Topological synchronization of chaotic systems

Abstract

A chaotic dynamics is typically characterized by the emergence of strange attractors with their fractal or multifractal structure. On the other hand, chaotic synchronization is a unique emergent self-organization phenomenon in nature. Classically, synchronization was characterized in terms of macroscopic parameters, such as the spectrum of Lyapunov exponents. Recently, however, we attempted a microscopic description of synchronization, called topological synchronization, and showed that chaotic synchronization is, in fact, a continuous process that starts in low-density areas of the attractor. Here we analyze the relation between the two emergent phenomena by shifting the descriptive level of topological synchronization to account for the multifractal nature of the visited attractors. Namely, we measure the generalized dimension of the system and monitor how it changes while increasing the coupling strength. We show that during the gradual process of topological adjustment in phase space, the multifractal structures of each strange attractor of the two coupled oscillators continuously converge, taking a similar form, until complete topological synchronization ensues. According to our results, chaotic synchronization has a specific trait in various systems, from continuous systems and discrete maps to high dimensional systems: synchronization initiates from the sparse areas of the attractor, and it creates what we termed as the 'zipper effect', a distinctive pattern in the multifractal structure of the system that reveals the microscopic buildup of the synchronization process. Topological synchronization offers, therefore, a more detailed microscopic description of chaotic synchronization and reveals new information about the process even in cases of high mismatch parameters.

Figure 1

Generalized fractal dimension for slightly mismatched Rössler systems. General dimension $D_q$ as a function of parameter q for the master (black) and slave for coupling strengths $\sigma =0.07$ (blue solid) and $\sigma =0.12$ (red dashed). Insets are blow-ups for the master and slave curves for $\sigma =0.12$ in the $q<0$ (bottom left) and $q>0$ (top right) regions. Topological synchronization occurs as the $D_q$ curve of the slave matches the $D_q$ curve of the master.

Figure 2

Microscopic build-up of synchronization for the Logistic map system. (a) Synchronization error E as function of coupling strength k. Complete synchronization $E\sim 0$ is reached around $k\sim 0.9$. (b)–(e) Topological synchronization and the zipper effect. General dimension $D_q$ as function of the parameter q of master (blue) and slave (red) attractors. As the coupling k increases a zipper effect from the negative ($q\le 0$) to the positive ($q>0$) part of $D_q$ is seen.

Figure 3

Distance between the master and slave’s general dimensions for the mismatched Logistic map systems. Upper panel, color map denoting the distance between the $D_q$ curves of the master and the slave, $\mathrm{\Delta }{D}_q$ as a function of the parameter q (y axis) and the coupling strength k (x axis). Vertical dashed lines show the negative and positive zipper effect regions (as $\mathrm{\Delta }{D}_q$ goes to zero). Bottom panel, distance between the master and slave $D_q$ curves calculated as the norm between them, $||\mathrm{\Delta }{D}_q||$ as function of k in the $q\le 0$ (red dashed curve), $q>0$ (yellow dot-dashed curve) intervals and in the whole range of q (blue solid line). At $k\sim 0.33$, $D_{q<0}$ of the slave has completed its synchronization with the master and its $D_{q>0}$ starts a gradual approach to the master curve. The zipper effect is completed around $k\sim 0.9$. Vertical dashed lines mark the negative and positive zipper effect regions.

Figure 4

Microscopic build-up of synchronization for high mismatched Rössler systems. (a) Synchronization error E as function of coupling strength $\mathrm{\sigma }$. We obtain a window of approximate complete synchronization, in which $E\ll 1$, between ${\mathrm{\sigma }}_{\mathrm{CS}}^1\sim 3$ and ${\mathrm{\sigma }}_{\mathrm{CS}}^2\sim 7$. (b)–(d) Topological synchronization and the zipper effect. General dimension $D_q$ as function of the parameter q of master (blue) and slave (red) attractors. As coupling $\mathrm{\sigma }$ increases a zipper effect from the negative to the positive part of $D_q$ can be seen. (c) At low couplings, the slaves’ negative $D_q$ part syncs and reaches a minimal stable distance from the master. (d) The positive $D_q$ part synchronizes only after the negative part has completed to synchronize and reaches a stable minimal distance from the master as well.

Figure 5

Distance between the general dimension of master and slave for high mismatched Rössler systems. Upper panel, color map denoting the normalized distance between the $D_q$ curves of the master and the slave, $\mathrm{\Delta }{D}_q^{\ast }$ as a function of the parameter q (y axis) and the coupling strength $\mathrm{\sigma }$ (x axis). Vertical dashed lines show the negative and positive zipper effect regions (as distance decreasing to a minimum fixed value). Bottom panel, the normalized distance between the $D_q$ curves of the master and the slave calculated as the norm between them, $||\mathrm{\Delta }{D}_q^{\ast }||$ as a function of $\mathrm{\sigma }$ in the $q\le 0$ (red dashed curve), $q>0$ (yellow dot-dashed curve) intervals and in the whole range of q (solid blue line). At $\mathrm{\sigma }\sim 3$, the negative part of $D_q$ has completed its synchronization, and the positive part starts a gradual approach to the master curve. The zipper effect completes around $\mathrm{\sigma }\sim 6$. At $\mathrm{\sigma }\sim 7$ the system exits the synchronization window with a reverse zipper effect. The positive part of $D_q$ gradually separates from the master curve (and thus the distance from the master increases) while the negative part of $D_q$ remains with the same close distance to the master. Vertical dashed lines show the negative and positive zipper effect regions.

Figure 6

Distance between the general dimension of master and slave for high dimensional Mackey–Glass system. The normalized distance between the $D_q$ curves of the master and the slave calculated as the norm between them, $||\mathrm{\Delta }{D}_q^{\ast }||$ as a function of $\mathrm{\sigma }$ in the $q\le 0$ (red triangles), $q>0$ (yellow rectangles) intervals and in the whole range of q (blue circles) and the synchronization error parameter E multiplied by 10 (purple dashed line). At $\mathrm{\sigma }\sim 0.5$, the negative part of $D_q$ has completed its synchronization, and the positive part starts to sync.