Rigorous verification of Hopf bifurcations in functional differential equations of mixed type
The paper is concerned with the development of a numerical method to prove the existence of Hopf bifurcations in simple functional differential equations of mixed type, otherwise known as advance-delay equations, or sometimes as forward-backward equations. The question of interest is to consider the properties of the eigenvalues, and use is made of the Newton-Kantorovich theorem. The authors prove the existence of Hopf bifurcations in the Lasota-Wazewska-Czyzewska model and the existence of periodic traveling waves in the Fisher equation with nonlocal reaction. The overall objective of the work is to `develop numerical methods which can lead to computer-assisted proofs of existence of different type of dynamical objects arising in the study of differential equations.' Consequently, there is a section that discusses computer-assisted proofs of some of the theorems presented and links are provided to the relevant code.
scientific article; zbMATH DE number 7479416
scientific article