Table Of Content.A CLASSICAL NONLINEAR DYNAMICS
DNA SOAHC
Entropy and Volume as Measures of Orbit Complexity
by
Sheldon E. Newhouse
Mathematics Department
University of North Carolina
Chapel Hill, North Carolina 27514
Abstract: Topological entropy and volume growth of smooth
disks are considered as measures of the orbit complexity of a
smooth dynamical system: In many cases, topological entropy
can De estimated vla volume growth. This gives methods of
estimating dynamical invariants of transient and attracting
sets and may apply to time series.
.I Introduction.
A basic problem in the theory of dynamical systems is to
understand chaotic motion. One wants to attach numerical
invariants to a system which measure the amount of chaos in the
system. A natural invariant of a continuous or discrete system
is the so-called topological entropy. This is a non-negative
number which gives a crude quantitative measure of the orbit
complexity of the system. The definition of the topological
entropy is not very amenable to its calculation. Recently,
results in the theory of smooth dynamical systems have related
the topological entropy to the maximum volume growth of smooth
disks in the phase space. Preliminary numerical studies
indicate that in many cases volume growth rates may be
estimated easily, and, hence the entropy itself may be
estimated.
A system with positive topological entropy may have no
complicated attracting sets. That is, the entropy may be given
by the orbit structure on non-attracting (i.e., transient) sets.
Typical orbits may spend varying amounts of time near these
transient sets and then wind toward periodic attracting orbits
(we are here, of course, thinking of dissipative dynamics).
The entropy can give information about transient behavior, but
it is interesting to ask how relevant it is for understanding
asymptotic behavior. In this connection a simple example will
be useful.
Consider the mapping fr(x)=rx(l-x) from the unit interval
0,1 to itself, where r is a real number in 0,4. It is known
that for r=0.25 and r=3.83 almost all orbits are asymptotic to
periodic sinks. In the first case the sink is a fixed point
while in the second case it is a periodic point of period
three. Suppose we ask how much of a movement in r is necessary
for the mapping fr to have a set of positive measure whose
orbits are not asymptotic to sinks. The answer is that much
more is required for r=0.25 than for r=3.83. Is there some way
of knowing this from fr itself? We suggest that the
topological entropy provides a clue. Indeed, the entropy for
r=0.25 is zero while the entropy for r=3.83 is log(~--)
0.481 If one can estimate the entropy and its modulus of
continuity, then one can get a predictive tool for the
appearance of chaotic attractors.
.2 Topological Entropy.
Let M be a smooth manifold and let f:M ~ M be a smooth
self-map. We get f as either a discrete dynamical system or as
a time-t map of a flow on M. Let d be a distance function on M
induced by a smooth Riemannian metric. Let ~ > 0 and let n be
a positive integer. A set E is (n,6)-separated if whenever x~y
in E there is an integer j e 0,n) such that d(fJx,fJy) > 6.
Letting r(n,6,f) denote the maximum possible number of elements
of any (n,6)-separated set E, it easy to show that r(n,~,f) ~ C
e na for some C > 0 and a > 0 The best such a is
r(c,f)= limsup I/n log r(n,6,f).
n+~
The number h(f) = lim r(6,f) is the topoZo~icul entropy of .f
6~0
Properties of h(f):
.I h(f n) = nh(f) for n ~ 0.
.2 h(¢f~ -I) = h(f) if ~ is a continuous change of
coordinates (i.e., h(f) is a topological invariant).
.3 h(f) = h(f -I) if f is a homeomorphism.
4. h(f) = sup {hp(f): p e M(f)),
where M(f) is the set of f-invariant probability
measures on M and hp(f) is the measure-theoretic
entropy of f with respect to p.
Note that if h(f) is positive, then f has invariant
probability measures with positive entropy so f has some
chaotic dynamics.
Examples:
.I Let f(z)=P(z)/Q(z) be a rational function in
one complex variable z,where P and Q have no common
factors.. Consider f as a mapping on the Riemann
sphere S 2. It can be shown that
h(f) = log(topological degree of )f
= log (max(degree P,degree Q))
(seeL,Nl).
.2 Let A be an integer N × N matrix with determinant one, and
let A be the induced linear automorphism of the
N-dimensional torus. Then, h(f) = ~ log llI
I is an
eigenvalue
of A with
Ikl>l
(seeB).
3. Let J={I,...,N} and let A be an N × N matrix of zeroes
and ones. Let ZE= J and let ~A = {~ e ~: Aa(i)a(i+l)=l
for all i}. Let °:~A ~ ~A be the shift map. Then
h(o) is the logarithm of the largest modulus of the
eigenvalues of A.
