Partial synchronization on a network with different classes of oscillators

Complete and partial synchronization have been largely studied on networks of identical coupled oscillators. However, we study a network in which not all oscillators when uncoupled show the same dynamics and nonetheless the network shows partial synchronization. Our system is composed by four Rössler oscillators diffusively coupled in a ring. Oscillators 1 and 3 are identical, as 2 and 4 are also. In short, the network is said to be composed of different classes of oscillators (in our example, two classes with two oscillators each). Primary synchronization is defined as the case when all oscillators on the same class are identically synchronized, for all classes. Secondary synchronization is related to the other possible cases of partial synchronization. Both are achieved for the system we have chosen, shown by means of direct integration and transverse Lyapunov exponent computation. Furthermore, evidence of riddled basins of attraction is presented.

Figure 1

The Lyapunov exponents related to the ABAB pattern. Both exponents have the same value at $c=0$ and the transverse Lyapunov exponent decreases as an approximately affine function of $c$. For $c\gtrsim 0.055$, the system achieves primary synchronization.

Figure 2

The Lyapunov exponents related to the ABCB pattern. The transverse Lyapunov exponent becomes negative for nearly the same range of $c$ on which the system achieves primary synchronization. In this case, the ABCB pattern is degenerated to ABAB.

Figure 3

The Lyapunov exponents related to the ABAC pattern. Secondary synchronization is finally achieved for the range $0.042\lesssim c\lesssim 0.055$. The existence of two basins of attraction appears for $c\gtrsim 0.055$ between ABAC and ABAB patterns.

Figure 4

The mean distance of the system dynamics from the synchronization manifolds is calculated varying $c$ for random initial conditions. The region of coexistence of ABAB and ABAC patterns is easily identified. The complete absence of ABCB pattern confirms our early predictions.

Figure 5

Evidence of riddled basins of attraction at $c=0.62$. Black points represent initial conditions leading to ABAC pattern, while white points lead to ABAB pattern. ${\Delta }_{\parallel }$ $\left({\Delta }_{\perp }\right)$ is the parallel (perpendicular) component to the ABAB manifold.