- Research Article
- Open Access

# Global Stability Analysis for Periodic Solution in Discontinuous Neural Networks with Nonlinear Growth Activations

- Yingwei Li
^{1}and - Huaiqin Wu
^{2}Email author

**2009**:798685

https://doi.org/10.1155/2009/798685

© Y.Li and H.Wu. 2009

**Received:**30 December 2008**Accepted:**18 March 2009**Published:**31 March 2009

## Abstract

This paper considers a new class of additive neural networks where the neuron activations are modelled by discontinuous functions with nonlinear growth. By Leray-Schauder alternative theorem in differential inclusion theory, matrix theory, and generalized Lyapunov approach, a general result is derived which ensures the existence and global asymptotical stability of a unique periodic solution for such neural networks. The obtained results can be applied to neural networks with a broad range of activation functions assuming neither boundedness nor monotonicity, and also show that Forti's conjecture for discontinuous neural networks with nonlinear growth activations is true.

## Keywords

- Neural Network
- Periodic Solution
- Global Asymptotical Stability
- Measurable Selection
- Global Exponential Stability

## 1. Introduction

The stability of neural networks, which includes the stability of periodic solution and the stability of equilibrium point, has been extensively studied by many authors so far; see, for example, [1–15]. In [1–4], the authors investigated the stability of periodic solutions of neural networks with or without time delays, where the assumptions on neuron activation functions include Lipschitz conditions, bounded and/or monotonic increasing property. Recently, in [13–15], the authors discussed global stability of the equilibrium points for the neural networks with discontinuous neuron activations. Particularly, in [14], Forti conjectures that all solutions of neural networks with discontinuous neuron activations converge to an asymptotically stable limit cycle whenever the neuron inputs are periodic functions. As far as we know, there are only works of Wu in [5, 7] and Papini and Taddei in [9] dealing with this conjecture. However, the activation functions are required to be monotonic in [5, 7, 9] and to be bounded in [5, 7].

In this paper, without assumptions of the boundedness and the monotonicity of the activation functions, by the Leray-Schauder alternative theorem in differential inclusion theory and some new analysis techniques, we study the existence of periodic solution for discontinuous neural networks with nonlinear growth activations. By constructing suitable Lyapunov functions we give a general condition on the global asymptotical stability of periodic solution. The results obtained in this paper show that Forti's conjecture in [14] for discontinuous neural networks with nonlinear growth activations is true.

For later discussion, we introduce the following notations.

Let , where the prime means the transpose. By (resp., ) we mean that (resp., ) for all . denotes the Euclidean norm of . denotes the inner product. denotes 2-norm of matrix , that is, , where denotes the spectral radius of .

where is the set of Lebesgue measure zero where does not exist, and is an arbitrary set with measure zero.

The rest of this paper is organized as follows. Section 2 develops a discontinuous neural network model with nonlinear growth activations, and some preliminaries also are given. Section 3 presents the proof on the existence of periodic solution. Section 4 discusses global asymptotical stability of the neural network. Illustrative examples are provided to show the effectiveness of the obtained results in Section 5.

## 2. Model Description and Preliminaries

where is the vector of neuron states at time ; is an matrix representing the neuron inhibition; is an neuron interconnection matrix; , , represents the neuron input-output activation and is the continuous -periodic vector function denoting neuron inputs.

Throughout the paper, we assume that

: has only a finite number of discontinuity points in every compact set of . Moreover, there exist finite right limit and left limit at discontinuity point .

where , are constants, and .

: for all , where is a constant.

Under the assumption , is undefined at the points where is discontinuous. Equation (2.1) is a differential equation with a discontinuous right-hand side. For (2.1), we adopt the following definition of the solution in the sense of Filippov [17] in this paper.

Definition 2.1.

is an -periodic solution of (2.1). Hence, for the neural network (2.1), finding the periodic solutions is equivalent to finding solutions of (2.5).

Definition 2.2.

The periodic solution with initial value of the neural network (2.1) is said to be globally asymptotically stable if is stable and for any solution , whose existence interval is , we have .

Lemma 2.3.

- (a)
the set is unbounded;

- (b)
the has a fixed point in , that is, there exists such that .

then is a class of norms of , , and are Banach space under the norm .

If is (i) regular in [16]; (ii) positive definite, that is, for , and ; (iii) radially unbounded, that is, as , then is said to be C-regular.

