Thomas C. SiderisAtlantis Studies in Differential EquationsOrdinary Differential Equations and Dynamical Systems201310.2991/978-94-6239-021-8_1

© Atlantis Press and the authors 2013

1. Introduction

Thomas C. Sideris¹  

(1)

Department of Mathematics, University of California, Santa Barbara, CA, USA

Thomas C. Sideris

Email: sideris@math.ucsb.edu

Abstract

The most general nth order ordinary differential equation (ODE) has the form

The most general $$n\mathrm{{th}}$$ order ordinary differential equation (ODE) has the form

where $$F$$ is a continuous function from some open set $$\varOmega \subset {\mathbb {R}}^{n+2}$$ into $${\mathbb {R}}$$ . An $$n$$ times continuously differentiable real-valued function $$y(t)$$ is a solution on an interval $$I$$ if

A necessary condition for existence of a solution is the existence of points $$p=(t,y_1,\ldots ,y_{n+1})\in {\mathbb {R}}^{n+2}$$ such that $$F(p)=0$$ . For example, the equation

has no (real) solutions, because $$F(p)=y_2^2+y_1^2+1=0$$ has no real solutions.

If $$F(p)=0$$ and $$\frac{\partial F}{\partial y_{n+1}}(p)\ne 0$$ , then locally we can solve for $$y_{n+1}$$ in terms of the other variables by the implicit function theorem

and so locally we can write our ODE as

This equation can, in turn, be written as a first order system by introducing additional unknowns. Setting

we have that

Therefore, if we define $$n$$ -vectors

we obtain the equivalent first order system

(1.1)

The point of this discussion is that there is no loss of generality in studying the first order system (1.1), where $$f(t,x)$$ is a continuous function (at least) defined on some open region in $${\mathbb {R}}^{n+1}$$ .

A fundamental question that we will address is the existence and uniqueness of solutions to the initial value problem (IVP)

for points $$(t_0,x_0)$$ in the domain of $$f(t,x)$$ . We will then proceed to study the qualitative behavior of such solutions, including periodicity, asymptotic behavior, invariant structures, etc.

In the case where $$f(t,x)=f(x)$$ is independent of $$t$$ , the system is called autonomous. Every first order system can be rewritten as an autonomous one by introducing an extra unknown. If

then from (1.1) we obtain the equivalent autonomous system

Suppose that $$f(x)$$ is a continuous map from an open set $$U\subset {\mathbb {R}}^n$$ into $${\mathbb {R}}^n$$ . We can regard a solution $$x(t)$$ of an autonomous system

(1.2)

as a curve in $${\mathbb {R}}^n$$ . This gives us a geometric interpretation of (1.2). If the vector $$x'(t)\ne 0$$ , then it is tangent to the solution curve at $$x(t)$$ . The Eq. (1.2) tells us what the value of this tangent vector must be, namely, $$f(x(t))$$ . So if there is one and only one solution through each point of $$U$$ , we know just from the Eq. (1.2) its tangent direction at every point of $$U$$ . For this reason, $$f(x)$$ is called a vector field or direction field on $$U$$ .

The collection of all solution curves in $$U$$ is called the phase diagram of $$f(x)$$ . If $$f\ne 0$$ in $$U$$ , then locally, the curves are parallel. Near a point $$x_0\in U$$ where $$f(x_0)=0$$ , the picture becomes more interesting.

A point $$x_0\in U$$ such that $$f(x_0)=0$$ is called, interchangably, a critical point, a stationary point, or an equilibrium point of $$f$$ . If $$x_0\in U$$ is an equilibrium point of $$f$$ , then by direct substitution, $$x(t)=x_0$$ is a solution of (1.2). Such solutions are referred to as equilibrium or stationary solutions.

To understand the phase diagram near an equilibrium point we are going to attempt to approximate solutions of (1.2) by solutions of an associated linearized system. Suppose that $$x_0$$ is an equilibrium point of $$f$$ . If $$f\in C^1(U)$$ , then Taylor expansion about $$x_0$$ yields

when $$x-x_0$$ is small. The linearized system near $$x_0$$ is

An important goal is to understand when $$y$$ is a good approximation to $$x-x_0$$ . Linear systems are simple, and this is the benefit of replacing a nonlinear system by a linearized system near a critical point. For this reason, our first topic will be the study of linear systems.

Thomas C. SiderisAtlantis Studies in Differential EquationsOrdinary Differential Equations and Dynamical Systems201310.2991/978-94-6239-021-8_2

© Atlantis Press and the authors 2013

2. Linear Systems

Thomas C. Sideris¹  

(1)

Department of Mathematics, University of California, Santa Barbara, CA, USA

Thomas C. Sideris

Email: sideris@math.ucsb.edu

Abstract

Let be a continuous map from an open set in to .

