Floquet theory and applications.

Thesis Overview

<p> </p><p>Introduction and Generalities<br>Introduction<br>In physical sciences (e.g, elasticity, astronomy) and natural sciences (e.g, ecology) among others,<br>the consideration of periodic environmental factor in the dynamics of multi-phenomena<br>interractions (or multi-species interractions) leads to the study of dierential systems with<br>periodic data. Therefore, it is worth investigating, the fundamental questions inherent in systems<br>of periodic (ordinary) dierential equations such as: existence, uniqueness and stability.<br>This work is specially devoted to one of the celebrated tools which is crucial in the analysis<br>and control of time-periodic systems, namely Floquet Theory. This basic theory which is<br>about a systematic study of linear systems of ordinary dierential equations with periodic<br>coecients, has had a rich history since the pioneering work of Floquet (1883) followed by the<br>contribution of Lyapunov (1892). Futhermore it gives a change of variables that transforms a<br>linear dierential system with periodic coecients, into a system with constant coecients,<br>and also provides a representation formula of the solutions of (1).<br>The aim of this project is to present Floquet theory and to use it to assess stability of<br>periodic solution of periodic dierential systems (linear or nonlinear).<br>The work is organized as follows:<br>First, we review some basic notions from Analysis and Linear algebra that will be used in<br>the subsequent chapters.<br>In chapter 2, we present the basic notions and theorems of the general theory of Ordinary<br>Dierential Equations (ODE).<br>Finally in chapter 3, we state Floquet’s theorem, prove it and use it to address stability of<br>periodic solutions of linear as well as nonlinear periodic systems. Furthermore, we illustrate<br>our method by studying Hill’s equation which is a generalization of Mathieu equation.<br>1<br>Generalities<br>The purpose of this chapter is to give some concepts and theorems of linear algebra and<br>Mathematical analysis that will be of importance in the subsequent chapters<br>1.1 Basic concepts from linear functional analysis<br>let’s recall some denitions and results from linear functional analysis<br>Denition 1.1.1 Let X be a linear space over a eld K; where K holds either for Rn or C.<br>A mapping k:k: X ô€€€! R is called a norm provided that the following conditions hold:<br>i) kxk 0; for all x 2 X and kxk= 0 , x = 0<br>ii) kxk= jjkxk for all 2 K; x 2 X<br>iii) kx + yk kxk+kyk for arbitrary x; y 2 X<br>If X is a linear space and k:k is a norm on X; then the pair (X; k:k) is called a normed<br>linear space over K.<br>Should no ambiguity arise about the norm, we simply abbreviate this pair by saying that X<br>is a normed linear space over K.<br>Example 1. Each of the following expressions denes on the vector space Rn a norm which<br>is in common use.<br>i) The absolute norm :<br>kxk1 =<br>Xn<br>i=1<br>jxij ; for every x = (x1; : : : ; xn) 2 Rn:<br>ii) The euclidean norm :<br>kxk2 =</p><p>Xn<br>i=1<br>jxij2<br>!1<br>2<br>; for every x = (x1; : : : ; xn) 2 Rn:<br>iii) The maximum norm :<br>kxk1 = max<br>1in<br>jxij; for every x = (x1; : : : ; xn) 2 Rn:<br>2<br>Example 2. Let X = C([0; 1]) be the space of all real-valued continuous functions on [0; 1].<br>Each of the following expressions denes on the vector space C([0; 1]) a norm which is in<br>common use.<br>i) kfk1=<br>R 1<br>0 jf(t)jdt for every f 2 C([0; 1]).