On fractional difference logistic maps: Dynamic analysis and synchronous control
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ORIGINAL PAPER
On fractional difference logistic maps: Dynamic analysis and synchronous control Yupin Wang · Shutang Liu
· Hui Li
Received: 1 January 2020 / Accepted: 29 August 2020 © Springer Nature B.V. 2020
Abstract This paper investigates a logistic map derived from a difference equation in the framework of discrete fractional calculus. Through the Poincaré plots and Julia sets, the map’s chaotic and fractal characteristics are studied comparing with those of a quadratic map to be proposed. The memory effect of fractional difference maps is reflected in these dynamics, and some reasonable explanations are given by combining with quantitative analysis. A coupled controller is designed to realize synchronization between fractional difference logistic map and fractional difference quadratic map.
1 Introduction
Keywords Discrete fractional calculus · Caputo delta difference · Logistic map · Chaos · Fractal · Synchronization
dx = r x(1 − x), dt
Mathematics Subject Classification 26A33 · 34D06 · 39A12 · 39A33 · 92D25 Y. Wang Institute of Marine Science and Technology, Shandong University, Qingdao 266237, Shandong, People’s Republic of China e-mail: [email protected] S. Liu (B) · H. Li School of Control Science and Engineering, Shandong University, Jinan 250061, Shandong, People’s Republic of China e-mail: [email protected] H. Li e-mail: [email protected]
The logistic map is often cited as an archetypal example of complex and chaotic behavior that can arise from very simple nonlinear dynamical systems. Mathematically, its standard form is xn+1 = r xn (1 − xn ).
(1)
The map, popularized by the biologist May (cf. [1]), is a discrete-time form of the logistic equation
where x ∈ [0, 1], which is a standardized version of the demographic model created by Verhulst (cf. [2,3]). Since the end of the 20th century, the logistic map (1) and its various generalizations have aroused more and more concerns in the field of nonlinear science, especially chaos and fractals. With the flourishing of fractional system modeling from the beginning of this century, various forms of the fractionalization of the classic logistic map (cf. [4]) and equation (cf. [5]) gradually appeared and attracted the scholars’ attention. In 2012, Edelman introduced the notions of the Riemann-Liouville and Caputo αfamilies of maps and studied their dynamic behavior taking the fractional logistic map as an example (cf. [6]). It is found that the existence of the cascade of
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bifurcations type trajectories is a general property of fractional dynamical systems (cf. [7]). In the last decade, discrete fractional calculus has increasingly become another research hotspot in mathematics besides conventional fractional calculus whose development is relatively mature. Various logistic-like equations with fractional difference operators have also begun to come into view and aroused widespread interest. In 2014, Wu et al. (cf. [8]) proposed a fractional difference equation (FDE, for short) as C
Δaα x(t) = r x(t + α − 1)[1 − x(t + α −
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