Multistability Analysis of Fractional-Order State-Dependent Switched Competitive Neural Networks With Sigmoidal Activation Functions
研究了具有S型激活函数和状态依赖切换的分数阶竞争神经网络的多稳定性，给出了平衡点个数和局部稳定性的充分条件，发现切换能提高存储容量，并通过数值例子和关联记忆应用验证了结果。
This work explores the issue of multistability for a competitive neural network (NN) class with sigmoidal activation functions (AFs) involving state-dependent switching and fractional-order derivative. Specifically, first, we consider three different switching point locations, and establish some sufficient criteria ensuring that NNs with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$</tex-math> </inline-formula> -neurons can have, and only have, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$5^{n_{1}}\cdot 3^{n_{2}}$</tex-math> </inline-formula> equilibrium points (EPs) with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n_{1}+n_{2}=n$</tex-math> </inline-formula> , by utilizing the geometric features of the sigmoidal functions, the fixed point theorem, the Filippov’s EP definition, and the contraction mapping theorem. Then, based on novel Lyapunov functions and by applying the fractional-order calculus theory, it is demonstrated that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3^{n_{1}}\cdot 2^{n_{2}}$</tex-math> </inline-formula> out of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$5^{n_{1}}\cdot 3^{n_{2}}$</tex-math> </inline-formula> total EPs are locally stable. This work’s investigation reveals that competitive NNs with switching afford more storage capacity compared to the nonswitching case. Additionally, our results are valid for the integer-order and fractional-order switched NNs, improving and generalizing current works. Furthermore, two numerical examples and an application example of associative memory are provided to validate the effectiveness of the theoretical findings, and the way various fractional orders affect the NNs’ convergence speed is shown through simulations.