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This paper is concerned with the problem of coexistence and dynamical behaviors of multiple equilibrium points for complex-valued competitive neural networks with discontinuous non-monotonic piecewise nonlinear activation functions. Without assuming the linearity or monotonicity of the activation functions, by virtue of the fixed point theorem and other analytical tools, several new sufficient conditions are developed to guarantee that the discontinuous complex-valued competitive neural networks have at least 16ⁿ equilibrium points, among which 9ⁿ are locally stable. In addition, some criteria for assuring the coexistence and local stability of multiple equilibria for real-valued competitive neural networks are established, which also show that the number of stable equilibria for the complex-valued neural networks is larger than the real-valued ones. A numerical simulation is conducted to illustrate the applicability …