Logistic Map Potentials

Abstract

We develop and illustrate methods to compute all single particle potentials
that underlie the logistic map, x --> sx(1-x) for 0<s<=4. We show that the
switchback potentials can be obtained from the primary potential through
functional transformations. We are thereby able to produce the various branches
of the corresponding analytic potential functions, which have an infinite
number of branch points for generic s>2. We illustrate the methods numerically
for the cases s=5/2 and s=10/3.

Full text

Page 1

arXiv:1005.5030v1 [math-ph] 27 May 2010
CERN-PH-TH/2010-123 and UMTG - 15
Logistic Map Potentials
Thomas Curtright§and Andrzej Veitia♯
Department of Physics, University of Miami, Coral Gables, FL 33124-8046, USA
Abstract
We develop and illustrate methods to compute all single particle potentials that underlie the logistic map, x ?→ sx(1 − x)
for 0 < s ≤ 4.We show that the switchback potentials can be obtained from the primary potential through functional
transformations. We are thereby able to produce the various branches of the corresponding analytic potential functions, which
have an infinite number of branch points for generic s > 2. We illustrate the methods numerically for the cases s = 5/2 and
s = 10/3.
I.INTRODUCTION
In two previous papers [1] it was shown how functions defined on a discrete lattice of time points may be smoothly
interpolated in t, for a continuum of time points, through the use of solutions to Schr¨ oder’s nonlinear functional
equation [2]. If the effect of the first discrete time step is given as the map
x ?→ f1(x,s) ,(1)
for some parameter s, then Schr¨ oder’s functional equation is
sΨ(x,s) = Ψ(f1(x,s),s) ,(2)
with Ψ to be determined. So, f1(x,s) = Ψ−1(sΨ(x,s),s). A continuous interpolation between the integer lattice
of time points is then, for any t,
ft(x,s) = Ψ−1?stΨ(x,s),s?
.(3)
This can be a well-behaved, analytic and single-valued function of both x and t provided that Ψ−1(x,s) is a well-
behaved, analytic, single-valued function of x, even though Ψ(x,s) might be, and typically is, multi-valued. In this
sense, analyticity in x leads to analyticity in t.
As discussed in [1], the interpolation can be envisioned as the trajectory of a particle passing through the initial x,
x(t) = ft(x,s) ,(4)
where the particle is moving according to Hamiltonian dynamics under the influence of a potential, V . Up to additive
and multiplicative constants, at various times during the evolution of the particle, we have
V (x(t)) = −
?dx(t)
dt
?2
.(5)
At t = 0 this becomes V (x) = −?ln2s??
At other times the x dependence of the potential also follows from that of the velocity profile of the interpolation,
dx(t)/dt, when the latter is expressed as a function of x(t). In general, this will exhibit the branches of the underlying
analytic potential function. But more importantly for our purposes here, the various branches of the potential can
also be determined directly from the functional equation V inherits from Ψ. This functional equation is
Ψ(x,s)
dΨ(x,s)/dx
?2
. Thus, V may inherit multi-valuedness from Ψ [1].
V (f1(x,s),s) =
?d
dxf1(x,s)
?2
V (x,s) .(6)
If the map (1) possesses a fixed point, we may attempt to solve this functional equation for V by series in x about
that fixed point. We shall discuss in some detail the circumstances for which this series method is successful in the
context of the logistic map x ?→ sx(1 − x). In general, if the series can be constructed, it will of course have a finite
radius of convergence. However, the series result can then be continued to other x by making use of the functional
equation itself (a technique very familiar, e.g., for the Γ and ζ functions) and also by exploiting other special features
for specific maps (cf. s → 2 − s duality for the logistic map, discussed in Appendix A). These additional techniques
will allow us to construct convergent series approximations for all branches of the potential in those situations where
V is multi-valued.The net result is a family of potential sequences that encode for the corresponding continuous
particle trajectories the various fixed points, bifurcations, limit cycles, and chaotic behavior of the discrete logistic
map, for all s of interest.
§curtright@miami.edu
♯aveitia@physics.miami.edu
1

