Enhancing master-slave synchronization: The effect of using a dynamic coupling

This paper introduces a modified master-slave synchronization scheme for dynamical systems. In contrast to the standard configuration, the slave system does not receive any driving signal from the master, but rather the interaction is through a linear dynamical system. The key feature of the proposed coupling scheme is that it induces synchronization in certain systems that cannot be synchronized when using the classical static interconnection. Likewise, the dynamic coupling achieves synchronization for arbitrarily large coupling strength values in certain systems for which the classical configuration is applicable only within a narrow interval of coupling strength values. The performance of the synchronization scheme is illustrated in pairs of identical chaotic and mechanical oscillators.

Figure 1

Proposed dynamic master-slave configuration.

Figure 2

The largest transverse Lyapunov exponent ${\lambda }_{max}^{\perp }$ computed as a function of the coupling strength $k$. The solid line indicates ${\lambda }_{max}^{\perp }$ for the proposed coupling scheme, whereas the dashed line shows ${\lambda }_{max}^{\perp }$ for the standard master-slave configuration. Clearly, the proposed configuration induces synchronization in the slave system for $k>8.4$, whereas for the standard configuration ${\lambda }_{max}^{\perp }$ stays positive for all $k>0$ and consequently, the synchronous solution is always unstable.

Figure 3

${\lambda }_{max}^{\perp }$ computed as a function of the coupling strength $k$ (on the logarithmic scale). The proposed configuration successfully extends the onset of synchronization beyond the narrow interval of the standard configuration. The solid and dashed lines have the same meaning as in Fig. 2.

Figure 4

Representative time series for two coupled Rössler systems for different coupling strengths. Left: Proposed coupling scheme. Right: Classical master-slave configuration. For (a) and (b): $k=5$. For (c) and (d): $k=1000$. The proposed coupling scheme induces synchronization, whereas the classical master-slave approach fails to synchronize them. $x_{1m}$: solid; $x_{1s}$: dashed.

Figure 5

Largest transverse Lyapunov exponent ${\lambda }_{max}^{\perp }$ as a function of the coupling strength $k$. The dashed line: ${\lambda }_{max}^{\perp }$ when $h_1$ is used as the coupling signal. The solid line: ${\lambda }_{max}^{\perp }$ when the coupling signal is $h_2$. In both cases, synchronization is achieved for large $k$. However, a smaller coupling strength is required when the coupling signal $h_2$ is used.

Figure 6

Largest transverse Lyapunov exponent on the $({\gamma }_1,k)$ plane. There exist several values of ${\gamma }_1$ for which synchronization can be observed besides the choice given in Eq. (31).

Figure 7

Largest transverse Lyapunov exponent ${\lambda }_{max}^{\perp }$ for different dynamical systems computed as a function of the coupling strength $k$ (on the logarithmic scale). (a) Chua's circuit. (b) The Hindmarsh-Rose neuronal model. (c) The forced van der Pol system. (d) The forced Duffing equation. The details for each system are given in Table 1. In all panels, the dashed lines show ${\lambda }_{max}^{\perp }$ for the traditional (static) configuration, whereas the solid lines denote ${\lambda }_{max}^{\perp }$ for the proposed configuration. In all cases, the proposed configuration extends the onset of synchronization beyond the narrow interval observed in the classical configuration.

Figure 8

${\lambda }_{max}^{\perp }$ as a function of $k$ for the first order dynamic master-slave configuration (43, 44, 45). Synchronization can possibly be achieved in the interval $-0.92<k<-0.37$. Compare to Fig. 2 (solid line).