Journal of Applied Mathematics and Physics, 2014, 2, 219224 Published Online April 2014 in SciRes. http://www.scirp.org/journal/jamp http://dx.doi.org/10.4236/jamp.2014.25027 How to cite this paper: Das, P. and Kundu, A. (2014) Bifurcation and Chaos in Delayed Cellular Neural Network Model. Journal of Applied Mathematics and Physics, 2, 219224. http://dx.doi.org/10.4236/jamp.2014.25027 Bifurcation and Chaos in Delayed Cellular Neural Network Model Pritha Das, Amitava Kundu Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah, India Email: prithadas01@yahoo.com, kundoo_22@yahoo.co.in Received February 2014 Abstract This paper deals with control of chaotic behavior of a delayed Cellular Neural Network (DCNN) model which is a onedimensional regular array of four cells with continuous activation function. We investigate different dynamical behaviors including limit cycle, torus, and chaos for different range of weight parameters of the system. Regarding synaptic weight as parameter, Hopf bifurca tions are obtained in the system without delay. In the delayed model condition for the Global as ympto ti c stability of the equilibrium point is presented. Numerical simulation and results are given to show the role of delay in chaos control of the CNNs. Keywords Delayed Cellular Neural Networks, Chaos, Bifurca tion, Global Stability 1. Introduction An artificial neural network (ANN) is a mathematical model that can mimic the biological process of human brain. An ANN can be presented by a directed graph composed of neurons as nodes and synapses or nerves as edges with an algorithm that show how impulses are conducted through the network. There are various types of network architecture. Cellular Neural Net wo rk s (CNNs) introduced by Chua and Yang [1] is one dimensional regular array of locally connected circuits (called cells), they are able to display various dynamics, such as limit cycle, torus, especially chaos. Neural oscillation can arise from interactions between different brain areas. Time delays can play an important role here because nerve impulse takes a time to travel the length of the axon to the target neuron which in turn takes the time to summate their inputs and produce response. Brain areas are as sumed to be bidirectionally (BAM) coupled forming delayed feedback loops. An example of such a feedback loop is the connection between the thalamus and cortex [2] [3]. The malfunctioning of the neural system is often related to changes in the delay parameter causing unmanageable shifts in the phases of the neural signals. Some of the visible mechanical consequences could be explained: such as tremor in the fingers, difficulties in balanc ing; the increased danger of falling over for elderly people or even motion disorders in the case of bursts of epi lepsy, Parkinson disease and so on [4]. Physiological experiments suggest that the main components of neural activity in olfactory systems are chao tic. In the olfactory system, the phase transition has the appearance of a change in the EEG from a chaotic, ape riodic fluctuation to a more regular nearly periodic oscillation. Chaotic activity enables the rapid state transitions
P. Das, A. Kundu essential for information processing. Without this ability, the brain could not quickly concern itself with a new task. Thus, we can assume chaos for the rapid transitions between perceptual states. Without it, observation could be extremely slow. It is supposed that brains injured by Alzheimer’s disease have electrophysiologically inactive neurons and/or synapses and show decreased chaotic behavior. Results of various research works in this area support the assumption that chaos plays an important role in brain function, for instance, learning and memory [5][13]. This paper is structured as follows: In Section 2 we analyze a new family of simplified 4CNNs, as shown in Figure 1. In Section 3, we investigate different dynamical behaviors including chaos for different range of weight parameters of the system. Using Matlab software we can observe limit cycle, torus and chaos in a new family of 4CNNs for different range of adjustable parameters of main diagonal of the weight matrix. Numerical results are discussed showing changes of dynamics of the system from unstable to stable (chaos control) one through bifurcation due to variation in the individual weight parameter. In Section 4, a delayed Cellular Neural Networks (DCNNs) is considered where the chaotic dynamics is controlled [see Figure 2]. A sufficient condition for the Global asymptotic stability of the equilibrium point in the delayed model is found. Finally some concluding remarks have been drawn on the implication of our results in the context of related work mentioned above. 2. Limit Cycle, Torus and Chaos in 4CNNs In this section, the dynamics of this system can be described by the following ordinary differential equations: ( )( ) 4 1 1,2,3,0 4, ii iijji j xcxtW fxitc = = =−+ > ∑ (1) where is a weight matrix describing the strength of the connection from th neuron on the th neuron. is the output state of the th neuron at time t . For numerical simulation we consider , and the weight matrix to be 3 2.1 2.1700 3.51 13.11 0 0 31.1 0020 30 Wp − = −− (2) where is an adjustable parameter. The phase portraits (Figures 35) show the evidence that there exist limit cycles, torus and chaotic attractors with different values of . In Figure 6, chaotic nature is controlled by taking which is also confirmed by bifurcation diagram (see Figure 7) with respect to parameter . Figure 1. Architecture of the network model 4cell Cellular Neural Networks. Figure 2. Architecture of the network model 4cell delayed Cellular Neural Networks.
P. Das, A. Kundu Figure 3. Solution trajectory showing periodic behavior and corresponding phase portrait in space with parameter . Figure 4. Solution trajectory showing chaotic behavior and corresponding phase portrait in space with parameter p3 = 4.4. . Figure 5. Solution trajectory showing chaotic behavior and corresponding phase portrait in space with parameter .
