Multistability of switched neural networks with sigmoidal activation functions under state-dependent switching

Journal Pre-proof Multistability of switched neural networks with sigmoidal activation functions under state-dependent switching Zhenyuan Guo, Shiqin Ou, Jun Wang

PII: DOI: Reference:

S0893-6080(19)30334-X https://doi.org/10.1016/j.neunet.2019.10.012 NN 4303

To appear in:

Neural Networks

Received date : 28 June 2019 Revised date : 4 September 2019 Accepted date : 17 October 2019 Please cite this article as: Z. Guo, S. Ou and J. Wang, Multistability of switched neural networks with sigmoidal activation functions under state-dependent switching. Neural Networks (2019), doi: https://doi.org/10.1016/j.neunet.2019.10.012. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Elsevier Ltd. All rights reserved.

Journal Pre-proof Multistability of Switched Neural Networks with Sigmoidal Activation Functions under State-dependent Switching I Zhenyuan Guoa , Shiqin Oua , Jun Wangb,c,∗ a College

of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China of Computer Science and School of Data Science, City University of Hong Kong, Kowloon, Hong Kong c Shenzhen Research Institute, City University of Hong Kong, Shenzhen 518057, China

pro of

b Department

Abstract

This paper presents theoretical results on the multistability of switched neural networks with commonly used sigmoidal activation functions under state-dependent switching. The multistability analysis with such an activation function is difficult because state-space partition is not as straightforward as that with piecewise-linear activations. Sufficient conditions are derived for ascertaining the existence and stability of multiple equilibria. It is shown that the number of stable equilibria of an n-neuron switched neural networks is up to 3n under given conditions. In contrast to existing multistability results with piecewise-linear activation functions, the results herein are also applicable to the equilibria at switching points. Four examples are discussed to substantiate the theoretical results.

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Keywords: Multistability, switched neural network, state-dependent, sigmoidal activation function. 1. Introduction

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urn a

lP

As known widely nowadays, neural networks play important roles in many technical areas, such as pattern recognition (Kwan & Cai, 1994, Suganthan et al., 1995, Zeng et al., 2005), associative memory (Isokawa et al., 2008, Zeng & Wang, 2008, 2009), and other areas (Bao et al., 2018a, 2016). In recent decades, various neural network models, including Hopfield-type neural network model (Huang & Cao, 2010, Zeng et al., 2016), cellular neural networks with memristors (Bao & Zeng, 2013, Duan et al., 2015, Di et al., 2017), Cohen-Grossberg neural network model (Wang & Zou, 2002, Wang et al., 2006, Gang et al., 2007, Zhang & Wang, 2008), switched neural network model (Li & Cao, 2007, Li et al., 2009, Liu et al., 2013, Zhao et al., 2016, Zhao & Zhao, 2017, Bao et al., 2018b), have been developed and analysed. Because of the complexity of actual environment, the connections of neurons in a network may change frequently. Due to the link failure or a new creation of the connection topology in the network, the switching between some different connection topologies are inevitable (Daafouz et al., 2002, Zhao et al., 2009). To describe this switching phenomenon in the network, switched system is introduced. A switched system, consisting of a number of subsystems or configurations modelled by using differential or difference equations, operates in a mode switching among these subsystems. Generally speaking, two categories of switching

modes are usually considered (Persis et al., 2003, Liberzon & Morse, 2015). One is time-dependent switching, i.e. the switching rule is controlled only by time. For example, a continuous switched system is characterized by the following differential equation:

I The work described in the paper was supported in part by the Research Grants Council, Hong Kong, under Grant 416812, National Natural Science Foundation of China under Grant 61573003. ∗ The corresponding author

Preprint submitted to Neural Networks

dx(t) = fσ(t,x) (x(t)). dt

Let F = {fp (x), p ∈ P }, the parameter p takes the value in the index set P , each map fp (x) : Rn → Rn in F is assumed to be locally Lipschitz, and σ(t, x) = σ(t) : [0, +∞) → P is the switching signal. The other type is state-dependent switching. In this case, the switching signal σ(t, x) = σ(x) : Rn → P depends only on state x. Compared with the switched systems under timedependent switching, switched systems under state-dependent switching with different initial values may take on different equations at the same instant. The dynamical analysis of switched systems is much more complicated and challenging than that of conventional ones. Due to the significant values of switched systems in both theory and practice, stability analysis of switched systems is an attractive topic; e.g.,Li et al. (2005), Yuan et al. (2006), Wu et al. (2011), Zhang et al. (2013). Multistability is a notion to characterize the coexistence of multiple stable equilibrium points or periodic solutions (Cheng et al., 2006, Wang et al., 2009, 2010, Wang & Chen, 2012, Wang, 2014, Nie & Zheng, 2015, Liu et al., 2016a,b, 2017a,b,c, 2018). It is well recognized that multistability analysis of neural networks depends critically upon the type of activation functions. With different activation functions, different multistability stability criteria September 4, 2019

Journal Pre-proof can be derived. For example, in an n-neuron Hopfield neural network with Mexican-hat-type activation functions, there exist at most 3n equilibrium points and at most 2n of them are locally stable and others are unstable (Wang & Chen, 2012, Wang, 2014). In a neural system with Gaussian activation functions, there exist 3k equilibrium points, in which 2k are exponentially stable, where 1 ≤ k ≤ n (Liu et al., 2017a). In a neural system with piecewise-linear nondecreasing activation functions, there exist (2r + 1)n equilibria, in which (r +1)n are locally exponentially stable, where r is the number of pieces in the piecewise-linear activation functions (Wang et al., 2010). In a neural network with discontinuous Mexican-hat-type activation functions under reasonable conditions, there exist 5n equilibrium points in Rn and 4n of them are located at the continuous part of the activation functions (Nie & Zheng, 2015). Numerous results are available for the stability analysis of the switched systems, e.g., Branicky (1998), Song et al. (2006), Long & Wei (2007), Li & Cao (2007), Chen et al. (2010), Yu et al. (2011), Singh & Sukavanam (2012), Long et al. (2013), Guo et al. (2014), Lian & Wang (2015), Hu et al. (2015), Nie & Cao (2015), Tang & Zhao (2017), Niu et al. (2017), Guo et al. (2018, 2019), Kahloul & Sakly (2018), Song et al. (2018). The global stability of switched neural networks and the control of switched systems are addressed in Branicky (1998), Song et al. (2006), Li & Cao (2007), Long & Wei (2007), Chen et al. (2010), Yu et al. (2011), Singh & Sukavanam (2012), Long et al. (2013), Hu et al. (2015), Tang & Zhao (2017), Niu et al. (2017), Kahloul & Sakly (2018), Song et al. (2018). In contrast to global stability, few results are available on the multistability of switched neural networks, especially under state-dependent switching, e.g., Guo et al. (2014), Nie & Cao (2015), Guo et al. (2018, 2019). The multistability results of memristive neural networks with non-monotonic piecewise-linear activation functions are presented in Nie & Cao (2015). The existence and attractivity of multiple equilibria of memristor-based cellular neural networks with piecewise-linear activation functions are addressed in Guo et al. (2014). The multistability results of switched neural networks with piecewise-linear radial basis functions and monotonic piecewise-linear functions under state-dependent switching are presented in Guo et al. (2018, 2019). As is well known, state-space partition is an important step for multistability analysis. In all existing results on multistability of switched neural networks, the activation functions considered are piecewise linear. The state-space is partitioned at the breakpoints of piecewise linear activation functions. However, state space partition is not straightforward for switched neural networks with smooth activation functions. The difficulties of multistability analysis are summarized as follows:

nontrivial. (ii) Due to the discontinuity of switched neural networks, the upper and lower bound functions are also discontinuous, which leads to the difficulty on extreme point analysis.

pro of

(iii) Due to the discontinuity of switched neural networks, the existing analysis methods on the multistability are invalid. The existence and stability of equilibria with at least one component at a switching point need to be considered. This paper addresses the multistability of switched neural networks with sigmoidal activation functions under statedependent switching. By combing Brouwer fixed theorem, local linearized method, state space partition and discussion of switching threshold, sufficient conditions are derived for ascertaining the existence and stability of multiple equilibria. Compared with related existing works, the main contributions of this paper are listed as follows: (i) There is no multistability results for switched neural networks with sigmoidal activation functions.

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(ii) The number of stable equilibria of an n-neuron switched neural networks is up to 3n under given conditions.

(iii) Sufficient conditions are derived for ascertaining the existence and stability of the maximal number and some numbers of equilibria on and off switching points.

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(iv) A switching rule is given for resulting in desirable number of stable equilibria.

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The rest of paper is organized as follows. In Section 2, a switched neural network model with sigmoidal activation function under state-dependent switching is proposed. Then, some definitions and lemmas are presented. In Section 3, the multistability results of switched neural systems are addressed. In Section 4, four examples are given to substantiate the effectiveness of the obtained criteria. In Section 5, concluding remarks are given. 2. Preliminaries

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In this section, the switched neural network model under state-dependent switching is introduced. Then, some necessary definitions and useful lemmas are also briefly outlined. N atation: Given the n-vector x ∈ Rn , k x k∞ = maxi=1,2,··· ,n |xi | denotes the ∞-norm of x. |N | denotes the number of elements in set N . For a function f (x), denote df (x0 )/dx = df (x)/dx |x=x0 .

(i) State-space partition is an important step for multistability analysis. For switched neural networks with smooth activation functions, state-space partition is 2

Journal Pre-proof 0

1

00

where T ∈ R is the switching threshold, bi > 0, bi > 0, 0 00 0 00 wij and wij are constants. Let ˇbi = min{bi , bi }, w ˇij = 0 00 0 00 0 00 min{wij , wij }, ˆbi = max{bi , bi }, and w ˆij = max{wij , wij }.

y

y  g ( )

2.2. Definitions and Lemmas Some useful definitions and lemmas are given for the analysis of main results. In view of the discontinuity of system (1), the definition for solutions of system (1) in the sense of Filippov is needed to be introduced. 



