Multistability and Multiperiodicity for a General Class of Delayed Cohen-Grossberg Neural Networks with Discontinuous Activation Functions

A general class of Cohen-Grossberg neural networks with time-varying delays, distributed delays, and discontinuous activation functions is investigated. By partitioning the state space, employing analysis approach and Cauchy convergence principle, sufficient conditions are established for the existence and locally exponential stability of multiple equilibrium points and periodic orbits, which ensure that n-dimensional Cohen-Grossberg neural networks with k-level discontinuous activation functions can have kn equilibrium points or kn periodic orbits. Finally, several examples are given to illustrate the feasibility of the obtained results.


Introduction
In recent years, great attention has been paid to the neural networks due to their applications in many areas such as signal processing, associative memory, pattern recognition, parallel computation, and optimization.It should be pointed out that the successful applications heavily rely on the dynamic behaviors of neural networks.Stability, as one of the most important properties for neural networks, is crucially required when designing neural networks.As models of human brains, neural networks have memory function that is, the state of the present time is related to the state of the past.Time delay provides information of history.In addition, neural networks have recently been implemented on electronic chips.In electronic implementation of neural networks, time delays are unavoidably encountered during the processing and transmission of signals.As in many other dynamical systems, it is well known that delays may result in oscillation and instability.Hence, to study delayed neural networks, one must address the problem of how to remove this destabilizing effect.There have been a great number of results on stability of delayed neural networks reported in the past two decades.Most of them focus on the uniqueness and global stability (or attractiveness) of the equilibrium, and periodic (or almost-periodic) trajectory of the delayed neural networks.Readers can refer to [1][2][3][4][5][6] and many others.On the other hand, in the applications of associative memory storage, pattern recognition, and decision making, the addressable memories or patterns or decisions are stored as stable equilibria or stable periodic orbits.So it is necessary that there exist multiple stable equilibria or periodic orbits for neural networks, which are usually referred to as multistability or multiperiodicity, respectively.In the last few years, the multistability and multiperiodicity of neural networks have been reported in [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] and the references therein.In particular, [14][15][16][17][18][19][20][21] investigated the multistability or multiperiodicity of neural networks with sigmoidal activation functions or nondecreasing saturated activation functions.In [22], sufficient conditions of the multistability were presented for a class of neural networks with Mexican-hat-type activation functions.[23] studied some multistability properties for a class of bidirectional associative memory recurrent neural networks with unsaturating piecewise linear transfer functions.In order to increase storage capacity, [24,25] investigated the multistability of two classes of neural networks with piecewise linear activation functions.Stability of multiple equilibria of neural networks with time-varying delays and concaveconvex characteristics was formulated in [26].Note that most of the results above were based on the assumption that the activation functions are continuous, whereas, as mentioned in [27] the neural networks with discontinuous activation functions are important and frequently encountered in applications when dealing with dynamical systems possessing high-slope nonlinear elements.For this reason, much effort has been devoted to analyzing the dynamic behavior of neural networks with discontinuous activation functions (see [28][29][30][31]).In [29], the authors considered a recurrent neural network with a special class of discontinuous activation functions which is piecewise constants in the state space.In [30,31], the multistability issues were discussed for two classes of recurrent neural networks with -level discontinuous activation functions.Sufficient conditions are established to ensure the existence of   locally exponentially stable equilibria.
We point out that the majority of the references mentioned above studied the Hopfield neural networks.As ones of the important neural networks, Cohen-Grossberg neural networks (CGNNs) include a lot of famous ecological systems and neural networks as special cases such as the Lotka-Volterra system, the Gilpia-Analg competition system, the Eingen-Schuster system, and the Hopfield neural networks, cellular neural networks, and other neural networks.CGNNs have aroused a tremendous surge of investigation in these years.On the other hand, neural networks usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths.It is desired to model them by introducing continuously distributed delays over a certain duration of time such that the distant past has less influence compared with the recent behavior of the state, whereas, to the best of our knowledge, the multistability and multiperiodicity issues are seldom considered for the CGNNs with distributed delays and discontinuous activation functions.
Motivated by the works of [30,31] and the discussions above, the purpose of this paper is to explore the multistability and multiperiodicity of CGNNs with time-varying delays, distributed delays, and discontinuous activation functions.To this end, we consider the following differential equations: where   () denotes the state of the th neuron at time ;   (  ()) denotes the amplification function;   > 0;   (  ()) denotes the activation function;   () denotes the time-varying delay associated with the th neuron, which is variable with time due to the finite switching speed of amplifiers and satisfies 0 ≤   () ≤ ;   () > 0 represents the delay kernel function, which is a real-valued continuous function;   ,   , and   denote the connection strengths;   is the external input bias.The characteristics of activation functions have a major impact on the existence and stability of stationary patterns for neural networks.In this paper, we consider the following -level discontinuous activation functions: The organization of this paper is as follows.In Section 2, some preliminaries are given.In Section 3, through decomposing state space into multiple positively invariant sets, we derive sufficient conditions for the existence and exponential stability of equilibria in any designated region.As an extension, similar results are presented for delayed CGNNs with periodic time-varying delays and external inputs in Section 4. In Section 5, four numerical examples are provided to illustrate the feasibility of the obtained results.A concluding remark is given in Section 6 to end this work.
Since   (⋅) is allowed to have points of discontinuity, we consider the solution of system (1) in the sense of Filippov [32].
Definition 3. Let  be a subset of   . is said to be a positively invariant set of system (1) if and only if for any initial condition Φ ∈ C([ 0 − ,  0 ], ), we have  Φ () ∈  for all  ≥  0 .

