Article Contents
Article Contents

# From approximate synchronization to identical synchronization in coupled systems

• * Corresponding author: Chih-Wen Shih

The authors are supported in part by the Ministry of Science and Technology of Taiwan.

• We establish a framework to investigate approximate synchronization of coupled systems under general coupling schemes. The units comprising the coupled systems may be nonidentical and the coupling functions are nonlinear with delays. Both delay-dependent and delay-independent criteria for approximate synchronization are derived, based on an approach termed sequential contracting. It is explored and elucidated that the synchronization error, the distance between the asymptotic state and the synchronous set, decreases with decreasing difference between subsystems, difference between the row sums of connection matrix, and difference of coupling time delays between different units. This error vanishes when these factors decay to zero, and approximate synchronization becomes identical synchronization for the coupled system comprising identical subsystems and connection matrix with identical row sums, and with identical coupling delays. The application of the present theory to nonlinearly coupled heterogeneous FitzHugh-Nagumo neurons is illustrated. We extend the analysis to study approximate synchronization and asymptotic synchronization for coupled Lorenz systems and show that for some coupling schemes, the synchronization error decreases as the coupling strength increases, whereas in another case, the error remains at a substantial level for large coupling strength.

Mathematics Subject Classification: Primary: 34D06, 92B25; Secondary: 34D05, 34C15.

 Citation:

• Figure 1.  Plot of $(u_1(t),v_1(t))$ for the solution $(u_i(t),v_i(t))_{i = 1, 2, 3}$ of system (69), starting from $(1.3,-1.2,1.5,-1.5,1.4,-1.3)$ at $t_0 = 0$, and evolution of $Err(t): = \max_{1\leq i \leq2} \{ \max\{|u_{i}(t)-u_{i+1}(t)|, |v_{i}(t)-v_{i+1}(t)|\}\}$, for various $(\delta^\ast_{F},\delta^\ast_\kappa,\delta^\ast_\tau)$ in Example 1

Figure 2.  Evolutions of $Err(t): = \max_{1\leq k \leq 3}\{|x_{1,k}(t)-x_{2,k}(t)|\}$ for the solutions $(x_{i,j}(t))$ of system (99) with $\vartheta = 1$, and (a) $c = 64$, (b) $c = 640$, (c) $c = 6400$, in Example 2, starting from $(14,12,-23,15,13,-22)$ at $t_0 = 0$

Figure 3.  Evolutions of $Err(t): = \max_{1\leq k \leq 3}\{|x_{1,k}(t)-x_{2,k}(t)|\}$ for the solutions $(x_{i,j}(t))$ of system (114) with $\vartheta = 1$ and (a) $c = 3014$, (b) $c = 6028$, and (c) $c = 12056$, starting from $(14,12,-23,15,13,-22)$ at $t_0 = 0$, in Example 3

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