceeding 1.009 are indicated by thin dotted segments. Solutions that also have real multiplier pairs $(1/\mu ,\mu )$ with $\mu <-1$ are along the thick dotted parts. Solid segments indicate the portions where there are no exponential instabilities, and the maximum modulus of the Floquet multipliers is below 1.009. Here and in the remainder of this subsection we have $\alpha =5$, $K_s=0.02$, $K_{\theta }=0.01$, $N=200$, and ${\varphi }_0=8\pi /180$.

Figure 20

(a) Energy $H$ as a function of frequency $\omega$ along the blue symmetric solution branch. (b) Strain and angle variables for the solutions at points $A$, $B$, and $C$ in (a). (c) $H\left(\omega \right)$ along the red symmetric solution branch. The inset showing Floquet multipliers illustrates the emergence of an exponential instability. A pair of complex Floquet multipliers (red crosses), associated with a solution before the transition, collides to form two positive real multipliers (blue crosses) associated with the solution after the collision. The corresponding symmetric multipliers inside the unit circle are not shown. (d) Strain and angle variables for the solutions at points $A$, $B$, and $C$ in (c). (e) $H\left(\omega \right)$ along the green symmetric solution branch. The inset shows the enlarged view near the end of the computed branch. (f) Strain and angle variables for the solutions at points $A$, $B$, and $C$ in (e). All solution profiles are shown at the time instances of maximal amplitude.