CONTROL AND SYNCHRONIZATION OF JULIA SETS GENERATED BY A CLASS OF COMPLEX TIME-DELAY RATIONAL MAP

Da Wang; Shutang Liu; Kexin Liu; Yang Zhao

doi:10.11948/2016068

2016 Volume 6 Issue 4

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Da Wang, Shutang Liu, Kexin Liu, Yang Zhao. CONTROL AND SYNCHRONIZATION OF JULIA SETS GENERATED BY A CLASS OF COMPLEX TIME-DELAY RATIONAL MAP[J]. Journal of Applied Analysis & Computation, 2016, 6(4): 1049-1063. doi: 10.11948/2016068

Citation:

Da Wang, Shutang Liu, Kexin Liu, Yang Zhao. CONTROL AND SYNCHRONIZATION OF JULIA SETS GENERATED BY A CLASS OF COMPLEX TIME-DELAY RATIONAL MAP[J]. Journal of Applied Analysis & Computation, 2016, 6(4): 1049-1063. doi: 10.11948/2016068

CONTROL AND SYNCHRONIZATION OF JULIA SETS GENERATED BY A CLASS OF COMPLEX TIME-DELAY RATIONAL MAP

1 College of Control Science and Engineering, Shandong University, Jinan, 250061, China;
2 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

Fund Project:

Abstract

Abstract

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.
- Time-delay complex system /
- julia set /
- control /
- synchronization
MSC: 34H05;93C28

