Multistability of Switched Neural Networks With Piecewise Linear Activation Functions Under State-Dependent Switching

This paper is concerned with the multistability of switched neural networks with piecewise linear activation functions under state-dependent switching. Under some reasonable assumptions on the switching threshold and activation functions, by using the state-space decomposition method, contraction mapping theorem, and strictly diagonally dominant matrix theory, we can characterize the number of equilibria as well as analyze the stability/instability of the equilibria. More interesting, we can find that the switching threshold plays an important role for stable equilibria in the unsaturation regions of activation functions, and the number of stable equilibria of an <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-neuron switched neural network with state-dependent parameters increases to <inline-formula> <tex-math notation="LaTeX">$3^{n}$ </tex-math></inline-formula> from <inline-formula> <tex-math notation="LaTeX">$2^{n}$ </tex-math></inline-formula> in the conventional one. Furthermore, for two-neuron switched neural networks, the precise attraction basin of each stable equilibrium point can be figured out, and its boundary is composed of the stable manifolds of unstable equilibrium points and the switching lines. Two simulation examples are discussed in detail to substantiate the effectiveness of the theoretical analysis.

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