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### 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/
```
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