Multistability for the CGNNs
In this section, we consider the existence and exponential stability of equilibrium points for system (1) in each division region.
From Case I to Case III, we know that  Φ () ∈ Λ for all  ≥  0 that is, Λ is a positively invariant set of system (1).The proof is complete.Theorem 7. Assume that assumptions (H1)and (H2) and the inequalities in (10) hold.For each division region Λ ∈ Ω, system (1) has a unique equilibrium located in Λ which is locally exponentially stable.
Remark 9.It is noted that the number of such region Λ is   in Ω.Hence, under conditions of Theorem 7, there exist   locally exponentially stable equilibrium points for system (1) with activation functions (2).Neural networks with this class of activation functions can store many more patterns than those with sigmoidal activation functions and nondecreasing activation functions with saturation (see, e.g., [14][15][16][17][18][19][20][21]) in practical applications.
Remark 10.An important application of multistability of recurrent neural networks is to implement pattern memory (see [34,35]).A recalling probe, which is sufficiently close to the pattern to be retrieved, is set as an initial state and the state variables converge to the locally stable equilibrium point, which corresponds to the pattern to be retrieved.Hence, our results are useful for associative memories since they provide new criteria to guarantee the coexistence of encoded patterns and their local attractivity.
Remark 11.In [30], the authors investigated the multistability for a class of Hopfield neural networks with time-varying delays and discontinuous activation functions.By applying the principle of compressed mapping, sufficient conditions are established for the existence of multiple equilibrium points.In contrast, our multistability results are derived by means of analysis approach and Cauchy convergence principle.When   (  ()) = 1,   = 1,   = 0 and  1 () =  2 () = ⋅ ⋅ ⋅ =   (), system (1) reduces to system (1) in [30].
Theorem 14.Under assumptions (H1) and (H2), for each division region Λ ∈ Ω, Λ is a positively invariant set of system (32), and system (32) has a unique periodic orbit located in Λ which is locally exponentially stable, if the following conditions are satisfied: where We can choose a positive integer  such that (  /  ) − ≤ ] < 1 for all 1 ≤  ≤ .() for all  ≥  0 .This shows that  Φ * () is a periodic orbit of (32) located in Λ.The locally exponential stability can be similarly proved by Theorem 7.This completes the proof.

Illustrative Examples
In this section, four examples are presented to illustrate our main results derived in Sections 3 and 4.
For simplicity, we choose the activation functions as It is not difficult to verify that system (1) with ( 34) and (35) satisfies assumptions (H1) and (H2) and the inequalities in (10).According to Theorems 6 and 7, system (1) has nine locally exponentially stable equilibrium points: (36) The dynamics of this system are depicted in Figure 1(a), where evolutions of 120 initial conditions have been tracked.
We can verify that system (1) with (37) satisfies assumptions (H1) and (H2) and the inequalities in (10).According to Theorems 6 and 7, system (1) has eight locally exponentially stable equilibrium points: The dynamics of this system are depicted in Figure 2(a).

Conclusions
In this paper, we performed multistability and multiperiodicity analyses for a general class of CGNNs with time-varying delays, distributed delays, and discontinuous activation functions.Sufficient conditions were established to ensure that there are   locally exponential stable equilibrium points for the -dimensional CGNNs with -level discontinuous activation functions.Also, we derived sufficient conditions for the existence and global exponential stability of equilibrium point for the system.As an extension of multistability, sufficient conditions were established to ensure that the system has   locally exponentially stable periodic orbits when time-varying delays and external inputs are periodic.Numerical simulations demonstrated the main results.The existing multistability results in [30] were improved and extended.Our results provide new criteria to guarantee the coexistence of encoded patterns and their local attractivity and are useful for associative memories.