Lemma 2.4 (Chain Rule [15]).

## 3. Existence of Periodic Solution

Theorem 3.1.

If the assumptions and hold, then for any , (2.1) has at least a solution defined on with the initial value .

Proof.

By the assumption , it is easy to get that : is an upper semicontinuous set-valued map with nonempty, compact, and convex values. Hence, by Definition 2.1, the local existence of a solution for (2.1) on , , with , is obvious [17].

Therefore, let , then, by (3.3), it follows that on . This means that the local solution is bounded. Thus, (2.1) has at least a solution with the initial value on . This completes the proof.

Theorem 3.1 shows the existence of solutions of (2.1). In the following, we will prove that (2.1) has an -periodic solution.

Let for all , then is a linear operator.

Proposition 3.2.

is bounded, one to one and surjective.

Proof.

this implies that is bounded.

Hence . It follows . This shows that is one to one.

in (3.12), then (3.12) is the solution of (3.10). This shows that is surjective. This completes the proof.

By the Banach inverse operator theorem, is a bounded linear operator.

Then has the following properties.

Proposition 3.3.

has nonempty closed convex values in and is also upper semicontinuous from into endowed with the weak topology.

Proof.

Noting that is an upper semicontinuous set-valued map with nonempty closed convex values on for a.e. , . Therefore, . This shows that is nonempty.

This implies , that is, is closed in . The proof is complete.

Theorem 3.4.

Under the assumptions and , there exists a solution for the boundary-value problem (2.5), that is, the neural network (2.1) has an -periodic solution.

Proof.

is a bounded subset of . Since is a bounded linear operator, is a bounded subset of . Noting that is compactly embedded in , is a relatively compact subset of . Hence by Proposition 3.3, is the upper semicontinuous set-valued map which maps bounded sets into relatively compact sets.

This shows that is a bounded subset of .

By the definition of , . Moreover, by Definition 2.1 and (3.29), is a solution of the boundary-value problem (2.5), that is, the neural network (2.1) has an -periodic solution. The proof is completed.

## 4. Global Asymptotical Stability of Periodic Solution

Theorem 4.1.

Suppose that and the following assumptions are satisfied.

where is the minimum eigenvalues of symmetric matrix , , for all . Then the neural network (2.1) has a unique -periodic solution which is globally asymptotically stable .

Proof.

where , , and . Obviously, is a solution of (4.6).

where . Thus, the solution of (4.6) is globally asymptotically stable, so is the periodic solution of the neural network (2.1). Consequently, the periodic solution is unique. The proof is completed.

- (1)
If is nondecreasing, then the assumption obviously holds. Thus the assumption is more general.

- (2)In [14], Forti et al. considered delayed neural networks modelled by the differential equation

has a unique periodic solution and all solutions converge to the asymptotically stable limit cycle when is a periodic function. When , the neural network (4.11) changes as the neural network (2.1) without delays. Thus, without assumptions of the boundedness and the monotonicity of the activation functions, Theorem 4.1 obtained in this paper shows that Forti's conjecture for discontinuous neural networks with nonlinear growth activations and without delays is true.

## 5. Illustrative Example

Example 5.1.

All the assumptions of Theorem 4.1 hold and the neural network in Example has a unique -periodic solution which is globally asymptotically stable.

## Declarations

### Acknowledgments

The authors are extremely grateful to anonymous reviewers for their valuable comments and suggestions, which help to enrich the content and improve the presentation of this paper. This work is supported by the National Science Foundation of China (60772079) and the National 863 Plans Projects (2006AA04z212).

