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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 857341, 19 pages
Exponential Stability of Periodic Solutions for Inertial Type BAM Cohen-Grossberg Neural Networks
Department of Mathematics, Shaoxing University, Shaoxing, Zhejiang 312000, China
Received 6 February 2014; Revised 22 March 2014; Accepted 3 April 2014; Published 19 May 2014
Academic Editor: Zidong Wang
Copyright © 2014 Chunfang Miao and Yunquan Ke. 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.
The existence and exponential stability of periodic solutions for inertial type BAM Cohen-Grossberg neural networks are investigated. First, by properly choosing variable substitution, the system is transformed to first order differential equation. Second, some sufficient conditions that ensure the existence and exponential stability of periodic solutions for the system are obtained by constructing suitable Lyapunov functional and using differential mean value theorem and inequality technique. Finally, two examples are given to illustrate the effectiveness of the results.
The Cohen-Grossberg-type BAM neural networks model is initially proposed by Cohen and Grossberg , has their promising potential for the tasks of parallel computation, associative memory, and has great ability to solve difficult optimization problems. Thus, the analysis of the dynamical behaviors of bidirectional associative memory neural networks and Cohen-Grossberg neural networks is important and necessary. In recent years, many researchers have studied the stability and other dynamical behaviors of the Cohen-Grossberg-type BAM neural networks; see [2–10].
On the other hand, some authors studied neural networks, added the inertia, and obtained some results. For example, Li et al.  added the inertia to a delay differential equation which can be described by and obtained obvious chaotic behavior. Liu et al. [12, 13] found chaotic behavior of the inertial two-neuron system with time through numerical simulation and gave that the system will lose its stability when the time delay is increased and will rise a quasiperiodic motion and chaos under the interaction of the periodic excitation. Wheeler and Schieve  added the inertia to a continuous-time Hopfield effective-neuron system which is shown to exhibit chaos. They explain that the chaos is confirmed by Lyapunov exponents, power spectra, and phase space plotsthis system is described by Babcock and Westervelt  studied the electronic neural networks with added inertia and found that when the neuron couplings are of an inertial nature, the dynamics can be complex, in contrast to the simpler behavior displayed when they of the standard resistor-capacitor variety. For various values of the neuron gain and the quality factor of the couplings, they find ringing about the stationary points, instability and spontaneous oscillation, intertwined basins of attraction, and chaotic response to a harmonic drive. Ge and Xu  considered an inertial four-neuron delayed bidirectional associative memory model. Weak resonant double Hopf bifurcations are completely analyzed in the parameter space of the coupling weight and the coupling delay by the perturbation-incremental scheme. Others, Liu et al. [17, 18], investigated the Hopf bifurcation and dynamics of an inertial two-neuron system or in a single inertial neuron mode. Zhao et al.  investigated the stability and the bifurcation of a class of inertial neural networks. The authors Ke and Miao [20, 21] investigated stability of equilibrium point and periodic solutions in inertial BAM neural networks with time delays, respectively. From the above, the inertia can be considered a useful tool that is added to help in the generation of chaos in neural systems. Horikawa and Kitajima  investigated a kinematical description of traveling waves of the oscillations in neural networks with inertia. When the inertia is below a critical value and the state of each neuron is overdamped, properties of the networks are the same as those without inertia. The duration of the transient oscillations increases with inertia, and the increasing rate of the logarithm of the duration becomes more than double. When the inertia exceeds a critical value and the state of each neuron becomes underdamped, properties of the networks qualitatively change. The periodic solution is stabilized through the pitchfork bifurcation as inertia increases. More bifurcations occur so that various periodic solutions are generated, and the stability of the periodic solutions changes alternately. Ke and Miao  investigated the stability of inertial Cohen-Grossberg-type neural networks with time delays. To the best of our knowledge, the question on the periodic solutions of inertial type BAM Cohen-Grossberg neural networks with time delays is still open. To provide the theoretical basis of practical application, this paper is devoted to present a sufficient criterion to ensure the existence and exponential stability of periodic solutions for inertial type BAM Cohen-Grossberg neural networks with time delays.
