The Prüfer System

For linear second order ordinary differential equations, the phase plane method is achieved by the so-called Prüfer substitution. It yields the phase and the amplitude of the sought after solution to the Sturm-Liouville equation.

The method to be developed applies to any differential equation having the form

$$\frac{d}{dx} \left( P(x)\frac{du}{dx}\right) +Q(x)u = 0~~\qquad~~a<x<b$$ (320)

Here $0<P(x)$ , $P'(x)$ , and $Q(x)$ are continuous.

We are interested in asking and answering the following questions:

1. How often does a solution oscillate in the interval $a<x<b$ ; i.e., how many zeroes does it have?
2. How many maxima and minima does it have between a pair of consecutive zeroes?
3. What happens to these zeroes when one changes $P(x)$ and $Q(x)$ ?

The questions can be answered by considering for this equation its phase portrait in the Poincaré phase plane. We do this by introducing the ``phase'' and the ``radius'' of a solution $u(x)$ . This is done in three steps.

A) First apply the Prüfer substitution

$$\boxed{P(x)u'(x)=r(x)\cos\theta (x);~~\quad~~u(x)=r(x)\sin\theta (x)}$$

to the quantities in Eq.(3.20). Do this by introducing the new dependent variable $r$ and $\theta$ as defined by the formulae

$$r^2=u^2+P^2(u')^2\,;~~\qquad~~\theta =\arctan\frac{u}{Pu'}\,.$$

Figure 3.7: The Poincaré phase plane of the second order linear differential equation is spanned by the amplitude $u$ and its derivative $u'$ (multiplied by the positive coefficient $P$ ). A solution to the differential equation is represented by an $x$ -parametrized curve. The (Prüfer) phase is the polar angle $\theta$ .

(Without loss of generality one may always assume that $u(x)$ is real. Indeed, if $u(x)$ were a complex solution, then it would differ from a real one by a mere complex constant.) A solution $u(x)$ can thus be pictured in this Poincaré plane as a curve parametrized by the independent variable $x$ .

The transformation

$$(Pu',u)\leftrightarrow (r,\theta )$$

is non-singular for all $r\not= 0$ . Furthermore, we always have $r>0$ for any non-trivial solutions. Why? Because if $r(x)=0$ , i.e., $u(x)=0$ and $u'(x)=0$ for some particular $x$ , then by the uniqueness theorem for second order linear o.d.e. $u(x)=0$ $\forall\,x$ , i.e., we have the trivial solution.

B) Second, obtain a system of first order o.d.e. which is equivalent to the given differential Eq.(3.20).

(i) Differentiate the relation

$$\cot\theta = \frac{Pu'}{u}~~.$$

(Side Comment: If $u=0$ , then we differentiate $\tan\theta = u /Pu'$ instead. This yields the same result.)
One obtains

$$-\csc^2\theta \frac{d\theta}{dx} = \frac{(Pu')'}{u} - \frac{Pu'}{u^2} u'$$

$$= -Q - \frac{1}{P}\frac{\cos^2\theta}{\sin^2\theta}~~,$$

or

$$\boxed{ \frac{d\theta}{dx} = Q(x)\sin^2\theta +\frac{1}{P(x)}\cos^2\theta \equiv F(x,\theta )\,.}$$ (321)

This is Prüfer's differential equation for the phase, the Prüfer phase.

(ii) Differentiate the relation

$$r^2=u^2+(Pu')^2$$

and obtain

$$r\frac{dr}{dx} = uu'+(Pu')(Pu')'$$

$$= \frac{u}{P}Pu'-Pu' Qu$$

$$= \frac{r\sin\theta}{P} r\cos\theta -r\cos\theta Qr\sin\theta$$

or

$$\boxed{\frac{dr}{dx} = \frac{1}{2}\left[ \frac{1}{P(x)} - Q(x)\right] r \sin 2\theta\,.}$$ (322)

This is Prüfer's differential equation for the amplitude.

C) Third, solve the system of Prüfer equations (3.21) and (3.22). Doing so is equivalent to solving the originally given equation 3.20. Any solution to the Prüfer system determines a unique solution to the equation (3.20), and conversely.

Of the two Prüfer equations, the one for the phase $\theta (x)$ is obviously much more important: it determines the qualitative, e.g. oscillatory, behavior of $u(x)$ . The feature which makes the phase equation so singularly attractive is that it is a first order equation which also is independent of the amplitude $r(x)$ . The amplitude $r(x)$ has no influence whatsoever on the phase function $\theta (x)$ . Consequently, the phase function is governed by the simplest of all possible non-trivial differential equations: an ordinary first order equation. This simplicity implies that rather straight forward existence and uniqueness theorems can be brought to bear on this equation. They reveal the qualitative nature of $\theta (x)$ (and hence of $u(x)$ ) without having to exhibit detailed analytic or computer generated solutions.

(1) One such theorem says that for any initial value

$$(a,\gamma )$$

$\exists$ a unique solution $\theta (x)$ which satisfies

$$\frac{d\theta}{dx} = F(x,\theta )$$

and

$$\theta (a) = \gamma ~~,$$

provided $P$ and $Q$ are continuous at $a$ . See Figure 3.8.

Figure 3.8: The phase function $\theta (x)$ is that unique solution to the Prüfer equation $d\theta /dx~ = ~ F(\theta ,x)$ whose graph passes through the given point $(a,\gamma )$ .

Existence and uniqueness of $\theta (x)$ prevails even if $P(x)$ and $Q(x)$ have finite jump discontinuities at $x\ne a$ .

(2) Once $\theta (x)$ is known, the Prüfer amplitude function $r(x)$ is determined by integrating Eq.(3.22). One obtains

$$r(x)=K\exp \int^x_a \frac{1}{2}\left[\frac{1}{P(x)} -Q(x)\right]\sin2\theta (x)\,dx$$

where $K=r(a)$ is the initial amplitude.

(3) Each solution to the Prüfer system, Eqs.(3.21) and (3.22), depends on two constants:

$$\textrm{the~initial~amplitude} K = r(a)$$

$$\textrm{the~initial~phase} \gamma = \theta (a)$$

Note the following important fact: Changing the constant $K$ just multiplies the solution $u(x)$ by a constant factor. Thus the zeroes of $u(x)$ can be located by studying only the phase d.e.

$$\boxed{\frac{d\theta}{dx} = F(x,\theta )\,.}$$

This is a major reason why we shall now proceed to study this equation very intensively.

Vibrations, oscillations, wiggles, rotations and undulations are all characterized by a changing phase. If the independent variable is the time, then this time, the measure of that aspect of change which permits an enumeration of states, manifests itself physically by the advance of the phase of an oscillating system.

Lecture 23

Summary. The phase of a system is the most direct way of characterizing its oscillatory nature. For a linear $2^{\textrm{nd}}$ order o.d.e., this means the Prüfer phase $\theta (x)$ , which obeys the first order d.e.

$$\frac{d \theta}{dx} = Q(x)\sin^2\theta +\frac{1}{P(x)} \cos^2\theta \equiv F(x,\theta )~~.$$ (323)

It is obtained from the second order equation

$$\left[\frac{d}{dx} P(x) \frac{d}{dx} +Q(x)\right] u(x) =$$ 0 (324)

by the Prüfer substitution

$$u(x)=r(x)\sin\theta (x)~\qquad~ Pu'(x) = r(x)\cos\theta (x)\,.$$

These equations make it clear that the zeroes and the oscillatory behavior of $u(x)$ are controlled by the phase function $\theta (x)$ .