Intermingled basins in coupled Lorenz systems

We consider a system of two identical linearly coupled Lorenz oscillators presenting synchronization of chaotic motion for a specified range of the coupling strength. We verify the existence of global synchronization and antisynchronization attractors with intermingled basins of attraction such that the basin of one attractor is riddled with holes belonging to the basin of the other attractor and vice versa. We investigated this phenomenon by verifying the fulfillment of the mathematical requirements for intermingled basins and obtained scaling laws that characterize quantitatively the riddling of both basins in this system.

Figure 1

Section at $x_1=x_2=y_1=y_2=1.0$ of the basins of synchronization (white pixels) and antisynchronization (black pixels) attractors of the coupled Lorenz system, for $\varepsilon =$ (a) $1.0$, (b) 2.0, (c) 2.5, and (d) 2.8.

Figure 2

Section at $z_1=z_2=22.0$ and (a) $y_1=y_2=1.0$, $x_1,x_2$ random in $[-1,3)$, (b) $x_1=x_2=1.0$, $y_1,y_2$ random in $[-1,3)$ of the basins of synchronization (white pixels) and antisynchronization (black pixels) attractors of the coupled Lorenz system for $\varepsilon =2.0$.

Figure 3

Trajectories of the coupled Lorenz system for the same initial condition ($x_1=y_1=z_1=1.0$ and $x_2=y_2=z_2=0.5$) up to $t=100$ and different coupling values: (a) $\varepsilon =0.5$, (b) $1.0$. In each case, the time evolution of the differences of coordinates are also shown (c), (d) for $\varepsilon =0.5$ and (e), (f) for $\varepsilon =1.0$.

Figure 4

Difference between the coordinates (a) $x_1-x_2$, (b) $y_1-y_2$, (c) $z_1-z_2$ of two coupled Lorenz systems at time $t={10}^3$, as a function of the coupling strength. One hundred initial conditions were randomly chosen (as in Fig. 1) for each value of $\varepsilon$ (varied in steps of 0.01).

Figure 5

(a) Fraction of initial conditions (over a total of ${10}^4$) yielding trajectories asymptoting synchronized $f_s$ or antisynchronized $f_a$ states as a function of time for $\varepsilon =2.0$. We also indicated the fraction of trajectories reaching either state ($f_s+f_a$) or none of them ($1-f_s+f_a$). (b) Fraction of initial conditions not reaching these states for different values of the coupling strength.

Figure 6

The three largest Lyapunov exponents of the coupled Lorenz system as a function of the coupling strength. Gray symbols correspond to the algorithm by Wolf et al., whereas the thick black curve is the result of the variational equations (6). (Inset) The largest exponent for a wider range of $\varepsilon$.

Figure 7

Probability density functions of the time-30 largest transversal Lyapunov exponent for different values of $\varepsilon$. Initial conditions were taken as in Fig. 1. The full lines are Gaussian fits.

Figure 8

Positive fraction of the largest time-$t$ transversal Lyapunov exponent as a function of the coupling strength for different values of $t$.

Figure 9

(a) Largest transversal Lyapunov exponent, as a function of $\varepsilon$, for the particular unstable periodic orbits (UPOs) embedded in the Lorenz chaotic attractor (to period 5), indicated on the figure by means of the sequence of symbols A, B denoting the turns around each unstable fixed points C${}^{+}$ and C${}^{-}$ of the Lorenz system. The curve for typical chaotic trajectories in the attractor is also shown. (b) Magnification of panel (a). (c) Stability intervals for each UPO in order of increasing stability at $\varepsilon =0$ from top to bottom: stable (full segment) and unstable (dotted). The vertical lines indicate ${\varepsilon }_1$ and ${\varepsilon }_2$. The symbols delimiting the stability intervals correspond to $\mu =1$ (full) and $-1$ (hollow).

Figure 10

(a) Schematic figure showing the structure of riddled basins near the invariant subspace that contains a chaotic attractor. (b) Fraction of trajectories $P_{☆}$ that asymptote to the antisynchronized state as a function of the distance $l=2\left|z\right|$ to the synchronized state for different values of the coupling strength $\varepsilon$. The full lines are least-squares fits. (c) $P_{☆}$ for $\varepsilon =1.5$ and different orientations of the deviation $l$.

Figure 11

Scaling exponent $\eta$ for the fraction of trajectories that asymptote to the antisynchronized state obtained by a numerical experiment (solid circles). For comparison, the theoretical values, given by $\eta =|{\lambda }_{\perp }|/D$, are also plotted (open squares).

Figure 12

Time decay of the variance of the finite-time largest transversal Lyapunov exponent for different values of the coupling strength $\varepsilon$. The lines are least-squares fits of the function $f\left(t\right)=2D/t$ to the numerical points, for large $t$, allowing estimation of the diffusion coefficient $D$.

Figure 13

(a) Schematic figure showing the numerical determination of the uncertainty fraction. (b) Fraction of uncertain initial conditions as a function of the uncertainty level for different values of the coupling strength. The solid lines are least-squares fits.