A Fractional-Order Sinusoidal Discrete Map
In this paper, a novel fractional-order discrete map with a sinusoidal function possessing typical nonlinear features, including chaos and bifurcations, is proposed. Firstly, the basic properties involving the stability of the equilibrium points and the symmetry of the map are studied by theoretical...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-02-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/24/3/320 |
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| author | Xiaojun Liu Dafeng Tang Ling Hong |
| author_facet | Xiaojun Liu Dafeng Tang Ling Hong |
| author_sort | Xiaojun Liu |
| collection | DOAJ |
| description | In this paper, a novel fractional-order discrete map with a sinusoidal function possessing typical nonlinear features, including chaos and bifurcations, is proposed. Firstly, the basic properties involving the stability of the equilibrium points and the symmetry of the map are studied by theoretical analysis. Secondly, the dynamics of the map in commensurate-order and incommensurate-order cases with initial conditions belonging to different basins of attraction is investigated by numerical simulations. The bifurcation types and influential parameters of the map are analyzed via nonlinear tools. Hopf, period-doubling, and symmetry-breaking bifurcations are observed when a parameter or an order is varied. Bifurcation diagrams and maximum Lyapunov exponent spectrums, with both a variation in a system parameter and an order or two orders, are shown in a three-dimensional space. A comparison of the bifurcations in fractional-order and integral-order cases shows that the variation in an order has no effect on the symmetry-breaking bifurcation point. Finally, the heterogeneous hybrid synchronization of the map is realized by designing suitable controllers. It is worth noting that the increase in a derivative order can promote the synchronization speed for the fractional-order discrete map. |
| first_indexed | 2024-03-09T19:52:41Z |
| format | Article |
| id | doaj.art-ef46ad67fdba411ea0485d5ca85d4e99 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-09T19:52:41Z |
| publishDate | 2022-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-ef46ad67fdba411ea0485d5ca85d4e992023-11-24T01:06:45ZengMDPI AGEntropy1099-43002022-02-0124332010.3390/e24030320A Fractional-Order Sinusoidal Discrete MapXiaojun Liu0Dafeng Tang1Ling Hong2School of Sciences, Xi’an University of Posts and Telecommunications, Xi’an 710061, ChinaSchool of Automation, Xi’an University of Posts and Telecommunications, Xi’an 710061, ChinaState Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, ChinaIn this paper, a novel fractional-order discrete map with a sinusoidal function possessing typical nonlinear features, including chaos and bifurcations, is proposed. Firstly, the basic properties involving the stability of the equilibrium points and the symmetry of the map are studied by theoretical analysis. Secondly, the dynamics of the map in commensurate-order and incommensurate-order cases with initial conditions belonging to different basins of attraction is investigated by numerical simulations. The bifurcation types and influential parameters of the map are analyzed via nonlinear tools. Hopf, period-doubling, and symmetry-breaking bifurcations are observed when a parameter or an order is varied. Bifurcation diagrams and maximum Lyapunov exponent spectrums, with both a variation in a system parameter and an order or two orders, are shown in a three-dimensional space. A comparison of the bifurcations in fractional-order and integral-order cases shows that the variation in an order has no effect on the symmetry-breaking bifurcation point. Finally, the heterogeneous hybrid synchronization of the map is realized by designing suitable controllers. It is worth noting that the increase in a derivative order can promote the synchronization speed for the fractional-order discrete map.https://www.mdpi.com/1099-4300/24/3/320a fractional-order discrete mapchaosbifurcationsynchronization |
| spellingShingle | Xiaojun Liu Dafeng Tang Ling Hong A Fractional-Order Sinusoidal Discrete Map Entropy a fractional-order discrete map chaos bifurcation synchronization |
| title | A Fractional-Order Sinusoidal Discrete Map |
| title_full | A Fractional-Order Sinusoidal Discrete Map |
| title_fullStr | A Fractional-Order Sinusoidal Discrete Map |
| title_full_unstemmed | A Fractional-Order Sinusoidal Discrete Map |
| title_short | A Fractional-Order Sinusoidal Discrete Map |
| title_sort | fractional order sinusoidal discrete map |
| topic | a fractional-order discrete map chaos bifurcation synchronization |
| url | https://www.mdpi.com/1099-4300/24/3/320 |
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