<br>ii) kfk2=<br>R 1<br>0 (jf(t)j)<br>1<br>2 dt<br>1<br>2 for every f 2 C([0; 1]).<br>iii) kfk1= max</p><p>jf(t)j: t 2 [0; 1]</p><p>.<br>Denition 1.1.2 (Equivalent norms)<br>Two norms k:k1 and k:k2 dened on a normed linear space X are said to be equivalent if<br>there exists &gt; 0 and &gt; 0 constants such that<br>kxk1 kxk2 kxk1 8x 2 X:<br>Theorem 1.1.3 In a nite dimensional normed linear space, all the norms are equivalent.<br>Denition 1.1.4 Every normed linear space E is canonically endowed with a metric d de-<br>ned on E E by<br>d(x; y) = jjx ô€€€ yjj 8 x; y 2 E:<br>Denition 1.1.5 (Cauchy sequence)<br>A sequence (xn)n1 of elements of a normed vector space X is a Cauchy sequence if<br>lim<br>n;m!1<br>kxn ô€€€ xmk= 0:<br>That is, for any &gt; 0 there is an interger N = N() such that kxn ô€€€ xmk&lt; whenever<br>n N and m N.<br>Remark. In a normed linear sapce, every Cauchy sequence (xn)n1 is bounded; i.e, there<br>exists a constant M 0 such that jjxnjj M ; 8n 1: (See also Denition 1.1.11 below)<br>Denition 1.1.6 (convergent sequence)<br>A sequence (xn)n1 of elements of a normed vector space X converges to an element x 2 X<br>if<br>lim<br>n!1<br>kxn ô€€€ xk= 0:<br>In such a case, we say that (xn)n1 is a convergent sequence.<br>Remark. In a normed linear space, every convergent sequence is a Cauchy sequence.<br>Denition 1.1.7 A normed linear space is complete if every Cauchy sequence in X has the<br>limit in X. A complete normed linear space is called a Banach space.<br>Remark. The notion of completeness is also dened for metric spaces which need not have<br>any linear structure.<br>Example (Banach space). The normed linear space<br>ô€€€<br>C([0; 1]); k k1</p><p>is a Banach space.<br>3<br>Denition 1.1.8 (Open sets and closed sets)<br>Let X be a normed linear space. We dene open (respectively closed) ball with center at a<br>point x 2 X and radius r &gt; 0 by<br>Br(x) = fx 2 X : kxk&lt; rg (respectively Br(x) = fx 2 X : kxk rg ) :<br>A nonempty subset A of a normed linear space X is said to be open if for all x 2 A; there<br>exists r &gt; 0 such that Br(x) A: And A is said to be closed if XnA is open.<br>Proposition 1.1.9 A subset A of a normed linear space is closed if and only if every con-<br>vergent sequence (an)n1 of elements of A has its limit in A:<br>Denition 1.1.10 (Closure and interior) Let A be a subset of a normed linear space X.<br>The interior of A denoted by intA is dened as the union of all open sets contained in A<br>and the closure of A denoted by cl(A) or A is dened as the intersection of all closed sets<br>containing A:<br>Theorem 1.1.11 Let A be a subset of a normed linear space X and x 2 X then,<br>a) x 2 intA if and only if 9 r &gt; 0 : B(x; r) A:<br>b) x 2 clA if and only if 8 r &gt; 0; B(x; r) A 6= ?:<br>Remark. Given a subset A of a normed linear space X; we have :<br>x 2 A () 9 (an)n A such that lim<br>n!+1<br>an = x:<br>Denition 1.1.12 Let X be a normed linear space, x 2 X and let V be a subset of X<br>containing x: We say that V is a neighbourhood of x if there exists an open set U of X<br>containing x and contained in V: We denote by N(x) the collection of all neighbourhoods of<br>x:<br>Denition 1.1.13 A subset of a normed linear space X is said to be bounded if it can be<br>included in some ball.<br>Theorem 1.1.14 (Riesz/ Heine-Borel) A normed linear space is nite dimensional if<br>and only if its closed unit ball is compact, i.e., every bounded sequence of the closed unit ball,<br>has a convergent subsequence.<br>1.1.1 Linear operators<br>In this section X and Y are normed linear spaces over K.