scholarly journals A Quadratic Fractional Map without Equilibria: Bifurcation, 0–1 Test, Complexity, Entropy, and Control

Electronics ◽  
2020 ◽  
Vol 9 (5) ◽  
pp. 748 ◽  
Author(s):  
Adel Ouannas ◽  
Amina-Aicha Khennaoui ◽  
Shaher Momani ◽  
Giuseppe Grassi ◽  
Viet-Thanh Pham ◽  
...  
Keyword(s):  
Lyapunov Exponent ◽  
Test Phase ◽  
Research Topic ◽  
Chaotic Attractors ◽  
Maximum Lyapunov Exponent ◽  
Bifurcation Diagrams ◽  
Hidden Attractors ◽  
Discrete Time Systems ◽  
And Control ◽  
Time Systems

Fractional calculus in discrete-time systems is a recent research topic. The fractional maps introduced in the literature often display chaotic attractors belonging to the class of “self-excited attractors”. The field of fractional map with “hidden attractors” is completely unexplored. Based on these considerations, this paper presents the first example of fractional map without equilibria showing a number of hidden attractors for different values of the fractional order. The presence of the chaotic hidden attractors is validated via the computation of bifurcation diagrams, maximum Lyapunov exponent, 0–1 test, phase diagrams, complexity, and entropy. Finally, an active controller with the aim for stabilizing the proposed fractional map is successfully designed.

Degenerate Chenciner Bifurcation Revisited

2021 ◽  
Vol 31 (10) ◽  
pp. 2150160
Author(s):  
G. Tigan ◽  
O. Brandibur ◽  
E. A. Kokovics ◽  
L. F. Vesa
Keyword(s):  
Discrete Time ◽  
Two Dimensional ◽  
Bifurcation Diagrams ◽  
Discrete Time Systems ◽  
Time Systems ◽  
Independent Parameters

Generic results for degenerate Chenciner (generalized Neimark–Sacker) bifurcation are obtained in the present work. The bifurcation arises from two-dimensional discrete-time systems with two independent parameters. We define in this work a new transformation of parameters, which enables the study of the bifurcation when degeneracy occurs. By the four bifurcation diagrams we obtained, new behaviors hidden by the degeneracy are brought to light.

scholarly journals Design of Positive, Negative, and Alternating Sign Generalized Logistic Maps

2015 ◽  
Vol 2015 ◽  
pp. 1-23 ◽  
Author(s):  
Wafaa S. Sayed ◽  
Ahmed G. Radwan ◽  
Hossam A. H. Fahmy
Keyword(s):  
Lyapunov Exponent ◽  
Bifurcation Diagram ◽  
Degrees Of Freedom ◽  
Chaotic Maps ◽  
Logistic Map ◽  
Maximum Lyapunov Exponent ◽  
Bifurcation Diagrams ◽  
Financial Modeling ◽  
Design Flexibility ◽  
Special Case

The discrete logistic map is one of the most famous discrete chaotic maps which has widely spread applications. This paper investigates a set of four generalized logistic maps where the conventional map is a special case. The proposed maps have extra degrees of freedom which provide different chaotic characteristics and increase the design flexibility required for many applications such as quantitative financial modeling. Based on the maximum chaotic range of the output, the proposed maps can be classified as positive logistic map, mostly positive logistic map, negative logistic map, and mostly negative logistic map. Mathematical analysis for each generalized map includes bifurcation diagrams relative to all parameters, effective range of parameters, first bifurcation point, and the maximum Lyapunov exponent (MLE). Independent, vertical, and horizontal scales of the bifurcation diagram are discussed for each generalized map as well as a new bifurcation diagram related to one of the added parameters. A systematic procedure to design two-constraint logistic map is discussed and validated through four different examples.

AN OPTIMIZATION APPROACH TO THE DESIGN OF CONSTRAINED REGULATORS FOR UNCERTAIN DISCRETE-TIME SYSTEMS

2006 ◽  
Vol 15 (03) ◽  
pp. 373-387
Author(s):  
M. VASSILAKI ◽  
G. BITSORIS
Keyword(s):  
Discrete Time ◽  
Sufficient Conditions ◽  
Optimization Approach ◽  
Control Constraints ◽  
Existence Of A Solution ◽  
Discrete Time Systems ◽  
Necessary And Sufficient ◽  
And Control ◽  
Time Systems ◽  
State And Control Constraints

In this paper the regulation problem of linear discrete-time systems with uncertain parameters under state and control constraints is studied. In the first part of the paper, two theorems concerning necessary and sufficient conditions for the existence of a solution to this problem are presented. Due to the constructive form of the proof of these theorems, these results can be used to the development of techniques for the derivation of a control law transferring to the origin any state belonging to a given set of initial states while respecting the state and control constraints.

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