Inverse problem with final overdetermination for time-fractional differential equation in a Banach space

We consider in a Banach space E the inverse problem(\mathbf{D}_{t}^{\alpha}u)(t)=Au(t)+\mathcal{F}(t)f,\quad t\in[0,T],u(0)=u^{0}% ,u(T)=u^{T},\,0<\alpha<1with operator A, which generates the analytic and compact α-times resolvent family {\{S_{\alpha}(t,A)\}_{t\geq 0}}, the function {\mathcal{F}(\,\cdot\,)\in C^{1}[0,T]} and {u^{0},u^{T}\in D(A)} are given and {f\in E} is an unknown element. Under natural conditions we have proved the Fredholm solvability of this problem. In the special case for a self-adjoint operator A, the existence and uniqueness theorems for the solution of the inverse problem are proved. The semidiscrete approximation theorem for this inverse problem is obtained.

Improved Homoclinic Predictor for Bogdanov–Takens Bifurcation

An improved homoclinic predictor at a generic codim 2 Bogdanov–Takens (BT) bifucation is derived. We use the classical "blow-up" technique to reduce the canonical smooth normal form near a generic BT bifurcation to a perturbed Hamiltonian system. With a simple perturbation method, we derive explicit first- and second-order corrections of the unperturbed homoclinic orbit and parameter value. To obtain the normal form on the center manifold, we apply the standard parameter-dependent center manifold reduction combined with the normalization, that is based on the Fredholm solvability of the homological equation. By systematically solving all linear systems appearing from the homological equation, we remove an ambiguity in the parameter transformation existing in the literature. The actual implementation of the improved predictor in MatCont and numerical examples illustrating its efficiency are discussed.