Floquet Theory and Applications

Floquet Theory and Applications

TABLE OF CONTENTS

Epigraph ii
Preface iii
Acknowledgement iv
Dedication v
1 Introduction and Generalities 1
1.1 Basic concepts from linear functional analysis . . . . . . . . . . . . . . . . . 2
1.1.1 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Matrix calculus and basic Operator theory . . . . . . . . . . . . . . . . . . . 5
1.2.1 Matrices, eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . 5
1.2.2 Limits of sequences of operators . . . . . . . . . . . . . . . . . . . . . 8
1.3 Review of calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 The mean value theorem . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.2 Integration in Banach space . . . . . . . . . . . . . . . . . . . . . . . 12
2 Basic notions of ordinary differential equations 13
2.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Existence and Uniqueness of Solutions to a System . . . . . . . . . . . . . . 15
2.2.1 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Linear systems of ODE . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Stability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.1 Phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.2 General definition of stability . . . . . . . . . . . . . . . . . . . . . . 26
2.3.3 Stability of linear systems . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Floquet theory: Presentation and stability of periodic solutions 32
3.1 Linear systems with periodic coefficients: Floquet theory . . . . . . . . . . . 32
3.1.1 Non-homogeneous linear systems . . . . . . . . . . . . . . . . . . . . . 36
3.2 Stability of periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.1 Autonomous systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.2 Non-autonomous systems . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Bibliography 45
vii

CHAPTER ONE

Introduction and Generalities

Introduction

In physical sciences (e.g, elasticity, astronomy) and natural sciences (e.g, ecology) among others, the consideration of periodic environmental factor in the dynamics of multi-phenomena interactions (or multi-species interactions) leads to the study of differential systems with periodic data. Therefore, it is worth investigating, the fundamental questions inherent in systems of periodic (ordinary) differential equations such as: existence, uniqueness and stability.

This work is specially devoted to one of the celebrated tools which is crucial in the analysis and control of time-periodic systems, namely Floquet Theory. This basic theory which is about a systematic study of linear systems of ordinary differential equations with periodic coefficients, has had a rich history since the pioneering work of Floquet (1883) followed by the contribution of Lyapunov (1892). Furthermore it gives a change of variables that transforms a linear differential system with periodic coefficients, into a system with constant coefficients, and also provides a representation formula of the solutions of (1).

The aim of this project is to present Floquet theory and to use it to assess stability of periodic solution of periodic differential systems (linear or nonlinear).

The work is organized as follows:

First, we review some basic notions from Analysis and Linear algebra that will be used in the subsequent chapters.

In chapter 2, we present the basic notions and theorems of the general theory of Ordinary Differential Equations (ODE).

Finally in chapter 3, we state Floquet’s theorem, prove it and use it to address stability of periodic solutions of linear as well as nonlinear periodic systems. Furthermore, we illustrate our method by studying Hill’s equation which is a generalization of Mathieu equation.

Generalities

The purpose of this chapter is to give some concepts and theorems of linear algebra and Mathematical analysis that will be of importance in the subsequent chapters

1.1 Basic concepts from linear functional analysis

let’s recall some definitions and results from linear functional analysis

Definition 1.1.1 Let X be a linear space over a eld K; where K holds either for Rn or C.

A mapping k:k: X ?! R is called a norm provided that the following conditions hold:

i) kxk 0; for all x 2 X and kxk= 0 , x = 0

ii) kxk= jjkxk for all 2 K; x 2 X

iii) kx + yk kxk+kyk for arbitrary x; y 2 X

If X is a linear space and k:k is a norm on X; then the pair (X; k:k) is called a normed linear space over K.
Should no ambiguity arise about the norm, we simply abbreviate this pair by saying that X is a normed linear space over K.

Example 1. Each of the following expressions denes on the vector space Rn a norm which is in common use.
i) The absolute norm :

kxk1 =
Xn
i=1
jxij ; for every x = (x1; : : : ; xn) 2 Rn:

ii) The euclidean norm :
kxk2 =

Xn
i=1
jxij2
!1
; for every x = (x1; : : : ; xn) 2 Rn:

iii) The maximum norm :
kxk1 = max
1in
jxij; for every x = (x1; : : : ; xn) 2 Rn:
Example 2. Let X = C([0; 1]) be the space of all real-valued continuous functions on [0; 1].

Each of the following expressions denes on the vector space C([0; 1]) a norm which is in common use.
i) kfk1=
R 1
0 jf(t)jdt for every f 2 C([0; 1]).
ii) kfk2=
R 1
0 (jf(t)j)
1
2 dt
1
2 for every f 2 C([0; 1]).
iii) kfk1= max

jf(t)j: t 2 [0; 1]

.
Definition 1.1.2 (Equivalent norms)

Two norms k:k1 and k:k2 dened on a normed linear space X are said to be equivalent if there exists > 0 and > 0 constants such that kxk1 kxk2 kxk1 8x 2 X:

Theorem 1.1.3 In a nite dimensional normed linear space, all the norms are equivalent.