2.1 Definition of a Linear System

Let $$f(t,x)$$ be a continuous map from an open set in $${\mathbb R}^{n+1}$$ to $${\mathbb R}^n$$ . A first order system

will be called linear when

Here $$A(t)$$ is a continuous $$n\times n$$ matrix valued function and $$g(t)$$ is a continuous $${\mathbb R}^n$$ valued function, both defined for $$t$$ belonging to some interval in $${\mathbb R}$$ .

A linear system is homogeneous when $$g(t)=0$$ . A linear system is said to have constant coefficients if $$A(t)=A$$ is constant.

In this chapter, we shall study linear, homogeneous systems with constant coefficients, i.e. systems of the form

where $$A$$ is an $$n\times n$$ matrix (with real entries).

2.2 Exponential of a Linear Transformation

Let $$V$$ be a finite dimensional normed vector space over $${\mathbb R}$$ or $${\mathbb C}$$ . $$L(V)$$ will denote the set of linear transformations from $$V$$ into $$V$$ .

Definition 2.1.

Let $$A\in L(V)$$ . Define the operator norm

Properties of the operator norm:

$$\Vert A\Vert <\infty $$ , for every $$A\in L(V)$$ .

$$L(V)$$ with the operator norm is a finite dimensional normed vector space.

Given $$A\in L(V)$$ , $$\Vert Ax\Vert \le \Vert A\Vert \Vert x\Vert $$ , for every $$x\in V$$ , and $$\Vert A\Vert $$ is the smallest number with this property.

$$\Vert AB\Vert \le \Vert A\Vert \Vert B\Vert $$ , for every $$A,B\in L(V)$$ .

Definition 2.2.

A sequence $$\{A_n\}$$ in $$L(V)$$ converges to $$A$$ if and only if

With this notion of convergence, $$L(V)$$ is complete.

All norms on a finite dimensional space are equivalent, so $$A_n\rightarrow A$$ in the operator norm implies componentwise convergence in any coordinate system.

Definition 2.3.

Given $$A\in L(V)$$ , define $$\displaystyle \mathrm{\mathrm{{exp}} }\;A=\sum _{k=0}^\infty \frac{1}{k!}A^k$$ .

Lemma 2.1.

Given $$A,B\in L(V)$$ , we have the following properties:

1.

$$\mathrm{\mathrm{{exp}} }\;At$$ is defined for all $$t\in {\mathbb R}$$ , and $$\Vert \mathrm{\mathrm{{exp}} }\;At\Vert \le \mathrm{\mathrm{{exp}} }\;\Vert A\Vert |t|$$ .

2.

, provided $$AB=BA$$ .

3.

, for all $$t,s\in {\mathbb R}$$ .

4.

$$\mathrm{\mathrm{{exp}} }\;At$$ is invertible for every $$t\in {\mathbb R}$$ , and $$(\mathrm{\mathrm{{exp}} }\;At)^{-1}=\mathrm{\mathrm{{exp}} }\;(-At)$$ .

5.

$$\displaystyle \frac{d}{dt}\mathrm{\mathrm{{exp}} }\;At = A\mathrm{\mathrm{{exp}} }\;At= \mathrm{\mathrm{{exp}} }\;At\; A$$ .

Proof.

The exponential is well-defined because the sequence of partial sums

is a Cauchy sequence in $$L(V)$$ and therefore converges. Letting $$m

From this, we see that $$S_n$$ is Cauchy. It also follows that $$\Vert \mathrm{\mathrm{{exp}} }\;A\Vert \le \mathrm{\mathrm{{exp}} }\;\Vert A\Vert $$ .

To prove proeperty (2), we first note that when $$AB=BA$$ the binomial expansion is valid:

Thus, by definition

The rearrangements are justified by the absolute convergence of all series.

Property (3) is a consequence of property (2), and property (4) is an immediate consequence of property (3).

Property (5) is proven as follows. We have

This last expression tends to 0 as $$\Delta t\rightarrow 0$$ . Thus, we have shown that $$\displaystyle \frac{d}{dt}\mathrm{\mathrm{{exp}} }\;At = \mathrm{\mathrm{{exp}} }\;At\; A$$ . This also equals $$ A\mathrm{\mathrm{{exp}} }\;At$$ because $$A$$ commutes with the partial sums for $$\mathrm{\mathrm{{exp}} }\;At$$ and hence with $$\mathrm{\mathrm{{exp}} }\;At$$ itself. $$\square $$

2.3 Solution of the Initial Value Problem for Linear Homogeneous Systems

Theorem 2.1.

