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Based on the centre manifold theorem for impulsive delay differential equations, we derive impulsive evolution equations and boundary conditions associated to a concrete representation of the centre manifold in Euclidean space, as well as finite-dimensional impulsive differential equations associated to the evolution on these manifolds. Though the centre manifolds are not unique, their Taylor expansions agree up to prescribed order, and we present an implicit formula for the quadratic term using a variation of the method of characteristics. We use our centre manifold reduction to derive analogues of the saddle-node and Hopf bifurcation for impulsive delay differential equations, and the latter leads to a novel bifurcation pattern to an invariant cylinder. Examples are provided to illustrate the correctness of the bifurcation theorems and to visualize the geometry of centre manifolds in the presence of impulse effects. [ABSTRACT FROM AUTHOR]

We propose a rigorously validated numerical method to prove the existence of Hopf bifurcations in functional differential equations of mixed type. The eigenvalue transversality and steady state conditions are verified using the Newton–Kantorovich theorem. The non-resonance condition and simplicity of the critical eigenvalues are verified by either computing a pair of generalized Morse indices of the step map, or by applying the argument principle to the characteristic equation and a suitable contour in the complex plane, computing the contour integral using the equivalence with a winding number. As a first application and test problem, we prove the existence of Hopf bifurcations in the Lasota–Wazewska–Czyzewska model and a pair of two such coupled equations. We then use our method to prove the existence of periodic traveling waves in the Fisher equation with nonlocal reaction. These periodic traveling waves are solutions of an ill-posed functional differential equation of mixed type. • Presents computer-assisted proofs of Hopf bifurcation in functional DE of mixed-type. • Bifurcation of periodic traveling waves in Fisher equation with nonlocal reaction. • Application to Hopf bifurcations in coupled Lasota–Wazewska–Czyzewska equations. [ABSTRACT FROM AUTHOR]