Bifurcation theory of impulsive dynamical systems / Kevin E. M. Church, Xinzhi Liu.

Format:

Book

Author/Creator:

Church, Kevin E. M., author.

Contributor:

Liu, Xinzhi, 1956- editor.

Series:

IFSR international series on systems science and engineering ; Volume 34.

IFSR international series on systems science and engineering ; Volume 34

Language:

English

Subjects (All):

Bifurcation theory.

System analysis.

Physical Description:

1 online resource (xvii, 388 pages) : illustrations.

Place of Publication:

Cham, Switzerland : Springer, [2021]

Summary:

This monograph presents the most recent progress in bifurcation theory of impulsive dynamical systems with time delays and other functional dependence. It covers not only smooth local bifurcations, but also some non-smooth bifurcation phenomena that are unique to impulsive dynamical systems. The monograph is split into four distinct parts, independently addressing both finite and infinite-dimensional dynamical systems before discussing their applications. The primary contributions are a rigorous nonautonomous dynamical systems framework and analysis of nonlinear systems, stability, and invariant manifold theory. Special attention is paid to the centre manifold and associated reduction principle, as these are essential to the local bifurcation theory. Specifying to periodic systems, the Floquet theory is extended to impulsive functional differential equations, and this permits an exploration of the impulsive analogues of saddle-node, transcritical, pitchfork and Hopf bifurcations. Readers will learn how techniques of classical bifurcation theory extend to impulsive functional differential equations and, as a special case, impulsive differential equations without delays. They will learn about stability for fixed points, periodic orbits and complete bounded trajectories, and how the linearization of the dynamical system allows for a suitable definition of hyperbolicity. They will see how to complete a centre manifold reduction and analyze a bifurcation at a nonhyperbolic steady state.

Contents:

Intro

Preface

Reading Guide

Contents

I Impulsive Functional Differential Equations

I.1 Introduction

I.1.1 Nonautonomous Dynamical Systems

I.1.2 History Functions

I.1.3 The Space RCR of Right-Continuous Regulated...

I.1.4 Gelfand-Pettis Integration

I.1.5 Integral and Summation Inequalities

I.1.6 Comments

I.2 General Linear Systems

I.2.1 Existence and Uniqueness of Solutions

I.2.2 Evolution Families

I.2.2.1 Phase Space Decomposition

I.2.2.2 Evolution Families are (Generally) NowhereContinuous

I.2.2.3 Continuity under the L2 Seminorm

I.2.3 Representation of Solutions of the ...

I.2.3.1 Pointwise Variation-of-Constants Formula

I.2.3.2 Variation-of-Constants Formula in theSpace RCR

I.2.4 Stability

I.2.5 Comments

I.3 Linear Periodic Systems

I.3.1 Monodromy Operator

I.3.2 Floquet Theorem

I.3.3 Floquet Multipliers, Floquet Exponents and...

I.3.4 Computational Aspects in Floquet Theory

I.3.4.1 Floquet Eigensolutions

I.3.4.2 Characteristic Equations for Finitely ReducibleLinear Systems

I.3.4.3 Characteristic Equations for Systems withMemoryless Continuous Part

I.3.5 Comments

I.4 Nonlinear Systems and Stability

I.4.1 Mild Solutions

I.4.2 Dependence on Initial Conditions

I.4.3 The Linear Variational Equation and Linearized...

I.4.4 Comments

I.5 Existence, Regularity and Invariance of Centre Manifolds

I.5.1 Preliminaries

I.5.1.1 Spaces of Exponentially Weighted Functions

I.5.1.2 η-Bounded Solutions from Inhomogeneities

I.5.1.3 Substitution Operator and Modification ofNonlinearities

I.5.2 Fixed-Point Equation and Existence...

I.5.2.1 A Remark on Centre Manifold Representations:Graphs and Images

I.5.3 Invariance and Smallness Properties

I.5.4 Dynamics on the Centre Manifold

I.5.4.1 Integral Equation.

I.5.4.2 Abstract Ordinary Impulsive DifferentialEquation

I.5.4.3 A Remark on Coordinates and Terminology

I.5.5 Reduction Principle

I.5.5.1 Parameter Dependence

I.5.6 Smoothness in the State Space

I.5.6.1 Contractions on Scales of Banach Spaces

I.5.6.2 Candidate Differentials of the SubstitutionOperators

I.5.6.3 Smoothness of the Modified Nonlinearity

I.5.6.4 Proof of Smoothness of the Centre Manifold

I.5.6.5 Periodic Centre Manifold

I.5.7 Regularity of Centre Manifolds...

