# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Synchronization in complex dynamical networks with nonsymmetric coupling. (English) Zbl 1178.34056

Subject of the paper are networks of coupled systems of the form

${x}_{i}^{\text{'}}\left(t\right)=f\left({x}_{i}\left(t\right)\right)+\sum _{j=1}^{N}{g}_{ij}A{x}_{j}\left(t\right),\phantom{\rule{1.em}{0ex}}i=1,\cdots ,N,$

where ${x}_{i}\left(t\right)\in {ℝ}^{n}$ is the state variable of one node of the network, $G=\left({g}_{ij}\right)$ is the coupling matrix. The authors extend the master stability method [L. Pecora, T. Carroll, G. Johnson, D. Mar, and J. Heagy, Chaos 7, 520–543 (1997; Zbl 0933.37030)] to obtain criteria for global synchronization, i.e. global stability of the invariant subspace $\left\{\left({x}_{1},\cdots ,{x}_{N}\right)|\phantom{\rule{4pt}{0ex}}{x}_{1}={x}_{2}=\cdots ={x}_{N}\right\}$. Using the technique of T. Nishikawa and A. E. Motter [Physica D 224, No. 1–2, 77–89 (2006; Zbl 1117.34048)], the authors include also the case, when the coupling matrix is nondiagonalizable.

##### MSC:
 34D05 Asymptotic stability of ODE 34C15 Nonlinear oscillations, coupled oscillators (ODE) 34D23 Global stability of ODE