Let $$A$$ be an $$n\times n$$ matrix over $${\mathbb R}$$ , and let $$x_0\in {\mathbb R}^n$$ . The initial value problem

(2.1)

has a unique solution defined for all $$t\in {\mathbb R}$$ given by

(2.2)

Proof.

We use the method of the integrating factor. Multiplying the system (2.1) by $$\mathrm{\mathrm{{exp}} }\;(-At)$$ and using Lemma 2.1, we see that $$x(t)$$ is a solution of the IVP if and only if

Integration of this identity yields the equivalent statement

which in turn is equivalent to (2.2). This establishes existence, and uniqueness. $$\square $$

2.4 Computation of the Exponential of a Matrix

The main computational tool will be reduction to an elementary case by similarity transformation.

Lemma 2.2.

Let $$A,S\in L(V)$$ with $$S$$ invertible. Then

Proof.

This follows immediately from the definition of the exponential together with the fact that $$(SAS^{-1})^k= SA^kS^{-1}$$ , for every $$k\in {\mathbb N}$$ . $$\square $$

The simplest case is that of a diagonal matrix $$D=\mathrm{diag }[\lambda _1,\ldots ,\lambda _n]$$ . Since $$D^k=\mathrm{diag }(\lambda _1^k,\ldots ,\lambda _n^k)$$ , we immediately obtain

Now if $$A$$ is diagonalizable, i.e. $$A=SDS^{-1}$$ , then we can use Lemma 2.2 to compute

An $$n\times n$$ matrix $$A$$ is diagonalizable if and only if there is a basis of eigenvectors $$\{v_j\}_{j=1}^n$$ . If such a basis exists, let $$\{\lambda _j\}_{j=1}^n$$ be the corresponding set of eigenvalues. Then

where $$D=\mathrm{diag }[\lambda _1,\ldots ,\lambda _n]$$ and $$S=[v_1\cdots v_n]$$ is the matrix whose columns are formed by the eigenvectors. Even if $$A$$ has real entries, it can have complex eigenvalues, in which case the matrices $$D$$ and $$S$$ will have complex entries. However, if $$A$$ is real, complex eigenvectors and eigenvalues occur in conjugate pairs.

In the diagonalizable case, the solution of the initial value problem (2.1) is

where the coefficients $$c_j$$ are the coordinates of the vector $$c=S^{-1}x_0$$ . Thus, the solution space is spanned by the elementary solutions $$\mathrm{\mathrm{{exp}} }\;\lambda _jt\;v_j$$ .

There are two important situations where an $$n\times n$$ matrix can be diagonalized.

$$A$$ is real and symmetric, i.e. $$A=A^T$$ . Then $$A$$ has real eigenvalues and there exists an orthonormal basis of real eigenvectors. Using this basis yields an orthogonal diagonalizing matrix $$S$$ , i.e. $$S^T=S^{-1}$$ .

$$A$$ has distinct eigenvalues. For each eigenvalue there is always at least one eigenvector, and eigenvectors corresponding to distinct eigenvalues are independent. Thus, there is a basis of eigenvectors.

An $$n\times n$$ matrix over $${\mathbb C}$$ may not be diagonalizable, but it can always be reduced to Jordan canonical (or normal) form. A matrix $$J$$ is in Jordan canonical form if it is block diagonal

and each Jordan block has the form

Since $$B$$ is upper triangular, it has the single eigenvalue $$\lambda $$ with multiplicity equal to the size of the block $$B$$ .

Computing the exponential of a Jordan block is easy. Write

where $$N$$ has $$1$$ ’s along the superdiagonal and $$0$$ ’s everywhere else. The matrix $$N$$ is nilpotent. If the block size is $$d\times d$$ , then $$N^d=0$$ . We also clearly have that $$\lambda I$$ and $$N$$ commute. Therefore,

The entries of $$\mathrm{\mathrm{{exp}} }\;Nt$$ are polynomials in $$t$$ of degree at most $$d-1$$ .

Again using the definition of the exponential, we have that the exponential of a matrix in Jordan canonical form is the block diagonal matrix

The following central theorem in linear algebra will enable us to understand the form of $$\mathrm{\mathrm{{exp}} }\;At$$ for a general matrix $$A$$ .

Theorem 2.2.