For more information on topological entropy, see DGS.
3. Volume growth and its relation to topological entropy.
Let D k be the closed unit ball in ~k A C k disk in M is a C k
map 7:D k ÷ M. .For such a O k disk ~ with k a ,i let 171 denote
its k-dimensional volume with multiplicities. This is defined
by
171 = JDklAkT7 I dl,
where T7 is the derivative of 7, AkT7 is the k th exterior power
of TT, and dl is Lebesgue volume on D k. When k=l, 171 is the
length of the curve 7. When k=2, it is the surface area of 7,
etc.
Given C k f:M ~ M with k > i and 7 as above, let
G(y,f) = limsup I/n log + Ifn-loTl.
~n
Here, log + is the positive part of the natural logarithm
function. Thus, G(7,f) is the volume growth rate of 7 by .f
Let G(7,f) = lim i/n log + Ifn-l¢71 when the limit exists.
n+~
Let G(f) = sup {G(7,f): 7 is any smooth disk in M}, and let
S(f) = sup {S(y,f): ~ is any smooth disk in M}. We emphasize
that the disks in the definitions of G(f) and G(f) have their
dimensions varying from I through dim M.
Theorem 1 NIl. Let f:M ÷ M be a C k self-map of the
compact manifold M with k > .I Then, h(f) ~ G(f).
Theorem 2 Y. Let f:M ~ M be a C ~ self-map of the
compact manifold M. Then, h(f) a G(f)
Actually, the techniques in NIl and N2 can be combined
with those in Y to prove the following sharper result.
Theorem 3.
.i Let f:M ~ M be a C ~ self-map of the compact manifold
M. Then, h(f) = G(f) and the supremum in G(f)
is actually assumed by some disk F.
2. The map f ~ h(f) is uppersemicontinuous on the space
of C ~ self-maps of M with the C ~ topology.
.3 For a fixed C ~ map ,f the mapping ~ ~ hp(f) is
uppersemicontinuous on the space of f-invariant
measures on M. In particular, every C ~ map has
measures of maximal entropy.
4. The map f ~ h(f) is continuous on the space of C ~
diffeomorphisms of a compact two-dimensional
manifold M .2 (The lowersemicontinuity of f ~ h(f)
for C l+a diffeomorphisms of surfaces was proved by
Katok ).K
.5 Let f:M 2 , M 2 be a C ~ diffeomorphism from the
compact two-manifold with boundary M 2 into its
interior. Assume that f is weakly dfssipative in
the sense that there is an integer T > 0 such that
the Jacobian determinant of f~ is less than one at
each point in M 2. Let OM 2 denote the boundary of
M 2. Then, h(f) = G(0M2,f).
Note that statement 5 of Theorem 3 applies to many forced
oscillations. To compute the entrop~ one only needs to compute
the growth rate of the length of the boundary.
4. Numerical results.
We considered several H6non mappings Xl=l + y - a x 2,
Yl=bX as a test for computing length growth for systems with
two degrees of freedom.
Figure 1 below shows a plot of the log of the length of
the n-th iterate of a certain line segment y as a function of n
for 5 ~ n s 2000 with a=1.4, b=0.3. The x-units are in
multiples of 5. The best least-squares line is also computed.
The average of the entropies is actually the average of the
IfJ~J
quantities 1/~ log for 5 s ~ ~ 2000. We take the least-
squares slope, LS, as an estimate of h(f). Note that LS is
approximately 0.45
Figure 2 shows a similar plot for a=1.27, b=0.3. Note
that there are two positive slopes. The first one is 0.35 and
the second one is 0.09. This indicates the presence of a
transient chaotic set with entropy ~ 0.35 and a strange
attractor with entropy ~ 0.09. A plot of the iterates of a
single orbit (not shown here) shows that the strange attractor
has seven pieces. Its characteristic exponent is ~ 0.084.
Figure 3 shows the log-of-length versus length plot for a=1.28,
b=O.3. The transient chaotic set seems to have merged with the
strange attractor to produce a single attractor with entropy
0.30 and characteristic exponent ~ 0.258.
8,998
t "~'
"*-a.
".a..
.t.
){ h. ".~.~....j..,
~0,8
"'/"~. r ~C ~-/~.e "9'e~O
".-~. " e~
"~.-~. _~c ~
• ..e. ~" c~ ~
"w.w.
"/-er.
