Defining Positive ZeroDate: 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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