IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i3p653-d1048776.html
   My bibliography  Save this article

On Cauchy Problems of Caputo Fractional Differential Inclusion with an Application to Fractional Non-Smooth Systems

Author

Listed:
  • Jimin Yu

    (School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
    College of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065, China)

  • Zeming Zhao

    (School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China)

  • Yabin Shao

    (School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China)

Abstract

In this innovative study, we investigate the properties of existence and uniqueness of solutions to initial value problem of Caputo fractional differential inclusion. In the study of existence problems, we considered the case of convex and non-convex multivalued maps. We obtained the existence results for both cases by means of the appropriate fixed point theorem. Furthermore, the uniqueness corresponding to both cases was also determined. Finally, we took a non-smooth system, the modified Murali–Lakshmanan–Chua (MLC) fractional-order circuit system, as an example to verify its existence and uniqueness conditions, and through several sets of simulation results, we discuss the implications.

Suggested Citation

  • Jimin Yu & Zeming Zhao & Yabin Shao, 2023. "On Cauchy Problems of Caputo Fractional Differential Inclusion with an Application to Fractional Non-Smooth Systems," Mathematics, MDPI, vol. 11(3), pages 1-18, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:653-:d:1048776
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/3/653/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/3/653/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Srinivasan, K. & Chandrasekar, V.K. & Venkatesan, A. & Raja Mohamed, I., 2016. "Duffing–van der Pol oscillator type dynamics in Murali–Lakshmanan–Chua (MLC) circuit," Chaos, Solitons & Fractals, Elsevier, vol. 82(C), pages 60-71.
    2. Fu, Shihui & Liu, Yuan, 2020. "Complex dynamical behavior of modified MLC circuit," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    3. Mikhail Kamenskii & Valeri Obukhovskii & Garik Petrosyan & Jen-Chih Yao, 2021. "On the Existence of a Unique Solution for a Class of Fractional Differential Inclusions in a Hilbert Space," Mathematics, MDPI, vol. 9(2), pages 1-19, January.
    4. Li, Chunguang & Chen, Guanrong, 2004. "Chaos and hyperchaos in the fractional-order Rössler equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 341(C), pages 55-61.
    5. Juan J. Nieto & Abdelghani Ouahab & P. Prakash, 2013. "Extremal Solutions and Relaxation Problems for Fractional Differential Inclusions," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-9, September.
    6. Vasily E. Tarasov, 2020. "Mathematical Economics: Application of Fractional Calculus," Mathematics, MDPI, vol. 8(5), pages 1-3, April.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ge, Zheng-Ming & Yi, Chang-Xian, 2007. "Chaos in a nonlinear damped Mathieu system, in a nano resonator system and in its fractional order systems," Chaos, Solitons & Fractals, Elsevier, vol. 32(1), pages 42-61.
    2. Tam, Lap Mou & Si Tou, Wai Meng, 2008. "Parametric study of the fractional-order Chen–Lee system," Chaos, Solitons & Fractals, Elsevier, vol. 37(3), pages 817-826.
    3. Lu, Jun Guo & Chen, Guanrong, 2006. "A note on the fractional-order Chen system," Chaos, Solitons & Fractals, Elsevier, vol. 27(3), pages 685-688.
    4. Lijuan Chen & Mingchu Yu & Jinnan Luo & Jinpeng Mi & Kaibo Shi & Song Tang, 2024. "Dynamic Analysis and FPGA Implementation of a New Linear Memristor-Based Hyperchaotic System with Strong Complexity," Mathematics, MDPI, vol. 12(12), pages 1-17, June.
    5. Zheng, Yongai & Ji, Zhilin, 2016. "Predictive control of fractional-order chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 307-313.
    6. Laarem, Guessas, 2021. "A new 4-D hyper chaotic system generated from the 3-D Rösslor chaotic system, dynamical analysis, chaos stabilization via an optimized linear feedback control, it’s fractional order model and chaos sy," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    7. Gao, Xin & Yu, Juebang, 2005. "Synchronization of two coupled fractional-order chaotic oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 141-145.
    8. Sheu, Long-Jye & Chen, Hsien-Keng & Chen, Juhn-Horng & Tam, Lap-Mou & Chen, Wen-Chin & Lin, Kuang-Tai & Kang, Yuan, 2008. "Chaos in the Newton–Leipnik system with fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 36(1), pages 98-103.
    9. Vasily E. Tarasov, 2021. "Integral Equations of Non-Integer Orders and Discrete Maps with Memory," Mathematics, MDPI, vol. 9(11), pages 1-12, May.
    10. Peng, Guojun & Jiang, Yaolin & Chen, Fang, 2008. "Generalized projective synchronization of fractional order chaotic systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(14), pages 3738-3746.
    11. Petráš, Ivo, 2008. "A note on the fractional-order Chua’s system," Chaos, Solitons & Fractals, Elsevier, vol. 38(1), pages 140-147.
    12. Silva-Juárez, Alejandro & Tlelo-Cuautle, Esteban & de la Fraga, Luis Gerardo & Li, Rui, 2021. "Optimization of the Kaplan-Yorke dimension in fractional-order chaotic oscillators by metaheuristics," Applied Mathematics and Computation, Elsevier, vol. 394(C).
    13. Xu, Fei & Lai, Yongzeng & Shu, Xiao-Bao, 2018. "Chaos in integer order and fractional order financial systems and their synchronization," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 125-136.
    14. Paula Morales-Bañuelos & Sebastian Elias Rodríguez Bojalil & Luis Alberto Quezada-Téllez & Guillermo Fernández-Anaya, 2025. "A General Conformable Black–Scholes Equation for Option Pricing," Mathematics, MDPI, vol. 13(10), pages 1-29, May.
    15. Zúñiga-Aguilar, C.J. & Gómez-Aguilar, J.F. & Escobar-Jiménez, R.F. & Romero-Ugalde, H.M., 2019. "A novel method to solve variable-order fractional delay differential equations based in lagrange interpolations," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 266-282.
    16. Deng, Hongmin & Li, Tao & Wang, Qionghua & Li, Hongbin, 2009. "A fractional-order hyperchaotic system and its synchronization," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 962-969.
    17. Lu, Jun Guo, 2006. "Nonlinear observer design to synchronize fractional-order chaotic systems via a scalar transmitted signal," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 359(C), pages 107-118.
    18. Deepika, Deepika & Kaur, Sandeep & Narayan, Shiv, 2018. "Uncertainty and disturbance estimator based robust synchronization for a class of uncertain fractional chaotic system via fractional order sliding mode control," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 196-203.
    19. Wang, Lingyu & Sun, Kehui & Peng, Yuexi & He, Shaobo, 2020. "Chaos and complexity in a fractional-order higher-dimensional multicavity chaotic map," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    20. Fiaz, Muhammad & Aqeel, Muhammad & Marwan, Muhammad & Sabir, Muhammad, 2022. "Integer and fractional order analysis of a 3D system and generalization of synchronization for a class of chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:653-:d:1048776. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.