Optimizing the maximum Lyapunov exponent and phase space portraits in multi-scroll chaotic oscillators

This investigation introduces the application of the nondominated sorting genetic algorithm to optimize two characteristics of multiscroll chaotic oscillators: (a) Maximizing the values of the maximum Lyapunov exponent (MLE), and (b) minimizing the dispersions of the phase space portraits (PSP) among all scrolls in an attractor. As shown in this study, these two oscillator’s characteristics are in conflict and must be considered at the same time. The cases of study are two multiscroll chaotic oscillators based on piecewise-linear functions, namely: saturated function series and Chua’s diode (negative slopes). Basically, a very new procedure to measure the PSP coverture among all generated scrolls is introduced in the optimization loop for each feasible solution maximizing the MLE. The best optimized results are compared with traditional values of the coefficients of the equations describing the oscillators. Finally, we list the values of the optimized MLE and their corresponding PSP when generating from 2- to 6-scroll attractors.

[1]  Guanrong Chen,et al.  Generating Multiscroll Chaotic Attractors: Theories, Methods and Applications , 2006 .

[2]  Kalyanmoy Deb,et al.  A fast and elitist multiobjective genetic algorithm: NSGA-II , 2002, IEEE Trans. Evol. Comput..

[3]  C. Wu,et al.  A radial-basis-function network-based method of estimating Lyapunov exponents from a scalar time series for analyzing nonlinear systems stability , 2012 .

[4]  R. M. López-Gutiérrez,et al.  Synchronization of multi-scroll chaos generators: application to private communication , 2008 .

[5]  Carlos Sánchez-López,et al.  Integrated circuit generating 3- and 5-scroll attractors , 2012 .

[6]  R. Aguilar-López,et al.  Chaotic Systems Synchronization Via High Order Observer Design , 2011 .

[7]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[8]  E. Tlelo-Cuautle,et al.  Optimizing the positive Lyapunov exponent in multi-scroll chaotic oscillators with differential evolution algorithm , 2013, Appl. Math. Comput..

[9]  C. Sánchez López.  A 1.7 MHz Chua's circuit using VMs and CF+s , 2012 .

[10]  L. C. Gómez-Pavón,et al.  Synchronization of PWL function-based 2D and 3D multi-scroll chaotic systems , 2012 .

[11]  M. Cao,et al.  Synthesis process and growth mechanism of γ′-Fe4N nanoparticles by phase-transformation , 2003 .

[12]  Francisco Balibrea,et al.  Stability of orbits via Lyapunov exponents in Autonomous and nonautonomous Systems , 2013, Int. J. Bifurc. Chaos.

[13]  Ricardo Suyama,et al.  Characterization of multiscroll attractors using Lyapunov exponents and Lagrangian coherent structures. , 2013, Chaos.

[14]  Luca Dieci,et al.  Jacobian Free Computation of Lyapunov Exponents , 2002 .

[15]  Sheng-Chieh Huang,et al.  Generating tri-chaos attractors with three positive Lyapunov exponents in new four order system via linear coupling , 2012, Nonlinear Dynamics.

[16]  Caijin Yang,et al.  Approximate and efficient calculation of dominant Lyapunov exponents of high-dimensional nonlinear dynamic systems , 2013, Commun. Nonlinear Sci. Numer. Simul..

[17]  Simin Yu,et al.  New results of study on generating multiple-scroll chaotic attractors , 2007, Science in China Series F: Information Sciences.

[18]  Henry Leung,et al.  Experimental verification of multidirectional multiscroll chaotic attractors , 2006, IEEE Transactions on Circuits and Systems I: Regular Papers.

[19]  Gene H. Golub,et al.  Matrix computations , 1983 .

[20]  Esteban Tlelo-Cuautle,et al.  A survey on the integrated design of chaotic oscillators , 2013, Appl. Math. Comput..

[21]  Julien Clinton Sprott,et al.  Amplitude control approach for chaotic signals , 2013 .

[22]  Serdar Çiçek,et al.  Simulation and Circuit Implementation of Sprott Case H Chaotic System and its Synchronization Application for Secure Communication Systems , 2013, J. Circuits Syst. Comput..

[23]  Kim-Fung Man,et al.  A Systematic Approach to Generating n-scroll attractors , 2002, Int. J. Bifurc. Chaos.