This paper focuses on the multistability analysis of delayed fuzzy cellular neural network (DFCNN) with discontinuous sawtooth-type activation function (DSAF). By applying Brouwer’s fixed-point theorem and the geometric characteristics of the DSAF, the existence and stability of multiple equilibrium points (EPs) are established. It can be demonstrated that DFCNN with DSAF has at least <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$7^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>7</mn> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> EPs, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$6^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>6</mn> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> of which are located at the points of continuity (POCs) of the DSAF, and the remaining EPs are located at the points of discontinuity (PODs) of the DSAF. Moreover, sufficient conditions for the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$4^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>4</mn> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> EPs located at the POCs of the DSAF to exhibit local exponential stability have been provided. Secondly, increasing the number of peak points can extend DSAF to more general situations. It can be demonstrated that the n-DFCNN has at least <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(2h+3)^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>h</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> EPs with the DSAF having h peak points, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(h+2)^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mi>h</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> of them are locally exponentially stable. Moreover, the method of increasing the number of peak points increases the number of total/stable EPs without altering the sufficient conditions or the computational complexity. Finally, the availability of the results in this paper is verified by two numerical examples.

Multistability of delayed fuzzy cellular neural networks with discontinuous sawtooth-type activation functions

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2026年4月7日

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Yang Liu,
Weixin Yan,
Zhen Wang

摘要

This paper focuses on the multistability analysis of delayed fuzzy cellular neural network (DFCNN) with discontinuous sawtooth-type activation function (DSAF). By applying Brouwer’s fixed-point theorem and the geometric characteristics of the DSAF, the existence and stability of multiple equilibrium points (EPs) are established. It can be demonstrated that DFCNN with DSAF has at least $7^n$ $7^{n}$ EPs, $6^n$ $6^{n}$ of which are located at the points of continuity (POCs) of the DSAF, and the remaining EPs are located at the points of discontinuity (PODs) of the DSAF. Moreover, sufficient conditions for the $4^n$ $4^{n}$ EPs located at the POCs of the DSAF to exhibit local exponential stability have been provided. Secondly, increasing the number of peak points can extend DSAF to more general situations. It can be demonstrated that the n-DFCNN has at least $(2h+3)^n$ ${(2 h + 3)}^{n}$ EPs with the DSAF having h peak points, $(h+2)^n$ ${(h + 2)}^{n}$ of them are locally exponentially stable. Moreover, the method of increasing the number of peak points increases the number of total/stable EPs without altering the sufficient conditions or the computational complexity. Finally, the availability of the results in this paper is verified by two numerical examples.