Poincare maps for detecting chaos in fractional-order systems with hidden attractors for its Kaplan-Yorke dimension optimization

被引：13

作者：

Clemente-Lopez, Daniel ^{[1
]}

Tlelo-Cuautle, Esteban ^{[1
]}

de la Fraga, Luis-Gerardo ^{[2
]}

de Jesus Rangel-Magdaleno, Jose ^{[1
]}

Manuel Munoz-Pacheco, Jesus ^{[3
]}

机构：

[1] Inst Nacl Astrofis Opt & Elect INAOE, Dept Elect, Luis Enrique Erro 1, Puebla 72840, Mexico

[2] CINVESTAV, Comp Sci Dept, Av IPN 2508, Mexico City 07360, DF, Mexico

[3] Benemerita Univ Autonoma Puebla, Fac Ciencias Elect, Ciudad Univ,18 Sur & Ave San Claudio San Manuel, Puebla 72592, Mexico

来源：

AIMS MATHEMATICS | 2022年 / 7卷 / 04期

关键词：

chaos; fractional calculus; Poincare map; differential evolution algorithm; accelerated particle swarm optimization; Kaplan-Yorke dimension; PERIODIC-SOLUTIONS; ALGORITHM; DYNAMICS; DESIGN;

D O I：

10.3934/math.2022326

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

The optimization of fractional-order (FO) chaotic systems is challenging when simulating a considerable number of cases for long times, where the primary problem is verifying if the given parameter values will generate chaotic behavior. In this manner, we introduce a methodology for detecting chaotic behavior in FO systems through the analysis of Poincare maps. The optimization process is performed applying differential evolution (DE) and accelerated particle swarm optimization (APSO) algorithms for maximizing the Kaplan-Yorke dimension (D-KY) of two case studies: a 3D and a 4D FO chaotic systems with hidden attractors. These FO chaotic systems are solved applying the Grunwald-Letnikov method, and the Numba just-in-time (jit) compiler is used to improve the optimization process's time execution in Python programming language. The optimization results show that the proposed method efficiently optimizes FO chaotic systems with hidden attractors while saving execution time.

引用

页码：5871 / 5894

页数：24

共 54 条

[1] The Grunwald-Letnikov Fractional-Order Derivative with Fixed Memory Length [J].

Abdelouahab, Mohammed-Salah ;

Hamri, Nasr-Eddine .

MEDITERRANEAN JOURNAL OF MATHEMATICS, 2016, 13 (02) :557-572

[2] An Enhanced Image Encryption Algorithm Using Fractional Chaotic Systems [J].

Ahmad, Musheer ;

Shamsi, Uzair ;

Khan, Imran Raza .

3RD INTERNATIONAL CONFERENCE ON RECENT TRENDS IN COMPUTING 2015 (ICRTC-2015), 2015, 57 :852-859

[3] Cryptanalysis of a Classical Chaos-Based Cryptosystem with Some Quantum Cryptography Features [J].

Arroyo, David ;

Hernandez, Fernando ;

Orue, Amalia B. .

INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, 2017, 27 (01)

[4] A Simple Technique for Studying Chaos Using Jerk Equation with Discrete Time Sine Map [J].

Arshad, Muhammad Haseeb ;

Kassas, Mahmoud ;

Hussein, Alaa E. ;

Abido, Mohammad A. .

APPLIED SCIENCES-BASEL, 2021, 11 (01) :1-16

[5] On the dynamics of fractional maps with power-law, exponential decay and Mittag-Leffler memory [J].

Avalos-Ruiz, L. F. ;

Gomez-Aguilar, J. F. ;

Atangana, A. ;

Owolabi, Kolade M. .

CHAOS SOLITONS & FRACTALS, 2019, 127 :364-388

[6] Single channel secure communication scheme based on synchronization of fractional-order chaotic Chua's systems [J].

Bettayeb, Maamar ;

Al-Saggaf, Ubaid Muhsen ;

Djennoune, Said .

TRANSACTIONS OF THE INSTITUTE OF MEASUREMENT AND CONTROL, 2018, 40 (13) :3651-3664

[7]

Bolotin Y, 2009, UNDERST COMPLEX SYST, P1, DOI 10.1007/978-3-642-00937-2_1

[8]

Carpinteri A., 2014, FRACTALS FRACTIONAL, V378

[9] Design of fractional-order hyper-chaotic multi-scroll systems based on hysteresis series [J].

Chen, Liping ;

Pan, Wei ;

Tenreiro Machado, J. A. ;

Lopes, Antnio M. ;

Wu, Ranchao ;

He, Yigang .

EUROPEAN PHYSICAL JOURNAL-SPECIAL TOPICS, 2017, 226 (16-18) :3775-3789

[10] Color maps of the Kaplan-Yorke dimension in optically driven lasers: Maximizing the dimension and almost-Hamiltonian chaos [J].

Chlouverakis, KE .

INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, 2005, 15 (09) :3011-3021

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