Complex Dynamics of BRD Sets
The intent of the paper is to study the dynamics of the Mandelbrot like Baker-Rippon-Devaney (BRD) sets for complex exponential family under Mann iterates.
- PDF / 986,157 Bytes
- 6 Pages / 439.37 x 666.142 pts Page_size
- 27 Downloads / 224 Views
Abstract The intent of the paper is to study the dynamics of the Mandelbrot like Baker-Rippon-Devaney (BRD) sets for complex exponential family under Mann iterates. Keywords Mandelbrot set • BRD set • Complex exponential function • Mann iteration
1 Introduction The Julia and Mandelbrot sets are of vital importance in the study of the complex dynamics of functions. A lot of work has been done on the structures of these sets of the complex analytic functions such as polynomial, rational and exponential functions. The importance of the transcendental functions lies in the fact that the Mandelbrot and Julia sets for such functions have an alternative characterisation suitable for easier computations. Such functions are studied by authors of this paper in [1, 2]. Misiurewicz [3] was the first to explore the mathematical aspects of the complex exponential maps of the type z n+1 = e z n, Baker and Rippon [4] studied the complex exponential family eλz and Devaney [5–7] extensively studied the family λe z of maps. Romera et al. [8] noticed that the two exponential families z n+1 = eλz n and z n+1 = λe z n have the same Mandelbrot-like sets. Thereafter, they call these sets as Baker-Rippon-Devaney (BRD) sets. They studied these BRD and 2 Julia sets for the complex families λe z , e z +λ and e z/λ from graphical point of view and obtained interesting results. The sequences {z n+1 } defined above are generated iteratively using Picard iteration scheme. Many authors have studied these families
B. Prasad (*) · K. Katiyar Department of Mathematics, Jaypee Institute of Information Technology, A-10, Sector-62, Noida 201307, UP, INDIA e-mail: [email protected] K. Katiyar e-mail: [email protected]
S. Sathiyamoorthy et al. (eds.), Emerging Trends in Science, Engineering and Technology, Lecture Notes in Mechanical Engineering, DOI: 10.1007/978-81-322-1007-8_64, © Springer India 2012
683
B. Prasad and K. Katiyar
684
by using different iteration procedures. Prasad and Katiyar [2] used the Mann iteration scheme in their bifurcation analysis related results for the complex exponential family λe z. Further, they used Ishikawa iteration scheme to study the fractal patterns of contractive maps (see [9]). In this paper, our aim is to study the structures of Mandelbrot like sets for the complex exponential family λe z and e z/λ by using the Mann iterative scheme.
2 Preliminaries In this section, we present the basic definitions and concepts required for our study. Definition 2.1. The Mandelbrot-like set of a family of complex maps z n+1 = f λ (z n ) for the initial value z0 (usually the one corresponding to the critical point of the family of maps) is defined as the set of λ ∈ C (the set of complex numbers) for which the n-th iteration of the function f λn (z 0 ) does not tend to ∞ as n tends to ∞
M = {λ ∈ C| f λn (z 0 )K∞ when n → ∞} We denote the complex exponential family of the map z n+1 = f λ (z n ) by E λ (z) . Definition 2.2. Let X be a non-empty set and f: X → X. The orbit of a point z in X is defi
Data Loading...
