SICNNs with Li-Yorke chaotic outputs on a time scale

The existence of Li-Yorke chaos in the dynamics of shunting inhibitory cellular neural networks (SICNNs) on time scales is investigated. It is rigorously proved by taking advantage of external inputs that the outputs of SICNNs exhibit Li-Yorke chaos. The theoretical results are supported by simulations, and the controllability of chaos on the time scale is demonstrated by means of the Pyragas control technique. This is the first time in the literature that the existence as well as the control of chaos are provided for neural networks on time scales. HighlightsChaotic dynamics of SICNNs on time scales is considered.The results are based on the Li-Yorke definition of chaos.Controllability of the chaos on time scales is numerically demonstrated.Simulations which support the theoretical results are depicted.This is the first paper which considers chaos for neural networks on time scales.

[1]  Yu Wang,et al.  Mittag-Leffler stability of fractional-order neural networks with impulses , 2016 .

[2]  Marat Akhmet,et al.  Attraction of Li-Yorke chaos by retarded SICNNs , 2015, Neurocomputing.

[3]  Marat Akhmet,et al.  Li-Yorke chaos generation by SICNNs with chaotic/almost periodic postsynaptic currents , 2016, Neurocomputing.

[4]  Stephen Grossberg,et al.  Absolute stability of global pattern formation and parallel memory storage by competitive neural networks , 1983, IEEE Transactions on Systems, Man, and Cybernetics.

[5]  B. Evans.  What Does Brain Damage Tell us about the Mechanisms of Sleep? , 2002, Journal of the Royal Society of Medicine.

[6]  Eckehard Schöll,et al.  Handbook of Chaos Control , 2007 .

[7]  Bahram Nabet,et al.  Analysis and analog implementation of directionally sensitive shunting inhibitory neural networks , 1991, Defense, Security, and Sensing.

[8]  Period-doubling cascades galore , 2009, Ergodic Theory and Dynamical Systems.

[9]  S. Jeeva Sathya Theesar,et al.  Synchronization of chaotic nonlinear continuous neural networks with time-varying delay , 2011, Cognitive Neurodynamics.

[10]  Yongqing Yang,et al.  Existence and global exponential stability of anti-periodic solutions for competitive neural networks with delays in the leakage terms on time scales , 2014, Neurocomputing.

[11]  Homoclinic orbits and periodic solutions for a class of Hamiltonian systems on time scales , 2014 .

[12]  Li Yang,et al.  Existence and exponential stability of periodic solution for stochastic Hopfield neural networks on time scales , 2015, Neurocomputing.

[13]  Zhang Yifeng,et al.  A secure communication scheme based on cellular neural network , 1997, 1997 IEEE International Conference on Intelligent Processing Systems (Cat. No.97TH8335).

[14]  Walter J. Freeman,et al.  Chaos and the new science of the brain , 1990 .

[15]  Masaharu Adachi,et al.  Response to external input of chaotic neural networks based on Newman-Watts model , 2012, The 2012 International Joint Conference on Neural Networks (IJCNN).

[16]  John E. Dowling.  Neurons and Networks: An Introduction to Behavioral Neuroscience , 2001 .

[17]  Edward Ott,et al.  Controlling chaos , 2006, Scholarpedia.

[18]  Jinde Cao,et al.  Synchronization of complex dynamical networks with discrete time delays on time scales , 2015, Neurocomputing.

[19]  Weigao Ge,et al.  Periodic solution and chaotic strange attractor for shunting inhibitory cellular neural networks with impulses. , 2006, Chaos.

[20]  F. R. Marotto.  Snap-back repellers imply chaos in Rn , 1978 .

[21]  Marat Akhmet,et al.  Differential equations on variable time scales , 2009 .

[22]  Alvin Shrier,et al.  Chaos in neurobiology , 1983, IEEE Transactions on Systems, Man, and Cybernetics.

[23]  Ke Qin,et al.  Projective synchronization of different chaotic neural networks with mixed time delays based on an integral sliding mode controller , 2014, Neurocomputing.

[24]  V. Lakshmikantham,et al.  Dynamic systems on measure chains , 1996 .

[25]  Marat Akhmet,et al.  Impulsive SICNNs with chaotic postsynaptic currents , 2016 .

[26]  Pagavathigounder Balasubramaniam,et al.  Feedback synchronization of the fractional order reverse butterfly-shaped chaotic system and its application to digital cryptography , 2013, Nonlinear Dynamics.

[27]  Robert B. Pinter,et al.  Luminance adaptation of preferred object size in identified dragonfly movement detectors , 1989, Conference Proceedings., IEEE International Conference on Systems, Man and Cybernetics.

