By Guanrong Chen, Xiaofan Wang, Xiang Li, Jinhu Lü (auth.), Kyandoghere Kyamakya, Wolfgang A. Halang, Herwig Unger, Jean Chamberlain Chedjou, Nikolai F. Rulkov, Zhong Li (eds.)

The chosen contributions of this booklet make clear a chain of attention-grabbing features concerning nonlinear dynamics and synchronization with the purpose of demonstrating a few of their attention-grabbing purposes in a chain of chosen disciplines. This e-book comprises 13th chapters that are equipped round 5 major components. the 1st half (containing 5 chapters) does specialise in theoretical features and up to date traits of nonlinear dynamics and synchronization. the second one half (two chapters) offers a few modeling and simulation matters via concrete software examples. The 3rd half (two chapters) is targeted at the program of nonlinear dynamics and synchronization in transportation. The fourth half (two chapters) provides a few purposes of synchronization in security-related approach innovations. The 5th half (two chapters) considers extra functions parts, i.e. development reputation and conversation engineering.

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**Example text**

O. Hongler necessarily connected nor undirected but with ak,j (t) > 0 for all t as entries of its adjacency matrix) lie on the left side of the C plan. Theorem 3 (Gerxgorin (1931)). Let L be a N × N matrix (with elements in R or C). If λ is an eigenvalue of L, then there exist k such that N |λ − lk,k | |lk,j |, j=k that is, all eigenvalues of A are in the union of the discs N Dk := λ | |λ − lk,k | |lk,j | . j=k For undirected N (t), L(t) is symmetric for all t and by the spectral theorem in R all its eigenvalues are real.

Proof. As before, linear analysis implies: ⎛ ⎞ ⎛ ⎞⎛ ⎞ ρ˙ −2 [η] [μ] O O ρ ⎝ τ˙ ⎠ = ⎝ O [K] L −Id ⎠ ⎝ τ ⎠ ˙ O −[K] L O (22) where [K] is the diagonal matrix with entries K1 , K2 , . . , KN . Again, as in 1 1 Theorem 1, radial perturbations decay exponentially. Let L(K) := K 2 L K 2 and consider S an orthogonal matrix such that S L(K)S = Λ(K), a diagonal matrix. The matrix L(K) is a left and right multiplication of L by an identical 1 positive deﬁnite diagonal matrix K 2 , hence the sign of its spectrum coincides with L, namely λk (K) < 0 for k = 2, .

Iii) Coupling dynamics, Cxk (x1 , . . , yN , Γ ) and Cyk (x1 , . . 3. 4. 1 Local Dynamics: Mixed Canonical-Dissipative systems N We introduce a collection of MCD oscillators {Ok }k=1 with dynamics: Ok ⎧ ⎪ ⎨ x˙ k = ⎪ ⎩ y˙ = k ωk ∂Hk ∂Hk (xk ,yk ) + ηk gk (Hk (xk ,yk ); μk ) (xk ,yk ), ∂yk ∂xk −ω k ∂Hk ∂Hk (xk ,yk ), (xk ,yk ) + ηk gk (Hk (xk ,yk ); μk ) ∂yk ∂xk canonical evolution (10) dissipative evolution where Hk : R2 −→ R+ and gk : R+ −→ R. e. e. energy). From now on, we assume that given μk > 0, Hk (x,y) = μk uniquely deﬁnes a set of closed, concentric curves Lk (μk ) in R2 that surround the origin.