PARAMETER ESTIMATION OF THE CLASSICAL FRACTAL MAP BASED ON A GIVEN JULIA SET’S SHAPE

Da Wang, Shicun Zhao, Ke Chen and Shutang Liu
Additional contact information
Da Wang: Research Center of Dynamics System and Control Science, Shandong Normal University, Ji’nan 250014, P. R. China
Shicun Zhao: Research Center of Dynamics System and Control Science, Shandong Normal University, Ji’nan 250014, P. R. China
Ke Chen: School of Electrical Engineering, Zhengzhou University, Zhengzhou 450001, P. R. China
Shutang Liu: School of Control Science and Engineering, Shandong University, Ji’nan 250061, P. R. China

FRACTALS (fractals), 2021, vol. 29, issue 08, 1-16

Abstract: Julia set is one of the most important branches in fractal theory, and has achieved much progress on both theoretical analysis and application research. Nevertheless, most of the existing investigations relied on the systems with known parameters. Few works have been done on the inverse problem about how to determine the system parameter based on a given Julia set’s shape. This work aims to address this issue by starting with the most classical polynomial map. First, by constructing a proper fitness function measuring the Julia sets’ area error, the Julia sets’ parameter estimation is formulated as a kind of optimization problem. According to the known conditions of Julia set to be estimated, the optimization problem is classified into two cases. For one case, when both the shape and size of Julia set are given, we only need to estimate the system’s own parameter c. For the other case, when the Julia set has only a shape, we redesign the problem into three dimensions by adding a new scaling factor parameter γ and applying the partial coverage principle. Second, a particle swarm optimization (PSO) approach including dynamically adjustment strategy is adopted to solve the proposed optimization problem. At last, we present seven groups of simulations in which both randomly generated Julia sets and those in existing literature (or on the internet) are included. The simulation results verify the effectiveness of the proposed parameter estimation scheme.

Keywords: Fractal; Parameter Estimation; Julia Set; Particle Swarm Optimization (search for similar items in EconPapers)
Date: 2021
References: Add references at CitEc
Citations:

Downloads: (external link)
http://www.worldscientific.com/doi/abs/10.1142/S0218348X21502479
Access to full text is restricted to subscribers

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:wsi:fracta:v:29:y:2021:i:08:n:s0218348x21502479

Ordering information: This journal article can be ordered from

DOI: 10.1142/S0218348X21502479

Access Statistics for this article

FRACTALS (fractals) is currently edited by Tara Taylor

More articles in FRACTALS (fractals) from World Scientific Publishing Co. Pte. Ltd.
Bibliographic data for series maintained by Tai Tone Lim ().