Attractors

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2 total messages Started by Sprott Fri, 16 Oct 1992 14:47

#102

Author: Sprott
Date: Fri, 16 Oct 1992 14:47

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Does anyone know of a mapping or system of ordinary differential equations
involving only polynomials whose solution attracts to:
     a) a 2-torus
     b) a 3-torus
     c) a Moebius strip?

Re: Attractors

#103

Author: Michael Cohen
Date: Sat, 17 Oct 1992 04:15

38 lines
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I wrote down a construction with just polynomials that is easily
capable of attracting to two and three Torus's.  I haven't figured
out how to attract to Mobius strips as yet.  A much more general
construction which does this and more is
written in my January 1992 paper in Neural Networks.

Briefly Let H = x^2 + y^2 
            G = z^2 + w^2

Let A(H) =( H -1)^2
    B(G) =( G -3)^2


Let b and c be positiveconstants whose ratio are irrational.  Let 

V(x,y,z,w) = [A'(H)H_x, A'(H)H_y, B'(G)G_z, B'(G)H_w] let
X = [x,y,z,w]  then a system which does the job (generates two torus's
as attractors using polynomials only is the following).  Three Torus's
are quite similar.

dX^t/dt =- QV^t + [bH_y, -bH_x, cG_w, -cG_z]^t where ^t denotes
transpose and Q is a Riemannian metric, a polynmomial matrix which is
globally positive definite.
                -mike









-- 
Boston University (617-353-7857) Email: mike@park.bu.edu
Smail: Michael Cohen                     111 Cummington Street, RM 242
       Center for Adaptive Systems        Boston, Mass 02215
       Boston University

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