Triple pendulum model involving fractional derivatives with different kernels

• Published in: Chaos, solitons and fractals Vol. 91; pp. 248 - 261
• Main Authors: Coronel-Escamilla, A., Gómez-Aguilar, J.F., López-López, M.G., Alvarado-Martínez, V.M., Guerrero-Ramírez, G.V.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.10.2016
• Subjects: Adams-Bashforth-Moulton method; Atangana-Baleanu derivative; Caputo-Fabrizio derivative; Fractional dynamics; Hamilton equations; Lagrangian dynamics; Liouville-Caputo derivative; Riemann-Liouville derivative; Lagrangian dynamics; 65R10; Caputo-Fabrizio derivative; Atangana-Baleanu derivative; 70Hxx; Fractional dynamics; Riemann-Liouville derivative; Hamilton equations; Liouville-Caputo derivative; 26A33; Adams-Bashforth-Moulton method
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2016.06.007

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.