<br>Denition 1.1.15 A K-linear operator T from X into Y is a map T : X ô€€€! Y such that<br>T(x + y) = Tx + Ty<br>for all ; 2 K and all x; y 2 X:<br>When Y = K, such a map is called a linear functional or a linear form.<br>4<br>Proposition 1.1.16 The set of K-linear operators from X into Y has a natural structure<br>of linear space over K and is denoted by L(X; Y ). Note that L(X;X) is simply denoted by<br>L(X).<br>Proposition 1.1.17 If Z is also a linear space, then<br>f 2 L(X; Y ) and g 2 L(Y;Z) =) gof 2 L(X;Z) :<br>Theorem 1.1.18 Let T 2 L(X; Y ). Then the following are equivalent<br>i) T is continuous at the origin (in the sense that if fxngn is a sequence in X such that<br>xn ! 0 as n ! 1, then T(xn) ! 0 in Y as n ! 1.<br>ii) T is Lipschitz, i.e., there exists a constant K 0 such that for every x 2 X,<br>jjT(x)jj Kjjxjj :<br>iii) The image of the closed unit ball, T<br>ô€€€<br>B1(0)</p><p>, is bounded.<br>Denition 1.1.19 A linear operator T : X ô€€€! Y is said to be bounded if there exist some<br>k 0 such that<br>kT(x)k kkxk<br>for all x 2 X:<br>If T is bounded, then the norm of T is dened by<br>kTk= inffk : kT(x)k kxk; x 2 Xg:<br>The set of bounded linear operators from X into Y is denoted B(X; Y ): If X = Y; one simply<br>writes B(X):<br>Proposition 1.1.20 Suppose X 6= f0g and T 2 B(X), then we have the following:<br>kTk= sup<br>kxk1<br>kT(x)k= sup<br>kxk=1<br>kT(x)k= sup<br>kxk6=0<br>kT(x)k<br>kxk<br>1.2 Matrix calculus and basic Operator theory<br>1.2.1 Matrices, eigenvalues and eigenvectors<br>Denition 1.2.1 An m n matrix A is a rectangular array of numbers, real or complex,<br>with m rows and n columns. We shall write aij for the number that appears in the ith row<br>and the jth column of A; this is called the (i; j) entry of A: We can either write A in the<br>extended form 0<br>[email protected]<br>a11 a12 a1n<br>a21 a22 a2n<br>…<br>…<br>. . .<br>…<br>am1 am2 amn<br>1<br>CCCA<br>5<br>or in the more compact form<br>(aij)m;n:<br>We will denoted A = (aij)m;n with 1 i m; 1 j n:<br>Associated with each matrix A a matrix At; known as the transpose of A, and obtained<br>from A by interchanging the rows and the columns of A. Thus, if A = (aij)m;n then At =<br>(aji)n;m: The trace denoted by tr of A is the sum of a diagonal elements of A. Two m n<br>matrices A and B are said to be equal if all corresponding elements are equal, that is , if<br>aij = dij for each i and j.<br>Let D = (dij)m;n, A = (aij)m;n and B = (bij)n;p be a matrices.<br>1) The sum of two m n matrices A and D is dened as matrix obtained by adding<br>corresponding elements:<br>D + A = (dij + aij)m;n:<br>Similarly, the dierence is<br>D ô€€€ A = (dij ô€€€ aij)m;n:<br>2) The multiplication of a matrix A by a scalar is dened as follows: A = (aij)m;n:<br>3) The product of a m n matrix A and a n p matrix B is<br>AB =</p><p>Xn<br>k=1<br>aikbkj<br>!<br>m;p<br>:<br>Special matrices<br>i) A row-vector or 1 m matrix: A = (a1; a2; :::; an) where the a0<br>is are scalars.<br>ii) A column-vector or n 1 matrix: A =<br>0<br>[email protected]<br>a1<br>…<br>an<br>1<br>CA<br>where the a0<br>is are scalar.<br>iii) A zero matrix is a matrix which all of whose entries are zero. The zero m n matrix<br>is denoted by 0m;n or simply 0:<br>iv) A square matrix is a matrix with the same number of rows and columns.<br>v) The identity matrix of order n has one on the principal diagonal, that is from top<br>left to bottom right, and zeros elsewhere; it is denoted by In = (ij) where ij is the<br>Kronecker’s symbol. From the denition of matrix multiplication we have<br>AI = IA = A<br>for any square matrix A.