0
2013
Impact Factor

    Generation of Robust Hyperbolic Chaos in CNN

    2019, Vol. 15, no. 2, pp.  109-124

    Author(s): Kuznetsov S. P.

    It is shown that on the basis of a cellular neural network (CNN) composed, e.g., of six cells, it is possible to design a chaos generator with an attractor being a kind of Smale – Williams solenoid, which provides chaotic dynamics that is rough (structurally stable), as follows from respective fundamental mathematical theory. In the context of the technical device, it implies insensitivity to small variations of parameters, manufacturing imperfections, interferences, etc. Results of numerical simulations and circuit simulation in the Multisim environment are presented. The proposed circuit is the first example of an electronic system where the role of the angular coordinate for the Smale – Williams attractor is played by the spatial phase of the sequence of patterns. It contributes to the collection of feasible systems with hyperbolic attractors and thus promotes filling with real content and promises practical application for the hyperbolic theory, which is an important and deep sector of the modern mathematical theory of dynamical systems.
    Keywords: cellular neural network, chaos, attractor, pattern, dynamics, structural stability
    Citation: Kuznetsov S. P., Generation of Robust Hyperbolic Chaos in CNN, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 2, pp.  109-124
    DOI:10.20537/nd190201


    Download File
    PDF, 2.57 Mb

    References

    [1] Chua, L. O. and Yang, L., “Cellular Neural Networks: Theory”, IEEE Trans. Circuits Syst., 35:10 (1988), 1257–1272  crossref  mathscinet  zmath
    [2] Chua, L. O. and Roska, T., “The CNN Paradigm”, IEEE Trans. Circuits Syst. I, 40:3 (1993), 147–156  crossref  zmath
    [3] Cimagalli, V., Balsi, M., and Caianiello, E., “Cellular Neural Networks: A Review”, Neural Nets WIRN Vietri'93: Proc. of 6th Italian Workshop (Salerno, 1993), World Sci., ed. E. R. Caianiello, 1993, 55–84  mathscinet
    [4] Chua, L. O. and Yang, L., “Cellular Neural Networks: Applications”, IEEE Trans. Circuits Syst., 35:10 (1988), 1273–1290  crossref  mathscinet
    [5] Chua, L. O., Hasler, M., Moschytz, G. S., and Neirynck, J., “Autonomous Cellular Neural Networks: A Unified Paradigm for Pattern Formation and Active Wave Propagation”, IEEE Trans. Circuits Syst. I, 42:10 (1995), 559–577  crossref  mathscinet
    [6] Hunt, K. J., Sbarbaro, D., Żbikowski, R., and Gawthrop, P. J., “Neural Networks for Control Systems: A Survey”, Automatica, 28:6 (1992), 1083–1112  crossref  mathscinet  zmath
    [7] Chua, L. O. and Roska, T., Cellular Neural Networks and Visual Computing: Foundations and Applications, Cambridge Univ. Press, Cambridge, 2002, 410 pp.
    [8] Shi, B. and Luo, T., “Spatial Pattern Formation via Reaction-Diffusion Dynamics in $32 \times 32 \times 4$ CNN Chip”, IEEE Trans. Circuits Syst. I, 51:5 (2004), 939–947  crossref
    [9] Gollas, F. and Tetzlaff, R., “Modeling Complex Systems by Reaction-Diffusion Cellular Nonlinear Networks with Polynomial Weight-Functions”, 9th Internat. Workshop on Cellular Neural Networks and Their Applications (Taiwan, 2005), 227–231
    [10] Pivka, L., “Autowaves and Spatio-Temporal Chaos in CNNs: 1. A Tutorial”, IEEE Trans. Circuits Syst. I, 42:10 (1995), 638–649  crossref
    [11] Chaotic Electronics in Telecommunications, eds. M. Kennedy, G. Setti, R. Rovatti, CRC, Boca Raton, Fla., 2000, 464 pp.
    [12] Cuomo, K. M. and Oppenheim, A. V., “Circuit Implementation of Synchronized Chaos with Applications to Communications”, Phys. Rev. Lett., 71:1 (1993), 65–68  crossref  adsnasa
    [13] Dmitriev, A. S., Panas, A. I., and Starkov, S. O., “Experiments on Speech and Music Signals Transmission Using Chaos”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5:4 (1995), 1249–1254  crossref  zmath
    [14] Bollt, E. M., “Review of Chaos Communication by Feedback Control of Symbolic Dynamics”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13:2 (2003), 269–285  crossref  mathscinet  zmath
