Abstract
A complex map can give rise to two kinds of fractal sets: the Julia sets and the parameters sets (or the connectivity loci) which represent different connectivity properties of the corresponding Julia sets. In the significative results of (Int. J. Bifurc. Chaos, 2009, 19:2123–2129) and (Nonlinear. Dyn. 2013, 73:1155–1163), the authors presented the two kinds of fractal sets of a class of alternated complex map and left some visually observations to be proved about the boundedness and symmetry properties of these fractal sets. In this paper, we improve the previous results by giving the strictly mathematical proofs of the two properties. Some simulations that verify the theoretical proofs are also included.
Highlights
Julia set [1], one of the most attractive fractal sets, is proposed by Gaston Julia when he studied the following complex quadratic polynomialPc : zn+1 = zαn + c c ∈ C, α = 2 (1)Since it was proposed, Julia set has attracted significant interests in the topics of property analysis [2,3,4,5], superior iteration [6,7,8], time-delay [9], control [10,11] and so forth
Based on these graphical results, some observations about the boundedness and symmetry properties of these connectivity loci were summarized in Section 4 of [18]
For the systems like (3) which have more than one critical point, the results mainly focused on the properties of their Julia sets
Summary
Julia set [1], one of the most attractive fractal sets, is proposed by Gaston Julia when he studied the following complex quadratic polynomial. Julia sets were respectively denoted as the Connectedness–Locus, Disconnectedness–Locus and Totally connectedness–Locus [17,18] Based on these graphical results, some observations about the boundedness and symmetry properties of these connectivity loci were summarized in Section 4 of [18]. Few research papers have addressed the problems about the boundedness and symmetry properties of their connectivity loci (or Mandelbrot set) since all the critical orbits should be considered. As a supplementary research of [17,18], the purpose and achievement of this paper are no other than giving the mathematical proof about the two properties of the Connectedness–Locus and the Julia sets generated from (2) via analyzing all the critical orbits of system (3).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
