From approximate synchronization to identical synchronization in coupled systems

Abstract

3677 3678 CHIH-WEN SHIH AND JUI-PIN TSENG j∈NWe shall establish the ε-approximate synchronization of system (4) by showing lim supfor some ε ≥ 0, under some conditions. The error ε off the synchronous set S naturally depends on the difference of units, represented by δF i,k , and the structure of network which is imbedded inG i,k . For the factors from the network structure, the row sums κ i of the connection matrix, defined in (7), and coupling delays τ ij , defined in (3), play important roles. Sequential contracting is an iterative argument which can be used to conclude the asymptotic behavior of z i,k (t) satisfying (23). To analyze the difference-differential equation (23), we shall recompose the terms in H i,k so that (23) can be rewritten aṡ z i,k (t) = h i,k (x i,k (t), x i+1,k (t), t) +h i,k (x i,k (t − τ ii ), x i+1,k (t − τ ii )) + w i,k (t), (26)