We consider the following inertial type BAM Cohen-Grossberg neural networks with time delays: for , , where the second derivative is called an inertial term of system (3); , are constants; and are the states of the neuron from the neural field and the th neuron from the neural field at the time , respectively; , denote the activation functions of neuron from and the neuron from , respectively; weights the strength of the neuron on the neuron at the time ; weights the strength of the neuron on the neuron at the time ; and ; , denote the external inputs on the neuron from and the neuron from at the time , respectively; and represent amplification functions; and are appropriately behaved functions such that the solutions of model (3) remain bounded.
The initial conditions of system (3) are given by where , , , and are bounded and continuous functions.
This paper is organized as follows. Some preliminaries are given in Section 2. In Section 3, the sufficient conditions are derived which ensure the existence and exponential stability of periodic solutions for inertial Cohen-Grossberg-type BAM neural networks. In Section 4, two illustrative examples are given to show the effectiveness of the proposed theory.
Throughout this paper, we make the following assumptions.For each , , the functions , , , and are differentiable and satisfy for all , .For each , , the activation functions , satisfy Lipschitz condition, and there exist constants , , , and , such that For each , , , are continuously periodic functions defined on with common period and satisfy , .Let ; there exist constants and , such that Let ; there exist constants and , such that
and are continuously differentiable periodic functions, and there exist constants and , such that where , , and and express the derivative of and .
Definition 1. Let be an periodic solution of system (3) with initial value for every solution of system (3) with any initial value If there exist constants and , such that for , , and , then solutions , are said to be exponentially stable, where
3. Main Results
In this section, we can derive some sufficient conditions which ensure the existence and exponential stability of periodic solutions for system (3).
Theorem 2. For system (3), under the hypotheses , then , , , and are bounded, , , and .
Proof. If , then we have
if , then
Hence, . Similarly, we can get
Since are differentiable on , and then we have
where lies between and .
It follows from (3) that Similarly, we can obtain From (21), (22), we can obtain where and , are any real constants: where and , are any real constants.
Since , , we have , , , and , and formula (23) shows that all solutions to (3) are bounded for , .
Formula (24) shows that all solutions to (3) are bounded for , .
On the other hand, from (3) we also can obtain
Since , are bounded, we may assume that , , where , are constants, , .
From (25), we have Formula (27) shows that all solutions are bounded for , .
From (26), we have Formula (28) shows that all solutions are bounded for , .
Theorem 3. Under the hypotheses , if , , and for , , then system (3) has one -periodic solution, which is exponentially stable.
Proof. If , , , and are -periodic solution of (11), which are exponentially stable, then we can obtain that () are -periodic solution of system (3), which is exponentially stable. In the following we only prove that (11) has one -periodic solution, which is exponentially stable.
Let be solution of system (3) with initial value , and let be solution of system (3) with any initial value .
Let for , .
From (11), we can obtain for for .
Since functions and are differentiable, using differential mean value theorem, we have where and lie between and .
Since , if , then we have and .
From (33) we get From (36), we can obtain for , .
Similar to the above derivation, from (34) we can get for , .
We consider the Lyapunov functional where is a small number.
Calculating the upper right Dini-derivative of along the solution of (33) and (34), using (37) and (38), we have From condition of Theorem 3, we can choose a small such that From (40), we get and so , for all . From (39), we have where
Since , from (42), we obtain From (44), we obtain
For , , when , , , and are continuously periodic functions defined on with common period , if , are the solutions of (3), then for any natural number , , are the solutions of (3). Thus, from (45), there exist constants and , such that for , , .
It is noted that, for any natural number , Thus
Since is bounded, it follows from (46) and (49) that uniformly converges to a continuous functions on any compact set of .
Similarly, since is bounded, from (47), uniformly converges to a continuous function on any compact set of .
When , , , and are bounded, , , and we can obtain that , are bounded. Similarly, from (44), they can be proved that , uniformly converge to continuous functions and on any compact set of , respectively.
Now we will show that is the -periodic solution of system (3).