T

ai

0





ci

Definition 1 A function x(t) = (x1 (t), x2 (t), · · · , xn (t))T is a solution in the sense of Filippov of system (1), if x(t) is absolutely continuous on any closed subinterval of [0, +∞) and satisfies the following differential inclusion

pro of



ai





Fig. 1. A unipolar sigmoid activation function g with  = 0.5.

dxi (t) ∈K[Φi (xi )], dt for i = 1, 2, · · · , n, where

2.1. Model Description Consider the following switched neural network model under state-dependent switching: n X

dxi (t) = − bi (xi (t))xi (t) + ωij (xi (t))gj (xj (t)) dt j=1

(1)

i = 1, 2, · · ·,n,

re-

+ ui ,

1 , 1 + e−η/ε

Φi (xi ) = −bi (xi (t))xi (t) + 0

0

Φi (xi ) = −bi xi (t) +

lP

where x(t) = (x1 (t), x2 (t), · · · , xn (t))T ∈ Rn is the state vector, B(x) = diag{b1 (x1 (t)), b2 (x2 (t)), · · · , bn (xn (t))} is the real diagonal positive-define matrix, u = (u1 , u2 , · · · , un ) is the input or bias vector; W (x) = [wij (xi (t))]n×n is the inter-neuron connection weight matrix, gj is a smooth activation function usually being sigmoidal or monotonically nondecreasing with saturations. Here gi (.) is the typical logistic or Fermi function (see Figure 1) defined by: gj (η) = g(η) =

 0  {Φi (xi )}, 00 K[Φi (xi )] = {Φi (xi )},  ˇ ˆ i (T )], [Φi (T ), Φ

00

00

Φi (xi ) = −bi xi (t) + 00

xi (t) < T, xi (t) > T, xi (t) = T,

(7)

wij (xi (t))g(xj (t)) + ui ,

j=1

n X

0

wij g(xj (t)) + ui ,

j=1

n X

00

wij g(xj (t)) + ui ,

j=1

ˇ i (T ) = min{Φ (T ), Φ (T )}, Φ ˆ i (T ) = max{Φ0 (T ), Φ00 (T )}, Φ i i i i 0

0

n X

(6)

0

00

00

Φi (T ) = limxi →T − Φi (xi ), and Φi (T ) = limxi →T + Φi (xi ).

where ε > 0 is a constant. In general, the sigmoidal activation function gi (η) is defined as follows:  0  < ni , gi (η) > 0,  mi < gi (η) 00 (η − σi )gi (η) < 0, (3)   lim gi (η) = ni , lim gi (η) = mi ,

Lemma 1 There is at least one solution to system (1) with the initial condition x(t0 ) = x0 in the sense of Filippov on [t0 , +∞).

urn a

(2)

η→+∞

The above lemma can be proved in a similar way as Lemma 1 in Guo et al. (2019). The solution of system (1) with initial condition x(t0 ) = x0 is denoted as x(t; t0 , x0 ) (or x(t) for short).

η→−∞

for η ∈ R and i = 1, 2, · · · , n, where mi , ni , and σi are constants. In this paper, for convenience, a special activation function in (2) is used. In fact, similar results on the multistability of system (1) with the activation function (3) can be obtained. The switching rule is defined as follows:  0 bi , if xi (t) ≤ T, bi (xi (t)) = (4) 00 bi , if xi (t) > T,

0 ∈ K[Φi (x∗i )], i = 1, 2, · · · , n.

Jo

and

T

Definition 2 A constant vector x∗ = (x∗1 , x∗2 , · · · , x∗n ) is said to be an equilibrium of system (1), if

wij (xi (t)) =

(

0

wij , 00 wij ,

if xi (t) ≤ T, if xi (t) > T,

From (7), the equilibrium equation of system (1) is decomposed as follows:  0 Pn 0 0  Φi (xi ) = −bi xi + Pj=1 wij g(xj ) + ui = 0, xi < T, 00 00 00 n Φ (x ) = −bi xi + j=1 wij g(xj ) + ui = 0, xi > T,  i i 0 ∈ K[Φi (T )], xi = T, (8) where i = 1, 2, · · · , n.

(5) 3

Journal Pre-proof h

Definition 3 A set Ω is said to be an invariant set of system (1), if the solution x(t; t0 , x0 ) of system (1) with any initial condition x(t0 ) = x0 ∈ Ω satisfies x(t; t0 , x0 ) ∈ Ω for t ≥ t0 .

1/ 4   bi  / wii

Definition 4 The matrix A = [aij ]n×n is strictly diagonally dominant, if | aii |>

n X

j=1,j6=i

  bi  / wii

| aij |, i = 1, 2, · · · , n.

0

g (T ) y2

1/ 2

y3

pro of

In this case, denote A ∈ SD.

y1

Lemma 2 (Brouwer fixed point theorem) (Cheng et al., 2015) Every continuous function from a convex compact subset Ω of a Euclidean space to Ω itself has a fixed point.

y

1

0

Fig. 2. The graph for function h(y) = y − y 2 and y1 = g(pi ), 00 00 y2 = g(pi ), y3 = g(qi ).

Lemma 3 (Tong, 1977) Let A = [aij ]n×n be a real matrix and A ∈ SD. Then the following results hold: (1) If aii > (<)0, i = 1, 2, · · · , n, then all eigenvalues of A have positive (negative) real parts. (2) Let index sets N1 = {i | aii > 0, i = 1, 2, · · · , n} and N2 = {i | aii < 0, i = 1, 2, · · · , n}, and denote the numbers of elements in the sets N1 and N2 by |N1 | and |N2 | respectively. If |N1 | = s > 0, |N2 | = t > 0 and s + t = n, then A has s eigenvalues with positive real parts and t eigenvalues with negative real parts.

Proof. Consider the following equation: ( 0 0 −bi + wii dg(η) η < T, dfi (η) dη = 0, = 00 dg(η) 00 dη −bi + wii dη = 0, η > T.

re-

The above equation can be transformed to the following form:  0 bi η < T, 0 , dg(η)  wii = 00  bi00 , η > T. dη wii

From (2), we have

3. Main Results

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In this section, the multistability of switched neural network model (1) with activation function (2) under statedependent switching is analyzed. In terms of the value of switching threshold T , there are three possible cases for us to discuss: Case A: T ∈ (−∞, 0), Case B: T = 0 and Case C: T ∈ (0, ∞). To facilitate our formulation in the following discussions, a single neuron analogue (no interaction among neurons) is introduced as follows:  0 0 dη −bi η + wii g(η) + ui , η ≤ T, = fi (η) = 00 00 dt −bi η + wii g(η) + ui , η > T.

urn a

0

(10)

Therefore, gi is a strictly increasing function, and dgi (η)/η is concave down and has its maximal value at η = 0. For convenience, denote y = g(η). Then y ∈ (0, 1) and g(0) = 1/2. It follows from (9) that 1 1 1 dg(η) = y 2 ( − 1) = (y − y 2 ). dη ε y ε

Hence, we have

3.1. Multistability Analysis if T ∈ (−∞, 0) 0 00 0 00 For parameters bi , bi , wij , wij , let us propose the following condition:

h(y) = y − y 2 =

00

  

0

bi ε 0 , wii 00 bi ε 00 , wii

y < g(T ), y > g(T ).

Since T ∈ (−∞, 0), we have g(T ) ∈ (0, 1/2). Therefore, from Fig. 2, it follows that, for each i, there exist three 0 00 00 0 00 00 points pi , pi , qi , and pi < T < pi < 0 < qi , such 0 00 00 that dfi (pi )/dη = dfi (pi )/dη = dfi (qi )/dη = 0. Since 0 limη→−∞ fi (η) = +∞ and limη→+∞ fi (η) = −∞, pi and 00 00 pi are the minimum value points of fi , and qi is the maximum value point of fi . 

(9)

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bε b ε 1 0 < i0 < h(g(T )) < i 00 < , 4 wii wii

1 dg(η) = (1 + e−η/ε )−2 e−η/ε > 0. dη ε

where h = g(η) − g 2 (η).

Lemma 4 If condition (9) holds, then there exist three 0 00 00 0 00 00 vectors p , p and q with pi < T < pi < 0 < qi such that 0 00 00 0 00 dfi (pi )/dη = dfi (pi )/dη = dfi (qi )/dη = 0, pi and pi are 00 the minimum value points of fi , and qi is the maximum value point of fi for i = 1, 2, · · · , n.

Remark 1 For system (1) with activation function (3), similar conclusion to that in Lemma 1 can be derived. Of 4

Journal Pre-proof Theorem 1 Assume that T ∈ (−∞, 0) and |N1 | + |N2 | + |N3 | + |N4 | = n. If condition (9) holds, then there exist 5|N1 |+|N2 |+|N3 | ·3|N4 | equilibria of system (1). Furthermore, if n X 0 0 dgj (T ) bi > ωij , (14) dη j=1

course, with the change of the activation functions, the switching threshold may change accordingly. For example, consider T < σi in system (1) with activation function (3). Under the following condition:

<

dgi (η) dη

0

<

00

<

dgi (T ) dη

=

dgi (σi ) dη ,

00

and

0

bi >

there exist three points pi , pi and qi with pi < T < pi < 00 0 00 00 σi < qi such that dfi (pi )/dη = dfi (pi )/dη = dfi (qi )/dη = 0 00 00 0, pi and pi are the minimum value points of fi , and qi is the maximum value point of fi for i = 1, 2, · · · , n. 0

Proof. Theorem 1 is proven in the following three steps: Step 1: The existence of equilibria is analyzed, with all components not at the switching.