<br>6<br>iv) A square matrix A is regular or non-singular if its column vectors are linearly independant<br>or equivalently its determinant, det(A), is nonzero. Otherwise A is said to be<br>singular or degenerated when det(A) = 0. (See details/recalls below).<br>v) A square matrix in which all the non-zero elements lie on the principal diagonal is<br>called a diagonal matrix.<br>vi) An n n matrix N is said to be nilpotent, if there is a positive integer k such that<br>Nk = 0:<br>Denition 1.2.2 Let A and B be two n n matrices. We say that A and B are similar,<br>notation A B; if and only if there exists a nonsingular matrix T such that Tô€€€1AT = B<br>Eigenvalues and Eigenvectors<br>Denition 1.2.3 Let A be an nn square matrix. Then a scalar is called an eigenvalue<br>of A, if there exists a nonzero vector v 2 Rn such that Av = v:<br>In this case, the is called an eigenvalue of A and the v; eigenvector associated with :<br>The eigenvalues of A are also the roots of the characteristic polynomial p() = det(Aô€€€I)<br>with p() of degree n:<br>Theorem 1.2.4 Let A be a square matrix, if the eigenvalues of A are all distincts, A is<br>similar to a diagonal matrix (whose diagonal entries are the eigenvalues of A).<br>Now if the matrix A has repeated eigenvalues, then it is not possible to diagonalize it. In<br>this case, we introduce the concept of generalized eigenvector.<br>Denition 1.2.5 A nonzero vector v is called a generalized eigenvector of rank k of A;<br>associated with an eigenvalue if and only if<br>(A ô€€€ I)kv = 0 and (A ô€€€ I)kô€€€1v 6= 0:<br>Lemma 1.2.6 If v is a generalized eigenvector of rank k; then the vectors<br>v; (A ô€€€ I)v; :::; (A ô€€€ I)kô€€€1v are linearly independent.<br>Recall that a set of vectors v1; v2; : : : ; vk is linearly dependent if there exist scalars c1; c2; : : : ; ck<br>not all zero, such that<br>c1v1 + c2v2 + : : : + ckvk = 0:<br>A set of vectors v1; v2; : : : ; vk is linearly independent if it is not linearly dependent.<br>Jordan form<br>From the lemma above, we construct a new basis for Cn such that the matrix representation<br>of A with respect to this new basis is the one we call Jordan canonical form denoted by J:<br>7<br>Theorem 1.2.7 For every nn complex matrix A with eigenvalues 1; :::; s (not necessarily<br>distinct) of multiplicities n1; :::; ns respectively, there exists a nonsingular n n matrix P<br>such that<br>Pô€€€1AP = J = diag(J1; :::; Js);<br>where each of block matrices J1; :::; Js is of the form<br>Jk =<br>0<br>[email protected]<br>k 1 0 0<br>0 k 1 0</p><p>0 k 1<br>0 0 k<br>1<br>CCCCA<br>; k = 1; :::; s;<br>and<br>Ps<br>k=1 nk = n:<br>For the proof of this theorem, refer to Coddington and Levinson [C/L] or Hirsch and Smale<br>[H/S.]<br>The block matrices J1; :::; Js are called Jordan blocks, and J is called the Jordan canonical<br>form of A: Note, any Jordan block Jk() can be written as Jk = kI + Nk; where Nk is<br>nilpotent of order k and that A is similar to J:<br>1.2.2 Limits of sequences of operators<br>We introduce the concepts of limit in the norm and of strong limit of a sequence of operators.<br>We shall then introduce the derivative and integral of operators depending on a parameter<br>and shall discuss series of operators.<br>Let X be a Banach space and (An) a sequence of operators in L(X):<br>– We say that (An) converges in norm to the operator A 2 L(X); if<br>lim<br>n!1<br>jjAn ô€€€ Ajj = 0: (1.2.1)<br>– If, for each element x 2 X;<br>lim<br>n!