scavo@cie.uoregon.edu (Tom Scavo) (03/26/91)
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
scavo@cie.uoregon.edu (Tom Scavo) (03/27/91)
The following message attempts to make some of yesterday's statements more precise. In a previous article, 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. Rearranging terms, we see that x(t) = c - a x(t-1) - b x(t-1)^2 which amounts to iterating the quadratic P(x) = C - Ax - Bx^2 in which I've capitalized the coefficients only for convenience. This three-parameter quadratic is dynamically equivalent to the simpler Q(x) = x^2 + c as noted in the original post. Here's why... ------------------------------------------------------------------------ Suppose we have a pair of arbitrary quadratics given by p(x) = a'x^2 + b'x + c' , and q(x) = ax^2 + bx + c . It can be shown that the following diagram commutes q R ------> R ^ ^ L | | L | | R ------> R p where L is a homeomorphism. In particular, suppose L is linear. Composing functions and equating coefficients [Exercise] we find that L(x) = a'/a x + (b' - b)/(2a) . (1) Moreover, one finds that (b' - 1)^2 - 4a'c' = (b - 1)^2 - 4ac (2) which provides a useful relationship between the two quadratics. Now set p = P and q = Q , and note that in our case a = 1 , b = 0 , c = c , a' = -B , b' = -A , and c' = C . Substituting into (1) and (2) we obtain L(x) = -(Bx + A/2) (3) with (A + 1)^2 + 4BC = 1 - 4c . Solving for c , we get c = ( 1 - (A + 1)^2 - 4BC ) / 4 . (4) In passing, also note that L^(-1)(x) = -(2x+A)/(2B) from (3). ------------------------------------------------------------------------ Here are some "facts" about Q , all of which are shown to be true in the reference given yesterday (see ch.4 in particular): (i) For c > 1/4 , Q has no fixed points, and in fact Q^n escapes to infinity for all x . (ii) For c = 1/4 , Q has a single, nonhyperbolic fixed point, namely x0 = 1/2 , which is weakly attracting for x in (-1/2, 1/2). (iii) For -3/4 < c < 1/4 , Q has a pair of fixed points x1 and x2 given by 1 + sqrt(1 - 4c) 1 - sqrt(1 - 4c) x1 = ---------------- and x2 = ---------------- 2 2 with x1 repelling and x2 attracting; the basin of attraction for x2 is (-x1, x1). (iv) For -5/4 < c < -3/4 , there exists an attracting period two orbit (both fixed points are now repelling) given by the pair of values -1 +|- sqrt(1 - 4(c+1)) ----------------------- 2 (v) For c = -2 , the system displays fully developed chaos: there are 2^n periodic points of period n , all of which are repelling. And here's how these results translate back into P : (i) When c > 1/4 , (A+1)^2 + 4BC < 0 by (4), and in this case all orbits under iteration of P escape. (ii) When c = 1/4 , it must be true that (A+1)^2 + 4BC = 0 . For example, A = 3 , B = 2 , and C = -2 satisfy this constraint. The corresponding fixed point is L^(-1)(1/2) = -(A+1)/(2B) = -1 which is easily verified to be fixed by P . (iii) When -3/4 < c < 1/4 , 0 < (A+1)^2 + 4BC < 4 , and the fixed points are given by L^(-1)(x1) and L^(-1)(x2) . (iv) Similarly, assuming 4 < (A+1)^2 + 4BC < 6 , P has an attracting period two orbit which is easily computed via L^(-1) . (v) When c = -2 , (A+1)^2 + 4BC = 9 . For example, A = 4 , B = -2 , and c = 2 give a P which is chaotic on [0,2] . In other words, anything true of Q (in the dynamical sense, that is) is also true of some subfamily of P , and vice versa. So, we may as well study the the "simplest" quadratic we can get our hands on, namely Q . Tom Scavo scavo@cie.uoregon.edu