00

0

n X

00

0

| wij | +ui , ki − = −

j=1,j6=i

fi

fi

n X

j=1,j6=i

fˇi (xi ) ≤ Φi (xi ) ≤ fˆi (xi ),

(13)

urn a i

i

i

i

i

i

0

Jo

and

i

i

i

i

00

i

i

i

00

i

i





T

ai





ai





bi





bi





qi

ci





ci





fi



fi





0

0

i

i

i

iN1

00

i

00

0

00

00

00

0

fine 4 disjoint closed regions as follows: ΩliN2 = {x ∈ R | 0

i



bi

0

0

00

00

00

00

a ˇi ≤ x ≤ a ˆi }, ΩliN2 = {x ∈ R | a ˇi ≤ x ≤ a ˆi }, Ωm iN2 = 00 00 00 00 00 r ˆ ˇ {x ∈ R | bi ≤ x ≤ bi } and ΩiN2 = {x ∈ R | cˇi ≤ x ≤ cˆi }. 0 00 00 00 i Denote N i = iN2 and αi = l , l , m or r . Then, Ωα N i is

00

i



l m r of 5 closed regions ΩliN1 , Ωm iN1 , ΩiN1 , ΩiN1 and ΩiN1 . 0 00 0 00 When i ∈ N2 , from Lemma 4, fˇi , fˇi , fˆi and fˆi are depicted in Fig. 4. Compared with the case of N1 , zero 0 0 points ˆbi , ˇbi disappear, and other zero points remain. De-

i

N4 = {i :fˆi (T ) < 0, fˆi (T ) < 0, 0 0 00 00 00 00 fˆ (p ) < 0, fˆ (p ) < 0, fˇ (q ) > 0}. 00

pi



bi

ΩriN1 = {x ∈ R | cˇi ≤ x ≤ cˆi }, where “l”, “m” and “r” respectively mean “left”,“middle” and “right”; “0 ” and “00 ” are used to distinguish two different subsystems. Denote 0 0 00 00 00 i N i = iN1 and αi = l , m , l , m or r . Then, Ωα N i is one

0 00 N3 = {i :fˇi (T ) > 0, fˆi (T ) < 0, 0 0 00 00 00 00 fˆ (p ) < 0, fˆ (p ) < 0, fˇ (q ) > 0},

i



ˇi ≤ 5 disjoint closed regions as follows: ΩliN1 = {x ∈ R | a 00 0 0 0 0 m l x ≤ a ˆi }, ΩiN1 = {x ∈ R | ˆbi ≤ x ≤ ˇbi }, ΩiN1 = {x ∈ 00 00 00 00 00 R | a ˇ ≤ x ≤ a ˆ }, Ωm = {x ∈ R | ˆb ≤ x ≤ ˇb } and

0 00 N2 = {i :fˆi (T ) < 0, fˇi (T ) > 0, 0 0 00 00 00 00 fˆ (p ) < 0, fˆ (p ) < 0, fˇ (q ) > 0},

i



ai

0 00 00 0 When i ∈ N1 , from Lemma 4, fˇi , fˇi , fˆi and fˆi are 0 0 depicted in Fig. 3. Then, there exist zero points a ˆi , ˆbi of 00 00 00 00 0 0 0 00 00 00 00 0 ˇi , ˇbi , cˇi of fˇi with ˇi , ˇbi of fˇi , and a ˆi , ˆbi , cˆi of fˆi , a fˆi , a 00 00 00 00 00 00 0 0 0 0 ˆi < ˆbi < ˇbi < cˇi < cˆi ˇi < a ˆi < ˆbi < ˇbi < T < a a ˇi < a 00 00 00 00 00 00 0 0 0 0 ci ) = ai ) = fˆi (ˆbi ) = fˆi (ˆ ai ) = fˆi (ˆbi ) = fˆi (ˆ such that fˆi (ˆ 00 00 00 00 00 00 0 0 0 0 ˇ ˇ ˇ ˇ ˇ ˇ ci ) = 0. Define ai ) = fi (bi ) = fˇi (ˇ ai ) = fi (bi ) = fi (ˇ fi (ˇ

| wij | +ui .

0 00 N1 = {i :fˇi (T ) > 0, fˇi (T ) > 0, 0 0 00 00 00 00 fˆ (p ) < 0, fˆ (p ) < 0, fˇ (q ) > 0},

i



0 00 0 00 Fig. 3. fˆi ,fˆi ,fˇi and fˇi under condition (9) and i ∈ N1 .

for xi 6= T and i = 1, 2, · · · , n. To analyze the multistability of system (1), the following index sets are defined:

i



pi



ai

00

Since 0 ≤ gi ≤ 1 for all i, we have

i



lP

and ki− = −





re-

j=1,j6=i

(15)

then there exist exactly 5|N1 |+|N2 |+|N3 | · 3|N4 | equilibria.

Note that condition (9) implies wii > 0 and wii > 0. For i = 1, 2, · · · , n, the upper and lower bound functions are defined respectively as follows: ( 0 0 0 0 fˆi (η) = −bi η + wii g(η) + ki+ , η ≤ T, ˆ fi (η) = 00 00 00 00 fˆi (η) = −bi η + wii g(η) + ki + , η > T, (11) and ( 0 0 0 0 fˇi (η) = −bi η + wii g(η) + ki− , η ≤ T, ˇ fi (η) = 00 00 00 00 − ˇ fi (η) = −bi η + wii g(η) + ki , η > T, (12) where n n X X 0 00 0 00 ki + = | wij | +ui , ki + = | wij | +ui , j=1,j6=i

n

1 X 00 ω , 4 j=1 ij

00

00

pro of

<

00 bi 00 wii

0

bi 0 wii dgi (η) maxη∈R dη

0 = inf η∈R

i

0

00

00

00

r one of 4 closed regions ΩliN2 , ΩliN2 , Ωm iN2 and ΩiN2 .

00

00 where fˇi (T ) = limη→T + fˇi (η), fˆi (T ) = limη→T + fˆi (η).

5

Journal Pre-proof



fi



fi



fi







ai









pi

pi 

ai



fi







T

ai





ai



pi

pi 

bi





bi





qi

ci





ci



ai



fi



fi



ai





T

bi





bi



qi



ci





ci





 

pro of

fi

0 00 0 0 Fig. 4. fˆi ,fˆi ,fˇi and fˇi under condition (9) and i ∈ N2 .



fi

˜ i (xi ) = −b0 xi + w0 g(xi ) + h i ii





n X

wij g(˜ xj ) + ui = 0,

n X

wij g(˜ xj ) + ui = 0,

0

j=1,j6=i

and when xi > T ,

pi

pi ai





ai





bi





bi





T

bi





bi





qi

ci





ci

˜ i (xi ) = −b00 xi + w00 g(xi ) + h i ii





fi



fi

0

00

0

0



00

urn a

0

iN3

i

00

i

iN3

00 00 00 00 {x ∈ R | ˆbi ≤ x ≤ ˇbi } and ΩriN3 = {x ∈ R | cˇi ≤ x ≤ cˆi }. 0 0 00 00 i Denote N i = iN3 and αi = l , m , m or r . Then, Ωα N i is 0

0

00

00

m r one of 4 closed regions ΩliN3 , Ωm iN3 , ΩiN3 and ΩiN3 . 0 00 0 00 When i ∈ N4 , from Lemma 4, fˇi , fˇi , fˆi and fˆi are depicted in Fig. 6. Compared with the case of N1 , zero 0 0 00 00 points ˆbi , ˇbi , a ˆi , a ˇi disappear, and other zero points re0

i

Jo

main. Define 3 disjoint closed regions as follows: ΩliN4 = 00 0 0 00 00 {x ∈ R | a ˇ ≤x≤a ˆ }, Ωm = {x ∈ R | ˆb ≤ x ≤ ˇb } 00

i

00

iN4

i

i

00

Hα (˜ x) = x.

00

and ΩriN4 = {x ∈ R | cˇi ≤ x ≤ cˆi }. Denote N i = iN4 and 0 00 00 i αi = l , m or r . Then, Ωα N i is one of 3 closed regions 00

00

r Ωm iN1 and ΩiN1 , respectively; when i ∈ N2 , we can always ˜ i (xi ) = 0, which lie in the regions find four solutions to h 00 0 00 00 l m l ΩiN2 , ΩiN2 , ΩiN2 and ΩriN2 , respectively; when i ∈ N3 , we ˜ i (xi ) = 0, which lie in can always find four solutions to h 0 0 00 00 l m m the regions ΩiN3 , ΩiN3 , ΩiN3 and ΩriN3 , respectively; when ˜ i (xi ) = 0, i ∈ N4 , we can always find three solutions to h 0 00 00 l m r which lie in the regions ΩiN4 , ΩiN4 and ΩiN4 , respectively. Therefore, in any regions in Ω, we can find a solution of ˜ i (xi ) = 0. Let us pick the one lying in Ωα and denote it h as x = (x1 , x2 , · · · , xn )T . Next, we consider the mapping Hα : Ωα −→ Ωα defined by

Define 4 disjoint closed regions as follows: ΩliN3 = {x ∈ 0 00 0 0 0 0 R|a ˇ ≤x≤a ˆ }, Ωm = {x ∈ R | ˆb ≤ x ≤ ˇb }, Ωm = i

(16)

for xi 6= T and i = 1, 2, · · · , n, we have the following inclusions: when i ∈ N1 , we can always find five solutions 0 0 00 ˜ i (xi ) = 0, which lie in the regions Ωl , Ωm , Ωl , to h iN1 iN1 iN1

ro points a ˆi , a ˇi disappear, and other zero points remain. i

j=1,j6=i

˜ i (xi ) ≤ fˆi (xi ), fˇi (xi ) ≤ h



Fig. 5. fˆi ,fˆi ,fˇi and fˇi under condition (9) and i ∈ N3 . 00

00

where i = 1, 2, · · · , n. Since

lP



0



˜ ˜ 1 (x1 ), h ˜ 2 (x2 ), · · · , h ˜ n (xn ))T = (x1 , x2 , · · · , xn )T in h(x) = (h 0, that is to say: when xi < T ,

re-

fi

fi



0 00 0 0 Fig. 6. fˆi ,fˆi ,fˇi and fˇi under condition (9) and i ∈ N4 .

0 00 0 00 When i ∈ N3 , from Lemma 4, fˇi , fˇi , fˆi and fˆi are depicted in Fig. 5. Compared with the case of N1 , ze-





The mapping Hα is continuous, since each gi is continuous. It follows from Brouwer’s fixed point theorem that there exists one fixed point x ¯ = (¯ x1 , x ¯2 , · · · , x ¯n )T of Hα in Ωα which is also a zero solution of the function Φ. Therefore, there exist 5|N1 | 4|N2 |+|N3 | 3|N4 | zeros solutions for Φ.

00

r ΩliN4 , Ωm iN4 and ΩiN4 . i Denote Ω = {Ωα = Πni=1 Ωα N i }. Obviously, there are 5|N1 | 4|N2 |+|N3 | 3|N4 | regions in Ω. Let us take Ωα as any one of these regions. For any given x ˜ ∈ Ωα , solve for x =

6

Journal Pre-proof That is to say there exist 5|N1 | 4|N2 |+|N3 | 3|N4 | equilibria, all components of them do not locate at the switching. Step 2: The existence of equilibria with at least one component at the switching is analyzed. When i ∈ N1 , we have 0

0

00

00

5|N1 | 4|N2 |+|N3 | 3|N4 | P|N2 |+|N3 | k + k=1 C|N2 |+|N3 | 5|N1 | 4|N2 |+|N3 |−k 3|N4 | |N1 |+|N2 |+|N3 | |N4 | 3 , =5

0 < fˇi (T ) ≤ lim Φi (xi ) = Φi (T ) ≤ fˆi (T ),

there exist 5|N1 |+|N2 |+|N3 | 3|N4 | equilibria in all for system (1) under condition (9). Step 3: Let’s prove the uniqueness of the equilibrium point in each local region. Assume that there exist two equilibrium points x∗ and ∗ y of systems (1) in Ωα , and x∗ 6= y ∗ . If i ∈ Λ, then x∗i = yi∗ = T . If i ∈ / Λ, then ( Pn 0 0 −bi x∗i + j=1 ωij gj (x∗j ) + ui = 0, x∗i < T, Pn 00 00 −bi x∗i + j=1 ωij gj (x∗j ) + ui = 0, x∗i > T

xi →T −

and 00

0 < fˇi (T ) ≤ lim + Φi (xi ) = Φi (T ) ≤ fˆi (T ). xi →T

Then,

ˇ i (T ), Φ ˆ i (T )]. 0∈ / [Φ

Meanwhile, when i ∈ N4 , we have 0 0 0 fˇi (T ) ≤ Φi (T ) ≤ fˆi (T ) < 0,

and

and

(

00 00 00 fˇi (T ) ≤ Φi (T ) ≤ fˆi (T ) < 0.

