Chaotic chimera attractors in a triangular network of identical oscillators

A prominent type of collective dynamics in networks of coupled oscillators is the coexistence of coherently and incoherently oscillating domains known as chimera states. Chimera states exhibit various macroscopic dynamics with different motions of the Kuramoto order parameter. Stationary, periodic and quasiperiodic chimeras are known to occur in two-population networks of identical phase oscillators. In a three-population network of identical Kuramoto-Sakaguchi phase oscillators, stationary and periodic symmetric chimeras were previously studied on a reduced manifold in which two populations behaved identically [Phys. Rev. E 82, 016216 (2010)]. In this paper, we study the full phase space dynamics of such three-population networks. We demonstrate the existence of macroscopic chaotic chimera attractors that exhibit aperiodic antiphase dynamics of the order parameters. We observe these chaotic chimera states in both finite-sized systems and the thermodynamic limit outside the Ott-Antonsen manifold. The chaotic chimera states coexist with a stable chimera solution on the Ott-Antonsen manifold that displays periodic antiphase oscillation of the two incoherent populations and with a symmetric stationary chimera solution, resulting in tristability of chimera states. Of these three coexisting chimera states, only the symmetric stationary chimera solution exists in the symmetry-reduced manifold.

Figure 1

Top: Bifurcation diagram of DSD-type chimera states. Bottom: Enlargement of the gray box in the top panel. Dashed and solid lines indicate unstable and stable curves, respectively. Black: symmetric stationary DSD. Red: asymmetric stationary $\text{D}\text{S}{\text{D}}^{\prime }$. Green: symmetric and asymmetric breathing chimeras. Blue: antiphase $\text{D}\text{S}{\text{D}}^{\prime }$ chimera states.

Figure 2

(a) Radial variables of the 6D OA dynamics (red: first; blue: third population; and orange: second population) and (b) Lyapunov exponents along the antiphase chimera trajectory. (c) Radial variables of the 9D WS dynamics with uniform constants of motion and $N=20$ and (d) and the corresponding Lyapunov exponents. In subfigures: $A=0.35$ and all are measured after $t>{10}^5$.

Figure 3

Poincaré section of 9D WS dynamics with uniform constants of motion and ${\mathrm{\Psi }}_1\equiv 2\pi$ for (a) $N=30$ and (b) $N=100$. (c) The largest and the second largest LEs as a function of $N$ for 9D WS dynamics. (d) Real (orange) and imaginary (gray) parts of ${\gamma }_1$ as a function of time after $t>{10}^4$. All the results are obtained with uniform constants of motion of WS dynamics, and $A=0.35$.

Figure 4

(a) Moduli of the Kuramoto order parameters of the three populations for the microscopic dynamics (1) with $N=40$ after $t>{10}^5$. (b) Radial variables of the WS dynamics (6) with $N=40$ after $t>{10}^5$. (c) The largest and the second largest LEs as a function of $N$. The 9D WS dynamics is obtained with nonuniform constants of motion and the microscopic dynamics from a random initial condition. The parameter used here is $A=0.3$.

Figure 5

The number of dynamical states at $t=20000$ starting from 200 random initial conditions for different values of $A$. (a) 6D Ott-Antonsen dynamics. (b) 9D Watanabe-Strogatz macroscopic dynamics with $N=80$ and uniform constants of motion. (c) $3N$-dimensional microscopic dynamics with $N=80$.

Figure 6

Stationary SDS chimeras in the 6D OA dynamics. (a) Time evolution of the radial variables of 6D OA dynamics after a transient time of ${10}^5$ units for $A=0.35$. (b) The eigenvalues of the Jacobian matrix evaluated at the stationary SDS chimera fixed point solution shown in (a) in the complex plane. Red circles indicate the eigenvalues in the 6D OA dynamics, and the blue squares those in the 2D symmetry-reduced manifold. (c) Bifurcation diagram of SDS chimeras. The states are born in a limit point bifurcation (LP). The red dashed and solid curves indicate the location of unstable and stable stationary SDS chimeras, respectively. The green curve shows the maxima of the radial variable of a breathing chimera state emerging in a supercritical Hopf bifurcation (HB).

Figure 7

Stationary SDS chimera states in the 9D Watanabe-Strogatz dynamics. (a), (b) Time evolution of the radial macroscopic variables after a transient time of ${10}^5$ units for $A=0.35$. Blue and red lines indicate ${\rho }_1\left(t\right)={\rho }_3\left(t\right)=1$ and the orange line ${\rho }_2\left(t\right)<1$. The black line shows the modulus of Kuramoto order parameter $r_2\left(t\right)$ calculated from the relation between Kuramoto order parameter and the WS variables with uniformly distributed constants of motion: (a) $N=40$ and (b) $N=10$. The inset of (b) shows the radial variables and Kuramoto order parameter with nonuniform constants of motion with $N=10$. (c) Instantaneous phase velocities obtained from Eq. (A6) for $N=10$ with uniform constants of motion. (d) Period of the modulus of the Kuramoto order parameter as a function of system size $N$. The red circles are numerically obtained periods and the black solid line is the curve $\frac{2\pi }{|\tilde{\mathrm{\Omega }}|N}$. (e) Lyapunov exponents of the 9D macroscopic variables of the WS dynamics.

Figure 8

Stationary SDS chimera states in $3N$-dimensional microscopic dynamics. (a), (b) Time series of the moduli of the Kuramoto order parameters of the three populations with (a) PIC and (b) n-PIC for $N=10$ and $A=0.35$ after a transient time of ${10}^5$ units. (c) Lyapunov exponents of the SDS chimeras initiated from a PIC for $N=20$ and $A=0.35$.

Figure 9

Unstable stationary DSD chimera state in 6D OA dynamics. (a) Radial variables ${\rho }_2\left(t\right)=1$ (orange line), ${\rho }_1\left(t\right)={\rho }_3\left(t\right)={\rho }_0<1$ (red and blue lines) as a function of time for $A=0.55$. (b) Time series of the angular variables with the same color scheme as used in (a). In (a) and (b) the first ${10}^5$ time units were discarded. (c) Eigenvalues of the Jacobian matrix evaluated at the DSD solution in the rotating reference frame.

Figure 10

Unstable stationary DSD chimera states in 9D WS dynamics. (a)–(c) Time evolution of the 9D Watanabe-Strogatz macroscopic variables after a transient time of ${10}^5$ units with the same color scheme in Fig. 9. (d) Time evolution of the radial variables with a small perturbation on the specific initial condition. $A=0.55$ and $N=40$.

Figure 11

(a) Lyapunov exponents of the 9D WS dynamics for $N=40$ corresponding to Figs. 10–10. (b) Lyapunov exponents of the $3N$-dimensional microscopic dynamics for $N=40$. (c)–(d) Time evolution of the moduli of the Kuramoto order parameters for the $3N$-dimensional microscopic dynamics with $N=40$ starting from a PIC. $A=0.55$.