.W r
L,9
Figure 1 ~'~
a = 1.4, b = 0.3, x - unit = ,5 LS ~ 0.45 average of entropies = 0.454
~.442
L,,
~ ~ .,,r,~... ''~
I 55~ -ti. "°~ ~I,
-"'"" 0 9~ ~ "~
S.221
.~.o'., °°~ ~ e~ o
~., ~'~",,,,, c~ ~ ~
/ "
/.
/
/
/.
"/,/ / ~ 55.
1,2
2~
Figure 2
a = 1.27, b = 0.3, x - unit = 5
2,826
toi...
r#. "°
.°~
..~. -.
"o~.t.
"I.W°t • i~"
~,g13
SM, .....,-."'e.~'~ °t~
../P" ,~C,
.,.. ..... .°' .~..~ e.~
...t/.° • ' E~ "
o,.r •
1,2
5
2~
Figure 3
a = 1.28, b = 0.3, x - unit = 5
References
B R. Bowen, Entropy for Group Endomorphisms and Homogeneous
Spaces, Trans. Amer. Math. Soc. 153(1971), 401-414,
181(1973), 509-510.
DGS M. Denker, .C Grillengerger, and K. Sigmund, Ergodic
Theory on Compact Spaces, Lecture Notes in Math.
52? (1976). K A. Katok and L. Mendosa, to appear.
L M. Ljubich, Entropy properties of rational endomorphisms
of the Riemann Sphere, Jour. Ergodic Theory and Dyn. Sys.
3(1983), 351-387.
N1 S. Newhouse, Entropy and Volume, to appear in Jour.
Ergodic Theory and Dyn. Sys.
N2 S. Newhouse, Continuity Properties of Entropy, preprint,
Mathematics Department, University of North Carolina,
Chapel Hill, NC 27514, USA.
Y Y. Yomdin, Volume Growth and Entropy, and ck-resolution
of Semi-algebraic Mappings--Addendum to the Volume
Growth and Entropy, to appear in Israel J. of Math.
A CHAOTIC I-D GAS: SOME IMPLICATIONS
O.E. Rossler
Institute for Physical and Theoretical Chemistry,
University of Tubingen, 7400 Tubingen, West Germany
A I-D classical gas with maximal chaos is described. It supports many simplified
(color-coded) chemical reactions - including far-from-equilibrium dissipative
structures. A proposed example generates a limit cycle. Its excitable analogue
is a model observer. Gibbs sy~netry invariably gives rise to a substitute Hamil-
tonian. The resulting pseudo-reversibility implies, for the model observer, that
all external objects are subject to Nelson stochasticity and hence quantum mechanics.
The theory of classical solitons in i D a sharply protruding half circle). Then
is rich in implications. It was recent- with ~ --~ 0 a classical collision pro-
ly used to solve the relativistic no- blem, a point-shaped billiard on a 2-D
interaction problem I. It might also table that sports a protruding half-circle
help solve the problem of whether or on one side, results. Sinai's theorem
not quantum mechanics can be reduced to (see 2 ) which implies chaos applies.
classical mechanics. The above f ("smoothed tent") yields,
In the following, classical nonrel- with ~ = 0.002, the same result numeric-
ativistic particles in 1D are consid- ally for non-selected initial conditions.
ered. Two types of particles are as- Second, the system has to be shown to
sumed. The "rods" present in a horiz- remain "maximally chaotic" as more and
ontal frictionless tube pass freely more particles are added. Specifically,
through each other. They interact only adding a second horizontal particle (i =
with the curved "bullets" that, while 1,2) augments the right-hand side of H
running in a vertical tube of their own, by 4 terms. The corresponding collision
each may or may not protrude with problem now is a point-shaped billiard on
their heads into the horizontal tube. a 3-D table - with one side of the box
The Hamiltonian in the simplest case sporting two protruding half-cylinders in
(just 2 particles; imax=l, Jmax=l) the shape of a cross. Hence Sinai's
becomes theorem applies in two mutually independ-
ent directions, this time. And so forth.
Pi ~ + + ~ + Hence the number of positive Lyapunov
characteristic exponents remains n-I
2 ~--X ~---i +
i l ( "maximal chaos" .)
Similarly if more vertical particles
+ ~ + E (1)
are added. Each new j augments H by
yj L-fj(x i)-yj '
l+i(max) terms. The j types of f
functions differ only in the positions
of the protrusions on the x-axis. (To
where f is a bowler-hat shaped func- accomodate many vertical slots, smaller
tion, being and smaller half circles are needed if
the unit x-interval is retained. The y-
lengths, L , then have to be decreased
f = /(X-0.45)2÷I0 -6 + f(x-0.55)2+10 -6 proportionally. ) There is maximal chaos
again: Each horizontal particle inter-
acts chaotically with each vertical one.
