Design and analysis of a first order time-delayed chaotic system

The present paper reports the design and analysis of a new time-delayed chaotic system and its electronic circuit implementation. The system is described by a first-order nonlinear retarded type delay differential equation with a closed form mathematical function describing the nonlinearity. We carry out stability and bifurcation analysis to show that with the suitable delay and system parameters the system shows sustained oscillation through supercritical Hopf bifurcation. It is shown through numerical simulations that the system depicts bifurcation and chaos for a certain range of the system parameters. The complexity and predictability of the system are characterized by Lyapunov exponents and Kaplan–York dimension. It is shown that, for some suitably chosen system parameters, the system shows hyperchaos even for a small or moderate delay. Finally, we set up an experiment to implement the proposed system in electronic circuit using off-the-shelf circuit elements, and it is shown that the behavior of the time delay chaotic electronic circuit agrees well with our analytical and numerical results.

[1]  Salvatore Graziani,et al.  Stochastic Resonance: Theory and Applications , 2000 .

[2]  Keiji Konishi,et al.  Design and experimental verification of multiple delay feedback control for time-delay nonlinear oscillators , 2011, Nonlinear Dynamics.

[3]  Arünas Tamasevicius,et al.  Two-scroll Attractor in a Delay Dynamical System , 2007, Int. J. Bifurc. Chaos.

[4]  Luigi Fortuna,et al.  Implementation of chaotic circuits with a digital time-delay block , 2012 .

[5]  Nikolay V. Kuznetsov,et al.  Time-Varying Linearization and the Perron Effects , 2007, Int. J. Bifurc. Chaos.

[6]  Pérez,et al.  Extracting messages masked by chaos. , 1995, Physical review letters.

[7]  Serdar Ozoguz,et al.  n-scroll chaotic attractors from a first-order time-delay differential equation. , 2007, Chaos.

[8]  K. Ikeda,et al.  Optical Turbulence: Chaotic Behavior of Transmitted Light from a Ring Cavity , 1980 .

[9]  A. Sedra.  Microelectronic circuits , 1982 .

[10]  Junjie Wei,et al.  Stability and bifurcation analysis in the cross-coupled laser model with delay , 2011 .

[11]  Giovanni Muscato,et al.  Chaos Does Help Motion Control , 2007, Int. J. Bifurc. Chaos.

[12]  Julien Clinton Sprott,et al.  A simple chaotic delay differential equation , 2007 .

[13]  Li Changguo,et al.  Effect of delay on a predator–prey model with parasitic infection , 2011 .

[14]  Kestutis Pyragas,et al.  An electronic analog of the Mackey-Glass system , 1995 .

[15]  Furong Gao,et al.  Extraction of delay information from chaotic time series based on information entropy , 1997 .

[16]  Richard H. S. Taylor,et al.  El Niño and the delayed action oscillator , 2007 .

[17]  Luigi Fortuna,et al.  Design of Time-Delay Chaotic Electronic Circuits , 2011, IEEE Transactions on Circuits and Systems I: Regular Papers.

[18]  Tanmoy Banerjee,et al.  Single amplifier biquad based inductor-free Chua’s circuit , 2012, 1210.8409.

[19]  Unas Tama,et al.  TWO-SCROLL ATTRACTOR IN A DELAY DYNAMICAL SYSTEM , 2007 .

[20]  R. D. Driver,et al.  A neutral system with state-dependent delay☆ , 1984 .

[21]  A. Tamasevicius,et al.  Very high and ultrahigh frequency hyperchaotic oscillators with delay line , 2003 .

[22]  Behzad Razavi,et al.  Modeling Op Amp Nonlinearity in Switched-Capacitor Sigma-Delta Modulators , 2006, IEEE Custom Integrated Circuits Conference 2006.

[23]  Chuandong Li,et al.  Stability and bifurcation analysis in tri-neuron model with time delay , 2007 .

[24]  Wang Bing-wen.  Secure communication based on chaotic synchronization , 2007 .

[25]  Colin H. Hansen,et al.  Effect of External Excitations on a Nonlinear System with Time Delay , 2005 .

[26]  A. Tamasevicius,et al.  Delayed feedback chaotic oscillator with improved spectral characteristics , 2006 .

[27]  Yu. A. Kuznetsov,et al.  Applied nonlinear dynamics: Analytical, computational, and experimental methods , 1996 .

[28]  I. Raja Mohamed,et al.  Design of Time Delayed Chaotic Circuit with Threshold Controller , 2011, Int. J. Bifurc. Chaos.

[29]  A.M. Davis,et al.  Microelectronic circuits , 1983, Proceedings of the IEEE.

[30]  Junjie Wei,et al.  Bifurcation analysis in a scalar delay differential equation , 2007 .

[31]  Zhenya He,et al.  A chaos-generator: analyses of complex dynamics of a cell equation in delayed cellular neural networks , 1998 .

[32]  L. Glass,et al.  Oscillation and chaos in physiological control systems. , 1977, Science.

[33]  Ahmet Uçar,et al.  On the chaotic behaviour of a prototype delayed dynamical system , 2003 .

[34]  Furong Gao,et al.  Adaptive control of chaotic continuous-time systems with delay , 1998 .

[35]  Qingyun Wang,et al.  Bifurcation dynamics of the modified physiological model of artificial pancreas with insulin secretion delay , 2011 .

[36]  I. Raja Mohamed,et al.  Design of Threshold Controller Based Chaotic Circuits , 2010, Int. J. Bifurc. Chaos.

[37]  Zhenya He,et al.  Chaotic behavior in first-order autonomous continuous-time systems with delay , 1996 .

[38]  Peng,et al.  Synchronizing hyperchaos with a scalar transmitted signal. , 1996, Physical review letters.

[39]  Young-Jai Park,et al.  Synchronization of chaotic oscillators due to common delay time modulation. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  J. D. Farmer,et al.  Chaotic attractors of an infinite-dimensional dynamical system , 1982 .

[41]  Luigi Fortuna,et al.  Chaotic pulse position modulation to improve the efficiency of sonar sensors , 2003, IEEE Trans. Instrum. Meas..