Transition to weak generalized synchrony in chaotically driven flows

We study regimes of strong and weak generalized synchronization in chaotically forced nonlinear flows. The transition between these dynamical states can occur via a number of different routes, and here we examine the onset of weak generalized synchrony through intermittency and blowout bifurcations. The quantitative characterization of this dynamical transition is facilitated by measures that have been developed for the study of strange nonchaotic motion. Weak and strong generalized synchronous motion show contrasting sensitivity to parametric variation and have distinct distributions of finite-time Lyapunov exponents.

Figure 1

Schematic phase diagram of driven Duffing oscillator Eq. (5) in the parameter space: white corresponds to regions of GS where dynamics are stabilized and black corresponds to regions of asynchrony where dynamics is chaotic.

Figure 2

(Color online) Variation of Lyapunov exponents of the drive Eq. (3) (black) and response Eq. (5) [red (dark gray)] along the line $h=0.2$. MS indicates a small window of multistability, where there is more than one attractor, with different Lyapunov exponents.

Figure 3

The largest Lyapunov exponent (a) ${\lambda }_d$ of the driving Rössler oscillator Eq. (3) and (b) ${\lambda }_r$ of response Duffing oscillator Eq. (5). (c) Variance of ${\lambda }_r$ evaluated from ensembles of initial conditions. In (c) wGS and sGS corresponds to regime of weak and strong generalized synchronization, and $c_I$ indicates the bifurcation point in the drive and response systems, respectively.

Figure 4

Projection of the dynamics of the response Duffing oscillator Eq. (5) on Poincaré section of $\left(x,\theta \text{mod}2\pi \right)$ plane. (a) Intermittent limit set at $c=5.185$ in the regime of weak GS. (b) Smooth limit set $c=5.1851$ in the regime of strong GS.

Figure 5

Variation of (a) Lyapunov exponent ${\lambda }_r$ of response Eq. (7) (b) Variance of Lyapunov exponents ${\lambda }_r$ with $\kappa$ (${\kappa }_b$ is the point of blowout bifurcation). (c) Projection of trajectory on the stroboscopic section at $\kappa =4.5$ showing a nonsmooth limit set. (d) Trajectory of $y_n$ on the nonsmooth limit set showing on-off intermittency.

Figure 6

(Color online) Strange limit sets of the forced Duffing oscillator, Eq. (5) in the regime of (a) weak GS at $c=9.2\left({\lambda }_r\approx -0.0068\right)$ and (b) asynchrony at $c=9.79\left({\lambda }_r\approx 0.0066\right)$. (c) Trajectories at $c=9.2$ (weak GS) starting from different initial conditions synchronize after some initial transients. (d) At $c=9.79$ (asynchrony), two nearby trajectories get desynchronize.

Figure 7

(Color online) Parameter sensitivity exponents (left panel) and corresponding distribution of finite-time Lyapunov exponents (FTLEs) (right panel) of response dynamics for Eq. (5) in (a)–(d) and Eq. (7) in (e) and (f): (a) Strong GS [red (dark gray) line] and weak GS (black line); (b) strong GS (black line) and weak GS [red (dark gray)] line; (c) asynchrony [red (dark gray) line] and weak GS (black line); (d) asynchrony [red (dark gray) line] and weak GS (black line); (e) weak GS due to blowout bifurcation and its (f) FTLEs distribution.