Then,

ˇ i (T ), Φ ˆ i (T )]. 0∈ / [Φ

Then,

ˇ i (T ), Φ ˆ i (T )]. 0 ∈ [Φ

Meanwhile, when i ∈ N3 , we have

and

00

00

ren X j=1

According to the definition of equilibrium, the switching T can be the ith component of one equilibrium in system (1) if i ∈ N2 ∪ N3 . Assume that x∗ = (x∗1 , x∗2 , · · · , x∗n )T is the equilibrium, at least one component of them locates at the αi switching. Next, we use similar discussion for Πi∈Λ / ΩN i as α that for Ω in Step 1 by substituting x ˜i with T for all i ∈ Λ, where Λ = {i : i ∈ N2 ∪N3 , x∗i = T }. Obviously, |Λ| can be one value in set {1, 2, · · · , |N2 | + |N3 |}. For a fixed Λ with |Λ| = k, the number of equilibria is 5|N1 | 4|N2 |+|N3 |−k 3|N4 | . Therefore, the number of all equilibria, at least one component of them locates at the switching, are

k=1

0

|ωij |

j=1 n X j=1

0

dgj (ηj∗ ) ∗ 0 |xj − yj∗ |/bi dη

|ωij |

dgj (ηj∗ ) 0 /bi } k x − y k∞ dη

< k x − y k∞ , which is a contradiction. Case (ii) x∗i > T and yi∗ > T . From (15), we have n X

n 1 dgj (ηj∗ ) 00 X 00 dgj ( ) 00 2 |ωij | /bi ≤ |ωij | /bi = dη dη j=1 j=1

Jo

X

n X

≤{

ˇ i (T ), Φ ˆ i (T )]. 0 ∈ [Φ

|N2 |+|N3 |

n dgj (ηj∗ ) 0 X 0 dgj (T ) 0 /bi ≤ |ωij | /bi < 1. dη dη j=1

|x∗i − yi∗ | =

00

fˇi (T ) ≤ Φi (T ) ≤ fˆi (T ) < 0.

Then,

0

|ωij |

It follows that for i = 1, 2, · · · , n,

urn a

0 0 0 0 < fˇi (T ) ≤ Φi (T ) ≤ fˆi (T ),

yi∗ < T, yi∗ > T.

where ηj∗ is some number between x∗j and yj∗ . Let us divide the discussion into two cases: Case (i) x∗i < T and yi∗ < T . From (14), we have

lP

0 0 0 fˇi (T ) ≤ Φi (T ) ≤ fˆi (T ) < 0,

00 00 00 0 < fˇi (T ) ≤ Φi (T ) ≤ fˆi (T ).

Pn 0 0 −bi yi∗ + j=1 ωij gj (yj∗ ) + ui = 0, P 00 00 n −bi yi∗ + j=1 ωij gj (yj∗ ) + ui = 0,

By subtracting one equation from the other and using differential mean value theorem, we obtain ( Pn 0 0 dgj (η ∗ ) −bi (x∗i − yi∗ ) + j=1 ωij dη j (x∗j − yj∗ ) = 0, Pn 00 00 dgj (η ∗ ) −bi (x∗i − yi∗ ) + j=1 ωij dη j (x∗j − yj∗ ) = 0,

Therefore, the ith component of any equilibrium in system (1) can not be T if i ∈ N1 ∪ N4 . When i ∈ N2 , we have

and

pro of

0

Since

00

Pn

j=1

00

|ωij | 00

4bi

It follows that for i = 1, 2, · · · , n, n ∗ X 00 dgj (ηj ) 00 |x∗i − yi∗ | ≤{ |ωij | /bi } k x − y k∞ dη j=1

k C|N 5|N1 | 4|N2 |+|N3 |−k 3|N4 | . 2 |+|N3 |

< k x − y k∞ ,

7

< 1.

Journal Pre-proof which is a contradiction. Therefore, the equilibrium point of systems (1) in Ωα is unique.  Next, we address the stability of equilibrium for system (1). ˜ = {Ω ˜ α˜ = Qn Ω ˜ α˜ i } as follows: Define Ω ˜i i=1 N 0 0 00 00 ˜ i = iN1 , α ˜ l = {η ∈ When i ∈ N1 , N ˜ i = l , l or r , Ω

and 00 00 00 00 00 00 00 fˆi (ˆbi − 0 ) = − bi (ˆbi − 0 ) + wii g(ˆbi − 0 ) + ki + 00 00 00 00 = − bi (ˆbi − 0 ) + wii g(ˆbi − 0 ) (19) n X 00 + | wij | +ui < 0.

iN1

j=1,j6=i

00 00 dxi (T + 0 ) = − bi (T + 0 ) + wii g(T + 0 ) dt n X 00 + wij g(xj (T + 0 )) + ui

i

iN2

00

In addition, by | g(·) |≤ 1, we obtain that from (18) and (19)

˜ r = {η ∈ R | η ≥ ˇb00 + 0 }. and Ω i iN2 0 0 00 ˜ i = iN3 , α ˜ l = {η ∈ When i ∈ N3 , N ˜ i = l , T or r , Ω iN3 0 ˜ T = {η ∈ R | ˇb0 + 0 ≤ η ≤ ˆb00 − 0 } R | η ≤ ˆb − 0 }, Ω 00

i

i

iN3

j=1,j6=i 00

˜ r = {η ∈ R | η ≥ ˇb00 + 0 }. and Ω i iN3 0 0 00 i ˜ ˜ l = {η ∈ When i ∈ N4 , N = iN4 , α ˜ i = l or r , Ω iN4 00 ˜ r = {η ∈ R | η ≥ ˇb00 + 0 }. R | η ≤ T − 0 } and Ω i iN4 ˜ = 3|N1 |+|N2 |+|N3 | 2|N4 | . Obviously, |Ω|

j=1,j6=i

and

00 00 00 00 00 dxi (ˆbi − 0 ) = − bi (ˆbi − 0 ) + wii g(ˆbi − 0 ) dt n X 00 00 + wij g(xj (ˆbi − 0 )) + ui

0

+

i

ii

n X

i

0

j=1,j6=i

| wij | +ui < 0.

(17)

lP

i

re-

Lemma 5 For i ∈ N1 ∪ N3 , xi (t) with xi (0) ∈ (−∞, ˆbi − 0 0 ] can not escape from (−∞, ˆbi − 0 ]. 0 0 0 0 0 0 0 fˆi (ˆbi − 0 ) = − bi (ˆbi − 0 ) + wii g(ˆbi − 0 ) + ki+ 0 0 0 0 = − b (ˆb − 0 ) + w g(ˆb − 0 )

0 0 0 0 0 dxi (ˆbi − 0 ) = − bi (ˆbi − 0 ) + wii g(ˆbi − 0 ) dt n X 0 0 + wij gj (xj (ˆbi − 0 )) + ui

urn a

00 00 00 00 00 00 00 fˇi (ˇbi + 0 ) = − bi (ˇbi + 0 ) + wii g(ˇbi + 0 ) + ki − 00 00 00 00 = − bi (ˇbi + 0 ) + wii g(ˇbi + 0 ) (20) n X 00 − | wij | +ui > 0.

j=1,j6=i

Note that xi (0) ∈ (−∞, ˆbi − 0 ]. Hence, xi (t) would never 0 go out of (−∞, ˆbi − 0 ]. 

j=1,j6=i

In addition, by | g(·) |≤ 1, we obtain that from (20)

00

Lemma 6 For i ∈ N1 ∪N2 , xi (t) with xi (0) ∈ [T +0 , ˆbi − 00 0 ] can not escape from [T + 0 , ˆbi − 0 ].

Jo

j=1,j6=i

Proof. For i ∈ N1 ∪ N2 ∪ N3 ∪ N4 , from Lemma 4, we have that

0 0 0 0 ≤ − bi (ˆbi − 0 ) + wii g(ˆbi − 0 ) n X 0 + | wij | +ui < 0.

00 00 00 00 00 dxi (ˇbi + 0 ) = − bi (ˇbi + 0 ) + wii g(ˇbi + 0 ) dt n X 00 00 + wij g(xj (ˇbi + 0 )) + ui

Proof. For i ∈ N1 ∪ N2 , from Lemma 4, we have that 00 00 00 00 fˇi (T + 0 ) = − bi (T + 0 ) + wii g(T + 0 ) + ki −

j=1,j6=i 00

00

= − bi (T + 0 ) + wii g(T + 0 ) n X 00 − | wij | +ui > 0

00 00 00 00 ≤ − bi (ˆbi − 0 ) + wii g(ˆbi − 0 ) n X 00 + | wij | +ui < 0.

Lemma 7 For i ∈ N1 ∪ N2 ∪ N3 ∪ N4 , xi (t) with xi (0) ∈ 00 00 [ˇbi + 0 , +∞) can not escape from [ˇbi + 0 , +∞).

j=1,j6=i

00

j=1,j6=i

00 Note that xi (0) ∈ [T + 0 , ˆbi − 0 ]. Hence, xi (t) would 00  never go out of [T + 0 , ˆbi − 0 ].

In addition, by | g(·) |≤ 1, we obtain that from (17)

0

00

≥ − bi (T + 0 ) + wii g(T + 0 ) n X 00 − | wij | +ui > 0,

i

Proof. For i ∈ N1 ∪ N3 , from Lemma 4, we have that

pro of

00

0 ˜ l = {η ∈ R | T + 0 ≤ η ≤ ˆb00 − 0 }, R | η ≤ ˆbi − 0 }, Ω i iN1 00 ˜ r = {η ∈ R | η ≥ ˇb00 + 0 }. Herein, we choose 0 and Ω i iN1 small enough. 0 0 00 00 ˜ i = iN2 , α ˜ l = {η ∈ When i ∈ N2 , N ˜ i = l , l or r , Ω iN2 00 ˜ l = {η ∈ R | T + 0 ≤ η ≤ ˆb00 − 0 }, R | η ≤ T − 0 }, Ω

(18)

00

j=1,j6=i

j=1,j6=i

8

00

00

≥ − bi (ˇbi + 0 ) + wii g(ˇbi + 0 ) n X 00 − | wij | +ui > 0.

