Figure 1

Lorenz (response) system: (a) the largest conditional Lyapunov exponent (LCLE) and (b) the average synchronization error ($\Delta$) between response system and its auxiliary copy, with coupling parameter $\epsilon$.Reuse & Permissions

Figure 2

(a) Lorenz attractor before the onset of GS at coupling strength $\epsilon =0.74$; (b) the symmetric pair of coexisting attractors $A$ and $B$ in a response system for model Eqs. (5); (c, d) the expanded views of coexisting attractors $A$ and $B$. Coupling parameter $\epsilon =1.59$.Reuse & Permissions

Figure 3

Lorenz attractor: The evolution of a response system ($x$ variable) for two different sets of initial conditions at $\epsilon =1.59$, where these conditions converges to (a) the same attractor $B$ (CS, $x_2=x_3$)l (b) either of symmetric coexisting attractors $A$ and $B$ ($APS$, $x_2=-x_3$). The parameters are those of Fig. 2b.Reuse & Permissions

Figure 4

Lorenz (response) system in the symmetry-breaking case: (a) the largest conditional Lyapunov exponent (LCLE) and (b) average synchronization error ($\Delta$), with coupling parameter $\epsilon$.Reuse & Permissions

Figure 5

(a) Asymmetric pair of coexisting attractors $C$ and $D$ in ($x$ coupled) the symmetry-breaking case; (b) and (c) the expanded views of coexisting attractors $C$ and $D$. Coupling parameter $\epsilon =2.0$.Reuse & Permissions

Figure 6

Symmetry-breaking interaction: The evolution of response system ($x$ variable) for two different initial conditions, where these conditions converge to (a) the same attractor $C$ ($\Delta =0$); (b) either of asymmetric coexisting attractors $C$ and $D$ ($\Delta >0)$. Coupling parameter $\epsilon =2.0$.Reuse & Permissions

Figure 7

The left and right panels correspond to the symmetry-preserving and symmetry-breaking interaction, respectively, with noise ($\xi =10.0$). (a, b) The largest conditional Lyapunov exponents (LCLE); (c, d) the average synchronization error ($\Delta$) as a function of coupling strength, $\epsilon$. (e, f) The trajectories in the ($x_2$, $x_3$) plane, when the response and its auxiliary copy converge to the same attractors $B$ and $D$, respectively. (g, h) The trajectories in ($x_2$, $x_3$) plane, when the response and its auxiliary copy converge to the different attractors $A$ or $B$ and $C$ or $D$, respectively. The coupling parameter in (e–h) is $\epsilon =2$.Reuse & Permissions