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From: scavo@cie.uoregon.edu (Tom Scavo)
Newsgroups: sci.math,comp.theory.dynamic-sys
Subject: Re: Non-linear difference equation
Summary: this is a quadratic equation
Keywords: discrete dynamical systems, iterative mappings, chaos
Message-ID: <1991Mar25.215710.10367@ariel.unm.edu>
Date: 25 Mar 91 21:57:10 GMT
References: <1991Mar25.183530.23916@athena.mit.edu>
Reply-To: scavo@cie.uoregon.edu (Tom Scavo)
Followup-To: sci.math
Organization: University of Oregon Campus Information Exchange
Lines: 32

In article <1991Mar25.183530.23916@athena.mit.edu> armann@athena.mit.edu (Armann Ingolfsson) writes:
>I'm trying to find out something about the behavior of the sequence {x(t)}
>determined by
>
>x(t) + a x(t-1) + b x(t-1)^2 = c, for t = 1,2,...
>
>with x(0) given.  Any pointers about under what conditions the sequence
>converges, what it converges to, and at what rate would be useful to me.

Unless I'm misinterpreting your notation, this is just a quadratic
equation.  Since quadratics are conjugate to one another, it suffices
to consider  Q(x) = x^2 + c  which in some sense is the "simplest"
of all quadratics.  Note the dependence on the *single* parameter  c .
For an elementary discussion of the dynamics of this map and its
complex analogue, see

	Devaney, R.L.  _Chaos, Fractals, and Dynamics_.  Addison-
	Wesley, Menlo Park, CA, 1990.

Another much studied quadratic is the logistic equation given by 
F(x) = r x (1-x) .  Everything you ever wanted to know about this
dynamical system will be found in

	Devaney, R.L.  _An Introduction to Chaotic Dynamical
	Systems_ (second edition).  Addison-Wesley, Redwood City,
	CA, 1989.

For certain parameter values, these simple equations have very
complicated dynamics including period-doubling and chaotic behavior.

Tom Scavo
scavo@cie.uoregon.edu