Continuous Dynamical Systems

9.4. Continuous Dynamical Systems#

In this section, we will deal with similar problems as in Section 9.1. There, we were concerned with discrete time. That is, we assumed a certain initial state \(\vect{x}_{0}\) and then predicted the next state \(\vect{x}_{1}\). That approach yields models that are often very useful. But just as often we want to deal with continuous time. That is, there is no next state but rather a state for every positive real number. In order to deal with this new context, we need some preliminaries from calculus.

Proposition 9.4.1

We will denote the derivative of a function \(f\) by \(f'\). Let \(f\) and \(g\) be differentiable functions on \(\R\) and let \(c\) be a real number. Then:

\((f+g)'=f'+g'\),
\((cf)'=cf'\),
if \(f(t)=e^{\lambda t}\), then \(f'(t)=\lambda e^{\lambda t}\).

This last point means that the equation \(x'=\lambda x\) apparently has solution \(y(t)=e^{\lambda t}\). We can then conclude that in fact \(y(t)=ce^{\lambda t}\) is a solution for every real number \(c\). We want to generalise this idea. But first, we need some terminology.

Suppose we have differentiable functions \(x_{1},...,x_{n}\) and real numbers \(a_{ij}\) for \(1\leq i,j\leq n\). A system of equations

\[\begin{split}\begin{cases} x_{1}'&=a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}\\ x_{2}'&=a_{21}x_{2}+a_{22}x_{2}+\cdots +a_{2n}x_{n}\\ &\,\vdots\\ x_{n}'&=a_{n1}x_{1}+a_{n2}x_{2}+\cdots+a_{nn}x_{n} \end{cases}\end{split}\]

can be conveniently rewritten as \(\vect{x}'=A\vect{x}\) where

\[\begin{split} \vect{x}= \begin{bmatrix} x_{1}\\ x_{2}\\ \vdots\\ x_{n} \end{bmatrix}, \quad \vect{x}'= \begin{bmatrix} x_{1}'\\ x_{2}'\\ \vdots\\ x_{n}' \end{bmatrix} \quad\text{and}\quad A= \begin{bmatrix} a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\cdots&a_{nn} \end{bmatrix}. \end{split}\]

Definition 9.4.1

In this context, we call \(\vect{x}'=A\vect{x}\) a system of (linear) differential equations, \(\vect{x}\) a vector-valued function, \(\vect{x}'\) the derivative of \(\vect{x}\), and the \(x_{i}\)’s the component functions of \(\vect{x}\). Any \(\vect{y}\) for which \(\vect{y}'=A\vect{y}\) holds is called a solution to the system of differential equations.

The following proposition will be quite useful to us. It tells us that, in order to find the full (infinite) solution set, it suffices to find a (finite) basis of solutions.

Proposition 9.4.2

If \(\vect{y}\) and \(\vect{z}\) are solutions of \(\vect{x}'=A\vect{x}\) and \(c\) and \(d\) are arbitrary real numbers, then \(c\vect{y}+d\vect{z}\) is also a solution of \(\vect{x}=A\vect{x}'\).

Proof. Exercise.

But how do we find solutions? Let us first consider an example.

Example 9.4.1

Consider the following system of differential equations:

\[\begin{split} \vect{x}'=A\vect{x}\quad\text{where}\quad A= \begin{bmatrix} 1&4\\ 1&1 \end{bmatrix}. \end{split}\]

Consider the vector-valued function

\[\begin{split} \vect{y}(t)=\begin{bmatrix} 2e^{3t}\\ 1e^{3t} \end{bmatrix}=\vect{v}e^{3t},\end{split}\]

then

\[\begin{split}\quad \vect{y}'(t)= \begin{bmatrix} 6e^{3t}\\ 3e^{3t} \end{bmatrix}= \begin{bmatrix} 1&4\\ 1&1 \end{bmatrix}\begin{bmatrix} 2e^{3t}\\ 1e^{3t} \end{bmatrix} =A\vect{y}(t), \end{split}\]

so \(\vect{y}\) is a solution.

Why does the particular choice of \(\vect{v}\) and \(e^{3t}\) in example Example 9.4.1 yield a solution? Well, because \(3\) is an eigenvalue of \(A\) with associated eigenvector \(\vect{v}\). For let us assume that \(\lambda\) is an eigenvalue of \(A\) with associated eigenvector \(\vect{v}\) and put \(\vect{y}=\vect{v}e^{\lambda t}\). Then

\[ \vect{y}'=(\vect{v}e^{\lambda t})'=\vect{v}\lambda e^{\lambda t}=A\vect{v} e^{\lambda t}=A\vect{y}, \]

showing that \(\vect{y}\) is indeed a solution. This observation leads us to the following statement.

Proposition 9.4.3

If \(\lambda\) is an eigenvalue of \(A\) with associated eigenvector \(\vect{v}\), then \(\vect{y}=\vect{v}e^{\lambda t}\) is a solution of the system of linear differential equations \(\vect{x}'=A\vect{x}\).

We now know how to solve systems of linear differential equations. But we know more. We also know how a solution \(f(t)\) to such a system will behave as \(t\) goes to infinity. In practical applications, \(t\) usually is time, so this gives us predictions for what happens after a long time.

Example 9.4.2

Suppose some airborn disease is affecting a population. To keep matters simple, we will assume that the population is constant and that recovery grants full immunity. Let \(S(t)\) be the number of susceptible members and \(I(t)\) the number of infected members of the population at time \(t\). If \(\alpha>0\) is the recovery rate and \(\beta>0\) is the infection rate, then we find:

\[\begin{split} \begin{array} S'(t)&=&-\beta S(t)&\\ I'(t)&=&\beta S(t)&-\alpha I(t) \end{array} \end{split}\]

Define

\[\begin{split} \vect{y}=\begin{bmatrix} S(t)\\ I(t) \end{bmatrix} \quad\text{and}\quad A=\begin{bmatrix} -\beta&0\\ \beta &-\alpha \end{bmatrix}. \end{split}\]

Since \(A\) is an upper diagonal matrix, we can conclude that its eigenvalues are \(-\beta\) and \(-\alpha\). Therefore, a solution to the system of linear differential equations \(\vect{y}'=A\vect{y}\) is given by

\[ \vect{y}=c_{1}\vect{v}_{-\beta}e^{-\beta t}+c_{2}\vect{v}_{-\alpha}e^{-\alpha t} \]

where \(c_{1}\) and \(c_{2}\) are some constants while \(\vect{v}_{-\beta}\) and \(\vect{v}_{-\alpha}\) are the eigenvectors of \(A\) corresponding \(-\beta \) and \(-\alpha\), respectively. In particular, if \(t\) gets very large, we find very large but negative exponents on the right hand side. That is, both \(\lim_{t\to\infty}S(t)\) and \(\lim_{t\to\infty} I(t)\) are \(0\). This makes perfect sense intuitively, as we expect all members of the population to get infected and recover. After that, they are neither susceptible nor infected anymore.