Contents

1. Overview

Up to this point, we have dealt only with ODEs that have a scalar dependent variable. These are important to understand thoroughly, but in applications it is rare to have just one unknown. In this chapter we take our first steps to vector-valued ODEs in the form

\[\frac{d \bfx}{dt} = \bff(t,\bfx), \]

where \(\bfx\) is \(n\)-dimensional. An equivalent point of view and terminology is a coupled system of scalar ODEs,

\[\begin{split}\begin{split} \frac{dx_1}{dt} &= f_1(t,x_1,\ldots,x_n),\\ & \vdots \\ \frac{dx_n}{dt} &= f_n(t,x_1,\ldots,x_n). \end{split}\end{split}\]

Example

A famous system of three ODEs is the Lorenz system,

\[\begin{split}\begin{split} \dot{x} & = \sigma(y-x), \\ \dot{y} & = \rho x - y - x z, \\ \dot{z} & = -\beta z + x y, \end{split}\end{split}\]

where the dots indicate time derivatives and \(\sigma\), \(\rho\), and \(\beta\) are constant parameters. Conversion to the system notation follows from the definitions \(x_1=x\), \(x_2=y\), and \(x_3=z\), though of course the ordering is arbitrary.

We will be limited for now to linear problems,

\[\frac{d \bfx}{dt} = \bfA(t) \bfx + \bff(t),\]

and open the discussion on nonlinear problems in the next chapter.

Example

Consider two connected tanks holding brine. Tank 1 holds 100 L, has an input of 4 L/hr of brine of concentration 3 kg/L, and an output of 6 L/hr. Tank 2 holds 200 L, has an input of 7 L/hr of brine of concentration 5 kg/L, and an output of 5 L/hr. Tank 1 pumps 1 L/hr into tank 2, and tank 2 pumps 3 L/hr into tank 1.

Let \(x_i(t)\) be the mass of salt in tank \(i\). The statements above imply

\[\begin{align*} \dd{x_1}{t} & = 4\cdot 3 - 6\cdot \frac{x_1}{100} - 1\cdot \frac{x_1}{100} + 3 \cdot \frac{x_2}{200} \\ \dd{x_2}{t} & = 7\cdot 5 - 5\cdot \frac{x_2}{200} - 3\cdot \frac{x_2}{200} + 1\cdot \frac{x_1}{100}\\ \end{align*}\]

This is neatly expressed using linear algebra:

\[\dd{}{t} \twovec{x_1}{x_2} = \frac{1}{200}\twomat{-14}{3}{2}{-8} \twovec{x_1}{x_2} + \twovec{12}{35}.\]

1.1. Important terms

asymptotically stable

Equilibrium to which small enough perturbations always return as \(t\to\infty\).

center

Equilibrium for which the coefficient matrix has pure imaginary eigenvalues.

coefficient matrix

Matrix multiplying the dependent variable in a linear ODE system.

constant-coefficient

Linear ODE system in which the coefficient matrix does not depend on the independent variable.

equilibrium solution

Constant (time-invariant) solution of an ODE.

forcing function

Nonhomogeneous term in a linear ODE system.

fundamental matrix

Square time-dependent matrix whose columns form a basis for the homogeneous solutions of a linear ODE.

initial condition

Prescribed value of the ODE solution at a particular time.

linear ODE system

ODE for a vector-valued function that is linear in the dependent variable.

node

Equilibrium for which the coefficient matrix has real eigenvalues of the same sign.

phase plane

Graphical depiction of the solutions of a 2D system as trajectories parameterized by time.

saddle

Equilibrium for which the coefficient matrix has real eigenvalues of opposite sign.

spiral

Equilibrium for which the coefficient matrix has complex conjugate eigenvalues with nonzero real parts.

stable

Equilibrium from which perturbations do not grow.

unstable

Equilibrium from which perturbations diverge.

4. Convolution 2. Conversion of higher-order ODEs