View source for Universal behaviour in dynamical systems

In the late 1970's, P. Coullet and C. Tresser [[#References|[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|Routes to chaos]]). By the example of the family of quadratic mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u0956501.png" /> acting (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u0956502.png" />) on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u0956503.png" />, the period-doubling scenario is recalled here. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u0956504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u0956505.png" /> has periodic points of every (least) period. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u0956506.png" /> be the infimum of parameter values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u0956507.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u0956508.png" /> has a periodic orbit of least period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u0956509.png" />. Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565010.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565011.png" /></td> </tr></table>

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565012.png" />, the dynamics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565013.png" /> is described by statements i)–iii) below.

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565014.png" /> has precisely one periodic orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565015.png" /> of (least) period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565016.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565017.png" />, and no other periodic orbits;

ii) any pair of adjacent points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565018.png" /> is separated by a unique point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565019.png" />;

iii) with the exception of the (countably many) orbits which land on some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565021.png" />, and stay there, every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565022.png" />-orbit tends asymptotically to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565023.png" />.

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565024.png" /> (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565025.png" /> is sometimes called the Feigenbaum mapping), statement i) holds, but with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565026.png" /> ranging over all non-negative integers, and ii) holds for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565027.png" />; furthermore, the following analogue of iii) holds:

iv) (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565028.png" />) the closure of the orbit of the turning point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565029.png" /> is a Cantor set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565030.png" />, which is the asymptotic limit of every orbit not landing on one of the periodic orbits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565032.png" />. The restricted mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565033.png" /> is a minimal homeomorphism (the  "adding machine for chaos in a dynamical systemadding machine" ).

Finally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565034.png" /> is the threshold of  "chaos" , in the following sense:

v) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565036.png" /> 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 [[#References|[a12]]] and N. Metropolis, M.L. Stein and P.R. Stein [[#References|[a13]]]. They recognized as well that the combinatorial structure of the periodic orbits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565037.png" /> is rigidly determined by the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565038.png" /> is unimodal (cf. [[#References|[a14]]]). In essence, the statements above can be formulated for any family of unimodal mappings (cf. ). In fact, the (weak) monotonicity of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565039.png" />'s, together with the fact that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565040.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565041.png" /> must have periodic orbits of least period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565042.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565043.png" /> (some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565044.png" />) and no others, follows for any family of continuous mappings on the line from Sharkovskii's theorem [[#References|[a16]]], [[#References|[a2]]]; recent work has yielded a more general understanding of the combinatorial structure of periodic orbits for continuous mappings in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565045.png" /> (cf. [[#References|[a1]]]).

Coullet, Tresser and Feigenbaum added to the topological picture described above a number of analytic and geometric features:

vi) the convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565046.png" /> is asymptotically geometric:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565047.png" /></td> </tr></table>

vii) the periodic orbits scale: let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565048.png" /> denote the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565049.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565050.png" />; then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565051.png" /></td> </tr></table>

These statements, formulated for the particular family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565052.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565053.png" /> has only finitely many periodic orbits while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565054.png" /> has positive entropy) and smoothness (essentially that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565055.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565056.png" /> and each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565057.png" /> has a non-degenerate critical point). And, sensationally, the constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565059.png" /> are independent of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565060.png" />.

In [[#References|[a6]]] and

these assertions were reduced, using ideas from renormalization theory, to certain technical conjectures concerning a doubling operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565061.png" /> acting on an appropriate function space. O. Lanford

(cf. also [[#References|[a3]]], [[#References|[a5]]]) gave a rigorous, computer-assisted proof of the basic conjecture, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565062.png" /> has a saddle-type fixed point with one characteristic multiplier <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565063.png" /> (the same as in vi)) and stable manifold of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565064.png" />. D. Sullivan [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565065.png" /> unimodal mappings. In particular, the asymptotic geometry of the Cantor set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565066.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565067.png" />) and of analogous sets appearing at other  "threshold"  parameter values (the  "infinitely renormalizable mappings of bounded type" ) is universal; for example, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565068.png" /> always has [[Hausdorff dimension|Hausdorff dimension]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565069.png" />. Full expositions of this theory are provided in [[#References|[a18]]] and [[#References|[a7]]].

These ideas have been applied as well to circle diffeomorphisms [[#References|[a10]]],

and area-preserving planar diffeomorphisms [[#References|[a4]]], .