Page 2

II.FUNCTIONAL METHODS AND SERIES SOLUTIONS
Consider in detail the logistic map [3–7] on the unit interval, x ∈ [0,1],
x ?→ sx(1 − x) .(7)
For the most part, we restrict our considerations to parameter values s ∈ [0,4].
obtained from x = 1/2, so without loss of any essential features, we will also most often restrict x ∈ [0,s/4].
map has fixed points at x = 0 and at x∗= 1 − 1/s. Schr¨ oder’s equation for this map is
sΨ(x,s) = Ψ(sx(1 − x),s) ,
and from this follows the functional equation for the underlying potential,
The maximum of the map is s/4,
The
(8)
V (sx(1 − x),s) = s2(1 − 2x)2V (x,s) .(9)
Applying the method of series solution about x = 0, with initial conditions that correspond to those used in [1] for
the function Ψ = x +
1
s−1x2+ ···, namely, V (0,s) = 0, V′(0,s) = 0, and V′′(0,s) = −2ln2s, we find:
V (x,s) = −?ln2s?
U (x,s) = x2
1 +
an(s) xn
,a1=
U (x,s) ,(10)
?
∞
?
n=1
?
2
1 − s,
a2=
5 − 3s
(s − 1)2(s + 1)
,··· .(11)
The higher coefficients in the expansion are determined recursively by
an+2=
1
(1 − sn+2)

4an+1− 4an+

n+1
?
j=1+⌊n−1
2 ⌋
(−1)n−jajsj
?
j + 2
n + 2 − j
?



for n ≥ 1,(12)
where ⌊···⌋ is the floor function. In principle, this series solves (9) for any s, within its radius of convergence.
Based on numerical studies, we infer the radius of convergence [8] of the series depends on s as follows:
R(s) =
1
|an(s)|1/n? =
lim
n→∞sup
?



















1
2
if 0 < s ≤2
3,
????1 −1
4
s
????if
2
3≤ s ≤ 2 ,
s
if 2 ≤ s ≤ 4 .
(13)
For 0 < s ≤ 2/3, and also for 2 ≤ s ≤ 4, the |an| are monotonic for large n and the radius immediately follows
either from the limsup expression in (13) or from the simple ratio test, R(s) = lim
n→∞
|an−1(s)/an(s)|.But for
2/3 < s < 2, there is spiky behavior in |an−1/an| for intervals in n, which makes it difficult to use the simple ratio
test to determine R. This is because the |an(s)| are not monotonic functions of n for these values of s. Occasionally
the coefficients become small before changing sign. Fortunately, the limsup expression for R circumvents this spiky
behavior to yield the values given in (13) for all s.
It is not difficult to work out explicit series results for U (x,s), for generic s, say to O?x12?.
described.Based on those explicit results, we infer that the series involve numerator polynomials, pn(s), of order
1 + (n − 2)(n − 1)/2 in s, as well as “s-factorials” in the following form:
?
Such polynomials
in x are sufficient approximations to obtain the graphics to follow, when augmented with functional methods to be
(s − 1)2U (x,s) = x2
(s − 1)2− 2(s − 1)x +
∞
?
n=2
pn(s)
[n]s!
xn
?
.(14)
Here, deformed integers and factorials are defined by: [k]s=sk−1
the polynomials follows from that for an(s). It involves a mix of ordinary and deformed integers:
s−1, and [n]s! =?n
k=1[k]s. The recursion relation for
pn+2(s) =
1
1 − s

4pn+1(s) − 4[n + 1]spn(s) + [n + 1]s!

n+1
?
j=1+⌊n−1
2 ⌋
(−1)n−j
[j]s!
?
j + 2
n + 2 − j
?
sjpj(s)

 .