Let $$A$$ be an $$n\times n$$ matrix over $${\mathbb C}$$ . There exists a basis $$\{v_j\}_{j=1}^n$$ for $${\mathbb C}^n$$ which reduces $$A$$ to Jordan normal form $$J$$ . That is, if $$S=[v_1\cdots v_n]$$ is the matrix whose columns are formed from the basis vectors, then

The Jordan normal form of $$A$$ is unique up to the permutation of its blocks.

When $$A$$ is diagonalizable, the basis $$\{v_j\}_{j=1}^n$$ consists of eigenvectors of $$A$$ . In this case, the Jordan blocks are $$1\times 1$$ . Thus, each vector $$v_j$$ lies in the kernel of $$A-\lambda _jI$$ for the corresponding eigenvalue $$\lambda _j$$ .

In the general case, the basis $$\{v_j\}_{j=1}^n$$ consists of appropriately chosen generalized eigenvectors of $$A$$ . A vector $$v$$ is a generalized eigenvector of $$A$$ corresponding to an eigenvalue $$\lambda _j$$ if it lies in the kernel of $$(A-\lambda _j I)^k$$ for some $$k\in {\mathbb N}$$ . The set of generalized eigenvectors of $$A$$ corresponding to a given eigenvalue $$\lambda _j$$ is a subspace, $$E(\lambda _j)$$ , of $${\mathbb C}^n$$ , called the generalized eigenspace of $$\lambda _j$$ . These subspaces are invariant under $$A$$ . If $$\{\lambda _j\}_{j=1}^d$$ are the distinct eigenvalues of $$A$$ , then

is a direct sum, that is, every vector in $$x\in {\mathbb C}^n$$ can be uniquely written as a sum $$x=\sum _{j=1}^dx_j$$ , with $$x_j\in E(\lambda _j)$$ .

We arrive at the following algorithm for computing $$\mathrm{\mathrm{{exp}} }\;At$$ . Given an $$n\times n$$ matrix $$A$$ , reduce it to Jordan canonical form $$A=SJS^{-1}$$ , and then write

Even if $$A$$ (and hence also $$\mathrm{\mathrm{{exp}} }\;At$$ ) has real entries, the matrices $$J$$ and $$S$$ may have complex entries. However, if $$A$$ is real, then any complex eigenvalues and generalized eigenvectors occur in conjugate pairs. It follows that the entries of $$\mathrm{\mathrm{{exp}} }\;At$$ are linear combinations of terms of the form $$t^ke^{\mu t}\cos \nu t$$ and $$t^ke^{\mu t}\sin \nu t$$ , where $$\lambda =\mu \pm i \nu $$ is an eigenvalue of $$A$$ and $$k=0,1,\ldots ,p$$ , with $$p+1$$ being the size of the largest Jordan block for $$\lambda $$ .

2.5 Asymptotic Behavior of Linear Systems

Definition 2.4.

Let $$A$$ be an $$n\times n$$ matrix over $${\mathbb R}$$ . Define the complex stable, unstable, and center subspaces of $$A$$ , denoted $$E_s^{\mathbb C}$$ , $$E_u^{\mathbb C}$$ , and $$E_c^{\mathbb C}$$ , respectively, to be the linear span over $${\mathbb C}$$ of the generalized eigenvectors of $$A$$ corresponding to eigenvalues with negative, positive, and zero real parts, respectively.

Arrange the eigenvalues of $$A$$ so that $$\mathrm{Re }\;\,\lambda _1\le \ldots \le \mathrm{Re }\;\,\lambda _n$$ . Partition the set $$\{1,\ldots ,n\}=I_s\cup I_c \cup I_u$$ so that

Let $$\{v_j\}_{j=1}^n$$ be a basis of generalized eigenvectors corresponding to the eigenvalues $$\lambda _1,\ldots ,\lambda _n$$ . Then

In other words, we have

It follows that $${\mathbb C}^n=E_s^{\mathbb C}\oplus E_c^{\mathbb C}\oplus E_u^{\mathbb C}$$ is a direct sum. Thus, any vector $$x\in {\mathbb C}^n$$ is uniquely represented as

These subspaces are invariant under $$A$$ .

The maps $$P_s,P_c,P_u$$ are linear projections onto the complex stable, center, and unstable subspaces. Thus, we have

Since these subspaces are independent of each other, we have that

Since these subspaces are invariant under $$A$$ , the projections commute with $$A$$ , and thus, also with any function of $$A$$ , including $$\mathrm{\mathrm{{exp}} }\;At$$ .

If $$A$$ is real and $$v\in {\mathbb C}^n$$ is a generalized eigenvector with eigenvalue $$\lambda \in {\mathbb C}$$ , then its complex conjugate $$\bar{v}$$ is a generalized eigenvector with eigenvalue $$\bar{\lambda }$$ .