[28]  Kazuyuki Aihara,et al.  Controlling a chaotic neural network for information processing , 2013, Neurocomputing.

[29]  Li and Yorke chaos with respect to the cardinality of the scrambled sets , 2005 .

[30]  Ethan Akin,et al.  Li-Yorke sensitivity , 2003 .

[31]  Wei Jiang,et al.  Almost periodic solutions for neutral type BAM neural networks with distributed leakage delays on time scales , 2015, Neurocomputing.

[32]  R Hicks,et al.  Simple mechanisms organise orientation of escape swimming in embryos and hatchling tadpoles of Xenopus laevis. , 2000, The Journal of experimental biology.

[33]  Qiankun Song,et al.  Global exponential stability of neural networks with discrete and distributed delays and general activation functions on time scales , 2011, Neurocomputing.

[34]  A. Roberts,et al.  Sensory Activation and Role of Inhibitory Reticulospinal Neurons that Stop Swimming in Hatchling Frog Tadpoles , 2002, The Journal of Neuroscience.

[35]  Ying-Cheng Lai,et al.  Controlling chaos , 1994 .

[37]  Jonas Gloeckner,et al.  Impulsive Differential Equations , 2016 .

[38]  M. U. Akhmeta,et al.  The differential equations on time scales through impulsive differential equations , 2006 .

[39]  Abdesselam Bouzerdoum,et al.  A shunting inhibitory motion detector that can account for the functional characteristics of fly motion-sensitive interneurons , 1990, 1990 IJCNN International Joint Conference on Neural Networks.

[40]  M. Feigenbaum.  Universal behavior in nonlinear systems , 1983 .

[41]  Yu Fei,et al.  Periodic Solutions for Shunting Inhibitory Cellular Neural Networks of Neutral Type with Time-Varying Delays in the Leakage Term on Time Scales , 2014, J. Appl. Math..

[42]  Walter J. Freeman,et al.  TUTORIAL ON NEUROBIOLOGY: FROM SINGLE NEURONS TO BRAIN CHAOS , 1992 .

[43]  Li Yang,et al.  Anti-periodic solutions for a class of Cohen–Grossberg neural networks with time-varying delays on time scales , 2011, Int. J. Syst. Sci..

[44]  Elena G Tolkacheva,et al.  Intermittent electrical stimulation of the right cervical vagus nerve in salt-sensitive hypertensive rats: effects on blood pressure, arrhythmias, and ventricular electrophysiology , 2015, Physiological reports.

[45]  Yongkun Li,et al.  Almost periodic solutions of shunting inhibitory cellular neural networks on time scales , 2012 .

[46]  Pagavathigounder Balasubramaniam,et al.  Synchronization of a novel fractional order stretch-twist-fold (STF) flow chaotic system and its application to a new authenticated encryption scheme (AES) , 2014, Nonlinear Dynamics.

[47]  Chi-Bin Cheng,et al.  An asymmetric image cryptosystem based on the adaptive synchronization of an uncertain unified chaotic system and a cellular neural network , 2013, Commun. Nonlinear Sci. Numer. Simul..

[48]  Luigi Occhipinti,et al.  Cellular neural networks in secure transmission applications , 1996, 1996 Fourth IEEE International Workshop on Cellular Neural Networks and their Applications Proceedings (CNNA-96).

[49]  Huaiping Zhu,et al.  Periodic solution of single population models on time scales , 2010, Math. Comput. Model..

[50]  P. Davis,et al.  Chaotic wandering and search in a cycle-memory neural network , 1992 .

[51]  Shigetoshi Nara,et al.  CHAOTIC MEMORY DYNAMICS IN A RECURRENT NEURAL NETWORK WITH CYCLE MEMORIES EMBEDDED BY PSEUDO-INVERSE METHOD , 1995 .

[52]  Leon O. Chua,et al.  Cellular neural networks: applications , 1988 .

[53]  Changjin Xu,et al.  Existence and global exponential stability of anti-periodic solutions of high-order bidirectional associative memory (BAM) networks with time-varying delays on time scales , 2015, J. Comput. Sci..

[54]  François Blanchard,et al.  On Li-Yorke pairs , 2002, Journal für die reine und angewandte Mathematik (Crelles Journal).

[55]  Peter E. Kloeden,et al.  Li–Yorke chaos in higher dimensions: a review , 2006 .

[56]  Marat Akhmet,et al.  Li-Yorke Chaos in Hybrid Systems on a Time Scale , 2015, Int. J. Bifurc. Chaos.

[57]  Xinzhi Liu,et al.  Application of Impulsive Synchronization to Communication Security , 2003 .

[58]  Marat Akhmet,et al.  Li-Yorke chaos in the system with impacts , 2009 .

