Emergent hybrid synchronization in coupled chaotic systems

We evidence an interesting kind of hybrid synchronization in coupled chaotic systems where complete synchronization is restricted to only a subset of variables of two systems while other subset of variables may be in a phase synchronized state or desynchronized. Such hybrid synchronization is a generic emergent feature of coupled systems when a controller based coupling, designed by the Lyapunov function stability, is first engineered to induce complete synchronization in the identical case, and then a large parameter mismatch is introduced. We distinguish between two different hybrid synchronization regimes that emerge with parameter perturbation. The first, called hard hybrid synchronization, occurs when the coupled systems display global phase synchronization, while the second, called soft hybrid synchronization, corresponds to a situation where, instead, the global synchronization feature no longer exists. We verify the existence of both classes of hybrid synchronization in numerical examples of the Rössler system, a Lorenz-like system, and also in electronic experiment.

Figure 1

Degradation of CS for a parameter detuning in two coupled Rössler oscillators. $b=0.22$, $c=0.2$, and $d_1=12$.

Figure 2

Mean frequency difference $\Delta \Omega$ (see text for definition) as a function of $d_2$ and $b_{1,2}=b$.

Figure 3

The first and second largest Lyapunov exponents (dotted blue and dotted black line, respectively) and the mean frequency difference (red or dark gray line) of the coupled Rössler systems are plotted as a function of the parameter $d_2$. $b=0.24$. Inset: Plot of the SLE (dotted green line) and the frequency difference (red or dark gray line) in the proximity of the transition from HHS to SHS.

Figure 4

Mutually coupled LLS: three LLE and the mean frequency differences $\Delta \Omega$ are plotted as a function of the parameter $C_2$.

Figure 5

Coupled Rössler circuit: All Op-amps are LM348 including integrators ($U1,U3,U4$) and ($U5,U7,U8$). Multipliers AD633 ($A1,A2$), resistances ($R=\text{100}\text{k}\Omega$, $R_a=10\text{k}\Omega$), and capacitances (100 nF) are used to design the OS1 and OS2. The coupling circuit is composed of summing amplifiers, inverters, and resistances. Power supply is $\pm 12\mathrm{V}$. Components values are noted in the circuit ($1\%$ tolerance).

Figure 6

Oscilloscope pictures (Rössler): 2D attractors of oscillator 1 measured at X1 vs X2 for different values of resistance $R_d$ (in $\text{k}\Omega$) are 50, 40, 35.2, and 33 in the first row and 31.7, 28.3, 21.6, and 14.5 in the second row, respectively. Third row: Output voltage measured at X1 vs Y1 shown in the first panel and X2 vs Y2 shown in the second panel; both confirm CS. Output voltage measured at X3(0.2V/div) vs Y3(2V/div) shown in the third panel (HHS) and X3 vs Y3 shown in the fourth panel (SHS).