## Authors’ Affiliations

## References

- Di Marco M, Forti M, Tesi A:
**Existence and characterization of limit cycles in nearly symmetric neural networks.***IEEE Transactions on Circuits and Systems I*2002,**49**(7):979-992. 10.1109/TCSI.2002.800481View ArticleGoogle Scholar - Chen B, Wang J:
**Global exponential periodicity and global exponential stability of a class of recurrent neural networks.***Physics Letters A*2004,**329**(1-2):36-48. 10.1016/j.physleta.2004.06.072MATHView ArticleGoogle Scholar - Cao J:
**New results concerning exponential stability and periodic solutions of delayed cellular neural networks.***Physics Letters A*2003,**307**(2-3):136-147. 10.1016/S0375-9601(02)01720-6MATHMathSciNetView ArticleGoogle Scholar - Liu Z, Chen A, Cao J, Huang L:
**Existence and global exponential stability of periodic solution for BAM neural networks with periodic coefficients and time-varying delays.***IEEE Transactions on Circuits and Systems I*2003,**50**(9):1162-1173. 10.1109/TCSI.2003.816306MathSciNetView ArticleGoogle Scholar - Wu H, Li Y:
**Existence and stability of periodic solution for BAM neural networks with discontinuous neuron activations.***Computers & Mathematics with Applications*2008,**56**(8):1981-1993. 10.1016/j.camwa.2008.04.027MATHMathSciNetView ArticleGoogle Scholar - Wu H, Xue X, Zhong X:
**Stability analysis for neural networks with discontinuous neuron activations and impulses.***International Journal of Innovative Computing, Information and Control*2007,**3**(6B):1537-1548.Google Scholar - Wu H:
**Stability analysis for periodic solution of neural networks with discontinuous neuron activations.***Nonlinear Analysis: Real World Applications*2009,**10**(3):1717-1729. 10.1016/j.nonrwa.2008.02.024MATHMathSciNetView ArticleGoogle Scholar - Wu H, San C:
**Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses.***Applied Mathematical Modelling*2009,**33**(6):2564-2574. 10.1016/j.apm.2008.07.022MATHMathSciNetView ArticleGoogle Scholar - Papini D, Taddei V:
**Global exponential stability of the periodic solution of a delayed neural network with discontinuous activations.***Physics Letters A*2005,**343**(1–3):117-128.MATHView ArticleGoogle Scholar - Wu H, Xue X:
**Stability analysis for neural networks with inverse Lipschitzian neuron activations and impulses.***Applied Mathematical Modelling*2008,**32**(11):2347-2359. 10.1016/j.apm.2007.09.002MATHMathSciNetView ArticleGoogle Scholar - Wu H, Sun J, Zhong X:
**Analysis of dynamical behaviors for delayed neural networks with inverse Lipschitz neuron activations and impulses.***International Journal of Innovative Computing, Information and Control*2008,**4**(3):705-715.Google Scholar - Wu H:
**Global exponential stability of Hopfield neural networks with delays and inverse Lipschitz neuron activations.***Nonlinear Analysis: Real World Applications*2009,**10**(4):2297-2306. 10.1016/j.nonrwa.2008.04.016MATHMathSciNetView ArticleGoogle Scholar - Forti M, Nistri P:
**Global convergence of neural networks with discontinuous neuron activations.***IEEE Transactions on Circuits and Systems I*2003,**50**(11):1421-1435. 10.1109/TCSI.2003.818614MathSciNetView ArticleGoogle Scholar - Forti M, Nistri P, Papini D:
**Global exponential stability and global convergence in finite time of delayed neural networks with infinite gain.***IEEE Transactions on Neural Networks*2005,**16**(6):1449-1463. 10.1109/TNN.2005.852862View ArticleGoogle Scholar - Forti M, Grazzini M, Nistri P, Pancioni L:
**Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations.***Physica D*2006,**214**(1):88-99. 10.1016/j.physd.2005.12.006MATHMathSciNetView ArticleGoogle Scholar - Clarke FH:
*Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts*. John Wiley & Sons, New York, NY, USA; 1983:xiii+308.Google Scholar - Filippov AF:
*Differential Equations with Discontinuous Right-Hand Side, Mathematics and Its Applications (Soviet Series)*. Kluwer Academic Publishers, Boston, Mass, USA; 1984.Google Scholar - Aubin J-P, Cellina A:
*Differential Inclusions: Set-Valued Maps and Viability Theory, Grundlehren der Mathematischen Wissenschaften*.*Volume 264*. Springer, Berlin, Germany; 1984:xiii+342.Google Scholar - Aubin J-P, Frankowska H:
*Set-Valued Analysis, Systems and Control: Foundations and Applications*.*Volume 2*. Birkhäuser, Boston, Mass, USA; 1990:xx+461.Google Scholar - Dugundji J, Granas A:
*Fixed Point Theory. Vol. I, Monografie Matematyczne, 61*. Polish Scientific, Warsaw, Poland; 1982.Google Scholar - Papageorgiou NS:
**Convergence theorems for Banach space valued integrable multifunctions.***International Journal of Mathematics and Mathematical Sciences*1987,**10**(3):433-442. 10.1155/S0161171287000516MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.