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Bifurcations, stability analysis and complex dynamics of Caputo fractal-fractional cancer model

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  • Xuan, Liu
  • Ahmad, Shabir
  • Ullah, Aman
  • Saifullah, Sayed
  • Akgül, Ali
  • Qu, Haidong

Abstract

The association of cancer and immune cells has complex nature and produces chaotic behavior when it is simulated. The newly introduced operators which combine the fractal and fractional operators produce excellent and profound hidden attractors in a chaotic system which is sometimes not possible to get hidden attractors using integer order operators. The cancer model is considered under fractal fractional operator in Caputo sense. Linear stability of different equilibrium points is analyzed. The primary objective of the current paper is to analyze different bifurcations like pitch-fork, quasi, and inverse period-doubling bifurcations. Another important objective of this article is to study hidden limit cycle type chaotic structures of the cancer model via Caputo fractal-fractional operator. The existence and uniqueness of the solution and Ulam–Hyres (UH) stability are studied through the concepts of nonlinear analysis. The numerical solution is derived through the predictor-corrector method. The obtained results were presented and validated through numerical simulations. The lyapunov spectra of the state variables are presented through graphical illustration and table. Sensitivity of the state variables to the initial conditions are simulated for initial conditions 0.1 and 0.11. For various values of fractal dimensions and fractional orders, the time series oscillations and hidden limit cycles type chaotic attractors are graphically presented through MATLAB-17.

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  • Xuan, Liu & Ahmad, Shabir & Ullah, Aman & Saifullah, Sayed & Akgül, Ali & Qu, Haidong, 2022. "Bifurcations, stability analysis and complex dynamics of Caputo fractal-fractional cancer model," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
  • Handle: RePEc:eee:chsofr:v:159:y:2022:i:c:s096007792200323x
    DOI: 10.1016/j.chaos.2022.112113
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    References listed on IDEAS

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    1. Qureshi, Sania & Atangana, Abdon, 2020. "Fractal-fractional differentiation for the modeling and mathematical analysis of nonlinear diarrhea transmission dynamics under the use of real data," Chaos, Solitons & Fractals, Elsevier, vol. 136(C).
    2. Sayed Saifullah & Amir Ali & Muhammad Irfan & Kamal Shah, 2021. "Time-Fractional Klein–Gordon Equation with Solitary/Shock Waves Solutions," Mathematical Problems in Engineering, Hindawi, vol. 2021, pages 1-15, October.
    3. Atangana, Abdon & Qureshi, Sania, 2019. "Modeling attractors of chaotic dynamical systems with fractal–fractional operators," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 320-337.
    4. Atangana, Abdon, 2017. "Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 396-406.
    5. Sajjadi, Samaneh Sadat & Baleanu, Dumitru & Jajarmi, Amin & Pirouz, Hassan Mohammadi, 2020. "A new adaptive synchronization and hyperchaos control of a biological snap oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    6. Dumitru Baleanu & Arran Fernandez & Ali Akgül, 2020. "On a Fractional Operator Combining Proportional and Classical Differintegrals," Mathematics, MDPI, vol. 8(3), pages 1-13, March.
    7. Ahmad, Shabir & Ullah, Aman & Akgül, Ali, 2021. "Investigating the complex behaviour of multi-scroll chaotic system with Caputo fractal-fractional operator," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    8. Letellier, Christophe & Sasmal, Sourav Kumar & Draghi, Clément & Denis, Fabrice & Ghosh, Dibakar, 2017. "A chemotherapy combined with an anti-angiogenic drug applied to a cancer model including angiogenesis," Chaos, Solitons & Fractals, Elsevier, vol. 99(C), pages 297-311.
    9. Saifullah, Sayed & Ali, Amir & Franc Doungmo Goufo, Emile, 2021. "Investigation of complex behaviour of fractal fractional chaotic attractor with mittag-leffler Kernel," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    10. A. Akgãœl & N. Ahmed & A. Raza & Z. Iqbal & M. Rafiq & M. A. Rehman & D. Baleanu, 2021. "A Fractal Fractional Model For Cervical Cancer Due To Human Papillomavirus Infection," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(05), pages 1-11, August.
    11. Pu, Xueke, 2015. "Dynamics of a fractional hydrodynamical equation for the Heisenberg paramagnet," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 213-229.
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