Traveling phase waves in asymmetric networks of noisy chaotic attractors

We explore identical Rössler systems organized into two equally sized groups, among which differing positive and negative in- and out-coupling strengths are allowed. With this asymmetric coupling, we analyze patterns in the phase dynamics that coexist with chaotic amplitudes. We specifically investigate traveling phase waves where the oscillators settle on a new rhythm different from their own. We show that these waves are possible even without coherence in the phase angles. It is further demonstrated that the emergence of these incoherent traveling waves depends on the type of coupling, not on the individual dynamics of the Rössler systems. Together with the study of noise effects, our results suggest a promising new avenue toward the interplay of chaotic, noisy, coherent, and incoherent collective dynamics.

Figure 1

Snapshots of the oscillators' trajectories of the weighted $x$-coupled Rössler model [Eq. (1)] with phase-coherent attractor ($a=0.15,b=0.2$, and $c=10$). (a) $\pi$-state where $\delta =\pi$ with $K_0=0$ and $G_0=0$. (b) Lag sync state for which $\delta <\pi$ with $K_0=0$ and $G_0=0.2$. Gray line in the background corresponds to a trajectory of a randomly selected oscillator in order to depict typical dynamics. Remaining parameters: $N=200,\mathrm{\Delta }K=0.1,\mathrm{\Delta }G=1$, and $D=0.03$. See Supplemental Material [7] for the animated version of the figure and additional videos on other collective states.

Figure 2

(a) Map showing the regions with escaping trajectories for weighted $x$-coupled Rössler oscillators [Eq. (1)] in the phase-coherent regime considering $N=500,D=0,\mathrm{\Delta }K=0.1$, and $\mathrm{\Delta }G=1$. For each pair $(G_0,K_0)$ the initial conditions $[x_i\left(0\right),y_i\left(0\right),z_i\left(0\right)]\forall i$ are randomly drawn according to a uniform distribution in the range $[-1,1]$. Simulation time $T=1\times {10}^5$. Solid black curves fitting the boundaries of the escaping region are obtained through the relation $\langle \langle KG\rangle \rangle =c$, where $c_{\mathrm{I}}=c_{\mathrm{III}}=11.8$ and $c_{\mathrm{II}}=c_{\mathrm{IV}}=-0.66$. Square at the center depicts the parameter region of Fig. 3. Area of escaping region as function of (b) simulation time $T$ for fixed $D=0$ and (c) as function of $D$ for $T=5\times {10}^3$.

Figure 3

Color maps showing (a) order parameter $r$, (b) phase lag $\delta$, and (c) speed of phase wave $|\mathrm{\Omega }-{\omega }_0|$ for weighted $x$-coupled Rössler oscillators [Eq. (1)] with phase-coherent (upper triangles) and funnel (lower triangles) attractors considering $D=0.03$. (d) Area of TW state in which $|\mathrm{\Omega }-{\omega }_0|\ge \varepsilon$, where we consider $\varepsilon =0.3$; and $\pi$ state for $\delta \ge \varepsilon$ with $\varepsilon =3$. Remaining parameters: $\mathrm{\Delta }K=0.1,\mathrm{\Delta }G=1,N={10}^3,50\times 50$ grid, simulation time $T=5\times {10}^3$. Curves in the interface between incoherent and partial synchronized state for phase-coherent oscillators are given by the condition $2D_{\mathrm{eff}}=\langle \langle KG\rangle \rangle$, here with $D_{\mathrm{eff}}\approx 0$ [13]. Horizontal dashed lines correspond to the critical points $K_{0c}=\pm \mathrm{\Delta }K/2$. See also Appendix pp1 for diagrams with $\mathrm{\Delta }K=\mathrm{\Delta }G=0$.

Figure 4

Color maps for the phase-coupled model [Eq. (4)] showing (a) (total) order parameter $r$, (b) phase lag $\delta$, (c) the spontaneous drift measured by $|\mathrm{\Omega }-{\omega }_0|$, where $\mathrm{\Omega }$ is the wave speed [Eq. (3)], and ${\omega }_0$ the natural frequency. (d) Area of TW state in which $|\mathrm{\Omega }-{\omega }_0|\ge \varepsilon$, where we consider $\varepsilon =0.3$; and $\pi$ state for $\delta \ge \varepsilon$ with $\varepsilon =3$. For each point in the $70\times 70$ grid, simulations are performed by evolving Eqs. (4) considering initial conditions ${\rho }_i\left(0\right)$ and $z_i\left(0\right)$ randomly chosen from uniform distribution $[-1,1]$ and ${\varphi }_i\left(0\right)$ uniformly distributed in the range $[-\pi ,\pi ]$. Other parameters: $N={10}^3,\mathrm{\Delta }K=1.45,\mathrm{\Delta }G=1,D=0.03$, averaging over $t\in [4000,5000]$ and $dt=0.05$. Curves in the interface between incoherent and partial synchronized state are given by the condition $2D=\langle \langle KG\rangle \rangle$. Horizontal dashed lines correspond to the critical points $K_{0c}=\pm \mathrm{\Delta }K/2$. See Appendices pp4 and pp5 for additional diagrams of the phase-coupled model.