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Ll. Alsedà,   J. Llibre,   M. Misiurewicz,   "Combinatorial dynamics and entropy in one dimension"  (to appear)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Block,   J. Guckenheimer,   M. Misiurewicz,   L.-S. Young,   "Periodic points and topological entropy of one dimensional maps"  Z. Nitecki (ed.)  C. Robinson (ed.) , ''Global theory of dynamical systems (Proc. Northwestern Univ., 1979)'' , ''Lect. notes in math.'' , '''819''' , Springer  (1980)  pp. 18–34</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Campanino,   H. Epstein,   D. Ruelle,   "On the existence of Feigenbaum's fixed point"  ''Comm. Math. Phys.'' , '''79'''  (1981)  pp. 261–302</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Collet,   J.-P. Eckmann,   H. Koch,   "On universality for area-preserving maps of the plane"  ''Physica'' , '''3D'''  (1981)  pp. 457–467</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P. Collet,   J.-P. Eckmann,   O. Lanford,   "Universal properties of maps on an interval"  ''Comm. Math. Phys.'' , '''76'''  (1980)  pp. 211–254</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P. Coullet,   C. Tresser,   "Itérations d'endomorphismes et groupe de rénormalisation"  ''J. Phys.'' , '''C5'''  (1978)  pp. 25–28</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  W. de Mello,   S. van Strien,   "One-dimensional dynamics"  (to appear)</TD></TR><TR><TD valign="top">[a8a]</TD> <TD valign="top">  M. Feigenbaum,   "Quantitative universality for a class of non-linear transformations"  ''J. Stat. Phys.'' , '''19'''  (1978)  pp. 25–52</TD></TR><TR><TD valign="top">[a8b]</TD> <TD valign="top">  M. Feigenbaum,   "The universal metric properties of a non-linear transformation"  ''J. Stat. Phys.'' , '''21'''  (1979)  pp. 669–706</TD></TR><TR><TD valign="top">[a9a]</TD> <TD valign="top">  L. Jonker,   D. Rand,   "Bifurcations in one dimension"  ''Invent. Math.'' , '''62'''  (1981)  pp. 347–365</TD></TR><TR><TD valign="top">[a9b]</TD> <TD valign="top">  L. Jonker,   D. Rand,   "Bifurcations in one dimension"  ''Invent. Math.'' , '''63'''  (1981)  pp. 1–16</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  L. Jonker,   D. Rand,   "Universal properties of maps of the circle with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565070.png" />-singularities"  ''Comm. Math. Phys.'' , '''90'''  (1983)  pp. 273–292</TD></TR><TR><TD valign="top">[a11a]</TD> <TD valign="top">  O. Lanford,   "A computer-assisted proof of the Feigenbaum conjectures"  ''Bull. Amer. Math. Soc.'' , '''6'''  (1982)  pp. 427–434</TD></TR><TR><TD valign="top">[a11b]</TD> <TD valign="top">  O.E. Lanford,   "Computer assisted proofs in analysis"  A.M. Gleason (ed.) , ''Proc. Internat. Congress Mathematicians (Berkeley, 1986)'' , Amer. Math. Soc.  (1987)  pp. 1385–1394</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  P.J. Myrberg,   "Sur l'iteration des polynomes réels quadratiques"  ''J. Math. Pures Appl.'' , '''41'''  (1962)  pp. 339–351</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  N. Metropolis,   M.L. Stein,   P.R. Stein,   "On finite limit sets for transformations on the unit interval"  ''J. Comb. Theory'' , '''15A'''  (1973)  pp. 25–44</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a15a]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a15b]</TD> <TD valign="top">  D. Rand,   "Global phase space universality, smooth conjugacies and renormalisation: the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565071.png" /> case."  ''Nonlinearity'' , '''1'''  (1988)  pp. 181–202</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  A.N. Sharkovskii,   "Coexistence of cycles of a continuous map of the line into itself"  ''Ukrain. Mat. Zh.'' , '''16'''  (1964)  pp. 61–71  (In Russian)</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  D. Sullivan,   "Bounds, quadratic differentials, and renormalization conjectures" , ''Centennial Publ.'' , '''2''' , Amer. Math. Soc.  (1991)</TD></TR></table>