Universal behaviour in dynamical systems
In the late 1970's, P. Coullet and C. Tresser [a6] and M. Feigenbaum
independently found striking, unexpected features of the transition from simple to chaotic dynamics in one-dimensional dynamical systems (cf. also Routes to chaos). By the example of the family of quadratic mappings acting (for ) on the interval , the period-doubling scenario is recalled here. For , has periodic points of every (least) period. Let be the infimum of parameter values for which has a periodic orbit of least period . Then
For , the dynamics of is described by statements i)–iii) below.
i) has precisely one periodic orbit of (least) period for each , and no other periodic orbits;
ii) any pair of adjacent points in is separated by a unique point in ;
iii) with the exception of the (countably many) orbits which land on some , , and stay there, every -orbit tends asymptotically to .
For (when is sometimes called the Feigenbaum mapping), statement i) holds, but with ranging over all non-negative integers, and ii) holds for each ; furthermore, the following analogue of iii) holds:
iv) (for ) the closure of the orbit of the turning point is a Cantor set , which is the asymptotic limit of every orbit not landing on one of the periodic orbits , . The restricted mapping is a minimal homeomorphism (the "adding machine for chaos in a dynamical systemadding machine" ).
Finally, is the threshold of "chaos" , in the following sense:
v) for , has infinitely many distinct periodic orbits, and positive topological entropy.
Many features of this "topological" , or combinatorial picture were understood by early researchers in this area, specifically P.J. Myrberg [a12] and N. Metropolis, M.L. Stein and P.R. Stein [a13]. They recognized as well that the combinatorial structure of the periodic orbits is rigidly determined by the fact that is unimodal (cf. [a14]). In essence, the statements above can be formulated for any family of unimodal mappings (cf. ). In fact, the (weak) monotonicity of the 's, together with the fact that if , then must have periodic orbits of least period for (some ) and no others, follows for any family of continuous mappings on the line from Sharkovskii's theorem [a16], [a2]; recent work has yielded a more general understanding of the combinatorial structure of periodic orbits for continuous mappings in dimension (cf. [a1]).
Coullet, Tresser and Feigenbaum added to the topological picture described above a number of analytic and geometric features:
vi) the convergence is asymptotically geometric:
vii) the periodic orbits scale: let denote the orbit for ; then
These statements, formulated for the particular family of quadratic mappings, are technically interesting, but not so striking. However, they observed that v)–vii) hold for a very broad class of unimodal one-parameter families, subject only to trivial "fullness" conditions (essentially that has only finitely many periodic orbits while has positive entropy) and smoothness (essentially that is and each has a non-degenerate critical point). And, sensationally, the constants and are independent of the family .
In [a6] and
these assertions were reduced, using ideas from renormalization theory, to certain technical conjectures concerning a doubling operator acting on an appropriate function space. O. Lanford
(cf. also [a3], [a5]) gave a rigorous, computer-assisted proof of the basic conjecture, that has a saddle-type fixed point with one characteristic multiplier (the same as in vi)) and stable manifold of codimension . D. Sullivan [a17] showed the uniqueness of this fixed point in the space of "quadratic-like" mappings. The final conjecture, concerning transversality of the stable manifold with certain bifurcation submanifolds, remains unproved. Recently, Sullivan , introducing a number of new ideas, has circumvented this difficulty and provided a rather complete theory of universal features for families of unimodal mappings. In particular, the asymptotic geometry of the Cantor set (for ) and of analogous sets appearing at other "threshold" parameter values (the "infinitely renormalizable mappings of bounded type" ) is universal; for example, the set always has Hausdorff dimension . Full expositions of this theory are provided in [a18] and [a7].
These ideas have been applied as well to circle diffeomorphisms [a10],
and area-preserving planar diffeomorphisms [a4], .
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|[a12]||P.J. Myrberg, "Sur l'iteration des polynomes réels quadratiques" J. Math. Pures Appl. , 41 (1962) pp. 339–351 MR0161968 Zbl 0106.04703|
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|[a14]||W. Thurston, "On iterated maps of the interval" J.C. Alexander (ed.) , Dynamical Systems (Proc. Maryland, 1986–7) , Lect. notes in math. , 1342 , Springer (1988) pp. 465–563 MR0970571 Zbl 0664.58015|
|[a15a]||D. Rand, "Universality and renormalization in dynamical systems" T. Bedford (ed.) J. W. Swift (ed.) , New directions in dynamical systems , Cambridge Univ. Press (1987) pp. 1–56|
|[a15b]||D. Rand, "Global phase space universality, smooth conjugacies and renormalisation: the case." Nonlinearity , 1 (1988) pp. 181–202 MR928952|
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|[a17]||D. Sullivan, "Quasiconformal homeomorphisms in dynamics, topology and geometry" A.M. Gleason (ed.) , Proc. Internat. Congress Mathematicians (Berkeley, 1986) , Amer. Math. Soc. (1987) pp. 1216–1228 MR0934326 Zbl 0698.58030|
|[a18]||D. Sullivan, "Bounds, quadratic differentials, and renormalization conjectures" , Centennial Publ. , 2 , Amer. Math. Soc. (1991) MR1184622 Zbl 0936.37016|
Universal behaviour in dynamical systems. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Universal_behaviour_in_dynamical_systems&oldid=24587