Bifurcation Theory of Impulsive Dynamical Systems (IFSR International Series in Systems Science and Systems Engineering, 34)

✍ Scribed by Kevin E.M. Church, Xinzhi Liu

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 389
• Edition: 1st ed. 2021
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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 t→Yj(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)> 4α
IV.1.1.2 Upward Rest Position
Case 1: (α+1)2cosh2(ρT)4α
Case 2: (α+1)2cosh2(ρT)> 4α
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
IV.4.3.2 Stability
IV.4.4 Bifurcation at Extinction
IV.4.4.1 Calculation of the Matrix Yc(t)
Case 1: t[tk+τ,tk+1] for some kZ
Case 2: t(tk,tk+τ) for some kZ
Factoring Pc(t)χ0 and Identifying Yc(t)
IV.4.4.2 Centre Manifold Quadratic Dynamics andBifurcation
IV.4.5 Discussion
IV.5 Dynamics of an In-host Viral Infection Model with DrugTreatment
IV.5.1 Derivation of the Delayed Terms
IV.5.2 Existence of a Disease-free Periodic Solution and aDisease-free Attractor
IV.5.3 Well-Posedness and Boundedness
IV.5.4 Numerical Bifurcation Analysis: Preamble
IV.5.4.1 Model Transformation
IV.5.4.2 Monodromy Operator Discretization
IV.5.4.3 Parameters
IV.5.5 Transcritical Bifurcation from the Disease-free Periodic Solution
IV.5.5.1 Results
Bibliography
Index