Definition 1.1.4 Every normed linear space E is canonically endowed with a metric d dened on E E by
d(x; y) = jjx ? yjj 8 x; y 2 E:

Definition 1.1.5 (Cauchy sequence)

A sequence (xn)n1 of elements of a normed vector space X is a Cauchy sequence if lim
n;m!1
kxn ? xmk= 0:
That is, for any > 0 there is an interger N = N() such that kxn ? xmk< whenever n N and m N. Remark. In a normed linear sapce, every Cauchy sequence (xn)n1 is bounded; i.e, there exists a constant M 0 such that jjxnjj M ; 8n 1: (See also Denition 1.1.11 below) Denition 1.1.6 (convergent sequence) A sequence (xn)n1 of elements of a normed vector space X converges to an element x 2 X if lim n!1 kxn ? xk= 0: In such a case, we say that (xn)n1 is a convergent sequence. Remark. In a normed linear space, every convergent sequence is a Cauchy sequence. Denition 1.1.7 A normed linear space is complete if every Cauchy sequence in X has the limit in X. A complete normed linear space is called a Banach space. Remark. The notion of completeness is also dened for metric spaces which need not have any linear structure. Example (Banach space). The normed linear space ? C([0; 1]); k k1 is a Banach space. Definition 1.1.8 (Open sets and closed sets) Let X be a normed linear space. We dene open (respectively closed) ball with center at a point x 2 X and radius r > 0 by Br(x) = fx 2 X : kxk< rg (respectively Br(x) = fx 2 X : kxk rg ) : A nonempty subset A of a normed linear space X is said to be open if for all x 2 A; there exists r > 0 such that Br(x) A: And A is said to be closed if XnA is open.

Proposition 1.1.9 A subset A of a normed linear space is closed if and only if every convergent sequence (an)n1 of elements of A has its limit in A:

Definition 1.1.10 (Closure and interior) Let A be a subset of a normed linear space X.

The interior of A denoted by intA is dened as the union of all open sets contained in A and the closure of A denoted by cl(A) or A is dened as the intersection of all closed sets containing A:

Theorem 1.1.11 Let A be a subset of a normed linear space X and x 2 X then,

a) x 2 intA if and only if 9 r > 0 : B(x; r) A:

b) x 2 clA if and only if 8 r > 0; B(x; r) \ A 6= ?:

Remark. Given a subset A of a normed linear space X; we have :

x 2 A () 9 (an)n A such that lim
n!+1
an = x:

Definition 1.1.12 Let X be a normed linear space, x 2 X and let V be a subset of X containing x: We say that V is a neighbourhood of x if there exists an open set U of X containing x and contained in V: We denote by N(x) the collection of all neighbourhoods of x:

Definition 1.1.13 A subset of a normed linear space X is said to be bounded if it can be included in some ball.
Theorem 1.1.14 (Riesz/Heine-Borel) A normed linear space is nite dimensional if and only if its closed unit ball is compact, i.e., every bounded sequence of the closed unit ball, has a convergent subsequence.

1.1.1 Linear operators

In this section X and Y are normed linear spaces over K.

Definition 1.1.15 A K-linear operator T from X into Y is a map T : X ?! Y such that T(x + y) = Tx + Ty for all ; 2 K and all x; y 2 X:

When Y = K, such a map is called a linear functional or a linear form.

Proposition 1.1.16 The set of K-linear operators from X into Y has a natural structure of linear space over K and is denoted by L(X; Y ). Note that L(X;X) is simply denoted by L(X).

Proposition 1.1.17 If Z is also a linear space, then f 2 L(X; Y ) and g 2 L(Y;Z) =) gof 2 L(X;Z) :

Theorem 1.1.18 Let T 2 L(X; Y ). Then the following are equivalent

i) T is continuous at the origin (in the sense that if fxngn is a sequence in X such that xn ! 0 as n ! 1, then T(xn) ! 0 in Y as n ! 1.

ii) T is Lipschitz, i.e., there exists a constant K 0 such that for every x 2 X, jjT(x)jj Kjjxjj :

iii) The image of the closed unit ball, T
?
B1(0) , is bounded.

Definition 1.1.19 A linear operator T : X ?! Y is said to be bounded if there exist some k 0 such that
kT(x)k kkxk for all x 2 X:

If T is bounded, then the norm of T is dened by kTk= inffk : kT(x)k kxk; x 2 Xg:

The set of bounded linear operators from X into Y is denoted B(X; Y ): If X = Y; one simply writes B(X):

Proposition 1.1.20 Suppose X 6= f0g and T 2 B(X), then we have the following:

kTk= sup
kxk1
kT(x)k= sup
kxk=1
kT(x)k= sup
kxk6=0
kT(x)k
kxk

1.2 Matrix calculus and basic Operator theory

1.2.1 Matrices, eigenvalues and eigenvectors

Definition 1.2.1 An m n matrix A is a rectangular array of numbers, real or complex, with m rows and n columns. We shall write aij for the number that appears in the with row and the jth column of A; this is called the (i; j) entry of A: We can either write A in the extended form 0