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

[31] Computational techniques for highly oscillatory and chaotic wave problems with fractional-order operator [J].

Owolabi, Kolade M. .

EUROPEAN PHYSICAL JOURNAL PLUS, 2020, 135 (10)

[32] Modelling of Chaotic Processes with Caputo Fractional Order Derivative [J].

Owolabi, Kolade M. ;

Francisco Gomez-Aguilar, Jose ;

Fernandez-Anaya, G. ;

Lavin-Delgado, J. E. ;

Hernandez-Castillo, E. .

ENTROPY, 2020, 22 (09)

[33]

Petra I., 2011, FRACTIONAL ORDER NON

[34] Discrete direct methods in the fractional calculus of variations [J].

Pooseh, Shakoor ;

Almeida, Ricardo ;

Torres, Delfim F. M. .

COMPUTERS & MATHEMATICS WITH APPLICATIONS, 2013, 66 (05) :668-676

[35]

Sajid M., 2013, INT J THEORET APPL R, V2, P121

[36] Self-Reproducing Hidden Attractors in Fractional-Order Chaotic Systems Using Affine Transformations [J].

Sayed, Wafaa S. ;

Radwan, Ahmed G. .

IEEE OPEN JOURNAL OF CIRCUITS AND SYSTEMS, 2020, 1 :243-254

[37] The Grunwald-Letnikov method for fractional differential equations [J].

Scherer, Rudolf ;

Kalla, Shyam L. ;

Tang, Yifa ;

Huang, Jianfei .

COMPUTERS & MATHEMATICS WITH APPLICATIONS, 2011, 62 (03) :902-917

[38] Control of multistability in hidden attractors [J].

Sharma, P. R. ;

Shrimali, M. D. ;

Prasad, A. ;

Kuznetsov, N. V. ;

Leonov, G. A. .

EUROPEAN PHYSICAL JOURNAL-SPECIAL TOPICS, 2015, 224 (08) :1485-1491

[39] Optimization of the Kaplan-Yorke dimension in fractional-order chaotic oscillators by metaheuristics [J].

Silva-Juarez, Alejandro ;

Tlelo-Cuautle, Esteban ;

Gerardo de la Fraga, Luis ;

Li, Rui .

APPLIED MATHEMATICS AND COMPUTATION, 2021, 394

[40] Differential evolution algorithm for optimal design of water distribution networks [J].

Suribabu, C. R. .

JOURNAL OF HYDROINFORMATICS, 2010, 12 (01) :66-82

← 1 2 3 4 5 6 →