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## 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

 (320)

Here , , and are continuous.

1. How often does a solution oscillate in the interval ; 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 and ?

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 . This is done in three steps.

A) First apply the Prüfer substitution

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

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

The transformation

is non-singular for all . Furthermore, we always have for any non-trivial solutions. Why? Because if , i.e., and for some particular , then by the uniqueness theorem for second order linear o.d.e. , 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

(Side Comment: If , then we differentiate instead. This yields the same result.)
One obtains

or
 (321)

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

(ii) Differentiate the relation

and obtain

or
 (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 is obviously much more important: it determines the qualitative, e.g. oscillatory, behavior of . 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 . The amplitude has no influence whatsoever on the phase function . 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 (and hence of ) without having to exhibit detailed analytic or computer generated solutions.

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

a unique solution which satisfies

and

provided and are continuous at . See Figure 3.8.
Existence and uniqueness of prevails even if and have finite jump discontinuities at .

(2) Once is known, the Prüfer amplitude function is determined by integrating Eq.(3.22). One obtains

where is the initial amplitude.

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

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

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 order o.d.e., this means the Prüfer phase , which obeys the first order d.e.

 (323)

It is obtained from the second order equation
 0 (324)

by the Prüfer substitution

These equations make it clear that the zeroes and the oscillatory behavior of are controlled by the phase function .

Next: Qualitative Results Up: Phase Analysis of a Previous: Phase Analysis of a   Contents   Index
Ulrich Gerlach 2010-12-09