Transient Uncoupling Induces Synchronization

Finding conditions that support synchronization is a fertile and active area of research with applications across multiple disciplines. Here we present and analyze a scheme for synchronizing chaotic dynamical systems by transiently uncoupling them. Specifically, systems coupled only in a fraction of their state space may synchronize even if fully coupled they do not. While for many standard systems coupling strengths need to be bounded to ensure synchrony, transient uncoupling removes this bound and thus enables synchronization in an infinite range of effective coupling strengths. The presented coupling scheme therefore opens up the possibility to induce synchrony in (biological or technical) systems whose parameters are fixed and cannot be modified continuously.

Figure 1

Synchronization depends on coupling strength. Trajectories of the driving (solid line) and driven (dashed line) unit of two coupled chaotic oscillators for (a) $\alpha =0.05$, (b) $\alpha =1.5$, and (c) $\alpha =5$ as indicated in panel (d). (d) The maximum transverse Lyapunov exponent ${\lambda }_{\max }^{\perp }$ indicates synchronization for intermediate coupling only.

Figure 2

Transient uncoupling through state-space clipping. The dynamics of two synchronized chaotic oscillators in the $x\text{−}y$ plane with $x^{*}=({\mathbf{x}}_2^{*}{)}_1=1.20$, $\mathrm{\Delta }=4.16$, and $\alpha =7.0$ (driving: solid curve; driven: dashed curve). Coupling is only active in the interval $x_2\in [x^{*}-\mathrm{\Delta },x^{*}+\mathrm{\Delta }]$ (shaded in gray).

Figure 3

Synchronization induced by transient uncoupling. Maximum transverse Lyapunov exponent for two transiently uncoupled chaotic oscillators (for parameters see text) for $\alpha =5$ and clipping with $({\mathbf{x}}_2^{*}{)}_1=1.20$. Synchronization emerges for moderate clipping, i.e., for intermediate values of ${\mathrm{\Delta }}^{\prime }$, though not without clipping (${\mathrm{\Delta }}^{\prime }=1$).

Figure 4

Transient uncoupling induces synchronization in an infinite range of coupling strengths. Trajectories of the driving (solid line) and driven (dashed line) units for (a) $\alpha =0.05$, (b) $\alpha =1.5$, and (c) $\alpha =5$, the same as in Fig. 1. The clipping is given by Eqs. (5) and (7) with $({\mathbf{x}}_2^{*}{)}_1=1.20$ and $\mathrm{\Delta }=4.16$ as in Fig. 2. (d) Maximum transverse Lyapunov exponent ${\lambda }_{\max }^{\perp }$ as a function of the coupling strength $\alpha$; note the logarithmic scale. The gray line shows ${\lambda }_{\max }^{\perp }$ for normal, unclipped coupling. With transient uncoupling, synchronization is stable for arbitrarily large coupling strengths.

Figure 5

Extended synchronization range by transient uncoupling and optimal clipping. (a)–(c) Depending on the coupling strength $\alpha$ and the percentage (${\mathrm{\Delta }}^{\prime }=2\mathrm{\Delta }/\mathrm{\Omega }$) of the state space where coupling is active, the system may or may not synchronize. The dark area marks the parameters where the synchronized state is stable, i.e., ${\lambda }_{\max }^{\perp }<0$. Clipping is (a) in $x$ direction [$x^{*}=({\mathbf{x}}_2^{*}{)}_1=1.2$], (b) in $y$ direction [$y^{*}=({\mathbf{x}}_2^{*}{)}_2=-1.5$], and (c) in the direction $y\approx 3.1x$ (${\theta }^{*}=0.4\pi$). Accordingly, the direction of clipping can be optimized to achieve the largest possible clipping range. (d) Effectiveness $S(\theta )$ [Eq. (8)] of clipping along the direction $\theta$ for fixed $\alpha =10$.

Figure 6

Multiple switches between stability and instability depending on the coupling location. The color of the points indicates the maximum transverse Lyapunov exponent of the system with clipping to sets $A=A_{{\mathbf{x}}_2^{*},r}$ [Eq. (11)], indicating stable synchronization (light) and no synchronization (dark) depending on the coupling location on the attractor. Parameters are $\alpha =5$ and $f=0.05$.