Explosive transitions in coupled Lorenz oscillators

We study the transition to synchronization in an ensemble of chaotic oscillators that are interacting on a star network. These oscillators possess an invariant symmetry and we study emergent behavior by introducing the timescale variations in the dynamics of the nodes and the hub. If the coupling preserves the symmetry, the ensemble exhibits consecutive explosive transitions, each one associated with a hysteresis. The first transition is the explosive synchronization from a desynchronized state to a synchronized state which occurs discontinuously with the formation of intermediate clusters. These clusters appear because of the driving-induced multistability and the resulting attractors exhibit intermittent synchrony (antisynchrony). The second transition is the explosive death that occurs as a result of stabilization of the stable fixed points. However, if the symmetry is not preserved, the system again makes a first-order transition from an oscillatory state to death, namely, an explosive death. These transitions are studied with the help of the master stability functions, Lyapunov exponents, and the stability analysis.

Figure 1

Diagram showing the dynamical transitions in system Eqs. (1) and (2) with increasing coupling strength ${\varepsilon }_z$ for $N=500$. In (a), we plot the order parameter $R$ showing transition from the desynchronized state to synchronization and in (b) we plot the order parameter $A\left(\varepsilon \right)$ that shows transition from an oscillatory state ($A\left(\varepsilon \right)\ne 0$) to a steady state ($A\left(\varepsilon \right)=0).$

Figure 2

Multistable attractors (a) $A_0$, (b) $A_{-}$, (c) $A_{+}$, (d) $A_{i^{-}}$, and (e) $A_{i^{+}}$ observed in coupled Lorenz oscillators [Eqs. (1) and (2)] at coupling strength ${\varepsilon }_z=0.3$.

Figure 3

Dynamics of the system given by Eqs. (1) and (2) at ${\varepsilon }_z=0.3$ and $N=500$. In (a), we plot the time series for any two nodes for $j=121$ (red line) and $k=403$ (black line) showing intermittent synchrony which is confirmed in (c) where we plot the time series of the difference variables $x_j-x_k$. The intermittent antisynchrony is observed in the nodes $j=121$ (red line) and $m=204$ (blue line) for which we plot in (b) the time series of the $x$ variables of the two nodes and in (d) the time series of $x_j+x_m$. Further the dynamics of the ensemble is described by plotting (e) the space time evolution of the ensemble, where the green region represents $A_{-}$ attractors, the blue region corresponds to $A_{+}$ attractors. The region for $i\gtrsim 260$ shows the $A_0$ and $A_i$ attractors.

Figure 4

Parameter space ${\varepsilon }_z-{\gamma }_n$ at ${\gamma }_h=30$ and $N=100$ for the system given by Eqs. (1) and (2). DS indicates the region for which we have desynchorinzed state (cyan), S denotes the synchronized state (yellow), and D implies the region of amplitude death (pink). Hysteresis regions corresponding to ES and ED are denoted by $H{A}_1$ (green) and $H{A}_2$ (purple) respectively. $H{A}_3$ (grey) is the hysteresis area corresponding to the transition from DS to amplitude death. The region shown in red is the region of chimera states. Blue line indicates the parameter value for which we have plotted the order parameters $R$ and $A\left(\varepsilon \right)$ in Fig. 1, while the black lines mark the boundaries of the hysteresis regions.

Figure 5

For an ensemble of $N=100$ oscillators, we plot the parameter space for the system given by Eqs. (1) and (2). We plot the parameter space (a) ${\varepsilon }_z-{\alpha }_h$ for ${\alpha }_n=2$ and (b) ${\varepsilon }_z-{\alpha }_n$ at ${\alpha }_h=20$. DS indicates the desynchronized region (cyan), $S$ denotes the synchronized region (yellow) and the region marked in red color indicates chimera states. D denotes the region of amplitude death (pink). $H{A}_1$ (green) and $H{A}_2$ (purple) are the hysteresis areas corresponding to ES and ED, respectively, with their boundaries marked by the black lines.

Figure 6

Stability of various states for the system given by Eqs. (1) and (2). In (a), we plot the variation of the master stability function (${\mathrm{\Lambda }}_m$) for three attractors $A_0,A_{-},A_{+}$; (b) transversal Lyapunov exponent (${\lambda }_T$) for two oscillators for 20 initial conditions; and (c) largest Lyapunov exponent (${\lambda }_i$) for the forward (red line) and backward (blue) continuations.

Figure 7

The one parameter bifurcation diagram corresponding to the fixed points (a) $P_{+}=(x_{+}^{*},y_{+}^{*},z_{+}^{*})$ and (b) $P_{-}=(x_{-}^{*},y_{-}^{*},z_{-}^{*})$. The dashed line represents the unstable steady state whereas the solid red and blue lines represent stable steady state and periodic solutions, respectively. The Hopf bifurcation point is marked as HB and is denoted by a blue dot. In (c), we plot the real part of the eigenvalue [$\mathrm{Re}({\lambda }_m$)] of the Jacobain matrix Eq. (C1).

Figure 8

Dynamical transitions for the system given by Eqs. (1) and (2) for the symmetry-broken case. We plot in (a) the variation of the order parameter $A\left(\varepsilon \right)$ as a function of ${\varepsilon }_y$ showing explosive death for $N=500$ and (b) the bifurcation diagram for the case of two coupled oscillators using XPPAUT software, where HB denotes the Hopf bifurcation. The dashed line represents the unstable steady state whereas the solid red and blue lines represent stable steady state and periodic solutions, respectively. Since the transitions do not depend on ${\gamma }_n$, we show the entire parameter space as a function of ${\varepsilon }_y$ for ${\gamma }_h=30$, showing desynchorinzed state (DS) in cyan, hysteresis area (HA) (grey), and the region of amplitude death (D) (pink). In (c), we plot the largest Lyapunov exponent (${\lambda }_i$) for the forward (red line) and backward (blue line) continuations. In (d), we plot the real part of the largest eigenvalue [$\mathrm{Re}({\lambda }_m$)] of the Jacobian [Eq. (C1)] for the symmetry-breaking case.