I.5.7.1 A Coordinate System and Pointwise PC1,m-Regularity

I.5.7.2 Reformulation of the Fixed-Point Equation

I.5.7.3 A Technical Assumption on the Projections Pc(t) and Pu(t)

I.5.7.4 Proof of PC1,m-Regularity at Zero

I.5.7.5 The Hyperbolic Part Is Pointwise PC1,m-Regular at Zero

I.5.7.6 Uniqueness of the Taylor Coefficients

I.5.7.7 A Discussion on the Regularity of the Matrices tYj(t)

I.5.8 Comments

I.6 Computational Aspects of Centre Manifolds

I.6.1 Euclidean Space Representation

I.6.1.1 Definition and Taylor Expansion

I.6.1.2 Dynamics on the Centre Manifold in EuclideanSpace

I.6.1.3 An Impulsive Evolution Equation and Boundary Conditions

I.6.2 Approximation by the Taylor Expansion

I.6.2.1 Evolution Equation and Boundary Conditions for Quadratic Terms

I.6.2.2 Solution by the Method of Characteristics

I.6.3 Visualization of Centre Manifolds

I.6.3.1 An Explicit Scalar Example Without Delays

I.6.3.2 Two-Dimensional Example with QuadraticDelayed Terms

I.6.3.3 Detailed Calculations Associated withExample I.6.3.2

The u12 Coefficient

The u22 Coefficient

The u1u2 Coefficient

I.6.4 The Overlap Condition

I.6.4.1 Distributed Delays

I.6.4.2 Transformations that Enforce the OverlapCondition for Discrete Delays

I.6.5 Comments.

I.7 Hyperbolicity and the Classical Hierarchy of InvariantManifolds

I.7.1 Preliminaries

I.7.2 Unstable Manifold

I.7.3 Stable Manifold

I.7.4 Centre-Unstable Manifold

I.7.5 Centre-Stable Manifold

I.7.6 Dynamics on Finite-Dimensional...

I.7.7 Linearized Stability and Instability,Revisited

I.7.8 Hierarchy and Inclusions

I.8 Smooth Bifurcations

I.8.1 Centre Manifolds Depending Smoothly on Parameters

I.8.2 Codimension-One Bifurcations for Systems with a Single Delay: Setup

I.8.3 Fold Bifurcation

I.8.3.1 Example: Fold Bifurcation in a Scalar System with Delayed Impulse

I.8.3.2 Calculation of the Function Y11(t) forExample I.8.3.1

I.8.4 Hopf-Type Bifurcation and Invariant Cylinders

I.8.4.1 Example: Impulsive Perturbation from a HopfPoint

I.8.5 Calculations Associated to Example I.8.4.1

I.8.5.1 The Projection Pc(t) and Matrix (t)

I.8.5.2 Calculation of π(t) and the Matrices A(t)and B

I.8.5.3 Calculation of n0(t): A Numerical Routine

I.8.5.4 Calculation of h2

I.8.6 A Recipe for the Analysis of Smooth LocalBifurcations

I.8.7 Comments

II Finite-Dimensional Ordinary Impulsive Differential Equations

II.1 Preliminaries

II.1.1 Existence and Uniqueness of Solutions

II.1.2 Dependence on Initial Conditions...

II.1.3 Continuity Conventions: Right- andLeft-Continuity

II.1.4 Comments

II.2 Linear Systems

II.2.1 Cauchy Matrix

II.2.2 Variation-of-Constants Formula

II.2.3 Stability

II.2.4 Exponential Trichotomy

II.2.5 Floquet Theory

II.2.5.1 Homogeneous Systems

II.2.5.2 Periodic Solutions of Homogeneous Systems

II.2.5.3 Periodic solutions of Inhomogeneous Systems

II.2.5.4 Periodic Systems Are ExponentiallyTrichotomous

II.2.5.5 Stability

II.2.6 Generalized Periodic Changes of Variables

II.2.6.1 A Full State Transformation and ChainMatrices.

II.2.6.2 Real Floquet Decompositions

II.2.6.3 A Real T-Periodic Kinematic Similarity

II.2.7 Comments

II.3 Stability for Nonlinear Systems

II.3.1 Stability

II.3.2 The Linear Variational Equation...