/(x-0.5)2+10 -4
2 Third, a first implication. The pre-
-
sent I-D gas can be made the basis of
chemical interactions. Elskens 3 al-
in the simplest case, and L = 0.5 and ready considered reactions supported by
H = 1.8, for example. an underlying deterministic dynamics - a
First, it is to be shown that this I-D gas of the quasiperiodic type. The
two degrees of freedom system is chaot- simplest possibility is color coding:
.ci This is easy if the bowler-hat Colors (chemical identities) change in a
function is ideal (flat zero except for lawful manner under collisions 3. The
reaction energies are hereby shielded actions (that from d to a ) actually
from contributing to the mechanical has to be second-order in reality - in-
ones. This is an admissible idealiza- volving a constant-concentration, ener-
tion (to be relaxed as more realistic gy-rich reaction partner. This "fifth"
molecular-dynamics Hamiltonians be- color is formally included in the above
come available). Unlike bimolecular pseudo-collision convention for first-
reactions, monomolecular ones require order reactions. The system of Eq.(2)
a special convention. Making these produces, at the assumed parameter values
color changes contingent on arbitrary and with a+b+c+d = I0, a deterministic
"supra-threshold" collisions is one limit cycle. It will be interesting to
possibility. Easier to implement is reproduce this limit cycle with the above
an artificial convention: There is a I-D molecular dynamics scheme - with
"color-changing position" in every unit small values of n like i00 to i000,
interval L , both horizontally and perhaps.
vertically, and there is some clock Fifth, a variant to the reaction sy-
(some - any - particle being in a cer- stem of Eq. )2( is bound to produce "ex-
tain color-specific interval, some- citable" behavior - stability toward very
where) determining whether or not a small-amplitude perturbations but auto-
color change takes place. Another pro- catalytic instability toward somewhat
blem is that there are necessarily two larger ones (with subsequent re-excita-
subpopulations to each color, one among bility after a refractory period), .fc
the bullets and one among the rods. 4. The system in this case will con-
It is conceivable that even with -tenmrys stitute a "formal neuron." Of course,
ric initial numbers and large n , their if one such neuron can be implemented by
numbers might diverge under certain con- Eq.(1), so can I0 I0, say. That ,si a
ditions. This unlikely situation has full-fledged macroscopic observer (of
yet to be ruled out. well-stirred type) can be implemented -
Again, chemical reactions relaxing in principle.
toward equilibrium (cf. 3) can be stu- Sixth, a new question can therefore
died. The present gas has the asset of be posed. How must the world appear to
being strongly mixing so that some of such a (fully transparent, in principle)
the results can be expected to be even observer?
more realistic. The question can be approached using
Fourth, it deserves to be stressed the present excitable system. (One neu-
that far-from-equilibrium situations and ron is as good as many in principle, es-
even open conditions can be included. pecially so as arbitrary classical mea-
Such systems are able to generate non- suring devices may be provided tO the
trivial dissipative structures (like system.) At first sight, nothing unusu-
limit cycles or other attractors) 4. al is to be expected.
They only have to obeymass conserva- The situation changes if the fact
tion in the present context. A conven- that the observer contains equal-type
ient example is the following simple classical particles is taken into ac-
4-variable quadratic mass action system: count. Such particles, if really iden-
tical (that is, unlabellable), introduce
a nontrivial syrmmetry. Note in this con-
a = 0.O011d - ab text that classical solitons - which pro-
vide the motivation for the present par-
ticles - are indeed unlabelable.
= ab - bc - 0.05b + 10-5d At first the simpler case of the re-
action-free gas is to be considered.
Here Gibbs's early finding of a "reduced
phase space volume" 5 can be confirmed.
c = 0.002d - bc - 0.035c )2(
N indistinguishable (as far as their
material identities are concerned) par-
ticles reduce this volt, eh by a factor of
= 0.05b + 2bc + 0.035c -
!N 5. This is because the lack of
knowledge about their nmterial identi-
ties gives rise to N! equally eligible,
- (0.0011 + 10 -5 + 0.002)d.
mirror-symmetric trajectories once a
single one is unambiguously defined in
There is mass conservation (a+b+c+d = space-time. Mutual identification of
const.). One of the monomolecular re- all of them then leads to this reduction.
01