    [15] Baptista, M. S., “Cryptography with Chaos”, Phys. Lett. A, 240:1–2 (1998), 50–54  crossref  mathscinet  zmath  adsnasa
    [16] Kocarev, L., “Chaos-Based Cryptography: A Brief Overview”, IEEE Circuits Syst. Mag., 1:3 (2001), 6–21  crossref
    [17] Dachselt, F. and Schwarz, W., “Chaos and Cryptography”, IEEE Trans. Circuits Syst. I, 48:12 (2001), 1498–1509  crossref  mathscinet  zmath
    [18] Stojanovski, T. and Kocarev, L., “Chaos-Based Random Number Generators: Part 1. Analysis [Cryptography]”, IEEE Trans. Circuits Syst. I, 48:3 (2001), 281–288  crossref  mathscinet  zmath
    [19] Stojanovski, T., Pihl, J., and Kocarev, L., “Chaos-Based Random Number Generators: Part 2. Practical Realization”, IEEE Trans. Circuits Syst. I, 48:3 (2001), 382–385  crossref  mathscinet  zmath
    [20] Bakiri, M., Guyeux, C., Couchot, J. F., and Oudjida, A. K., “Survey on Hardware Implementation of Random Number Generators on FPGA: Theory and Experimental Analyses”, Comput. Sci. Rev., 27 (2018), 135–153  crossref  mathscinet  zmath
    [21] Verschaffelt, G., Khoder, M., and Van der Sande, G., “Random Number Generator Based on an Integrated Laser with On-Chip Optical Feedback”, Chaos, 27:11 (2017), 114310, 7 pp.  crossref  adsnasa
    [22] Harman, S. A., Fenwick, A. J., and Williams, C., “Chaotic Signals in Radar?”, Proc. of the 3rd European Radar Conference IEEE (Manchester, September 2006), 49–52
    [23] Liu, Z., Zhu, X., Hu, W., and Jiang, F., “Principles of Chaotic Signal Radar”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17:5 (2007), 1735–1739  crossref  zmath
    [24] Willsey, M. S., Cuomo, K. M., and Oppenheim, A. V., “Selecting the Lorenz Parameters for Wideband Radar Waveform Generation”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21:9 (2011), 2539–2545  crossref  zmath
    [25] Banerjee, S., Yorke, J. A., and Grebogi, C., “Robust Chaos”, Phys. Rev. Lett., 80:14 (1998), 3049–3052  crossref  mathscinet  zmath  adsnasa
    [26] Potapov, A. and Ali, M. K., “Robust Chaos in Neural Networks”, Phys. Lett. A, 277:6 (2000), 310–322  crossref  mathscinet  zmath  adsnasa
    [27] Elhadj, Z. and Sprott, J. C., “On the Robustness of Chaos in Dynamical Systems: Theories and Applications”, Front. Phys. China, 3:2 (2008), 195–204  crossref  adsnasa  elib
    [28] Elhadj, Z. and Sprott, J. C., Robust Chaos and Its Applications, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, 79, World Sci., Hackensack, N.J., 2011, 472 pp.  mathscinet
    [29] Gusso, A., Dantas, W. G., and Ujevic, S., “Prediction of Robust Chaos in Micro and Nanoresonators under Two-Frequency Excitation”, Chaos, 29:3 (2019), 033112  crossref  mathscinet  adsnasa
    [30] Shilnikov, L., “Mathematical Problems of Nonlinear Dynamics: A Tutorial”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 7:9 (1997), 1953–2001  crossref  mathscinet  zmath
    [31] Botella-Soler, V., Castelo, J. M., Oteo, J. A., and Ros, J., “Bifurcations in the Lozi Map”, J. Phys. A, 44:30 (2011), 305101, 14 pp.  crossref  mathscinet  zmath  elib
    [32] Elhadj, Z., Lozi Mappings: Theory and Applications, CRC, Boca Raton, Fla., 2013, 338 pp.  mathscinet
    [33] Belykh, V. N. and Belykh, I., “Belykh Map”, Scholarpedia, 6:10 (2011), 5545  crossref  adsnasa
    [34] Kuznetsov, S. P., “Belykh Attractor in Zaslavsky Map and Its Transformation under Smoothing”, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 26:1 (2018), 64–79 (Russian)  mathscinet; (2017), arXiv: 1710.07828 [nlin.CD] (English)
    [35] Anosov, D. V., “Dynamical Systems in the 1960s: The Hyperbolic Revolution”, Mathematical Events of the Twentieth Century, eds. A. A. Bolibruch, Yu. S. Osipov, Ya. G. Sinai, Springer, Berlin, 2006, 1–17  mathscinet  zmath
    [36] Smale, S., “Differentiable Dynamical Systems”, Bull. Amer. Math. Soc., 73:6 (1967), 747–817  crossref  mathscinet  zmath
    [37] Dynamical Systems 9: Dynamical Systems with Hyperbolic Behaviour, Encyclopaedia Math. Sci., 66, ed. D. V. Anosov, Springer, Berlin, 1995, 236 pp.
