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Attractors of Caputo fractional differential equations with triangular vector fields

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It is shown that the attractor of an autonomous Caputo fractional differential equation of order \(\alpha \in (0,1)\) in \({\mathbb {R}}^d\) whose vector field has a certain triangular structure and satisfies a smooth condition and dissipativity condition is essentially the same as that of the ordinary differential equation with the same vector field. As an application, we establish several one-parameter bifurcations for scalar fractional differential equations including the saddle-node and the pichfork bifurcations. The proof uses a result of Cong & Tuan [2] which shows that no two solutions of such a Caputo FDE can intersect in finite time.

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  1. Aguila-Camacho, N., Duar-Mermoud, M.A., Gallegos, J.A.: Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simulat. 19, 2951–2957 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  2. Cong, N.D., Tuan, H.T.: Generation of nonlocal dynamical systems by fractional differential equations. J. Integral Equations Appl. 29, 585–608 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  3. Diethelm, K.: On the separation of solutions of fractional differential equations. Fract. Calc. Appl. Anal. 11, 259–268 (2008)

    MathSciNet  MATH  Google Scholar 

  4. Diethelm, K.: The Analysis of Fractional Differential Equations. Springer Lecture Notes in Mathematics, Vol. 2004, Springer, Heidelberg (2010)

  5. Diethelm, K., Ford, N.J.: Volterra integral equations and fractional calculus: do neighboring solutions intersect? J. Integral Equations Appl. 24(1), 25–37 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  6. Doan, T.S., Kloeden, P.E.: Semi-dynamical systems generated by autonomous Caputo fractional differential equations. Vietnam J. of Math. 49(4), 1305–1315 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  7. Hale, J.K., Koçak, H.: Dynamics and Bifurcations, Springer-Verlag. New York (1991).

    Article  MATH  Google Scholar 

  8. Kloeden, P.E., Rasmussen, M.: Nonautonomous Dynamical Systems. American Mathematical Society, Providence (2011)

    Book  Google Scholar 

  9. Miller, R.K., Sell, G.R.: Volterra Integral Equations and Topological Dynamics. Ser. Memoirs Amer. Math. Soc., Vol. 102 (1970)

  10. Sell, G.R.: Topological Dynamics and Ordinary Differential Equations. Van Nostrand Reinhold Mathematical Studies, London (1971)

    MATH  Google Scholar 

  11. Wang, D., Xiao, A.: Dissipativity and contractivity for fractional-order systems. Nonlinear Dyn. 80, 287–294 (2015).

    Article  MathSciNet  MATH  Google Scholar 

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The work of Thai Son Doan is funded by Vietnam Academy of Science and Technology under grant number NCVCC01.11/22-23.

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Correspondence to Peter E. Kloeden.

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Doan, T.S., Kloeden, P.E. Attractors of Caputo fractional differential equations with triangular vector fields. Fract Calc Appl Anal 25, 720–734 (2022).

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