1.2 Definitions and notations

For any $x_{0} \in R^{n}$ and $t_{0} \in R$, the system

 

x' = f(t,x) (1.3)

together with the initial condition

 

x(t₀) = x₀ (1.4)

is defined to be an initial value problem (IVP).

A function y(t) is said to be a solution of the IVP (1.3)-(1.4) if y' (t) = f(t,y(t)) and y(t₀) = x₀. Of course, for an IVP, we are concerned with the problem of existence of a unique solution. This is guaranteed if the function f(t,x) is ``well-behaved'' or ``smooth enough''. You may have seen the requirements on f in the scalar case and we shall return to this point later.

If all the f_i (t,x) in (1.3) are independent of t, then the system (1.3) is said to be autonomous and in this case it is often written as

 

x' = f(x) (1.5)

Generally, system (1.5) is easier to deal with than that of (1.3). Of course, strictly speaking, all the systems should be nonautonomous since in the real world nothing is permanent except change. But in many situations, autonomous systems do give very good approximations and therefore we are often content with them. In this course, we shall devote our attention mainly on system (1.5) while the general system (1.3) will be treated in more advanced courses (e.g. AM 751). For autonomous system (1.5), the value of t₀ in the initial condition (1.4) is insignificant and usually set to be 0.

Higher order equations of the form

$$y^{(n)} = g(y, y', \ldots , y^{(n-1)})$$ (1.6)

can be reduced to a 1$^{\rm st}$-order system of the form

$$\left\{ \begin{array}{l} x_{1}' = x_{2}, \\ x_{2}' = x_{3}, ... ...ots \\ x_{n}' = g(x_{1}, \ldots , x_{n}), \end{array} \right.$$ (1.7)

by defining new variables

$$x_{1} = y, \, x_{2} = y', \ldots , x_{n} = y^{(n-1)} \, .$$

Example 1.2.1: Mass-spring system

Let y₁ be the displacement of mass m₁ from its equilibrium position, and let y₂ be the displacement of mass m₂ from its equilibrium position.

Assume Hooke's law for forces exerted by springs and assume there is no friction. Application of Newton's law gives

$$\left\{ \begin{array}{l} m_{1} y_{1}'' = - k_{1} y_{1} + k_{2... ... m_{2} y_{2}'' = - k_{2} (y_{2} - y_{1}). \end{array} \right.$$ (1.8)

Letting $x_{1} = y_{1}, \, x_{2} = y_{1}', \, x_{3} = y_{2}, \, x_{4} = y_{2}'$, then (1.8) reduces to

$$\left\{ \begin{array}{l} x_{1}' = x_{2}, \\ \\ x_{2}' = - {... ...m_{2}} x_{1} - \frac{k_{2}}{m_{2}}} x_{3}. \end{array} \right.$$ (1.9)

For a vector $x \in R^{n}$, we define the Euclidean ``norm'' $\parallel x \parallel$ by

$$\parallel x \parallel = \left( \sum_{i=1}^{n} x_{i}^{2} \right)^{\frac{1}{2}}.$$ (1.10)

The distance between any two points $x, y \in R^{n}$ is defined by

$$\parallel x - y \parallel = \left[ \sum_{i=1}^{n} (x_{i} - y_{i})^{2} \right]^{\frac{1}{2}} = d (x,y).$$ (1.11)

It is evident that (1.10) defines the distance between x and 0. Of course, there are other norms which could be used, e.g.

$$\parallel x \parallel = \sum_{i=1}^{n} \mid x_{i} \mid \, , \... ...llel = \max_{1 \leq i \leq n} \mid x _{i} \mid \, , {\rm etc.}$$

In general, we define a norm on Rⁿ to be any function $N : R^{n} \raro R$ satisfying the following conditions:

1.

$N(x) \geq 0$ and N(x) = 0 iff x =0;

2.

$N (x+y) \leq N(x) + N(y)$;

3.

$N(\alpha x) = \mid \alpha \mid \, N (x)$ $\alpha \in \Real$.

It should be noted that in finite dimensional spaces such as Rⁿ, all norms are equivalent. This is stated in the following Lemma.

Lemma 1.2.1     Let $N : R^{n} \raro R$ be any norm. Then there exist constants A > 0, B >0 such that

$$A \parallel x \parallel \leq N (x) \leq B \parallel x \parallel$$ (1.12)

for all x, where $\parallel x \parallel$ is the Euclidean norm.

Proof: (Hirsch & Smale, P. 78-9)

We say x converges to y, denoted as $x \raro y$ if $\parallel x -y \parallel \raro 0$. Clearly $x \raro y$ iff $x_{i} \raro y_{i}$, $\forall i$.