    [38] Katok, A. and Hasselblatt, B., Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl., 54, Cambridge Univ. Press, Cambridge, 1995, 802 pp.  mathscinet  zmath
    [39] Pugh, C. and Peixoto, M. M., “Structural Stability”, Scholarpedia, 3:9 (2008), 4008  crossref  adsnasa
    [40] Kuznetsov, S. P., “Example of a Physical System with a Hyperbolic Attractor of the Smale – Williams Type”, Phys. Rev. Lett., 95:14 (2005), 144101, 4 pp.  crossref  adsnasa  elib
    [41] Kuznetsov, S. P. and Pikovsky, A., “Autonomous Coupled Oscillators with Hyperbolic Strange Attractors”, Phys. D, 232:2 (2007), 87–102  crossref  mathscinet  zmath  elib
    [42] Wilczak, D., “Uniformly Hyperbolic Attractor of the Smale – Williams Type for a Poincaré Map in the Kuznetsov System: With Online Multimedia Enhancements”, SIAM J. Appl. Dyn. Syst., 9:4 (2010), 1263–1283  crossref  mathscinet  zmath  elib
    [43] Kuznetsov, S. P., “Dynamical Chaos and Uniformly Hyperbolic Attractors: From Mathematics to Physics”, Phys. Uspekhi, 54:2 (2011), 119–144  crossref  adsnasa  elib; Uspekhi Fiz. Nauk, 181:2 (2011), 121–149 (Russian)  mathnet  crossref
    [44] Kuznetsov, S. P., Hyperbolic Chaos: A Physicist's View, Springer, Berlin, 2012, 336 pp.  zmath  adsnasa
    [45] Kuznetsov, S. P. and Seleznev, E. P., “Strange Attractor of Smale – Williams Type in the Chaotic Dynamics of a Physical System”, J. Exp. Theor. Phys., 102:2 (2006), 355–364  crossref  mathscinet  adsnasa  elib; Zh. Èksper. Teoret. Fiz., 129:2 (2006), 400–412 (Russain)
    [46] Kuznetsov, S. P. and Ponomarenko, V. I., “Realization of a Strange Attractor of the Smale – Williams Type in a Radiotechnical Delay-Feedback Oscillator”, Tech. Phys. Lett., 34:9 (2008), 771–773  crossref  adsnasa  elib; Pisma Zh. Tekh. Fiz., 34:18 (2008), 1–8 (Russian)
    [47] Kuznetsov, S. P., Ponomarenko, V. I., and Seleznev, E. P., “Autonomous System Generating Hyperbolic Chaos: Circuit Simulation and Experiment”, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 21:5 (2013), 17–30 (Russian)
    [48] Isaeva, O. B., Kuznetsov, S. P., Sataev, I. R., Savin, D. V., and Seleznev, E. P., “Hyperbolic Chaos and Other Phenomena of Complex Dynamics Depending on Parameters in a Nonautonomous System of Two Alternately Activated Oscillators”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25:12 (2015), 1530033, 15 pp.  crossref  mathscinet  zmath
    [49] Kuptsov, P. V., Kuznetsov, S. P., and Pikovsky, A., “Hyperbolic Chaos of Turing Patterns”, Phys. Rev. Lett., 108:19 (2012), 194101, 4 pp.  crossref  adsnasa  elib
    [50] Isaeva, O. B., Kuznetsov, A. S., and Kuznetsov, S. P., “Hyperbolic Chaos of Standing Wave Patterns Generated Parametrically by a Modulated Pump Source”, Phys. Rev. E, 87:4 (2013), 040901(R), 4 pp.  crossref  adsnasa  elib
    [51] Kruglov, V. P., Kuznetsov, S. P., and Pikovsky, A., “Attractor of Smale – Williams Type in an Autonomous Distributed System”, Regul. Chaotic Dyn., 19:4 (2014), 483–494  mathnet  crossref  mathscinet  zmath  adsnasa  elib
    [52] Benettin, G., Galgani, L., Giorgilli, A., and Strelcyn, J.-M., “Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems: A Method for Computing All of Them: P. 1: Theory”, Meccanica, 15:1 (1980), 9–20  crossref  mathscinet  zmath  adsnasa
    [53] Shimada, I. and Nagashima, T., “A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems”, Progr. Theoret. Phys., 61:6 (1979), 1605–1616  crossref  mathscinet  zmath  adsnasa
    [54] Pikovsky, A. and Politi, A., Lyapunov Exponents: A Tool to Explore Complex Dynamics, Cambridge Univ. Press, Cambridge, 2016, 295 pp.  mathscinet  zmath
    [55] Kaplan, J. L. and Yorke, J. A., “Chaotic Behavior of Multidimensional Difference Equations”, Functional Differential Equations and Approximation of Fixed Points, Lecture Notes in Math., 730, eds. H.-O. Peitgen, H.-O. Walther, Springer, Berlin, 1979, 204–227  crossref  mathscinet
    [56] Farmer, J. D., Ott, E., and Yorke, J. A., “The Dimension of Chaotic Attractors”, Phys. D, 7:1–3 (1983), 153–180  crossref  mathscinet  zmath



    Creative Commons License
    This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License