By Chai Wah Wu
This publication brings jointly rising study parts: synchronization in coupled nonlinear structures and complicated networks, and research stipulations less than which a fancy community of dynamical structures synchronizes. whereas there are numerous texts that learn synchronization in chaotic platforms or houses of complicated networks, there are few texts that think of the intersection of those very energetic and interdisciplinary learn components. the most subject of this ebook is that synchronization stipulations may be relating to graph theoretical homes of the underlying coupling topology. The publication introduces principles from structures conception, linear algebra and graph thought and the synergy among them which are essential to derive synchronization stipulations. a number of the effects, that have been received relatively lately and feature earlier now not seemed in textbook shape, are provided with whole proofs. this article is appropriate for graduate-level research or for researchers who wish to be higher accustomed to the newest examine during this region.
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Extra resources for Synchronization in Complex Networks of Nonlinear Dynamical Systems
1) synchronizes if the following two conditions are satisfied: (1) limt→∞ ui − uj = 0 for all i, j, (2) There exists an n by n matrix U ∈ Ws such that (U ⊗ V )(G(t) ⊗ D(t) − I ⊗ Y (t)) 0 for all t. Proof: We prove this via Lyapunov’s second method. Construct the Lyapunov function g(x) = 12 xT (U ⊗ V )x.
2 Schematic diagram of a neural network. Transportation networks An excellent example of a man-made network is the transportation network, for instance a network of roads connecting cities, or the network of (passenger or freight) airplane flights between cities. Analyzing and understanding such networks is of great utility to modern society and several problems regarding such networks are notoriously hard. For instance, in the Traveling Salesman Problem (TSP) which is NP-complete, the goal is to find the shortest path which starts at one city, visits all cities once and return to the starting city.
However, for non-regular and the eigenvalues of L and L ˜ and L will have different properties. 3 this means that λ2 (L) ≤ 2n n 2 r(L) ≤ n 2 = 8(n−1) kn2 8 k(n+1) 8 n 8 1 n− n , n even 8 ≤ , n odd n− 1 n 8 , n even ≤ , n odd k(n + 1) The reason for studying the two cases is that C2k gives better bounds 1 when k grows slower than n 3 and the construction in [Atay et al. (2006)] 1 gives better bounds when k grows faster than n 3 . In either case, λ2 , r → 0 as n → ∞. 1). 37. In conclusion, whereas λ2 and r → 0 as n → ∞ for Construction 1, they remain bounded from below for Construction 2.
Synchronization in Complex Networks of Nonlinear Dynamical Systems by Chai Wah Wu