Acta Polytechnica


doi:10.14311/AP.2014.54.0130
Acta Polytechnica 54(2):130–132, 2014 © Czech Technical University in Prague, 2014

available online at http://ojs.cvut.cz/ojs/index.php/ap

CAN ONE REALLY STUDY CHAOS ANALYTICALLY?

M. Howard Leea, b
a Department of Physics and Astronomy, University of Georgia, Athens, GA 30602 USA
b Korea Institute for Advanced Study, Seoul 130-012, Korea
correspondence: MHLee@uga.edu
Abstract. One generally thinks that chaos can be studied only numerically by aid of the computer.
It is however suggested by the theorem of Sharkovskii and Li and Yorke that in Id continuous maps
analytical studies are possible. How one might achieve such a goal in one special map is described.
Keywords: chaos, logistic map, Sharkovskii theorem.

1. Introduction
When one thinks of chaos, what most likely come to
mind are beautiful pictures drawn by the computer. In
fact virtually all articles and books on chaos have some
such pictures illustrating unimagined properties of
nonlinear behavior [1,2]. Thus one tends to think that
an analytical approach to chaos is nearly unthinkable.
One might even ask why one should want to attempt
it. If it were, however, possible even in a special
case or cases, it could help provide perhaps a deeper
understanding. This attitude is to our mind something
that we do or try to do in physics.
Such a possibility is opened by a remarkable the-

orem proved by Sharkovskii [3], and later indepen-
dently by Li and Yorke [4]. This theorem applies to
one-dimensional continuous maps of real numbers in
a finite interval. The essence of the theorem reads
that if 3-cycle exists in one domain of such a map, it
implies the existence of all other cycles in the same
domain and hence it implies the existence of chaos
itself therein. What is especially significant, and not
yet widely recognized, is that by means of 3-cycle a
chaotic domain can be determined without introduc-
ing a working definition of chaos. This is contrary to
what is normally done in numerical studies. In these
studies, a domain is said to be chaotic if it meets
a certain criterion or a set of criteria, which repre-
sent working definitions of chaos. These definitions
are phenomenologically based, not derived from some
higher principles of chaos.
We will not go into why 3-cycle plays this pivotal

role, for this subject is now well known. What we
will do is to demonstrate the existence of 3-cycle in
an applicable map and how we can learn about chaos
from the content of the theorem. This article be-
ing intended as an overview, we will not present any
detailed derivations, just ideas of what can be done.

2. One-dimensional continuous
map of real numbers in an
interval

Let xi and xj be two real numbers in (0,1). If xi → xj
by an action f, we write the action as xj = f(xi),
calling it a map. If, for some value of xi, say x∗, it

goes into itself and goes nowhere else, it is termed a
fixed point of f, i.e., f(x∗) = x∗. To a fixed point, f
acts like an identity operation.
Let x1,x2,x3, .. be a set of points in (0,1), not

necessarily ordered. By the action of f, let f(x1) = x2,
f(x2) = x3, . . . ,f(xN ) = xN +1 etc., a process termed
an iteration. Suppose f(x3) = x1, i.e., x4 = x1. The
iteration has terminated in 3 steps giving the 3-cycle
condition: f(x1) = x2, f(x2) = x3, and f(x3) = x1,
which may be expressed as f(f(f(x)))≡f3(x) = x,
where x = x1,x2, or x3, and x1 6=x2 6=x3. These three
values are fixed points of f3, giving 3-cycle. Excluded
are fixed points of f, which are also fixed points of f3,
since they do not satisfy the 3-cycle condition.
If f(xN ) = x1, it defines N-cycle. It would be

termed periodic since, like an orbital motion, it returns
to its initial point after N steps. If, however, f(xN ) =
xN +1, N →∞, never returning to its initial point, it
would be chaotic, like a trajectory which never closes.
Even at this stage, one might ask whether there are
some properties of the interval of points, to which the
two fundamentally different kinds of trajectory could
be ascribed. A periodic trajectory has a finite set of
“frequencies”. They are probably connected one with
another through a sequence of rational numbers that
fill the interval. A chaotic trajectory goes from one
point to another in a finite interval unpredictably and
never returning. This kind of behavior suggests that
to describe its movement one would have to bring in
the irrational numbers that fill the interval.
Returning to the theorem of Sharkovskii and Li-

