Defining Positive Zero

Date: 03/07/2003 at 22:32:33
From: Kenneth Wong
Subject: Please explain the term positive zero

This term appears in the following statement: the definite integral 
of f(x) from 0 to x has exactly one positive zero at x = a.

The statement actually appears as one of the requirements for 
Lienard's theorem (nonlinear dynamics, limit cycle) to be true. The 
whole statement says: "The odd function F(x) = int(0,a) f(u)du had 
exactly one positive zero at x=a, is negative for 0<x<a, is positive 
for x>a and F(x) approaches infinity as x approaches infinity. 

This is taken from the book _Nonlinear Dynamics and Chaos_ by Steve 
Strogatz, p. 211.

Thank you very much.

Date: 03/08/2003 at 01:09:24
From: Doctor Douglas
Subject: Re: Please explain the term positive zero

Hi, Kenneth,

Thanks for submitting your question to the Math Forum.

I think it means that the integral is equal to zero only when the 
upper limit x is equal to a (where "positive" refers to the fact that 
a>0):

   Int{0,a} f(x) dx = 0

   Int(0,b} f(x) dx  is nonzero if b <> a.

I checked Strogatz's book and yes, it means that

  F(x=a) = 0 for a unique value of a, where a>0.

The word "positive" might be the confusing part - it just means that 
we're temporarily considering only positive values of x.

Roughly speaking, the graph of F must look something like this:

        F                      
   __   |              ___________/
  /  \_ |          ___/
       \|         /
 -------+--------+---
        |\_    _/|
        0  \__/  a

The function F crosses the x-axis exactly three times - once at the 
origin, once for x>0 (at x=a), and once at x=-a. F has three zeroes, 
but only one of them is positive.

More generally, we say that a function G has a zero at x=c if  
G(x=c) = 0. In nonlinear dynamics language, a fixed point is where
the vector field has a zero.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/