Asymptotic stability and fold bifurcation analysis in Caputo-Hadamard type fractional differential system

This paper mainly deals with the asymptotic stability and the fold bifurcation for Caputo-Hadamard type fractional differential system (C-HTFDS). The sufficient criteria of asymptotic stability for n-dimensional linear C-HTFDS are derived by the aid of the modified Laplace transform technique and asymptotic expansions of the Mittag-Leffler function. Additionally, comparative analysis indicates that the obtained results are comparable with the preceding conclusions of Caputo-Hadamard fractional differential system (C-HFDS). Not only that, it is also observed and demonstrated that the peculiar parameter mu affects the stability regions and phase transitions consistently. Besides, in light of the Taylor expansion and the implicit/inverse function theorem, the normal form of the fold bifurcation for C-HTFDS with parameters is computed and schemed, accompanying with topological equivalence. In addition, benefited from the stability results derived and the idea of linearization, we further detect the fold bifurcation conditions by varying the system parameters mu and eta.