<br>jjAnx ô€€€ Axjj = 0; (1.2.2)<br>we shall say that An converges strongly to the operator A 2 L(X):<br>It is immediate that (1.2.1) implies (1.2.2). The converse is not in general true. However, in<br>the case of nite dimensional Banach spaces it is true, for if we put x = e(i); we have from<br>(1.2.2)<br>lim<br>n!1<br>jj(a(1)<br>i1 ; :::; a(k)<br>ik )jj = 0 for i = 1; :::; k:<br>This ensures that all the components tend to 0; and since there are only a nite number of<br>them, the limit is uniform.<br>It is now possible to dene the meaning of the series<br>X1<br>s=1<br>As;<br>8<br>the series<br>1P<br>s=1<br>As is said to be convergent if the sequence made by the partial sums<br>XN<br>s=1<br>As;<br>converges in L(X).<br>In this case of matrices,<br>NP<br>s=1<br>As corresponds to a matrix of which elements are<br>NP<br>s=1<br>a(s)<br>ij = aij :<br>– We say that the series of operators<br>1P<br>s=1<br>As converges absolutely, if the series<br>X1<br>s=1<br>jjAsjj<br>converges.<br>In the case of matrices, this happens if and only if, for every component,<br>X1<br>s=1<br>ja(s)<br>ij j<br>converges with As = (a(s)<br>ij ):<br>Proposition 1.2.8 Let X be a normed linear space. If A 2 L(X); then the series<br>1P<br>0<br>An<br>n! is<br>absolutely convergent.<br>Proof. If suces to show that the sequence of partial sums fSNg1N<br>=1 for the series<br>1P<br>n=0<br>kAnk<br>n! is a Cauchy sequence. Let us dene<br>NP<br>0<br>An<br>n! : Note that the partial sums of the convergent<br>series of real number<br>1P<br>0<br>kAkn<br>n! = ekAk form a cauchy sequence. Using this fact, it follows<br>that SN is a cauchy sequence in L(X).<br>Dene the exponential map exp : L(X) ! L(X) by exp(A) = eA =<br>1P<br>n=0<br>An<br>n! :<br>The main properties of exponential map are summarized in the following proposition<br>Proposition 1.2.9 Suppose that A;B 2 L(Rn):<br>i) If A 2 L(Rn; ) then eA 2 L(Rn):<br>ii) If B is nonsingular, then Bô€€€1eAB = eBô€€€1AB:<br>iii) eô€€€A = (eA)ô€€€1:<br>9<br>iv) keAk ekAk:<br>– Let A be an operator depending on a real parameter t with a t b: Let t0 2 [a; b]; if<br>h is suciently small, the operator<br>A(t0 + h) ô€€€ A(t0)<br>h<br>can be dened. If its limit as h ! 0 exists, we say that the operator A is dierentiable at t<br>with respect to t: The limiting operator is denoted by d<br>dtA(t): We thus have<br>lim<br>h!0</p><p>A(t0 + h) ô€€€ A(t0)<br>h<br>ô€€€<br>dA<br>dt</p><p>= 0:<br>In the case of matrices, the limits exists if and only if each component aij is dierentiable,<br>and we have<br>d<br>dt<br>A(t) =</p><p>d<br>dt<br>aij(t)</p><p>:<br>In general, if A and B are two dierentiable operators in L(X); the following product rule<br>is valid.<br>d<br>dt<br>(AB) =<br>dA<br>dt<br>B + A<br>dB<br>dt<br>:<br>We say that the series of operators depending on a parameter converges uniformly if, for<br>every &gt; 0; there exists a &gt; 0 such that for every t 2 [a; b] and every &gt;</p><p>X1<br>s=<br>As(t)</p><p>&lt; :<br>If the operator As(t) are dierntiable and the series<br>X1<br>s=0<br>d<br>dt<br>As(t)<br>converges uniformly, then also the operator A =<br>1P<br>s=0<br>As(t) is dierntiable, and we have<br>dA<br>dt<br>=<br>X1<br>s=0<br>d<br>dt<br>As:<br>We illustrate this notion with the following result: If A 2 L(X) then tA 2 L(X) for each<br>t 2 R so that the function t 7ô€€€! etA is dierentiable and<br>d<br>dt<br>(etA) = AetA:<br>If A is a nonsingular matrix, then logarithm of A denoted by ln(A) is well-dened matrix.<br>This important result is stated in the following theorem.