The Local Stochastic stability for Complex Networks under pinning Control

The local stochastic stability of complex networks under pinning control is studied, with stochastic perturbations to the coupling strengths. The nodes of complex network are modeled as second-order differential equations subject to stochastic parametric excitations. The complex network is first linearized at its trivial solution, and the resulting equations are reduced to independent subsystems by using a suitable linear transformation. Then the condition of the stochastic stability can be determined by Lyapunov exponents of the subsystems. The largest Lyapunov exponent of the subsystem can be expressed analytically, as a function of the eigenvalue of a matrix associated with the coupling matrix and pinning control matrix for a given system of parameters. And the stability region with respect to the eigenvalue can be obtained. It is pointed out that the local stochastic stability of the network is finally determined by the maximal and minimum eigenvalues of the matrix. Numerical results with positive and negative damping coefficients are given to illustrate the criterion. Moreover, for the positive damping coefficient case, the pinning control may destabilize the network; and for the negative damping coefficient case, the pinning control may stabilize the network.