Chaos and its synchronization in two-neuron systems with discrete delays. (English) Zbl 1048.37527
Summary: It is well known that complex dynamic behaviors exist in time-delayed neural systems. Infinite positive Lyapunov exponents can be found in time-delayed chaotic systems since the dimension of such systems is infinite. However, theoretical and experimental models studied thus far are low-dimensional systems with only one positive Lyapunov exponent. Consequently, messages masked by such chaotic systems are shown to be easily extracted in some cases. Therefore, communication system with a higher security level can be designed by means of the time-delayed neuron systems. We firstly investigate the dynamical behavior of two-neuron systems with discrete delays. Then, the chaos synchronization in time-delayed neuron system is studied based on the method of designing the coupled system and employing Krasovskii–Lyapunov theory to search the synchronization conditions. Numerical results illustrate the correctness of our theoretical analyses.
MSC:
| 37N25 | Dynamical systems in biology |
| 34K23 | Complex (chaotic) behavior of solutions to functional-differential equations |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 92C20 | Neural biology |
References:
| [1] | Pecora, L. M.; Carroll, T. L., Synchronization in Chaotic Systems, Phys. Rev. Lett., 64, 8, 821-824 (1990) · Zbl 0938.37019 |
| [2] | Ogorzalek, M. J., Taming chaos-part I :Synchronization, IEEE Trans. CAS-I, 40 (1993) · Zbl 0850.93353 |
| [3] | Kolumbán, G.; Kennedy, M. P.; Chua, L. O., The role of synchronization in digital communication using chaos-part II: Chaotic modulation and chaotic synchronization, IEEE Trans. CAS- I, 45, 11, 1129-1140 (1998) · Zbl 0991.93097 |
| [4] | Morgül, Ö.; Feki, M., A chaotic masking scheme by using synchronized chaotic systems, Phys. Lett. A, 251, 2, 169-176 (1999) |
| [5] | Grassi, G.; Mascolo, S., Synchronizing hyperchaotic system by observer design, IEEE Trans. CAS- II, 46, 4, 478-483 (1999) · Zbl 1159.94361 |
| [6] | Mensour, B.; Longtin, A., Synchronization of delay-differential equations with application to private communication, Phys. Lett. A, 244, 1, 59-70 (1998) |
| [7] | He, R.; Vaidya, P. G., Time delayed chaotic systems and their synchronization, Phys. Rev. E, 9, 7, 1051-4048 (1999) |
| [8] | Gopalsamy, K.; Leung, K. C., Convergence under dynamical thresholds with delays, IEEE Trans. NN, 8, 2, 341-348 (1997) |
| [9] | Pyragas, K., Synchronization of coupled time-delay systems: Analytical estimations, Phy. Rev. E, 58, 3, 3067-3071 (1998) |
| [10] | Eudo, T.; Chua, E. O., Synchronization of Chaos in Phase-Locked Loops, IEEE Trans. CAS, 38, 12, 1580-1588 (1991) |
| [11] | Rosenblum, M. G.; Pikovsky, A. S.; Kurths, J., Phase synchronization of chaotic oscillators, Phys. Rev. Lett., 76, 1804-1807 (1996) |
| [12] | Rosenblum, M. G.; Pikovsky, A. S.; Kurths, J., Phase synchronization in driven and coupled chaotic oscillators, IEEE Trans. CAS- I, 44, 10, 874-881 (1996) |
| [13] | Park, S. H.; Kim, S.; Pyo, H.-B., Effects of time-delayed interations on dynamic patterns in a coupled phase oscillater system, Phys. Rev., 60, 4, 4962-4965 (1999) |
| [14] | Dabrowski A, Galias Z, Ogorzalek MJ. Phase synchronization phenomena in generalized CNN composed of chaotic cells, (CNNA 2000). In Proceedings of the 6th IEEE International Workshop on Cellular Neural Networks and Their Applications, 2000: p. 253-8; Dabrowski A, Galias Z, Ogorzalek MJ. Phase synchronization phenomena in generalized CNN composed of chaotic cells, (CNNA 2000). In Proceedings of the 6th IEEE International Workshop on Cellular Neural Networks and Their Applications, 2000: p. 253-8 |
| [15] | Perez, G.; Cerdeira, H. A., Extracting messages masked by chaos, Phys. Rev. Lett., 74, 1970-1973 (1999) |
| [16] | Liao, X. F.; Wong, K. W.; Leung, C. S.; Wu, Z. F., Hopf bifurcation and chaos in a single delayed neural equation with non-monotonic activation function, Chaos, Solitons & Fractals, 12, 1535-1547 (1999) · Zbl 1012.92005 |
| [17] | Liao, X. F.; Wong, K. W.; Wu, Z.f., Asymptotic stability criteria for a two-neuron network with different time delays, IEEE Trans. NN, 14, 1, 222-227 (2003) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
