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Triple pendulum model involving fractional derivatives with different kernels

Author

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  • Coronel-Escamilla, A.
  • Gómez-Aguilar, J.F.
  • López-López, M.G.
  • Alvarado-Martínez, V.M.
  • Guerrero-Ramírez, G.V.

Abstract

The aim of this work is to study the non-local dynamic behavior of triple pendulum-type systems. We use the Euler-Lagrange and the Hamiltonian formalisms to obtain the dynamic models, based on the Riemann-Liouville, Liouville-Caputo, Caputo-Fabrizio and Atangana-Baleanu-Caputo fractional derivative definitions. In these representations, an auxiliary parameter σ is introduced, to define the equations in a fractal temporal geometry, which provides an entire new family of solutions for the dynamic behavior of the pendulum-type systems. The phase diagrams allow to visualize the effect of considering the fractional order approach, the classical behavior is recovered when the order of the fractional derivative is 1.

Suggested Citation

  • Coronel-Escamilla, A. & Gómez-Aguilar, J.F. & López-López, M.G. & Alvarado-Martínez, V.M. & Guerrero-Ramírez, G.V., 2016. "Triple pendulum model involving fractional derivatives with different kernels," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 248-261.
    • Handle: RePEc:eee:chsofr:v:91:y:2016:i:c:p:248-261
    DOI: 10.1016/j.chaos.2016.06.007
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    4. Stachowiak, Tomasz & Okada, Toshio, 2006. "A numerical analysis of chaos in the double pendulum," Chaos, Solitons & Fractals, Elsevier, vol. 29(2), pages 417-422.
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    Cited by:

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    7. Al-khedhairi, A. & Elsadany, A.A. & Elsonbaty, A., 2019. "Modelling immune systems based on Atangana–Baleanu fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 129(C), pages 25-39.
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    9. Atangana, Abdon & Gómez-Aguilar, J.F., 2017. "Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 285-294.
    10. Owolabi, Kolade M., 2019. "Mathematical modelling and analysis of love dynamics: A fractional approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 849-865.
    11. Shah, Syed Azhar Ali & Khan, Muhammad Altaf & Farooq, Muhammad & Ullah, Saif & Alzahrani, Ebraheem O., 2020. "A fractional order model for Hepatitis B virus with treatment via Atangana–Baleanu derivative," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 538(C).

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