Yorke, what is required is to obtain the 3 fixed points
of 3-cycle for a particular 1d continuous map. It
amounts to solving for three roots of f3(x) − x =
0, probably a cubic equation, which should pose no
special difficulty. Had the theorem called for 4-cycle,
presenting a quartic equation, it would still be no
challenge. Had it asked for 5-cycle, any attempt at
an analytical solution would have to be abandoned.
Fortuitously it is 3-cycle that the theorem asks for
and this is why an analytical study seems feasible.

But we cannot know for sure until the 3-cycle of an
applicable map is constructed. We shall now turn to
one such map to see what is to be realized.

130

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vol. 54 no. 2/2014 Can One Really Study Chaos Analytically?

3. Logistic map
Perhaps one of the simplest and no doubt the most
widely studied among all chaotic maps is the logistic
map. Its widespread appeal recalls the Ising model
of magnetism in statistical mechanics. Perhaps it is
not too far-fetched to call the logistic map the Ising
model of chaos.
The logistic map is a one-dimensional continuous

map of real numbers in the interval (0, 1), to which
the theorem of Sharkovskii is applicable. The usual
way to define the map is as x′ = f(x), where

f(x) = ax(1 −x), x = (0, 1), (1)

where a is a control parameter, 0 ≤ a ≤ 4. The fixed
points of f and f2 for (1) are obtained readily: By
solving f(x) − x = 0, we obtain x = x0 = 1 − a−1,
a > 1. Next by solving (f2 − x)/(f − x) = 0, we
obtain a quadratic equation, from which x = x1,2 =
(a + 1)/2a{1 ±

√
(a− 3)/(a + 1)}, a > 3. Now we

anticipate that (f3−x)/(f−x) = 0 is a cubic equation,
from which 3 roots are to be found.
We now construct Q(x) ≡ (f3 −x)/(f −x) = 0 in

a new variable t = ax for simplification [5]:

Q(t) = t6 − (3a + 1)t5 + (3a2 + 4a + 1)t4

− (a3 + 5a2 + 3a + 1)t3 + (2a3 + 3a2 + 3a + 1)t2

− (a3 + 2a2 + 2a + 1)t + (a2 + a + 1). (2)

To our surprise it is a sextic equation. Since there
are no known formulas for solving any equations of
degrees equal to or higher than 5, at a first glance we
seem to have arrived at a brick wall.
Perhaps Q(t) is not a general sextic equation but

a special one. If so, it might be solvable in some
special way. If Ek denotes an equation of degree k,
E6 would be solvable at least in principle if it could
be expressed as: E2 ×E2 ×E2 or E3 ×E3. Looking
at (2), we see at once that Q(t) must be real if t
is real since its coefficients are all real. This means
that the 6 roots can only be either 3 pairs of complex
conjugates or two sets of 3 real roots, corresponding
to two solvable forms for Q(t). This impels us to look
for an indication that Q(t) is not general. If (2) is
expressed as

Q(t) =
6∑

k=0
(−)kδktk, (3)

we find that ∆ ≡
∑5

k=0(−)
kδk = 0, referred to as the

delta sum rule. This sum rule is a clear indication
that Q(t) is not a general sextic equation.

