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 $f_{\mu }(x)=1-\mu x^2$ acting (for $0\le \mu \le 2$) on the interval $x\in [-1,1]$, the period-doubling scenario is recalled here. For $\mu =2$, $f_{\mu }$ has periodic points of every (least) period. Let ${\mu }_i$ be the infimum of parameter values $\mu$ for which $f_{\mu }$ has a periodic orbit of least period $2^i$. Then

$$0<{\mu }_0<{\mu }_1<\dots ,$$

and

$$sup{\mu }_i={\mu }_{\mathrm{\infty }}\sim 1.401155\dots .$$

For ${\mu }_i<\mu \le {\mu }_{i+1}$, the dynamics of $f_{\mu }$ is described by statements i)–iii) below.

i) $f_{\mu }$ has precisely one periodic orbit ${\mathrm{\Lambda }}_j$ of (least) period $2^j$ for each $j=0\dots i$, and no other periodic orbits;

ii) any pair of adjacent points in ${\mathrm{\Lambda }}_i$ is separated by a unique point in ${\cap }_{j<i}{\mathrm{\Lambda }}_j$;

iii) with the exception of the (countably many) orbits which land on some ${\mathrm{\Lambda }}_j$, $j<i$, and stay there, every $f_{\mu }$- orbit tends asymptotically to ${\mathrm{\Lambda }}_i$.

For $\mu ={\mu }_{\mathrm{\infty }}$( when $f_{\mu }$ is sometimes called the Feigenbaum mapping), statement i) holds, but with $j$ ranging over all non-negative integers, and ii) holds for each $i=0,1,\dots$; furthermore, the following analogue of iii) holds:

iv) (for $\mu ={\mu }_{\mathrm{\infty }}$) the closure of the orbit of the turning point $0$ is a Cantor set ${\mathrm{\Lambda }}_{\mathrm{\infty }}$, which is the asymptotic limit of every orbit not landing on one of the periodic orbits ${\mathrm{\Lambda }}_i$, $i=0,1,\dots$. The restricted mapping $f_{\mu }{\mid }_{{\mathrm{\Lambda }}_{\mathrm{\infty }}}$ is a minimal homeomorphism (the "adding machine for chaos in a dynamical systemadding machine" ).

Finally, $\mu ={\mu }_{\mathrm{\infty }}$ is the threshold of "chaos" , in the following sense:

v) for $\mu >{\mu }_{\mathrm{\infty }}$, $f_{\mu }$ 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 ${\mathrm{\Lambda }}_j$ is rigidly determined by the fact that $f_{\mu }$ 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 ${\mu }_i$' s, together with the fact that if $\mu <{\mu }_{\mathrm{\infty }}$, then $f_{\mu }$ must have periodic orbits of least period $2^j$ for $j=0\dots i$( some $i$) 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 $1$( cf. [a1]).

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

vi) the convergence ${\mu }_i\uparrow {\mu }_{\mathrm{\infty }}$ is asymptotically geometric:

$$\underset{i\to \mathrm{\infty }}{lim}\frac{{\mu }_{\mathrm{\infty }}-{\mu }_i}{{\mu }_{\mathrm{\infty }}-{\mu }_{i+1}}=\delta \sim 4.669\dots ;$$

vii) the periodic orbits scale: let ${\mathrm{\Lambda }}_i^{\ast }$ denote the orbit ${\mathrm{\Lambda }}_i$ for $\mu ={\mu }_{i+1}$; then

$$\underset{i\to \mathrm{\infty }}{lim}\frac{\mathrm{dist}(0,{\mathrm{\Lambda }}_i^{\ast })}{\mathrm{dist}(0,{\mathrm{\Lambda }}_{i+1}^{\ast })}=\alpha \sim 2.5029\dots .$$

These statements, formulated for the particular family $f_{\mu }$ 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 $f_0$ has only finitely many periodic orbits while $f_2$ has positive entropy) and smoothness (essentially that $(x,\mu )\to f_{\mu }(x)$ is $C^2$ and each $f_{\mu }$ has a non-degenerate critical point). And, sensationally, the constants $\delta$ and $\alpha$ are independent of the family $f_{\mu }$.