II.3.3 Comments

II.4 Invariant Manifold Theory

II.4.1 Existence and Smoothness

II.4.2 Invariance Equation for Nonautonomous...

II.4.3 Invariance Equation for Systems with...

II.4.4 Dynamics on Invariant Manifolds

II.4.5 Reduction Principle for the Centre Manifold

II.4.6 Approximation by Taylor Expansion

II.4.7 Parameter Dependence

II.4.7.1 Centre Manifolds Depending on a Parameter

II.4.8 Comments

II.5 Bifurcations

II.5.1 Reduction to an Iterated Map

II.5.2 Codimension-one Bifurcations

II.5.2.1 Fold Bifurcation

II.5.2.2 Period-Doubling Bifurcation

Special Case: q=1

II.5.2.3 Cylinder Bifurcation

II.5.3 Comments

III Singular and Nonsmooth Phenomena

III.1 Continuous Approximation

III.1.1 Introduction

III.1.1.1 Singular Unfolding of an Impulsive Differential Equation

III.1.1.2 Preliminaries

III.1.1.3 Time q Map

III.1.1.4 The Realization Problem

III.1.1.5 A Brief Discussion on the ContinuityConvention

III.1.2 Pointwise Convergence and the Candidate...

III.1.3 Smoothness of the Time q Map

III.1.4 Sensitivity and Realization

III.1.5 An Important Comment (Or Warning)...

III.1.6 Example: Continuous-Time Logistic...

III.2 Non-smooth Bifurcations

III.2.1 Overview

III.2.1.1 Bifurcations Involving Perturbations of Impulse Times

III.2.1.2 Bifurcations Involving Crossings of ImpulseTimes and Delays

III.2.2 Centre Manifolds Parameterized...

III.2.2.1 Dummy Matrix System and Robustness ofSpectral Separation

III.2.2.2 Centre Manifold Construction

III.2.3 Overlap Bifurcations

III.2.3.1 Floquet Spectrum.

III.2.3.2 Symmetries of Periodic Solutions

III.2.3.3 A State Transformation that Eliminatesthe Delay

III.2.3.4 Bifurcations of Periodic Solutions

III.2.3.5 The Introductory Example Revisited

III.2.4 Comments

IV Applications

IV.1 Bifurcations in an Impulsively Damped or Driven Pendulum

IV.1.1 Stability Analysis: The ModelWithout Delay

IV.1.1.1 Downward Rest Position

Case 1: (α+1)2cos2(ρT)4α

Case 2: (α+1)2cos2(ρT)&gt

4α

IV.1.1.2 Upward Rest Position

Case 1: (α+1)2cosh2(ρT)4α

Case 2: (α+1)2cosh2(ρT)&gt

IV.1.2 Stability Analysis: The Model withDelay

IV.1.2.1 Downward Rest Position

IV.1.2.2 Upward Rest Position

IV.1.3 Cylinder Bifurcation at the Downward...

IV.1.4 Cylinder Bifurcation at the Downward...

IV.1.4.1 Floquet Multiplier Transversality Condition

IV.1.4.2 Computation of the First LyapunovCoefficient

IV.2 The Hutchinson Equation with Pulse Harvesting

IV.2.1 Dummy Matrix System: Setup for the Non-smooth Centre Manifold

IV.2.2 Dynamics on the Centre Manifold

IV.2.3 The Transcritical Bifurcation

IV.3 Delayed SIR Model with Pulse Vaccination and TemporaryImmunity

IV.3.1 Introduction

IV.3.2 Vaccinated Component Formalism

IV.3.3 Existence of the Disease-free Periodic Solution

IV.3.4 Stability of the Disease-free Periodic Solution

IV.3.5 Existence of a Bifurcation Point

IV.3.6 Transcritical Bifurcation in Terms of Vaccine Coverage at R0=1 with One Vaccination Pulse Per Period

Linearization

Centre Fibre Bundle

Projection of χ0 Onto the Centre Fibre Bundle

Dynamics on the Centre Manifold and Bifurcation

IV.3.7 Numerical Bifurcation Analysis

IV.4 Stage-Structured Predator-Prey System with Pulsed Birth

IV.4.1 Model Derivation

IV.4.2 Stability of the Extinction Equilibrium

IV.4.3 Analysis of Predator-Free PeriodicSolution.

IV.4.3.1 Existence and Uniqueness of the Predator-Free Solution.

Notes:

Includes bibliographical references and index.

Description based on print version record.

ISBN:

3-030-64533-9

OCLC:

1243514212