Multistable dissipative breathers and collective states in SQUID Lieb metamaterials

A SQUID (Superconducting QUantum Interference Device) metamaterial on a Lieb lattice with nearest-neighbor coupling supports simultaneously stable dissipative breather families which are generated through a delicate balance of input power and intrinsic losses. Breather multistability is possible due to the peculiar snaking flux amplitude-frequency curve of single dissipative-driven SQUIDs, which for relatively high sinusoidal flux field amplitudes exhibits several stable and unstable solutions in a narrow frequency band around resonance. These breathers are very weakly interacting with each other, while multistability regimes with a different number of simultaneously stable breathers persist for substantial intervals of frequency, flux field amplitude, and coupling coefficients. Moreover, the emergence of chimera states as well as temporally chaotic states exhibiting spatial homogeneity within each sublattice of the Lieb lattice is demonstrated. The latter of the states emerge through an explosive hysteretic transition resembling explosive synchronization that has been reported before for various networks of oscillators.

Figure 1

(a) Schematic of a Lieb lattice; each unit cell (green square) has a corner SQUID (black) and two edge SQUIDs (red and blue). The nearest-neighbor coupling coefficients are indicated as ${\lambda }_x$ and ${\lambda }_y$. (b) Schematic of a single SQUID. (c) Equivalent electrical circuit for a dissipative-driven SQUID.

Figure 2

The snaking flux amplitude ${\varphi }_{\max }$-driving frequency $\mathrm{\Omega }$ curve for a single SQUID with ${\beta }_L=0.86$ and ${\varphi }_{ac}=0.05$ (blue curves). The green curves are calculated from Eq. (10). The vertical orange line is at frequency $\mathrm{\Omega }=1.01$ ($T=2\pi /\mathrm{\Omega }=6.22$). The red symbols superposed on some branches of the ${\varphi }_{\max }\text{−}\mathrm{\Omega }$ curve are the amplitudes of stable dissipative discrete breather families (except the ones indicated by the arrows; see text).

Figure 3

(a) The total energy $E_{\mathrm{tot}}=H$ of the SQUID Lieb metamaterial as a function of $\tau$ for $N_x=N_y=16$, ${\beta }_L=0.86$, ${\lambda }_x={\lambda }_y=-0.02$, $\gamma =0.01$, $\mathrm{\Omega }=1.01$, ${\varphi }_{ac}=0.05$, and four initial conditions-trivial breather configurations. Inset: The ratio $e_{\mathrm{DB}}\equiv H_{n=n_e,m=m_e}/H$ as a function of the steady-state dissipative breather amplitude ${\varphi }_{\max }$ for the four multistable dissipative breathers. The blue-dotted curve is a guide to the eye. (b) The energetic participation ratio $\mathit{epr}$ as a function of $\tau$ for the four initial conditions-trivial breather configurations. Inset: The $\mathit{epr}$ as a function of $\tau$ for the trivial breather configurations leading to the three more localized dissipative breathers. (c) The amplitude of the four multistable dissipative breathers ${\varphi }_{\max }$ as a function of $\tau$. The asymptotic values of ${\varphi }_{\max }$ have been used in the inset in (a).

Figure 4

The energy density $H_{n,m}$ of the SQUID Lieb metamaterial on the $n\text{−}m$ plane, in which four dissipative breathers exist simultaneously, for $N_x=N_y=16$, ${\beta }_L=0.86$, $\gamma =0.01$, ${\lambda }_x={\lambda }_y=-0.02$, $\mathrm{\Omega }=1.01$, ${\varphi }_{ac}=0.05$, and different separations. The central breather sites for ${\mathrm{DB}}_1$, ${\mathrm{DB}}_2$, ${\mathrm{DB}}_3$, and ${\mathrm{DB}}_4$, are located on a square with vertices, respectively, at (a) $(n_e,m_e)=(4,4),(4,12),(12,4),(12,12)$; (b) $(n_e,m_e)=(6,6),(6,10),(10,6),(10,10)$; (c) $(n_e,m_e)=(7,7),(7,9),(9,7),(9,9)$.