<br>10<br>Theorem 1.2.10 Let A be a nonsingular n n matrix, then there exists an n n matrix<br>B (called logarithm of A) such that A = eB:<br>Let A(t) be a matrix depending on a parameter t; we suppose that the component aij are<br>all integrable functions over the interval [t0; t]: We shall say that the matrix<br>Z t<br>t0<br>A( )d;<br>is the intergral of the matrix A(t) between t0 and t:<br>According to the following denition<br>d<br>dt<br>(aij)m;n =</p><p>daij<br>dt</p><p>m;n<br>;<br>we have that if the functions aij are continuous, then<br>d<br>dt<br>Z<br>A( )d = A(t):<br>If, nally A is an m n matrix, and L is a number greater than or equal to the absolute<br>values of aij ; then<br>Zt<br>t0<br>A( )d</p><p>p<br>m:n:Ljt ô€€€ t0j:<br>1.3 Review of calculus<br>We now give the general denition of dierentiability. Let U be a nonempty open subset of<br>a Banach space X, let Y denote a Banach space, and let the symbol k:k denote the norm in<br>both Banach spaces.<br>Denition 1.3.1 A function f : U ô€€€! Y is dierentiable at x 2 U if there is a map<br>A 2 L(X; Y ) such that<br>lim<br>h!0<br>kf(x + h) + f(x) ô€€€ Ahk<br>khk<br>= 0<br>Remark. If such a linear map exists, then it is unique and we write it as A = f0(x) called<br>the derivative of f at x:<br>Other common notations for the derivative are Df and fx<br>The following is a list of standard facts about the derivative. For the statement in the list,<br>the symbols X; Y; Xi; and Yi denoted Banach spaces.<br>i) If f : X ô€€€! Y is dierentiable at a 2 X, then f is continuous at a.<br>ii) If f : X ô€€€! Y1 ::: Yn is given by f(x) = (f1(x); :::fn(x)); and if fi is dierentiable<br>for each i, then so is f and,<br>Df(x) = (Df1(x); :::;Dfn(x)):<br>11<br>iii) If the function f : X1 X2 ::: Xn ô€€€! Y is given by (x1; x2; :::; xn) 7ô€€€!<br>f(x1; x2; :::; xn), then the ith partial derivative of f at (a1; a2:::; an) 2 X1X2:::Xn<br>is the derivative of the function g : Xi ô€€€! Y dened by g(xi) = f(a1; :::; aiô€€€1; xi; ai+1; :::; an).<br>This derivatives is denoted by Dif(a): If f is dierentiable, then all its partial derivatives<br>exist and, if we dene h = (h1; h2; :::; hn); we have<br>Df(x)h =<br>Xn<br>i=1<br>Dif(x)hi:<br>The converse is not true in general, but if all the partial derivatives of f exist and are<br>continuous in an open set U X1 X2 :::Xn then f is continuously dierentiable<br>in U.<br>1.3.1 The mean value theorem<br>Theorem 1.3.2 Suppose that [a; b] is a closed interval, and f : [a; b] ô€€€! Y is a continuous<br>function. If f is dierentiable on the open interval (a; b) and there is some number M &gt; 0<br>such that kf0(t)k M for all t 2 (a; b); then<br>kf(b) ô€€€ f(a)k M(b ô€€€ a)<br>Theorem 1.3.3 (Mean value theorem) Suppose that f : X ô€€€! Y is dierentiable on an<br>open set U X with a; b 2 U and a + t(b ô€€€ a) 2 U for 0 t 1: If there is some M &gt; 0<br>such that<br>sup<br>0t1<br>kDf(a + t(b ô€€€ a)k M;<br>then<br>kf(b) ô€€€ f(a)k Mkb ô€€€ ak:<br>1.3.2 Integration in Banach space<br>Let X be a Banach space and I = [a; b] R where a &lt; b.<br>Denition 1.3.4 If f : I ! X is continuous on I, we dene its integral in the sense of<br>Rieman by the following formula:<br>Z b<br>a<br>f(t)dt = lim<br>n!1<br>b ô€€€ a<br>n<br>Xnô€€€1<br>k=0<br>f(a + k<br>b ô€€€ a<br>n<br>):<br>It is easily seen that<br>Z b<br>a<br>f(t)dt</p><p>Z b<br>a<br>kf(t)kdt:<br>Theorem 1.3.5 Suppose that U is a nonempty open subset of X: If f : X ! Y is a<br>dierentiable function, and x + ty 2 U for 0 t 1; then<br>f(x + y) ô€€€ f(x) =<br>Z 1<br>0<br>Df(x + ty)ydt:<br>12</p> <br><p></p>