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References

[1]	J. Argyris, T. E. Karakasidis and I. Andreadis, On the Julia set of the perturbed Mandelbrot map, Chaos, Solitons & Fractals, 11(2000), 2067-2073. Google Scholar
[2]	J. Argyris, I. Andreadis and T. E. Karakasidis, On perturbations of the Mandelbrot map, Chaos, Solitons & Fractals, 11(2000), 1131-1136. Google Scholar
[3]	J. Argyris, T. E. Karakasidis and I. Andreadis, On the Julia sets of a noiseperturbed Mandelbrot map, Chaos, Solitons & Fractals, 13(2002), 245-252. Google Scholar
[4]	I. Andreadis and T. E. Karakasidis, On a topological closeness of perturbed Mandelbrot sets, Applied Mathematics and Computation, 215(2010), 3674-3683. Google Scholar
[5]	I. Andreadis and T. E. Karakasidis, On a topological closeness of perturbed Julia sets, Applied Mathematics and Computation, 217(2010), 2883-2890. Google Scholar
[6]	I. Andreadis and T. E. Karakasidis, On probabilistic Mandelbrot maps, Chaos, Solitons & Fractals, 42(2010), 1577-1583. Google Scholar
[7]	I. Andreadis and T. E. Karakasidis, On a closeness of the Julia sets of noiseperturbed complex quadratic maps, International Journal of Bifurcation and Chaos, 22(2012), pp.1250221. Google Scholar
[8]	I. Andreadis and T. E. Karakasidis, On numerical approximations of the area of the generalized Mandelbrot sets, Applied Mathematics and Computation, 219(2013), 10974-10982. Google Scholar
[9]	C. Beck, Physical meaning for Mandelbrot and Julia sets, Physica D:Nonlinear Phenomena, 125(1999), 171-182. Google Scholar
[10]	P. Blanchard, R. L. Devaney, A. Garijo, S. M. Marotta and E. D. Russell, The rabbit and other Julia sets wrapped in Sierpinski carpets, Complex Dynamics:Families and Friends, pp.277C296, 2009. Google Scholar
[11]	P. Blanchard, R. L. Devaney, A. Garijo and E. D. Russel, A generalized version of the McMullen domain, International Journal of Bifurcation and Chaos, 18(2008), 2309-2318. Google Scholar
[12]	G. R. Chen and S. T. Liu, On generalized synchronization of spatial chaos, Chaos, Solitons Fractals, 15(2003), 311-318. Google Scholar
[13]	B. Derrida, L. De Seze and C. Itzykson, Fractal structure of zeros in hierarchical models, Journal of Statistical Physics, 33(1983), 559-569. Google Scholar
[14]	I. D. Entwistle, Julia set art and fractals in the complex plane, Computers & Graphics, 13(1989), 389-392. Google Scholar
[15]	K. Falconer, Fractal Geometry:Mathematical Foundations and Applications, John Wiley Sons, 2013. Google Scholar
[16]	J.E. Fornaess, The Julia set of Henon maps, J. Math. Ann., 334(2006), 457-464. Google Scholar
[17]	C. Getz and J. M. Helmstedt, Graphics with Mathematica:fractals, Julia sets, patterns and natural forms, Elsevier, 2004. Google Scholar
[18]	G. Julia, Memoire sur literation des fonctions rationnelles, J. Math. Pures Appl., 4(1918), 47-245. Google Scholar
[19]	M. Levin, A Julia set model of field-directed morphogenesis:developmental biology and artificial life, Computer. applications in the biosciences, 10(1994), 85-105. Google Scholar
[20]	G. M. Levin, Symmetries on the Julia set, Mathematical Notes, 48(1990), 1126-1131. Google Scholar
[21]	P. Liu and S. T. Liu, Control and synchronization of Julia sets in coupled map lattice, Communications in Nonlinear Science and Numerical Simulation, 16(2011), 3344-3355. Google Scholar
[22]	S. T. Liu and F. F. Zhang, Complex function projective synchronization of complex chaotic system and its applications in secure communication, Nonlinear Dynamics, 76(2014), 1087-1097. Google Scholar
[23]	X. S. Luo, G. R. Chen, B. H. Wang and Q. F. Jin, Hybrid control of perioddoubling bifurcation and chaos in discrete nonlinear dynamical systems, Chaos, Solitons Fractals, 18(2003), 775-783. Google Scholar
[24]	B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM review, 10(1968), 422-437. Google Scholar
[25]	M. Morabito and R. L. Devaney, Limiting Julia sets for singularly perturbed rational maps, International Journal of Bifurcation and Chaos, 18(2008), 3175-3181. Google Scholar
[26]	N. S. Mojica, J. Navarro and P. C. Marijuan, Cellular bauplans:Evolving unicellular forms by means of Julia sets and Pickover biomorphs, Biosystems, 98(2009), 19-30. Google Scholar
[27]	W. Qiu, X. Wang and Y. Yin, Dynamics of McMullen maps,, Advances in Mathematics, 229(2012), 2525-2577. Google Scholar
[28]	M. Rani and R. Agarwal, Effect of stochastic noise on superior Julia sets, Journal of Mathematical Imaging and Vision, 36(2010), 63-68. Google Scholar
[29]	N. Saitoh, A. Shimizu and K. Yoshida, An analysis of a family of rational maps containing:integrable and non-integrable difference analogue of the logistic equation, J. Phys. A. Math. Gen., 29(1996), 1831-1840. Google Scholar
[30]	Y. Y. Sun, Z. Lu and P. Li, Complex time-delay dynamical systems of quadratic polynomials mapping, Nonlinear Dynamics, 79(2015), 369-375. Google Scholar
[31]	J. Sun, W. Qiao and S. T. Liu, New identification and control methods of sinefunction Julia sets, Journal of Applied Analysis and Computation, 5(2015), 220-231. Google Scholar
[32]	Y. Y. Sun and X. Y. Wang, Noise-perturbed quaternionic Mandelbrot sets, International Journal of Computer Mathematics, 86(2009), 2008-2028. Google Scholar
[33]	Y. Y. Sun, R. Xu, L. Chen, R. Kong and X. Hu, A Novel Fractal Coding Method Based on MJ Sets, PloS one, 9(2014), pp. e101697. Google Scholar
[34]	X. Y. Wang, and P. Chang, Research on fractal structure of generalized MJ sets utilized Lyapunov exponents and periodic scanning techniques, Applied mathematics and computation, 175(2006), 1007-1025. Google Scholar
[35]	X. Y. Wang, P. J. Chang and N. N. Gu, Additive perturbed generalized Mandelbrot-Julia sets, Applied Mathematics and Computation, 189(2007), 754-765. Google Scholar
[36]	X. Y. Wang, Z. Wang, Y. H. Lang and Z. F. Zhang, Noise perturbed generalized Mandelbrot sets, Journal of Mathematical Analysis and Applications, 347(2008), 179-187. Google Scholar
[37]	X. Y. Wang, R. H. Jia and Z. F. Zhang, The generalized Mandelbrot set perturbed by composing noise of additive and multiplicative, Applied Mathematics and Computation, 210(2009), 107-118. Google Scholar
[38]	X. Y. Wang, R. H. Jia and Y. Y. Sun, The generalized Julia set perturbed by composing additive and multiplicative noises, Discrete Dynamics in Nature and Society, 2009(2010), Article ID:781976. Google Scholar
[39]	X. Y. Wang and F. Ge, Quasi-sine Fibonacci M set with perturbation, Nonlinear Dynamics, 69(2012), 1765-1779. Google Scholar
[40]	D. Wang and S. T. Liu, Synchronization between the spatial Julia sets of complex Lorenz system and complex Henon map, Nonlinear Dynamics, 81(2015), 1197-1205. Google Scholar
[41]	Y. P. Zhang, Control and synchronization of Julia sets of the complex perturbed rational maps, International Journal of Bifurcation and Chaos, 23(2013), pp.1350083. Google Scholar
[42]	Y. P. Zhang and S. T. Liu, Gradient control and synchronization of Julia sets, Chinese Physics B, 17(2008), pp.543. Google Scholar