Journal Pre-proof 00

Note that xi (0) ∈ [ˇbi + 0 , +∞). Hence, xi (t) would never 00 go out of [ˇbi + 0 , +∞).  0

In addition, by | g(·) |≤ 1, we obtain that from (23) 0 0 dxi (T − 0 ) = − bi (T − 0 ) + wii g(T − 0 ) dt n X 0 + wij g(xj (T − 0 )) + ui

00

Lemma 8 For i ∈ N3 , xi (t) with xi (0) ∈ [ˇbi + 0 , ˆbi − 0 ] 0 00 can not escape from [ˇbi + 0 , ˆbi − 0 ]. Proof. For i ∈ N3 , from Lemma 4, we have that 0

0

0

0

0

ii

i

fˇi (ˇbi + 0 ) = − bi (ˇbi + 0 ) + wii g(ˇbi + 0 ) + ki− 0 0 0 0 = − b (ˇb + 0 ) + w g(ˇb + 0 ) i

−

i

n X

j=1,j6=i 0

(21)

j=1,j6=i

0

j=1,j6=i

| wij | +ui > 0

Note that xi (0) ∈ (−∞, T − 0 ]. Hence, xi (t) would never go out of (−∞, T − 0 ]. 

and 00 00 00 00 00 00 00 fˆi (ˆbi − 0 ) = − bi (ˆbi − 0 ) + wii g(ˆbi − 0 ) + ki + 00 00 00 00 = − bi (ˆbi − 0 ) + wii g(ˆbi − 0 ) (22) n X 00 + | wij | +ui < 0.

Theorem 2 Assume that T ∈ (−∞, 0) and |N1 | + |N2 | + ˜ α˜ in |N3 | + |N4 | = n. If condition (9) holds, then each Ω ˜ Ω is positively invariant. Furthermore, if 0

bi >

j=1,j6=i

and

0

00

bi >

0

0

| wij |

00

dg(ˆ aj ) , i = 1, 2, · · · , n dη

| wij | max{ 00

| wij |

00

00

00

cj ) dg(ˆ aj ) dg(ˇ dη , dη },

dg(ˇ cj ) dη ,

(24)

i ∈ N1 ∪ N2 ,

i ∈ N3 ∪ N4 ,

(25) then there exists an asymptotically stable equilibrium in ˜ α˜ for system (1), and the number of stable equilibria each Ω |N1 |+|N2 |+|N3 | |N4 | is 3 2 .

lP

0 0 0 0 ≥ − bi (ˇbi + 0 ) + wii g(ˇbi + 0 ) n X 0 | wij | +ui > 0 −

j=1,j6=i

and 00

˜ α˜ in Proof. According to Lemmas 5, 6, 7, 8, 9, each Ω ˜ is positively invariant. From Theorem 1, for each Ω ˜ α˜ , Ω there exists an equilibrium x ¯ = (¯ x1 , x ¯2 , · · · , x ¯n )T and a ˜ α˜ . region Ωα such that x ¯ ∈ Ωα ⊆ Ω Next, we will prove that equilibrium x ¯ is asymptotically stable in two cases. Case 1: x ¯i 6= T , i = 1, 2, · · · , n. The linearized system of (1) at equilibrium x ¯ is ( Pn 0 dg(¯ 0 x ) −bi xi + j=1 wij dη j xj , xi < T, dxi = P 00 00 dg(¯ x ) n dt −bi xi + j=1 wij dη j xj , xi > T.

urn a

00 00 00 00 dxi (ˆbi − 0 ) = − bi (ˆbi − 0 ) + wii g(ˆbi − 0 ) dt n X 00 00 + wij g(xj (ˆbi − 0 )) + ui

j=1,j6=i 00

00

00

≤ − bi (ˆbi − 0 ) + wii g(ˆbi − 0 ) n X 00 + | wij | +ui < 0. j=1,j6=i

That is

00 Note that xi (0) ∈ [ˇbi + 0 , ˆbi − 0 ]. Hence, xi (t) would 0 00 never go out of [ˇbi + 0 , ˆbi − 0 ]. 

Jo

00

00

00

00

00

x1 ) dg(¯ x2 ) dg(¯ xi ) dg(¯ xn ) or (wi1 dg(¯ dη , wi2 dη , · · · , wii dη −bi , · · · , win dη ). For j ∈ N1 ∪ N2 , since dg(η)/dη is positive, concave down and has its maximal value at η = 0, we have that 00 00 ˜ l j or Ω ˜r j, when x ¯j ∈ Ω N N

Proof. For i ∈ N2 ∪ N4 , from Lemma 4, we have that 0 0 0 0 fˆi (T − 0 ) = − bi (T − 0 ) + wii g(T − 0 ) + ki+ 0

= − bi (T − 0 ) + wii g(T − 0 ) n X 0 + | wij | +ui < 0.

dx(t) = Ax(t), dt

where x = (x1 , x2 , · · · , xn )T , A = (AT1 , AT2 , · · · , ATn )T , 0 0 dg(¯ 0 dg(¯ 0 0 dg(¯ x1 ) x2 ) xi ) xn )) Ai = (wi1 dg(¯ dη , wi2 dη , · · · , wii dη −bi , · · · , win dη

Lemma 9 For i ∈ N2 ∪ N4 , xi (t) with xi (0) ∈ (−∞, T − 0 ] can not escape from (−∞, T − 0 ].

0

j=1

Pn

j=1

j=1,j6=i

0

 Pn

re-

0 0 0 0 dxi (ˇbi + 0 ) = − bi (ˇbi + 0 ) + wii g(ˇbi + 0 ) dt n X 0 0 + wij g(xj (ˇbi + 0 )) + ui

n X j=1

In addition, by | g(·) |≤ 1, we obtain that from (21) and (22)

00

0

≤ − bi (T − 0 ) + wii g(T − 0 ) n X 0 + | wij | +ui < 0.

0

pro of

0

(23)

00

max{

j=1,j6=i

9

00

dg(ˆ aj ) dg(ˇ cj ) dg(¯ xj ) , }> . dη dη dη

Journal Pre-proof 00

˜r j, Similarly, for j ∈ N3 ∪ N4 , when x ¯j ∈ Ω N

For i ∈ N3 , we have 0 0 0 0 < fˇi (T ) ≤ lim − Φi (xi ) = Φi (T ) ≤ fˆi (T ),

00

dg(ˇ cj ) dg(¯ xj ) > . dη dη

xi →T

and

It follows from (25) that 00

bi >

j=1

00 00 00 fˇi (T ) ≤ lim Φi (xi ) = Φi (T ) ≤ fˆi (T ) < 0.

xi →T +

dg(¯ xj ) . dη

00

| wij |

Therefore, there exits a positive constant ˜ > 0 such that 0

0 < x˙ i = Φi (xi ), xi ∈ (T − ˜, T )

00

Besides, due to wii > 0 and dg(η)/dη > 0, we can obtain that: 00 dg(¯ xi ) − bi + wii dη

n X

00

dg(¯ xj ) < 0. | wij | dη 00

j=1,j6=i

and

(26)

Therefore, there exists a t1 such that xi (t; t0 , x0 ) = T, t ≥ t1 , x(t0 ) = x0 ∈ (T − ˜, T + ˜).

j=1,j6=i

Denote N31 = {i ∈ N3 , x ¯i 6= T }, N32 = {i ∈ N3 , x ¯i = T } and y¯ is new vector obtained by deleting all components x ¯i with i ∈ N32 from x ¯. Next, we consider the following system in the neighbourhood of the equilibrium y¯ for t ≥ t1

00 00 dg(¯ xi ) dg(¯ xj ) < bi − wii , dη dη

00

| wij |

and 00

00

−bi + wii

dg(¯ xi ) < 0. dη

dxi (t) = − bi (xi (t))xi (t) dt n X +

re-

n X

dg(η) dη

is positive, concave down and has its 0 ˜l j, maximal value at η = 0, we have that when x ¯j ∈ Ω N 0

dg(ˆ aj ) dg(¯ xj ) > . dη dη

0

bi >

n X j=1

dg(¯ xj ) | wij | . dη 0

0

wii

dg(¯ xi ) − bi + dη 0

n X

j=1,j6=i

0

| wij |

> 0, we have

| wij |

(28)

ωij (xi (t))gj (T )

dg(¯ xj ) < 0. dη

+ ui , i ∈ / N32 .

By similar discussion to that in Case 1, y¯ is asymptotically stable. Therefore, x ¯ is asymptotically stable. 

(27)

Remark 2 According to Theorem 2, if |Nk | = n (k = 1, 2, 3), there exist 5n equilibrium points in system (1) and 3n of them are asymptotically stable. If |N4 | = n, there exist 3n equilibrium points in system (1) and 2n of them are asymptotically stable. It means that the numbers of equilibrium points and stable equilibrium points depend on 0 00 0 00 the values of fˆi , fˆi , fˇi and fˇi at switching threshold T .

dg(¯ xi ) < 0. dη

0 0 00 00 Remark 3 If fˆi (T )fˇi (T ) < 0 or fˆi (T )fˇi (T ) < 0, it is hard to determine the number of equilibrium points and these cases are not addressed. However, it can be found from the proofs of Theorems 1 and 2 that the numbers of equilibrium points and stable equilibrium points do not exceed 5n and 3n , respectively.

Jo

0

−bi + wii

ωij (xi (t))gj (xj (t))

The linearized system of (28) at equilibrium y¯ is ( P 0 dg(¯ 0 xj ) −bi xi + j ∈N i∈ / N32 , xi < T, dxi / 32 wij dη xj , = P 00 dg(¯ 00 xj ) dt w −bi xi + j ∈N x , i ∈ / N32 , xi > T. j / 32 ij dη

0 0 dg(¯ dg(¯ xj ) xi ) < bi − wii , dη dη

and 0

0

j=1,j6=i

That is to say: n X

dg(η) dη

j∈N1 ∪N2 ∪N31 ∪N4 n X

j∈N32

urn a

Besides, due to wii > 0 and

+

lP

It follows from (24) that

0

00

x˙ i = Φi (xi ) < 0, xi ∈ (T, T + ˜).