[59]  A. Peterson,et al.  Dynamic Equations on Time Scales: An Introduction with Applications , 2001 .

[60]  Guanrong Chen,et al.  Discrete chaos in Banach spaces , 2005 .

[61]  Lin-Bao Yang,et al.  Cellular neural networks: theory , 1988 .

[62]  J J Hopfield,et al.  Neurons with graded response have collective computational properties like those of two-state neurons. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

[63]  M. U. Akhmet,et al.  Entrainment by Chaos , 2012, J. Nonlinear Sci..

[64]  Warren M Grill,et al.  Bladder emptying by intermittent electrical stimulation of the pudendal nerve , 2006, Journal of neural engineering.

[65]  J. Yorke,et al.  Period Three Implies Chaos , 1975 .

[66]  Abdesselam Bouzerdoum,et al.  Properties of shunting inhibitory cellular neural networks for colour image enhancement , 1999, ICONIP'99. ANZIIS'99 & ANNES'99 & ACNN'99. 6th International Conference on Neural Information Processing. Proceedings (Cat. No.99EX378).

[67]  L.O. Chua,et al.  Cellular neural networks , 1993, 1988., IEEE International Symposium on Circuits and Systems.

[68]  Kestutis Pyragas,et al.  Modification of delayed feedback control using ergodicity of chaotic systems , 2010 .

[69]  M. Akhmet.  Devaney’s chaos of a relay system , 2009 .

[70]  Marat Akhmet,et al.  Replication of Chaos in Neural Networks, Economics and Physics , 2015 .

[71]  V.M. Mladenov,et al.  Synchronization of Chaotic Cellular Neural Networks based on Rössler Cells , 2006, 2006 8th Seminar on Neural Network Applications in Electrical Engineering.

[72]  R. B. Pinter,et al.  Shunting inhibitory cellular neural networks: derivation and stability analysis , 1993 .

[73]  Abdesselam Bouzerdoum,et al.  Nonlinear Lateral Inhibition Applied to Motion Detection in the fly Visual System , 2018 .

[74]  Christopher C. Tisdell,et al.  Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling , 2008 .

[75]  Chao Wang,et al.  Almost periodic solutions of impulsive BAM neural networks with variable delays on time scales , 2014, Commun. Nonlinear Sci. Numer. Simul..

[76]  Guanrong Chen,et al.  Chaos of discrete dynamical systems in complete metric spaces , 2004 .

[77]  Johan A. K. Suykens,et al.  Coupled chaotic simulated annealing processes , 2003, Proceedings of the 2003 International Symposium on Circuits and Systems, 2003. ISCAS '03..

[78]  Yongkun Li,et al.  Anti-periodic solutions to impulsive shunting inhibitory cellular neural networks with distributed delays on time scales , 2011 .

[79]  J. Hale,et al.  Dynamics and Bifurcations , 1991 .

[80]  Marat Akhmet,et al.  Replication of chaos , 2013, Commun. Nonlinear Sci. Numer. Simul..

[81]  Ivan Zelinka,et al.  Evolutionary Algorithms and Chaotic Systems , 2010, Evolutionary Algorithms and Chaotic Systems.

[82]  Marat Akhmet,et al.  Principles of Discontinuous Dynamical Systems , 2010 .

[83]  Qiankun Song,et al.  Global stability of complex-valued neural networks with both leakage time delay and discrete time delay on time scales , 2013, Neurocomputing.

[84]  Wolfgang A. Halang,et al.  Li–Yorke chaos in a spatiotemporal chaotic system , 2007 .

[85]  B. Aulbach,et al.  On Three Definitions of Chaos , 2001 .

[86]  A. Londei,et al.  Synchronization phenomena in 2D chaotic CNN , 1994, Proceedings of the Third IEEE International Workshop on Cellular Neural Networks and their Applications (CNNA-94).

[87]  K. Yamashita,et al.  A chaotic neural network for reducing the peak-to-average power ratio of multicarrier modulation , 2003, Proceedings of the International Joint Conference on Neural Networks, 2003..

[88]  Kestutis Pyragas.  Continuous control of chaos by self-controlling feedback , 1992 .

[89]  M. O. Fen,et al.  Shunting inhibitory cellular neural networks with chaotic external inputs. , 2013, Chaos.

[90]  Zhang Lei,et al.  The Chaotic Cipher Based on CNNs and Its Application in Network , 2011, 2011 2nd International Symposium on Intelligence Information Processing and Trusted Computing.

[91]  Jitao Sun,et al.  Stationary oscillation for chaotic shunting inhibitory cellular neural networks with impulses. , 2007, Chaos.

[92]  A. Roberts,et al.  Spinal networks controlling swimming in hatchling Xenopus tadpoles , 1997 .