Titlebook: Bifurcation Theory of Impulsive Dynamical Systems; Kevin E.M. Church,Xinzhi Liu Book 2021 The Editor(s) (if applicable) and The Author(s),

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Nonlinear Systems and StabilityThis chapter contains a proof of the principle of linearized stability for nonlinear impulsive functional differential equations, in addition to some auxiliary results on smooth dependence on initial conditions.

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Computational Aspects of Centre ManifoldsA follow-up to Chapter ., this chapter is devoted to computational aspects of centre manifold theory. This includes a concrete representation of the centre manifold in Euclidean space, Taylor expansions, and an explicit ordinary impulsive differential equation for the dynamics on the manifold.

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Hyperbolicity and the Classical Hierarchy of Invariant ManifoldsWe discuss the existence and smoothness of unstable, stable and centre-stable manifolds, thereby establishing the classical hierarchy of invariant manifolds for impulsive functional differential equations.

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BifurcationsThis chapter presents analogues of the classical codimension-one bifurcations of flows and maps. Specifically, this includes analogues of the fold (saddle-node), period-doubling, and Hopf (Neimark-Sacker) bifurcation. Additionally, we present the transcritical and pitchfork bifurcations.

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Introductionocesses, these bursts of activity are sometimes intrinsic to the dynamics. For example, the Hodgkin–Huxley model [.] is a nonlinear ordinary differential equation that describes the propagation of action potentials of neurons; here, the bursts of activity correspond to the action potentials and are