That is to say:

Similarly, since

pro of

n X

By the definition of strictly diagonally dominant matrix, we can obtain that A is a strictly diagonally dominant matrix. From Lemma 3, all the eigenvalues of A have negative real parts. Therefore, equilibrium x ¯ is asymptotically stable. Case 2: there exists at least one component x ¯i = T . 10

Journal Pre-proof Theorem 3 Assume that T = 0 and |N1 | + |N2 | + |N3 | + |N4 | = n. If condition (29) holds, and for i = 1, 2, · · · , n,

Remark 4 As revealed in Cheng et al. (2006, 2007, 2015), Huang & Cao (2010), there are 3n equilibrium points in an n-neuron neural network model with sigmoidal activation function, and 2n of them are asymptotically stable. However, as shown in Theorem 2, more stable equilibrium points can be made by properly selecting switching threshold for an n-neuron neural network model with sigmoidal activation functions under state-dependent switching. If 0 00 0 00 bi = bi , wij = wij , for i, j = 1, 2, · · · , n, the multistability results are applicable for non-switching neural networks. In other words, traditional neural networks can be considered as special cases of switched neural networks under state-dependent switching.

0

bi >

j=1

and 00

bi >

0

00

00

0

fi (pi ) dη

=

00

fi (qi ) dη

00

0

00

T < qi such that 1, 2, · · · , n.

0

lP

0

00

0

00

there exist two points pi and qi with pi < σi < qi such dη

=

dη

= 0 for i = 1, 2, · · · , n.

Denote

i

0

i

<

and

i

0

i

i

i

i

<

00

bi 00 wii dgi (η) maxη∈R dη

0 dfi (pi )

dη

i

0

i

i

<

<

dgi (T ) dη

=

dgi (σi ) dη , 0

0

00

0

=

0 dfi (qi )

dη

=

00 dfi (pi )

dη

= 0 for i =

i

i

i

i

00

i

i

i

i

i

0 00 N3 = {i :fˇi (T ) > 0, fˇi (T ) < 0, 0 0 0 0 00 00 fˆ (p ) < 0, fˇ (q ) > 0, fˇ (q ) > 0},

i

i

i

i

i

i

i

and

00

i

= 0 for i =

N2 = {i :fˇi (T ) < 0, fˆi (T ) > 0, 0 0 0 0 00 00 fˆ (p ) < 0, fˇ (q ) > 0, fˇ (q ) > 0},

0

N4 = {i :fˆi (T ) < 0, fˆi (T ) < 0, 0 0 00 00 fˆ (p ) < 0, fˇ (q ) > 0}. i

dη

0 00 N1 = {i :fˆi (T ) > 0, fˆi (T ) > 0, 0 0 0 0 00 00 fˆ (p ) < 0, fˇ (q ) > 0, fˇ (q ) > 0},

0 00 N3 = {i :fˇi (T ) > 0, fˆi (T ) < 0, 0 0 00 00 fˆ (p ) < 0, fˇ (q ) > 0},

i

=

Denote

Jo

i

0 bi 0 wii

00

00

i

dη

dgi (η) dη

0

N2 = {i :fˆi (T ) < 0, fˇi (T ) > 0, 0 0 00 00 fˆ (p ) < 0, fˇ (q ) > 0}, i

=

dη

0 = inf η∈R

T < qi such that 1, 2, · · · , n.

0 00 N1 = {i :fˇi (T ) > 0, fˇi (T ) > 0, 0 0 00 00 fˆ (p ) < 0, fˇ (q ) > 0},

i

0

there exist three points pi , qi and qi with pi < σi < qi <

urn a

that

0

00 dfi (pi )

Remark 6 Consider T > σi in system (1) with activation function (3). Under the following conditions:

dgi (η) dgi (η) b dgi (σi ) 0 = inf < i00 < max = , η∈R η∈R dη dη dη wii 00 fi (qi )

00

0 dfi (qi )

Proof. By similar discussion to that in Lemma 4, we can obtain the conclusion of Lemma 11. 

Remark 5 Consider T = σi in system (1) with activation function (3). Under the following conditions: dgi (η) b dgi (σi ) dgi (η) 0 = inf < i0 < max = , η∈R η∈R dη dη dη wii

0 dfi (pi )

re-

Proof. By similar discussion to that in Lemma 4, we can obtain the conclusion of Lemma 10. 

0 fi (pi )

0

(30)

there exist three points pi qi and qi with pi < 0 < qi <

= 0 for i = 1, 2, · · · , n.

00

0

bε 1 b ε 0 < i 00 < h(g(T )) < i0 < , 4 wii wii

then there exist two points pi and qi with pi < 0 < qi such that

00

dg(ˇ cj ) | wij | , dη 00

3.3. Multistability Analysis When T ∈ (0, ∞) Lemma 11 If

(29)

0

dg(ˆ aj ) dη

then there exist 5|N3 | 3|N1 |+|N2 |+|N4 | equilibria for system (1), 3|N3 | 2|N1 |+|N2 |+|N4 | of them are asymptotically stable. Proof. By applying the same method with that in Theorems 1 and 2, we can obtain the conclusion of Theorem 3. 

00

1 1 b ε bi ε < , 0 < i 00 < , 0 4 4 wii wii

0

0

| wij |

pro of

0<

n X j=1

3.2. Multistability Analysis When T = 0 Lemma 10 If 0

n X

00

N4 = {i :fˆi (T ) < 0, fˇi (T ) < 0, 0 0 0 0 00 00 fˆ (p ) < 0, fˇ (q ) > 0, fˇ (q ) > 0}.

i

i

11

i

i

i

i

i

Journal Pre-proof Theorem 4 Assume that T ∈ (0, +∞) and |N1 | + |N2 | + |N3 | + |N4 | = n. If condition (30) holds, n X j=1

and 0

bi >

 Pn

j=1

Pn

j=1

00

00

| wij |

0

dg(ˇ cj ) , i = 1, 2, · · · , n dη

| wij | max{ 0

| wij |

0

0 dg(ˆ aj )

dη

dg(ˆ aj ) dη ,

,

0 dg(ˇ cj )

dη

b2 (t) = }, i ∈ N2 ∪ N4 ,

w11 (t) =

i ∈ N1 ∪ N3 ,

then there exist 5|N2 |+|N3 |+|N4 | 3|N1 | equilibrium points for system (1), 3|N2 |+|N3 |+|N4 | 2|N1 | of them are asymptotically stable.

w12 (t) =

Proof. By applying the same method with that in Theorems 1 and 2, we can obtain the conclusion of Theorem 4. 

and

Remark 7 According to Theorems 2, 3, and 4, the number of stable equilibrium points depends on the value of the switching threshold T or the upper and lower functions at T . For example, when |N1 | = n, the number of stable equilibria is 3n if T < 0 and 2n if T ≥ 0. Therefore, a switching rule can be designed for desirable number of stable equilibria.

if − ∞ < η ≤ pi , − pi ), if pi ≤ η ≤ qi , if qi ≤ η < ∞,

0.2, 0.1,

if x1 ≤ −1, if x1 > −1,

w21 (t) =



0.5, if x2 ≤ −1, 0.1, if x2 > −1

w22 (t) =



40, 12,

if x2 ≤ −1, if x2 > −1.

0

00

1 b ε 1 h(g(T )) = h(g(−1)) < 100 = < , 5 4 w11

re-

vi −ui qi −pi (η



30, if x1 ≤ −1, 10, if x1 > −1,

1 b ε < h(g(T )) = h(g(−1)) ≈ 0.105, 0 < 10 = 30 w11

0

b ε 1 0 < 20 = < h(g(T )) = h(g(−1)) ≈ 0.105 40 w22

and

(31)

where ui , vi , pi , qi are constants with ui < vi and pi < qi for i = 1, 2, · · · , n. Because the state space can be partitioned at the switching points and the breakpoints, we can still study the multistability without the differentiability of the breakpoints, see Guo et al. (2019). In the paper, only one switching threshold is considered. By similar analysis method, we can study the multistability of system (1) with more than one switching threshold. Then, more stable equilibrium points can be obtained by properly choosing switching thresholds.

00

5 b ε 1 h(g(T )) = h(g(−1)) < 200 = < , 24 4 w22

lP

  ui , ui + gi (η) =  vi ,



if x2 ≤ −1, if x2 > −1,

2, 5,

Since

Consider a piecewise-linear activation function defined as:



pro of

00

bi >

Example 1 Consider system (32) with the parameters as follows: u1 = −5, u2 = −6,  2, if x1 ≤ −1, b1 (t) = 4, if x1 > −1,

urn a

condition (9) holds. 6

4

2

x2(t)

0

−2

4. Numerical Simulation

−4

In this section, four illustrative examples are elaborated to substantiate the theoretical results. System (1) with two neurons has the following form:  x˙ 1 (t) = −b1 (x1 (t))x1 (t) + w11 (x1 (t))g((x1 (t))     +w12 (x1 (t))g((x2 (t)) + u1 , (32)  x ˙ (t) = −b 2 2 (x2 (t))x2 (t) + w21 (x2 (t))g((x1 (t))    +w22 (x2 (t))g((x2 (t)) + u2 ,

−6

−8 −8

−6

−4

−2

0

2

4

6

Jo

x1(t)

Fig. 7. Phase plot of state variable (x1 , x2 )T in Example 1.