4. Solving the sextic equation:
preliminaries

If a = 0 in (2), which still satisfies the delta sum
rule, the roots are: exp(±iπ/7), exp(±i3π/7) and
exp(±i5π/7), 3 pairs of complex conjugates lying on

the unit circle. If the roots are to be real, none of
them may lie on the negative real axis of t since the
coefficients of t in odd power have ‘−’ signs. The six
roots must all lie on the positive real axis. If a →∞,
one of the roots goes as 1/a asymptotically. Thus
as a increases from zero, the three pairs of complex
roots must move toward the positive real axis. At
some value of a = ã say, each complex-conjugate
pair becomes two identical real roots simultaneously.
Thereafter each degenerate real root splits into two
real roots. This overall behavior is consistent with
the two solvable forms for Q(t). We cannot yet know
however whether ã < 4 or > 4. The existence of
3-cycle in the logistic map requires that ã be less than
4 and also that at least one set of real roots lies in the
interval t = (0,a) or x = (0, 1). If otherwise, 3-cycle
does not exist in it.
If a > ã, there must be two sets of 3 positive real

roots. Hence, we shall write Q(t) = q(t)×q′(t), where
q and q′ are cubic equations with (as yet undeter-
mined) roots, resp., tk and t′k, k = 1, 2, 3. Thus one
can express q(t) as

q(t) = t3 −αt2 + βt−γ, (4)

where α = t1 +t2 +t3, β = t1t2 +t2t3 +t3t1, γ = t1t2t3,
and q′(t) similarly with primes written on. They (αβγ)
will be referred to as trigonals.

To understand why Q is special we look for some
structural relations between the trigonals of the same
kind contained in the structure of the logistic map.
Indeed we find that [7]: α−β+γ = 0 and α′−β′+γ′ =
0, together called the trigonal relation. To no surprise
the trigonal relation implies the delta sum rule. In
addition, β = (a+1)α−(a2 +a+1) and γ = aα−(a2 +
a + 1), leaving only α and α′ independent, leaving
only two unknowns to the problem. The transition
condition E2×E2×E2 = E3×E3, together with the
trigonal relation yields σ = 0, where σ = (a2 − 2a−
7)1/2. Hence, ã = 1 +

√
8 = 3.872281323 · · · as the

transition value. Since ã < 4, 3-cycle can exist in the
logistic map.
There are two sets of 3 positive real roots, or two

forms of 3-cycle, that had not been implied by the
theorem of Sharkovskii, but that evidently exist in the
logistic map. Unless their relationship is found, the
two cubic equations are still left with one unknown
each, α or α′. How to distinguish the two remains an
obstacle to the final solution.

5. Solving the sextic equation:
an internal degree of freedom

If there are two forms of 3-cycle, they must be dis-
tinguishable by some internal degree of freedom in
q and q′. It cannot appear in Q, meaning that
it cannot appear in δ′s, the coefficients of t in Q,
e.g. δ5 = 3a + 1 = α + α′. Since the two forms
must be equivalent, they are like two states of parity,

131



M. Howard Lee Acta Polytechnica

which could be represented by a double-valued func-
tion. For 3-cycle in the logistic map it happens to be
σ = (a2 − 2a− 7)1/2 [7]. Thus, we shall assume

α =
1
2
δ5 + Kσ, (5)

α′ =
1
2
δ5 −Kσ, (6)

where if K = 1/2 it yields all the coefficients (δ′s)
exactly. Therewith we finally obtain:

α =
1
2

(3a + 1 + σ), (7)

β =
1
2

(a2 + 2a− 1 + (a + 1)σ), (8)

γ =
1
2

(a2 −a− 2 + aσ), (9)

and α′,β′,γ′ by taking −σ in (7)–(9). Hence if q =
q(σ), q′ = q(−σ). It is sufficient to solve only one of
them.