In [a6] and

these assertions were reduced, using ideas from renormalization theory, to certain technical conjectures concerning a doubling operator $\mathcal{R}$ acting on an appropriate function space. O. Lanford

(cf. also [a3], [a5]) gave a rigorous, computer-assisted proof of the basic conjecture, that $\mathcal{R}$ has a saddle-type fixed point with one characteristic multiplier $\delta \sim 4.669\dots$( the same as in vi)) and stable manifold of codimension $1$. 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 $C^2$ unimodal mappings. In particular, the asymptotic geometry of the Cantor set ${\mathrm{\Lambda }}_{\mathrm{\infty }}$( for $\mu ={\mu }_{\mathrm{\infty }}$) and of analogous sets appearing at other "threshold" parameter values (the "infinitely renormalizable mappings of bounded type" ) is universal; for example, the set ${\mathrm{\Lambda }}_{\mathrm{\infty }}$ always has Hausdorff dimension $0.538045$. 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], .

References[edit]

[a1]	Ll. Alsedà, J. Llibre, M. Misiurewicz, "Combinatorial dynamics and entropy in one dimension" (to appear)
[a2]	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 MR0591173 Zbl 0447.58028
[a3]	M. Campanino, H. Epstein, D. Ruelle, "On the existence of Feigenbaum's fixed point" Comm. Math. Phys. , 79 (1981) pp. 261–302 MR612250
[a4]	P. Collet, J.-P. Eckmann, H. Koch, "On universality for area-preserving maps of the plane" Physica , 3D (1981) pp. 457–467 MR0631180 Zbl 1194.37050
[a5]	P. Collet, J.-P. Eckmann, O. Lanford, "Universal properties of maps on an interval" Comm. Math. Phys. , 76 (1980) pp. 211–254 MR0588048 Zbl 0455.58024
[a6]	P. Coullet, C. Tresser, "Itérations d'endomorphismes et groupe de rénormalisation" J. Phys. , C5 (1978) pp. 25–28 MR0512110
[a7]	W. de Mello, S. van Strien, "One-dimensional dynamics" (to appear)
[a8a]	M. Feigenbaum, "Quantitative universality for a class of non-linear transformations" J. Stat. Phys. , 19 (1978) pp. 25–52 MR501179
[a8b]	M. Feigenbaum, "The universal metric properties of a non-linear transformation" J. Stat. Phys. , 21 (1979) pp. 669–706 MR555919
[a9a]	L. Jonker, D. Rand, "Bifurcations in one dimension" Invent. Math. , 62 (1981) pp. 347–365 MR0608525 MR0604832 Zbl 0475.58015
[a9b]	L. Jonker, D. Rand, "Bifurcations in one dimension" Invent. Math. , 63 (1981) pp. 1–16 MR0608525 MR0604832 Zbl 0475.58015
[a10]	L. Jonker, D. Rand, "Universal properties of maps of the circle with -singularities" Comm. Math. Phys. , 90 (1983) pp. 273–292 MR714439
[a11a]	O. Lanford, "A computer-assisted proof of the Feigenbaum conjectures" Bull. Amer. Math. Soc. , 6 (1982) pp. 427–434 MR0648529 Zbl 0487.58017
[a11b]	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 MR0934342 Zbl 0676.65039
[a12]	P.J. Myrberg, "Sur l'iteration des polynomes réels quadratiques" J. Math. Pures Appl. , 41 (1962) pp. 339–351 MR0161968 Zbl 0106.04703
[a13]	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 MR0316636 Zbl 0259.26003
[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
[a16]	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) MR1415876 MR1361914
[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