From (11) and (12), we have

where the activation function g(·) is defined as (4) with  = 0.5. First, consider Case A: T ∈ (−∞, 0).

fˆ1 (x1 ) = 12



0 fˆ1 (x1 ) = −2x1 + 30g(x1 ) − 4.8, 00 fˆ (x1 ) = −4x1 + 10g(x1 ) − 4.9,

1

x1 ≤ −1, x1 > −1,

Journal Pre-proof

fˇ1 (x1 ) =



0 fˇ1 (x1 ) = −2x1 + 30g(x1 ) − 5.2, 00 fˇ1 (x1 ) = −4x1 + 10g(x1 ) − 5.1,

x1 ≤ −1, x1 > −1,

fˆ2 (x2 ) =



0 fˆ2 (x2 ) = −2x2 + 40g(x2 ) − 5.5, 00 fˆ (x2 ) = −5x2 + 12g(x2 ) − 5.9,

x2 ≤ −1, x2 > −1,

2

a ˇi -2.4995 -3.2180

0 fˇ2 (x2 ) = −2x2 + 40g(x2 ) − 6.5, 00 fˇ2 (x2 ) = −5x2 + 12g(x2 ) − 6.1,

00

ˇb00 i 0.1017 0.1021

a ˇi -0.9494 -0.8481

00

cˇi 0.8163 0.7211

0 0 0 fˆ1 (p1 ) = fˆ1 (−1.665) ≈ −0.433 < 0,

00 00 00 fˆ1 (p1 ) = fˆ1 (−0.481) ≈ −0.211 < 0, 00 00 00 fˇ1 (q1 ) = fˇ1 (0.481) ≈ 0.211 > 0,

x2 ≤ −1, x2 > −1.

pro of



ˇb0 i -1.0978 -1.0438

00 00 fˇ2 (T ) = fˇ2 (−1) ≈ 0.328 > 0,

and

fˇ2 (x2 ) =

Table 3. Zeros of fˇ1 and fˇ2

0

i 1 2

0 0 0 fˆ2 (p2 ) = fˆ2 (−1.818) ≈ −0.837 < 0,

00 00 00 fˆ2 (p2 ) = fˆ2 (−0.434) ≈ −0.182 < 0

and

10

x1(t)

5

Therefore, N1 = {1, 2}. Since

0 −5 −10

0

2

4

6

8

10

0

x2(t)

5

re-

10

00

00

b1 = 4 > 2.767 ≈| w11

0 −5

0

2

4

6

8

10

lP

t

Fig. 8. Transient behaviors of x1 and x2 in Example 1.

0 dg(ˆ a2 ) dg(ˆ a1 ) + | w12 | , dη dη 00

00

00

00

dg(ˆ a1 ) dg(ˇ c1 ) | max{ , } dη dη

00

+ | w12 | max{

0

0

0

0

b1 = 2 > 0.683 ≈| w11 |

t

−10

00 00 00 fˇ2 (q2 ) = fˇ2 (0.434) ≈ 0.182 > 0.

dg(ˆ a2 ) dg(ˇ c2 ) , }, dη dη

0

0

b2 = 2 > 0.406 ≈| w21 |

0

0 dg(ˆ a1 ) dg(ˆ a2 ) + | w22 | , dη dη

and

Table 1. Local extreme points of fˆi and fˇi

i 1 2

0

0

pi -1.665 -1.818

00

pi -0.481 -0.434

00

qi 0.481 0.434

urn a

i 1 2

Table 2. Zeros of fˆ1 and fˆ2

a ˆi -2.2278 -2.6508

ˆb0 i -1.2392 -1.2596

00

a ˆi -0.8163 -0.7211

ˆb00 i -0.1017 -0.1021

00

00

b2 = 5 > 3.739 ≈| w21 | max{ 00

+ | w22 | max{

00

00

00

00

c1 ) dg(ˆ a1 ) dg(ˇ , } dη dη dg(ˆ a2 ) dg(ˇ c2 ) , }, dη dη

conditions (24) and (25) hold. According to Theorem 2, there exist 32 (nine) stable equilibria of system (32). Fig. 7 shows a phase plot with 400 random initial values, where all trajectories converge to nine stable equilibria. Fig. 8 shows the transient behaviors of states x1 and x2 .

00

cˆi 0.9494 0.8481

00

0

00

0

00

0

00

Let b1 = b1 = 2, b2 = b2 = 2, w11 = w11 = 30, w12 = 00 0 00 0 w12 = 0.2, w21 = w21 = 0.5, w22 = w22 = 40, the above model degenerates into a neural network without switching. According to the results in Cheng et al. (2006, 2007, 2015), Huang & Cao (2010), there exist 22 (four) stable equilibria. Fig. 9 shows the phase plot with 100 random initial values, in contrast to nine stable equilibria in a switched neural network shown in Fig. 7. In comparison, the results herein generalize improve the existing ones.

Tables 1, 2 and 3 list all local extreme points and zeros of the upper functions and lower functions. Moreover, the values of the upper functions and lower functions at the switching threshold and all local extreme points can be computed as follows:

Jo

0

0 0 fˇ1 (T ) = fˇ1 (−1) ≈ 0.370 > 0, 00 00 fˇ1 (T ) = fˇ1 (−1) ≈ 0.090 > 0, 0 0 fˇ2 (T ) = fˇ2 (−1) ≈ 0.260 > 0,

13

Journal Pre-proof 20

10 5

x1(t)

15

0 −5

10

x2(t)

−10

0

20

40

60

80

100

80

100

t

5 10 0

x2(t)

5

−5

0

pro of

−5

−10

−10 −10

−5

0

5

10

15

0

20

40

60

t

x1(t) 00

0

00

Fig. 9. Phase plot of state variable (x1 , x2 )T when bi = bi , wij =

Fig. 11. Transient behaviors of x1 and x2 in Example 2.

0

wij in Example 1.

By similar computation as that in Example 1, we have that N3 = {1}, N4 = {2} and all conditions in Theorem 2 are satisfied. Therefore, system (32) has fifteen equilibria, six of them are stable. Fig. 10 shows a phase plot with 400 random initial values, where six stable equilibria are obvious. Fig. 11 shows the transient behaviors of states x1 and x2 .

re-

Example 2 Consider system (32) with the following parameters: u1 = −2.11, u2 = −4,  1, if x1 ≤ −1, b1 (t) = 1.5, if x1 > −1,  1, if x2 ≤ −1, b2 (t) = 2, if x2 > −1,  10, if x1 ≤ −1, w11 (t) = 5, if x1 > −1,  0.01, if x1 ≤ −1, w12 (t) = 0.001, if x1 > −1,  1, if x2 ≤ −1, w21 (t) = 0.5, if x2 > −1, 

8, 8,

6

4

2

2

x (t)

0

−6

−8 −8

−6

Jo

−2

−4

if x2 ≤ −1, if x2 > −1.

−4

lP

w22 (t) =

Example 3 Consider the system (32) with the parameters as follows: u1 = −1.6, u2 = −7,  1, if x1 ≤ 0, b1 (t) = 1.5, if x1 > 0,

urn a

and

Next, consider Case B: T = 0.

−2

0

2

4

Fig. 10. Phase plot of state variable (x1 , x2



w11 (t) = w12 (t) =





2, if x2 ≤ 0, 3, if x2 > 0, 4, if x1 ≤ 0, 4, if x1 > 0, 0.1, 0.2,

if x1 ≤ 0, if x1 > 0, if x2 ≤ 0, if x2 > 0

w21 (t) =



2, 1.5,

w22 (t) =



20, if x2 ≤ 0, 25, if x2 > 0.

and

By similar computation to that in Example 1, we have that N1 = {1, 2} and all conditions in Theorem 3 are satisfied. Therefore, system (32) has nine equilibria, four of them are stable. Figs. 12 and 13 show the simulation results with 400 random initial values, where four stable equilibria are obvious.

6

x1(t)

)T

b2 (t) =

Finally, consider Case C: T ∈ (0, +∞).

in Example 2.

14

Journal Pre-proof Example 4 Consider system (32) with the parameters as follows: u1 = −5, u2 = −7.5,  4, if x1 ≤ 1, b1 (t) = 1, if x1 > 1, b2 (t) =

8 6

w11 (t) =

4

w12 (t) =

0 −2 −4

and

−6 −8 −8

−6

−4

−2

0

2

4

6

x1(t)





6, if x2 ≤ 1, 1, if x2 > 1, if x1 ≤ 1, if x1 > 1,

10, 6.8,

0.1, if x1 ≤ 1, 0.01, if x1 > 1,

pro of

2

x (t)

2



w21 (t) =



0.1, if x2 ≤ 1, 0.05, if x2 > 1,

w22 (t) =



15, 9.5,

if x2 ≤ 1, if x2 > 1.

6

Fig. 12. Phase plot of state variable (x1 , x2 )T in Example 3.

re-

4

2

x2(t)

0

−2

lP

−4

10

−5

0

20

40

urn a

x1(t)

0

60

80

−4

−2

0

2

4

6

10 5

5 0 −5

0

20

40

60

80

x1(t)

10

x2(t)

−6

x1(t)

100

t

−10

−8 −8

Fig. 14. Phase plot of state variable (x1 , x2 )T in Example 4.

5

−10

−6

0 −5 −10

0

2

4

6

8

10

6

8

10

t 10

100

5

x2(t)

Jo

t

Fig. 13. Transient behaviors of x1 and x2 in Example 3.

0 −5 −10

0

2

4

t

Fig. 15. Transient behaviors of x1 and x2 in Example 4.

15

Journal Pre-proof By similar computation as that in Example 1, we have that N1 = {1, 2} and all conditions in Theorem 4 are satisfied. Therefore, there are 25 equilibria for system (32), and nine of them are stable. Fig. 14 shows a phase plot with 400 random initial values, where nine stable equilibria are obvious. Fig. 15 shows the transient behaviors of states x1 and x2 .