Figure 5

Color maps for weighted $x$-coupled Rössler oscillators with phase-coherent ($a=0.15,b=0.2$, and $c=10$) and funnel attractors ($a=0.2,b=0.2$, and $c=7$) in the absence of coupling mismatches, i.e., $\mathrm{\Delta }K=0,\mathrm{\Delta }G=0$. Parameters: $N={10}^3,dt=0.05$. For each $(G_0,K_0)$ in the $50\times 50$ grid, the quantities are averaged over t $\in [4700,5000]$. Initial conditions $(x_i\left(0\right),y_i\left(0\right),z_i\left(0\right))$ are selected from an uniform distribution in the range $[-1,1]$.

Figure 6

Dependence of incoherent TWs on the system size for $K_0=0.5,G_0=-0.5,\mathrm{\Delta }K=0.1,\mathrm{\Delta }G=1$, and $D=0.03$. Each point is an average over 100 different realizations. Size of error bars is smaller than the dots' sizes.

Figure 7

Examples of time evolution of order parameters for phase-coherent oscillators in (a) $\pi$ state $\left(K_0=G_0=0\right)$, (b) lag sync ($K_0=0.25,G_0=-0.1$); and oscillators with funnel attractor also in (c) $\pi$ state ($K_0=G_0=0$) and (d) lag sync $(K_0=0.03,G_0=0.5)$. Other parameters: $\mathrm{\Delta }K=0.1,\mathrm{\Delta }G=1,D=0.03$, and $N={10}^3$. Shaded area corresponds to the time-averaging window considered in Fig. 3.

Figure 8

Color maps for weighted $x$-coupled Rössler oscillators with phase-coherent attractor ($a=0.15,b=0.2$ and $c=10$). (a) $r_1$, (b) $r_2$, (c) $r$, (d) $|\mathrm{\Omega }-{\omega }_0|$, and (e) $\delta$. Remaining parameters: $\mathrm{\Delta }K=0.1,\mathrm{\Delta }G=1,D=0.03$, and $N={10}^3$. Notice in panel (d) that the white upper left and lower right regions are undefined since the phase lag is undefined. On the other hand, the frequency difference (e) defined via the Hilbert transform [Eq. (3)] indicates a strict acceleration.

Figure 9

Color maps for phase-coupled Rössler oscillators with phase-coherent attractor ($a=0.15,b=0.2$, and $c=10$) for $\mathrm{\Delta }K=1.45,\mathrm{\Delta }G=1,D=0.03$, and $N={10}^3$. Concerning (d), see the remark on the phase lag in Fig 8.

Figure 10

The absolute value of the difference between the mean phases of the two populations, $\delta$, in dependence on the mean in- and out-coupling strengths, $K_0$ and $G_0$, respectively. The two panels show zoomed regions from Fig. 4, with different transition scenarios into the TW states. In (a) the TW region is surrounded by zero-lag states, whereas in (b) the TW states are surrounded by $\pi$ states. Parameters: $N={10}^3,\mathrm{\Delta }K=1.45,\mathrm{\Delta }G=1,D=0.03,dt=0.05$. For each $(G_0,K_0)$ in the $100\times 100$ grid, the quantities averaged over $t\in [4000,5000]$. Dashed magenta line in panel (a) marking the interface between Incoherence and Zero-lag sync is obtained by the critical condition $2D=\langle \langle KG\rangle \rangle$, where $\langle \langle ...\rangle \rangle =\int d{K}^{\prime }\int d{G}^{\prime }...P(K^{\prime },G^{\prime })$ and $P(K,G)$ is the joint probability distribution of in- and out-couplings. Horizontal dashed lines in both panels correspond to the critical points $K_{0c}=\pm \mathrm{\Delta }K/2$.

Figure 11

Color maps for (a) $|\mathrm{\Omega }-{\omega }_0|$ and (b) $|{\langle \dot{\mathrm{\Theta }}\rangle }_t-{\omega }_0|$ of phase-coupled Rössler oscillators. Parameters: $N={10}^3,\mathrm{\Delta }K=1.45,\mathrm{\Delta }G=1,D=0.03$, averaging over $t\in [4000,5000]$ and $dt=0.05$.

Figure 12

Spontaneous drift measured by locking frequency ${\langle \dot{\mathrm{\Theta }}\rangle }_t$ and mean ensemble frequency $\mathrm{\Omega }$. Parameters: $N={10}^4,\mathrm{\Delta }K=1.45,\mathrm{\Delta }G=1,G_0=2.5,D=0.03$, averaging over $t\in [4000,5000]$ and $dt=0.05$.

Figure 13

Order parameter $r$ and mean ensemble frequency $\mathrm{\Omega }$ as functions of $G_0$ in regions of incoherence and zero-lag synchronization for oscillators with funnel attractors. Parameters: $N={10}^3,\mathrm{\Delta }K=0.1,\mathrm{\Delta }G=1,K_0=0.5$, and $D=0.03$.