Even beforehand, it is possible to prove that tk and
t′k,k = 1, 2, 3, all lie in the interval (0,a). The proof
goes as follows: Let tk = a− �. If ã ≤ a ≤ 4, tk > 0
as already proved. Hence a− � > 0. This gives � < a,
an upper bound on �. To determine a lower bound on
�, we consider the trigonal relation with tk = a−� for
any k = 1, 2 or 3. By (7)–(9) we find that

α−β + γ = −R(�)/(a− �), (10)

where
R(�) = �3 −A�2 + B�− 1, (11)

where A and B are real and positive if ã ≤ a ≤ 4.
The trigonal relation requires that R(�) = 0. This
is possible if and only if � > 0, which gives a lower
bound on �. Thus, 0 < � < a and this implies that
tk = a − � must lie in the interval t = (0,a). This
proof agrees with the cubic solutions obtained from
q = 0 and q′ = 0. We conclude that the 3-cycle exists
continuously in the logistic map from a = ã to a = 4.
Throughout this domain there exists chaos because
there are, according to the theorem of Sharkovskii,
infinitely many cycles of all kinds.

6. Conclusion: aleph cycle and
chaotic trajectories

The theorem of Sharkovskii asserts that if 3-cycle
exists in a domain, all other cycles exist in that domain.
For the logistic map we take this to mean that there

are infinitely many cycles of all varieties in the domain
ã ≤ a ≤ 4. Thus, in this domain the fixed points form
a dense set of points in x = (0, 1), a strip made up of
irrational points of all different kinds. It has a finite
measure. If x1 is an initial value that belongs to this
set of points of finite measure, fN (x1) = xN +1,N →
∞, termed an aleph cycle. Evidently an aleph cycle
describes a chaotic trajectory. If x0 is an initial value
that does not belong to this set, thus belongs to a set
of points of 0 measure, fM (x0) = x0, M < ∞, a finite
cycle, giving a periodic trajectory.

This assertion can be justified for some special val-
ues of a such as a = 4, for which one can obtain a
large number of fixed points from which a set of points
of finite measure can be constructed [6].
An aleph cycle has a very close connection to an

ergodic trajectory in statistical mechanics according to
the ergometic theory of the ergodic hypothesis [8]. It is
thus possible to gain via an aleph cycle an insight into
the ergodic theory of chaos that has been proposed [9].
As we have stated at the outset, while it would

be nearly impossible to study chaos analytically, the
theorem of Sharkovskii gives a window through which
something can be done analytically, as we have de-
scribed here in this short note.

Acknowledgements
I thank Dr Miloslav Znojil of the Doppler Institute for
Mathematical Physics and Applied Mathematics, Rez,
Czech Republic for suggesting that I write this article
based on a talk presented at Vila Lanna, Prague in October
2013.

References
[1] B. Hu, Phys. Rep. 91, 234 (1982)

doi: 10.1016/0370-1573(82)90057-6
[2] D. Gulick, Encounters with chaos (McGraw-Hill, NY
1992).

[3] A.N. Sharkovskii, Ukr. Math. Z. 16. 61 (1964).
[4] T-Y Li and J.A. Yorke, Am. Math. Monthly 82, 985
(1975).

[5] M.H. Lee, Acta Phys. Pol. B 42, 1071 (2011).
[6] M.H. Lee, Acta Phys. Pol. B 43,1053 (2012).
[7] M.H. Lee, Acta Phys. Pol. B 44, 925 (2013).
[8] M.H. Lee, Phys. Rev. Lett. 87, 250601 (2001)

doi: 10.1103/PhysRevLett.87.250601; Phys. Rev. Lett.
98, 110403 (2007) doi: 10.1103/PhysRevLett.98.110403

[9] J.-P. Ekmann and D. Rulle, Rev. Mod. Phys. 57, 617
(1985) doi: 10.1103/RevModPhys.57.617

132

http://dx.doi.org/10.1016/0370-1573(82)90057-6
http://dx.doi.org/10.1103/PhysRevLett.87.250601
http://dx.doi.org/10.1103/PhysRevLett.98.110403
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	Acta Polytechnica 54(2):130–132, 2014
	1 Introduction
	2 One-dimensional continuous map of real numbers in an interval
	3 Logistic map
	4 Solving the sextic equation: preliminaries
	5 Solving the sextic equation: an internal degree of freedom
	6 Conclusion: aleph cycle and chaotic trajectories
	Acknowledgements
	References