Gang, W., Zhang, H., & Dong, F. (2007). Stability analysis of stochastic Cohen-Grossberg neural networks with mixed time delays. Journal of Northe Astern University, 183 , 464–470. Guo, Z., Liu, L., & Wang, J. (2018). Multistability of recurrent neural networks with piecewise-linear radial basis functions and state-dependent switching parameters. IEEE Transactions on Systems, Man, and Cybernetics: Systems doi: 10.1109/TSMC.2018.2853138. Guo, Z., Liu, L., & Wang, J. (2019). Multistability of switched neural networks with piecewise-linear activation functions under statedependent switching. IEEE Transactions on Neural Networks and Learning Systems, 30 , 2052–2066. Guo, Z., Wang, J., & Yan, Z. (2014). Attractivity analysis of memristor-based cellular neural networks with time-varying delays. IEEE Transactions on Neural Networks and Learning Systems, 25 , 704–717. Hu, Q., Fei, Q., Ma, H., Wu, Q., & Geng, Q. (2015). Identification for switched systems. Ifac Papersonline, 48 , 514–519. Huang, G., & Cao, J. (2010). Delay-dependent multistability in recurrent neural networks. Neural Networks, 23 , 201–209. Isokawa, T., Nishimura, H., Kamiura, N., & Matsui, N. (2008). Associative memory in quaternionic Hopfield neural network. International Journal of Neural Systems, 18 , 135. Kahloul, A. A., & Sakly, A. (2018). Constrained parameterized optimal control of switched systems based on continuous Hopfield neural networks. International Journal of Dynamics and Control, 6 , 262–269. Kwan, H. K., & Cai, Y. (1994). A fuzzy neural network and its application to pattern recognition. IEEE Transactions on Fuzzy Systems, 2 , 185–193. Li, H. X., Huang, H., & Qu, Y. (2005). Robust stability analysis of switched Hopfield neural networks with time-varying delay under uncertainty. Physics Letters A, 345 , 345–354. Li, P., & Cao, J. (2007). Global stability in switched recurrent neural networks with time-varying delay via nonlinear measure. Nonlinear Dynamics, 49 , 295–305. Li, Q., Zhao, J., Dimirovski, G. M., & Liu, X. (2009). Tracking control for switched linear systems with time-delay: a statedependent switching method. Asian Journal of Control, 11 , 517– 526. Lian, J., & Wang, J. (2015). Passivity of switched recurrent neural networks with time-varying delays. IEEE Transactions on Neural Networks and Learning Systems, 26 , 357–366. Liberzon, D., & Morse, A. S. (2015). Basic problems in stability and design of switched systems. IEEE Control Systems Magazine, 19 , 59–70. Liu, H., Shen, Y., & Zhao, X. (2013). Finite-time stabilization and boundedness of switched linear system under state-dependent switching. Journal of the Franklin Institute, 350 , 541–555. Liu, P., Zeng, Z., & Wang, J. (2016a). Multistability analysis of a general class of recurrent neural networks with non-monotonic activation functions and time-varying delays. Neural Networks, 79 , 117–127. Liu, P., Zeng, Z., & Wang, J. (2016b). Multistability of recurrent neural networks with nonmonotonic activation functions and mixed time delays. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 46 , 512–523. Liu, P., Zeng, Z., & Wang, J. (2017a). Complete stability of delayed recurrent neural networks with gaussian activation functions. Neural Networks, 85 , 21–32. Liu, P., Zeng, Z., & Wang, J. (2017b). Multiple Mittag-Leffler stability of fractional-order recurrent neural networks. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47 , 2279– 2288. 24. Liu, P., Zeng, Z., & Wang, J. (2017c). Multistability of delayed recurrent neural networks with Mexican hat activation functions. Neural Computation, 29 , 423–457. Liu, P., Zeng, Z., & Wang, J. (2018). Multistability of recurrent neural networks with nonmonotonic activation functions and unbounded time-varying delays. IEEE Transactions on Neural Networks and Learning Systems, 29 , 3000–3010.

5. Concluding Remarks

pro of

This paper addresses the multistability of a switched neural network model with sigmoidal activation function under state-dependent switching. By analyzing the switching threshold, partitioning state space and linearizing the system, sufficient conditions are derived for ascertaining the coexistence and stability of equilibria. Future works will aim at multistability analysis of switched neural networks with time delays and non-monotone smooth activation functions. References

Jo

urn a

lP

re-

Bao, G., & Zeng, Z. (2013). Multistability of periodic delayed recurrent neural network with memristors. Neural Computing and Applications, 23 , 1963–1967. Bao, H., Cao, J., J, K., A, A., & B, A. (2018a). H state estimation of stochastic memristor-based neural networks with time-varying delays. Neural Networks, 99 , 79–91. Bao, H., Cao, J., & Kurths, J. (2018b). State estimation of fractionalorder delayed memristive neural networks. Nonlinear Dynamics, 94 , 1215–1225. Bao, H., Park, J. H., & Cao, J. (2016). Exponential synchronization of coupled stochastic memristor-based neural networks with timevarying probabilistic delay coupling and impulsive delay. IEEE Transactions on Neural Networks and Learning Systems, 27 , 190– 201. Branicky, M. S. (1998). Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Transactions on Automatic Control, 43 , 475–482. Chen, X., Long, F., & Fu, Z. (2010). Model reference adaptive neural network control for a class of switched nonlinear singular systems. In 3rd International Symposium on Systems and Control in Aeronautics and Astronautics (pp. 55–60). Cheng, C., Lin, K., & Shih, C. (2006). Multistability in recurrent neural networks. SIAM Journal on Applied Mathematics, 66 , 1301–1320. Cheng, C., Lin, K., Shih, C., & Tseng, J. (2015). Multistability for delayed neural networks via sequential contracting. IEEE Transactions on Neural Networks and Learning Systems, 26 , 3109– 3122. Cheng, C. Y., Lin, K. H., & Shih, C. W. (2007). Multistability and convergence in delayed neural networks. Physica D: Nonlinear Phenomena, 225 , 61–74. Daafouz, J., Riedinger, P., & Lung, C. (2002). Stability analysis and control synthesis for switched systems: A switched lyapunov function approach. IEEE Transactions on Automatica Control, 47 , 1883–1887. Di, M. M., Forti, M., & Pancioni, L. (2017). Convergence and multistability of nonsymmetric cellular neural networks with memristors. IEEE Transactions on Cybernetics, 47 , 2970–2983. Duan, S., Hu, X., Dong, Z., Wang, L., & Mazumder, P. (2015). Memristor-based cellular nonlinear neural network: design, analysis, and applications. IEEE Transactions on Neural Networks and Learning Systems, 26 , 1202–1213.

16

Journal Pre-proof Long, F., & Wei, W. (2007). On neural network switched stabilization of SISO switched nonlinear systems with actuator saturation. In 4th International Symposium on Neural Networks (pp. 292–301). Long, F., Xiao, Y., Chen, X., & Zhang, Z. (2013). Disturbance attenuation for nonlinear switched descriptor systems based on neural network. Neural Computing and Applications, 23 , 2211– 2219. Nie, X., & Cao, J. (2015). Multistability of memristive neural networks with non-monotonic piecewise linear activation functions. In 12th International Symposium on Neural Networks (pp. 182– 191). Nie, X., & Zheng, W. (2015). Multistability of neural networks with discontinuous non-monotonic piecewise linear activation functions and time-varying delays. Neural Networks, 71 , 65–79. Niu, B., Wang, D., Li, H., Xie, X., Alotaibi, N. D., & Alsaadi, F. E. (2017). A novel neural-network-based adaptive control scheme for output-constrained stochastic switched nonlinear systems. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 49 , 418–432. Persis, C. D., Santis, R. D., & Morse, A. S. (2003). Switched nonlinear systems with state-dependent dwell-time. Systems and Control Letters, 50 , 291–302. Singh, H. P., & Sukavanam, N. (2012). Simulation and stability analysis of neural network based control scheme for switched linear systems. Isa Transactions, 51 , 105–110. Song, S., Liu, J., & Wang, H. (2018). Adaptive neural network control for uncertain switched nonlinear systems with time delays. IEEE Access, 6 , 22899 – 22907. Song, Y., Xiang, Z., Chen, Q., & Hu, W. (2006). Robust reliable control of switched uncertain systems with time-varying delay. International Journal of Systems Science, 37 , 1077–1087. Suganthan, P. N., Teoh, E. K., & Mital, D. P. (1995). Pattern recognition by graph matching using the Potts MFT neural networks. Pattern Recognition, 28 , 997–1009. Tang, L., & Zhao, J. (2017). Neural network based adaptive prescribed performance control for a class of switched nonlinear systems. Neurocomputing, 230 , 316–321. Tong, W. (1977). On the distribution of eigenvalues of some matrices. Acta Math Sinica, 20 , 272–275. Wang, L. (2014). Stability analysis on neural networks with a class of Mexican-hat-type activation functions. In International Conference on Information Science, Electronics and Electrical Engineering (pp. 1151–1155). Wang, L., & Chen, T. (2012). Multistability of neural networks with Mexican-hat-type activation functions. IEEE Transactions on Neural Networks and Learning Systems, 23 , 1816–1826. Wang, L., Lu, W., & Chen, T. (2009). Multistability of neural networks with a class of activation functions. In 6th International Symposium on Neural Networks (pp. 323–332). Wang, L., Lu, W., & Chen, T. (2010). Coexistence and local stability of multiple equilibria in neural networks with piecewise linear nondecreasing activation functions. Neural Networks, 23 , 189– 200. Wang, L., & Zou, X. (2002). Exponential stability of CohenGrossberg neural networks. Neural Networks, 15 , 415–422. Wang, Z., Liu, Y., Li, M., & Liu, X. (2006). Stability analysis for stochastic Cohen-Grossberg neural networks with mixed time delays. IEEE Transactions on Neural Networks, 17 , 814–820. Wu, Z., Shi, P., Su, H., & Chu, J. (2011). Delay-dependent stability analysis for switched neural networks with time-varying delay. IEEE Transactions on Systems, Man, and Cybernetics, Part B , 41 , 1522–1530. Yu, L., Fei, S., Yu, L., & Fei, S. (2011). Adaptive neural-network tracking stabilization for switched nonlinear systems with disturbances. In Control and Decision Conference (pp. 1391–1394). Yuan, K., Cao, J., & Li, H. (2006). Robust stability of switched Cohen-Grossberg neural networks with mixed time-varying delays. IEEE Transactions on Systems, Man, and Cybernetics, Part B , 36 , 1356–1363. Zeng, Z., Huang, D., & Wang, Z. (2005). Memory pattern analysis

Jo

urn a

lP

re-

pro of

of cellular neural networks. Physics Letters A, 342 , 114–128. Zeng, Z., Huang, T., & Zheng, W. (2016). Multistability of recurrent neural networks with time-varying delays and the piecewise linear activation function. Neurocomputing, 216 , 135–142. Zeng, Z., & Wang, J. (2008). Design and analysis of high-capacity associative memories based on a class of discrete-time recurrent neural networks. IEEE Transactions on Systems, Man, and Cybernetics, Part B , 38 , 1525–1536. Zeng, Z., & Wang, J. (2009). Associative memories based on continuous-time cellular neural networks designed using spaceinvariant cloning template. Neural Networks, 22 , 651–657. Zhang, H., & Wang, Y. (2008). Stability analysis of Markovian jumping stochastic Cohen-Grossberg neural networks with mixed time delays. IEEE Transactions on Neural Networks, 19 , 366–370. Zhang, W., Tang, Y., Miao, Q., & Du, W. (2013). Exponential synchronization of coupled switched neural networks with modedependent impulsive effects. IEEE Transactions on Neural Networks and Learning Systems, 24 , 1316–1326. Zhao, J., Hill, D., & Liu, T. (2009). Synchronization of complex dynamical networks with switching topology: a switched system point of view. Automatica, 45 , 2502–2511. Zhao, X., Yin, Y., & Zheng, X. (2016). State-dependent switching control of switched positive fractional-order systems. ISA Transactions, 62 , 103–108. Zhao, X., & Zhao, J. (2017). Output feedback passification of saturated switched systems under the improved state-dependent switching. International Journal of Systems Science, 48 , 1356– 1366.

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Conflict of Interest

Declaration of interests X The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: