Asymptotic behaviour of non-autonomous Caputo fractional differential equations with a one-sided dissipative vector field

T. S. Doan ; P. E. Kloeden

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2024), pp. 44-52

Résumé

A non-autonomous Caputo fractional differential equation of order $\alpha\in(0,1)$ in $\mathbb{R}^d$ with a driving system $\{\vartheta_t\}_{t\in \mathbb{R}}$ on a compact base space $P$ generates a skew-product flow on $\mathfrak{C}_{\alpha}\times P$, where $\mathfrak{C}_{\alpha}$ is the space of continuous functions $f$ $:$ $\mathbb{R}^+$ $\to$ $\mathbb{R}^d$ with a weighted norm giving uniform convergence on compact time subsets. It was shown by Cui Kloeden [3] to have an attractor when the vector field of the Caputo FDE satisfies a uniform dissipative vector field. This attractor is closed, bounded and invariant in $\mathfrak{C}_{\alpha}\times P$ and attracts bounded subsets of $\mathfrak{C}_{\alpha}$ consisting of constant initial functions. The structure of this attractor is investigated here in detail for an example with a vector field satisfying a stronger one-sided dissipative Lipschitz condition. In particular, the component sets of the attractor are shown to be singleton sets corresponding to a unique entire solution of the skew-product flow. Its evaluation on $\mathbb{R}^d$ is a unique entire solution of the Caputo FDE, which is both pullback and forward attracting.