Dynamic analysis of a new two-dimensional map in three forms: integer-order, fractional-order and improper fractional-order

In this paper, a new 2-dimensional chaotic map with a simple algebraic form is proposed. And the numerical solution of the corresponding fractional-order map is derived. It is novel that the new map still exhibits chaotic behaviors when the new map is expanded to fractional-order and improper fracti...

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Published inThe European physical journal. ST, Special topics Vol. 230; no. 7-8; pp. 1945 - 1957
Main Authors Ma, Chenguang, Mou, Jun, Li, Peng, Liu, Tianming
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2021
Springer Nature B.V
Subjects
Algorithms
Atomic
Chaos theory
Circuit Application of Chaotic Systems: Modeling
Classical and Continuum Physics
Condensed Matter Physics
Dynamical Analysis and Control
Entropy
Integers
Liapunov exponents
Materials Science
Measurement Science and Instrumentation
Molecular
Optical and Plasma Physics
Permutations
Physics
Physics and Astronomy
Regular Article
Two dimensional analysis
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Summary:In this paper, a new 2-dimensional chaotic map with a simple algebraic form is proposed. And the numerical solution of the corresponding fractional-order map is derived. It is novel that the new map still exhibits chaotic behaviors when the new map is expanded to fractional-order and improper fractional-order. The dynamical characteristics of these three forms are detected through the bifurcation diagram, the maximum Lyapunov exponent spectrum, Kolmogorov entropy and attractor portraits. More interestingly, the new map has multiple coexisting attractors, but the multistability of the fractional-order and improper fractional-order is more complicated than the integer-order form. In addition, the Permutation entropy (PE) complexity algorithm and 2-dimensional maximum Lyapunov exponent diagram are used to explore the dynamic changes of three forms when the amplitude changes simultaneously. The analysis shows that under the appropriate order, the chaotic range of the fractional-order and improper fractional-order is larger than that of the integer order. This research provides guidance on the application and teaching of discrete fractional-order systems.
ISSN:1951-6355
1951-6401
DOI:10.1140/epjs/s11734-021-00133-w