Figure 5

(a) The four dissipative breather amplitudes ${\varphi }_{\max }$ as a function of the driving field amplitude ${\varphi }_{ac}$, for $N_x=N_y=16$, ${\beta }_L=0.86$, $\gamma =0.01$, $\mathrm{\Omega }=1.01$, and ${\lambda }_x={\lambda }_y=-0.02$. (b) The corresponding energetic participation ratios $epr$ as a function of ${\varphi }_{ac}$. Inset: Enlargement for low $epr$ values. (c) The corresponding magnitudes of the synchronization parameter averaged over the steady-state integration time ${\langle r\rangle }_{\mathrm{int}}$ as a function of ${\varphi }_{ac}$. Inset: Enlargement for values of ${\langle r\rangle }_{\mathrm{int}}\lesssim 1$.

Figure 6

(a) The flux amplitudes ${\varphi }_{\max }$ of the four dissipative breather families as a function of the coupling coefficients ${\lambda }_x={\lambda }_y=\lambda$, for $N_x=N_y=16$, ${\beta }_L=0.86$, $\gamma =0.01$, $\mathrm{\Omega }=1.01$, and ${\varphi }_{ac}=0.05$. (b) The corresponding energetic participation ratios $epr$ as a function of $\lambda$.

Figure 7

(a) The synchronization parameter averaged over the driving period $T$, ${\langle r\rangle }_T$, as a function of $\tau$ for $N_x=N_y=16$, $\beta =0.86$, $\gamma =0.01$, ${\lambda }_x={\lambda }_y=-0.02$, ${\varphi }_{ac}=0.1$, and $\mathrm{\Omega }=1.01$ (black); $\mathrm{\Omega }=1.02$ (red); $\mathrm{\Omega }=1.03$ (green); $\mathrm{\Omega }=1.04$ (blue); $\mathrm{\Omega }=1.05$ (orange). (b) The corresponding probability distribution functions $pdf\left({\langle r\rangle }_T\right)$ normalized to unity area. The actual peak of the orange curve, which is practically a $\delta$ function is at $pdf\left({\langle r\rangle }_T\right)=8000$.

Figure 8

The energy density $H_{n,m}$ of the SQUID Lieb metamaterial (energy per unit cell) on the $n\text{−}m$ plane, after integrating the dynamic equations for ${10}^{7}T$ time units, for $N_x=N_y=16$, ${\beta }_L=0.86$, $\gamma =0.01$, ${\lambda }_x={\lambda }_y=-0.02$, ${\varphi }_{ac}=0.1$, and (a) $\mathrm{\Omega }=1.01$; (b) $\mathrm{\Omega }=1.03$; (c) $\mathrm{\Omega }=1.05$. The value of the synchronization parameter averaged over the steady-state integration time is ${\langle r\rangle }_{\mathrm{int}}\sim 0.77$, $\sim 0.71$, and $\sim 0.59$, respectively.

Figure 9

(a) The flux oscillation amplitudes ${\varphi }_{\max }^A$, ${\varphi }_{\max }^B$, and ${\varphi }_{\max }^C$ of the SQUIDs of kind $A$ (blue), $B$ (red), and $C$ (green), respectively, of the $(n_e,m_e)\mathrm{th}$ unit cell as a function of the driving flux field amplitude ${\varphi }_{ac}$ for $N_x=N_y=16$, ${\beta }_L=0.86$, $\gamma =0.01$, $\mathrm{\Omega }=1.01$, and ${\lambda }_x={\lambda }_y=-0.02$. Inset: The magnitude of the synchronization parameter averaged over the steady-state integration time ${\langle r\rangle }_{\mathrm{int}}$ (red) and the total energy of the SQUID Lieb metamaterial divided by $E_0={10}^6$, $E_{\mathrm{tot}}/E_0$ (blue), as a function of ${\varphi }_{ac}$. Note the large hysteresis region in those curves. (b) Time dependence of ${\varphi }^A$ (blue), ${\varphi }^B$ (red), and ${\varphi }^C$ (green), for ${\varphi }_{ac}=0.2$ and the other parameters as in (a). (c) The fluxes ${\varphi }^A$ (green), ${\varphi }^B$ (red), and ${\varphi }^C$ (blue), on the $n\text{−}m$ plane for ${\varphi }_{ac}=0.2$ and the other parameters as in (a). (d) Stroboscopic plots of ${\varphi }^C\left(nT\right)-q^C\left(nT\right)$, with $q^C\equiv {\dot{\varphi }}^C$, for ${\varphi }_{ac}=0.1$ (blue) and ${\varphi }_{ac}=0.2$ (red). An enlargement of the period-1 attractor is shown in the inset. The red arrows are along the transient leading to the chaotic attractor.