P. Das, A. Kundu Figure 6. Solution trajectory showing stable behavior and corresponding phase portrait in space with pa rameter . (a) (b) Figure 7 . (a) Bifurcation diagram with respect to parameter , (b) The four Lyapunov exponents when . Again, if we consider all other selfsynaptic weights like as adjustable parameter, then we get sim ilar type of behavior which is shown in the following Figure 8. 3. Mathematical Model with Time Delay and Global Stabilit y We consider an artificial 4neuron network model of Cellular Neural Networks time delayed connections between the neurons by the delay differential equations: ( )( )() 4 1 ,0 1,2,3,4 ii iiiiijji j ij xcxt afxtbfxitc τ = ≠ = =−++− > ∑ (3) with is the activation state of th neuron at time t, is the output state of the th neuron at time t, is selfsynaptic weight, is the strength of the th neuron on the th neuron at time , is the signal transmission delay and (decay rate) is the rate with which the th neuron will reset its potential to the resting state in isolation when disconnected from the network. In the following, we assume that each of the relation between the output of the cell and the state of the cell possess following properties: (H1) is bounded on ;
P. Das, A. Kundu (H2) There is a number such that for any It is easy to find from (H2) that is a continuous function on . In particular, if output state of the cell is described by , then it is easy to see that the function clearly satisfy the hypotheses (H1) and (H2). Theorem 1 For the DCNN (3), suppose output of the cell satisfy the hypotheses (H1) and (H2) above and there exist constants , , , , , , , such that ** ** * 2 4 22 2 22 22 1 ( )2 ij ij j ij ij ijijji i ijiiij jii j ij a abbc η ηξ ξ αβ αβ µµ µµ = ≠ +++ < ∑ (4) , in which , , , , , , , are any real constants with , , , , is a constant number of the hypothesis (H2) above. Then the equilibrium of the DCNN (3) is also globally asymptotically stable independent of delays. 4. Conclusion This paper addresses chaos control by varying the adjustable parameter (selfconnection weights) and introducing time delay [see Figure 9] in original model of 4cell CNNs. These two approaches stabilize unstable Figure 8 . Bifurcation diagram with respect to parameter . Figure 9. Control of chaos: Solution trajectory showing periodic behavior corresponding phase portrait in space with parameter .
P. Das, A. Kundu Figure 10. Solution trajectory showing stable behavior satisfying globally asymptotically stable conditions. periodic orbits that a chaotic attractor encompasses. Furthermore, different from the work of Yang and Huang [14] where adjustable parameter lies off the main diagonal (selfconnection weights), we consider it on the main diagonal which seems more meaningful for controlling chaos in CNNs. Besides global stability criteria ensures asymptotically stable behavior for DCNN model [see Figure 10]. The results obtained enforce more restrictions on synaptic weight and decay parameters. References [1] Chua, L.O. and Yang, L. (1988) Cellular Neural Networks: Theory. IEEE Transactions on Circuits and Systems I, 35, 12571272. [2] Aihara, K., Ta k abe, T. and Toyoda, M. (1990) Chaotic Neural Networks. Physics Letter A, 6, 333 340. http://dx.doi.org/10.1016/03759601(90)90136C [3] P yragas, K. (1992) Continuous Control of Chaos, by SelfControlling Feedback. Physics Letters A, 170, 4 21 428. http://dx.doi.org/10.1016/03759601(92)907458 [4] Wu, J. (2001) Introduction to Neural Dynamics and Signal Transmission Delay. de Gruyter, New York. [5] Ott, E., Grebogi, C. and Yor ke, J.A. (1990) Controlling Chaos. Physical Review Letters, 64, 1196119 9 . http://dx.doi.org/10.1103/PhysRevLett.64.1196 [6] Babloyantz, A. and Destexhe, A. (1986) LowDimensional Chaos in an Instant of Epilepsy. Proceedings of the Na tional Academy of Sciences of the USA, 83, 3513 35 17 . http://dx.doi.org/10.1073/pnas.83.10.3513 [7] Free man, W.J. (1987) Simulation of Chaotic EEG Patterns with a Dynamic Model of the Olfactory System. Biological Cybernet ics, 56, 139 150. http://dx.doi.org/10.1007/BF00317988 [8] Jeong, J. and Kim, S.Y. (1997) Nonlinear Analysis of Chaotic Dynamics Underlying the Electroencephalogram in Pa tients with Alzheimer’s Disease. Journal of the Korean Physical Society, 30, 320327. [9] Skar da , C.A. and Freeman, W.J. (1987) How Brains Make Chaos in Order to Make Sense of the World. Behavioral and brain Sc., 19, 161195. [10] Skard a, C.A. an d F reeman, W.J. (1990) Chaos and the New Science of the Brain. Neurosci ence, 1, 275285. [11] Chen, L. and Aihara , K. (1999) Global Searching Ability of Chaotic Neural Networks. IEEE Transaction on Circuits and Systems I, 46, 974993. [12] Das, A., Roy, A.B. and Das, P. (2002) Chaos in a Three Dimensional General Model of Neural Network. International Journal of Bifurcation and Chaos, 12, 22712281 . http://dx.doi.org/10.1142/S0218127402005820 [13] Kundu, A., Das, P. and Roy, A. B. (2014) Complex Dynamics of a Four Neuron Network Model Having a Pair of ShortCut Connections with Multiple Delays. Nonlinear Dynamics, 72, 643 662. http://dx.doi.org/10.1007/s1107101207422 [14] Yang, X.S. and Huang, Y. (2007) Chaos and TwoTo r i in a New Family of 4CNNs. International Journal of Bifurcation and Chaos, 17, 953 963. http://dx.doi.org/10.1142/S0218127407017677
