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Article
Periodic orbits in Hořava–Lifshitz cosmologies
We consider spatially homogeneous Hořava–Lifshitz models that perturb General Relativity (GR) by a parameter \(v\in (0,1)\) ...
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Article
Eigenvalues and delay differential equations: periodic coefficients, impulses and rigorous numerics
We develop validated numerical methods for the computation of Floquet multipliers of equilibria and periodic solutions of delay differential equations, as well as impulsive delay differential equations. Using ...
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Book
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Chapter
Preliminaries
This chapter is devoted to some preliminary background on ordinary impulsive differential equations. This includes existence and uniqueness of solutions and dependence on initial conditions and parameters.
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Chapter
Stability for Nonlinear Systems
In this chapter we will discuss some methods of proving stability for nonlinear systems.
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Chapter
The Hutchinson Equation with Pulse Harvesting
We prove the existence of a transcritical bifurcation in the pulse-harvested Hutchinson equation. Specifically, we prove the bifurcation with respect to the period of impulse effect. The proof requires a reduc...
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Chapter
Hyperbolicity and the Classical Hierarchy of Invariant Manifolds
We 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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Chapter
Bifurcations
This 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) bifu...
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Chapter
Non-smooth Bifurcations
In this chapter we will be interested in bifurcations that result from two “non-smooth” phenomena:
perturbations in the sequence of impulses and
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Chapter
General Linear Systems
In this chapter, we cover existence and uniqueness of solutions, evolutions families, phase-space decompositions and the variation-of-constants formula for linear impulsive functional differential equations.
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Chapter
Stage-Structured Predator–Prey System with Pulsed Birth
We study bifurcations in a stage-structured predator prey system with pulsed birth. We determine the stability of the extinction equilibrium. Next, we establish the existence and uniqueness of a predator-free ...
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Chapter
Nonlinear Systems and Stability
This 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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Chapter
Computational Aspects of Centre Manifolds
A follow-up to Chapter 5, this chapter is devoted to computational aspects of centre manifold theory. This includes a concrete representation of the centre manifol...
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Chapter
Smooth Bifurcations
The centre manifold reduction provides a framework in which bifurcations of fixed points and periodic solutions can be studied. In this section we will explain how the centre manifold reduction can be adapted ...
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Chapter
Linear Systems
This chapter contains the essential theory of linear impulsive differential equations. This includes the variation-of-constants formula, Floquet theory, and stability.
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Chapter
Invariant Manifold Theory
This chapter will be devoted to the invariant manifold theory of impulsive differential equations. At the theoretical level, we will assume only that the reference bounded solution has exponential trichotomy, ...
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Chapter
Continuous Approximation
In the theory of impulsive dynamical systems, impulses are often interpreted as idealized discrete jumps associated with a process that is continuous in time but occurs on a negligibly small time scale. The in...
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Chapter
Bifurcations in an Impulsively Damped or Driven Pendulum
We study the dynamics resulting from impulsive dam** or driving forces applied to a classical rigid pendulum. The impulse effect adjusts the angular velocity at fixed times based on either a point measuremen...
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Chapter
Dynamics of an In-host Viral Infection Model with Drug Treatment
In this final chapter, we study an in-host viral infection model with impulsive drug treatment. We first restrict to the disease-free subspace and prove the existence of a disease-free periodic solution and a ...
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Chapter
Introduction
Many real-world processes exhibit continuous-time evolution with intermittent bursts of comparatively fast dynamics. In mathematical models of such processes, these bursts of activity are sometimes intrinsic t...
