Control and Synchronization of Julia Sets Generated by a Class of Complex Time-Delay Rational MAP

In this paper, a class of complex time-delay rational map is studied by analyzing the fractal and dynamical properties of its corresponding Julia sets (CTRM-Julia sets for short). By utilizing these given properties, a hybrid control method which contains both state feedback and parameters perturbation is applied to achieve the boundary control of CTRM-Julia set. Moreover, the synchronization of two different CTRM-Julia sets is also investigated by using coupling method. The synchronization index method is applied to demonstrate the relationship between the degree of synchronization and the coupling strength. Numerical examples are given to verify the effectiveness of control and synchronization methods.

[1]  Janet Helmstedt,et al.  Graphics with Mathematica: Fractals, Julia Sets, Patterns and Natural Forms , 2004 .

[2]  J. Argyris,et al.  On the Julia sets of a noise-perturbed Mandelbrot map , 2002 .

[3]  Xing-yuan Wang,et al.  Noise perturbed generalized Mandelbrot sets , 2008 .

[4]  Bernard Derrida,et al.  Fractal structure of zeros in hierarchical models , 1983 .

[5]  Ruihong Jia,et al.  The Generalized Julia Set Perturbed by Composing Additive and Multiplicative Noises , 2009 .

[6]  Mamta Rani,et al.  Effect of Stochastic Noise on Superior Julia Sets , 2009, Journal of Mathematical Imaging and Vision.

[7]  John Erik Fornaess.  The Julia set for Henon maps , 2005 .

[8]  J. Argyris,et al.  On the Julia set of the perturbed Mandelbrot map , 2000 .

[9]  Peng Li,et al.  Complex time-delay dynamical systems of quadratic polynomials mapping , 2015 .

[10]  Shutang Liu,et al.  Control and synchronization of Julia sets in coupled map lattice , 2011 .

[11]  Xingyuan Wang,et al.  Quasi-sine Fibonacci M set with perturbation , 2012 .

[12]  Shu-Tang Liu,et al.  New identification and control methods of sine-function Julia sets , 2015 .

[13]  Nelly Selem Mojica,et al.  Cellular "bauplans": Evolving unicellular forms by means of Julia sets and Pickover biomorphs , 2009, Biosyst..

[14]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[15]  Robert L. Devaney,et al.  Limiting Julia Sets for singularly perturbed Rational Maps , 2008, Int. J. Bifurc. Chaos.

[16]  Xingyuan Wang,et al.  Additive perturbed generalized Mandelbrot-Julia sets , 2007, Appl. Math. Comput..

[17]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[18]  Yongping Zhang,et al.  Control and Synchronization of Julia Sets of the Complex perturbed Rational Maps , 2013, Int. J. Bifurc. Chaos.

[19]  Xingyuan Wang,et al.  Research on fractal structure of generalized M-J sets utilized Lyapunov exponents and periodic scanning techniques , 2006, Appl. Math. Comput..

[20]  Satoru Saito,et al.  An analysis of a family of rational maps containing integrable and non-integrable difference analogue of the logistic equation , 1996 .

[21]  G. Julia.  Mémoire sur l'itération des fonctions rationnelles , 1918 .

[22]  Guanrong Chen,et al.  On generalized synchronization of spatial chaos , 2003 .

[23]  Guanrong Chen,et al.  Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems , 2003 .

[24]  Fangfang Zhang,et al.  Complex function projective synchronization of complex chaotic system and its applications in secure communication , 2013, Nonlinear Dynamics.

[25]  张永平,et al.  Gradient control and synchronization of Julia sets , 2008 .

[26]  Ioannis Andreadis,et al.  On numerical approximations of the area of the generalized Mandelbrot sets , 2013, Appl. Math. Comput..

[27]  Ian D. Entwistle,et al.  Julia set art and fractals in the complex plane , 1989, Comput. Graph..

[28]  X. Y. Wang,et al.  Noise-perturbed quaternionic Mandelbrot sets , 2009, Int. J. Comput. Math..

[29]  Shutang Liu,et al.  Synchronization between the spatial Julia sets of complex Lorenz system and complex Henon map , 2015 .

[30]  Yongcheng Yin,et al.  Dynamics of McMullen maps , 2012, 1204.1907.

[31]  M. Levin,et al.  A Julia set model of field-directed morphogenesis: developmental biology and artificial life , 1994, Comput. Appl. Biosci..

[32]  Robert L. Devaney,et al.  The Rabbit and Other Julia Sets Wrapped in Sierpi_ski Carpets , 2009 .

[33]  Xiaopeng Hu,et al.  A Novel Fractal Coding Method Based on M-J Sets , 2014, PloS one.

[34]  G. M. Levin,et al.  Symmetries on the Julia set , 1990 .

[35]  Zhang Zhenfeng,et al.  The generalized Mandelbrot set perturbed by composing noise of additive and multiplicative , 2009 .

[36]  Ioannis Andreadis,et al.  On probabilistic Mandelbrot maps , 2009 .

[37]  Ioannis Andreadis,et al.  On a Closeness of the Julia Sets of noise-perturbed Complex quadratic Maps , 2012, Int. J. Bifurc. Chaos.

[38]  Ioannis Andreadis,et al.  On perturbations of the Mandelbrot map , 2000 .

[39]  C. Beck.  Physical meaning for Mandelbrot and Julia sets , 1999 .

[40]  Ioannis Andreadis,et al.  On a topological closeness of perturbed Julia sets , 2010, Appl. Math. Comput..

[41]  Robert L. Devaney,et al.  A Generalized Version of the McMullen Domain , 2008, Int. J. Bifurc. Chaos.

[42]  John Erik Fornæss.  The Julia Set of Hénon Maps , 2005 .

[43]  Ioannis Andreadis,et al.  On a topological closeness of perturbed Mandelbrot sets , 2010, Appl. Math. Comput..