(15)
2

Page 3

This recursion relation is seeded by
p1(s) = 2(1 − s) ,
1−sprefactor in (15) is always canceled.
p2(s) = 5 − 3s .(16)
As written, it looks rather miraculous that the
follows from lims→1pn(s), but we have not yet determined an elegant proof of this fact.
As originally obtained by Schr¨ oder, there are three closed-form solutions known for Ψ, for s = −2, 2, and 4. These
are:
√3
62
Ψ(x,2) = −1
Ψ(x,4) =?arcsin√x?2,
Note that while the Ψ are multi-valued the inverse functions are all single-valued.
expressions for
1
Nevertheless, it is.This
Ψ(x,−2) =
?
2π − 3arccos
?
Ψ−1(x,2) =1
Ψ−1(x,4) =?sin√x?2.
x −1
??
,Ψ−1(x,−2) =1
2− cos
?2x
√3+π
3
?
,
2ln(1 − 2x) ,
2
?1 − e−2x?
,
(17)
The corresponding closed-form
U (x,s) =
?
Ψ(x,s)
dΨ(x,s)/dx
?2
(18)
are also multi-valued and follow immediately:
U (x,s = −2) =
U (x,s = 2) =1
U (x,s = 4) = x(1 − x)arcsin2√x .
1
36(1 + 2x)(3 − 2x)
?
2π − 3arccos
?
x −1
2
??2
, (19)
4(1 − 2x)2ln2(1 − 2x) ,(20)
(21)
Were we to start from these expressions for U, we could recover Ψ by solving and integrating (18). Another way to
express the result for s = 4 is similar in form to that for s = −2, namely,
U (x,s = 4) =1
4x(1 − x)(π − arccos(2x − 1))2.(22)
Indeed, it is well-known that the logistic maps for s = 4 and s = −2 are intimately related through the functional
conjugacy of the underlying Schr¨ oder equations. (See Appendix A.)
III.OBTAINING THE SWITCHBACK POTENTIALS
The sequence of switchback potentials, i.e. the various branches of the analytic potential function, can be obtained
from the functional equation for the potential. (The same procedure also works to give the branch structure of the
Schr¨ oder Ψ function. For an example, see Appendix B.) This follows from (8) and (18), namely,
U (sx(1 − x)) = s2(1 − 2x)2U (x) .
?
(23)
Next we write y = sx(1 − x), so then x± =
relation is just U (y) = s(s − 4y)U (x±). Now, we rename y → x to obtain
1
2
1 ±?1 − 4y/s
?
and (1 − 2x±)2= 1 − 4y/s.Thus the previous
x±=1
2±1
2
?
1 − 4x/s ,(24)
U±(x) ≡ s(s − 4x) U (x±) = s(s − 4x) U
?1
2±1
2
?
1 − 4x/s
?
.(25)
One of these potentials (U−) reproduces the original series for the primary potential expanded about x = 0, as may be
seen by direct comparison of the series or by numerical evaluation. Alternatively, the other (U+) gives the potential
on another sheet of the function’s Riemann surface, hence the first switchback potential for s > 2.
checked against the closed-form results for the s = 4 case.
Since U− has built into it zeroes at both x = 0 and xmax = s/4, it is actually a more useful form than just
the series about x = 0.When 2 < s ≤ 4, the series has radius of convergence R(s) = s/4.
This is easily
However, from
3

Page 4

using U− instead of the direct series, we have convergence over the whole closed interval, x ∈ [0,s/4], since then
0 ≤1
region of convergence of its series about zero.That is to say, first construct the series for U, and then from that
series build U−. Finally, identify this with U0, the primary potential in the sequence.
Similarly, U+may be identified with the first switchback in the sequence, U1, but for better convergence properties
it is useful to build U+from U0instead of U. By doing this when 2 < s ≤ 4, we have convergence over the whole
closed sub-interval, x ∈?1
?1
2−1
2
?1 − 4x/s ≤ 1/2 < R(s) when 2 < s ≤ 4. Thus we need only evaluate U appearing in U−within the
16s2(4 − s),s/4?, with zeroes of U1built-in at the end-points of the interval.
2−1
?1
In the first of these expressions, U is given by the direct series solution of the functional equation (23).
Now, go through this procedure all over again, beginning with U (sy(1 − y)) = s2(1 − 2y)2U (y).
z = sy (1 − y).
wherex±±
2
?
So the primary potential and the first switchback are well-represented by
U0(x) = s(s − 4x) U
2
?
?
1 − 4x/s
?
?
,(26)
U1(x) = s(s − 4x) U0
2+12
1 − 4x/s.(27)
Let
Then y± =
1
?
?
1
2
?
1 ±?1 − 4z/s
?
and U (z) = s(s − 4z)U (y±) = s(s − 4z)s(s − 4y±)U?(x±)±
any combinationof±s
?
?
2
?
=1 ±?1 − 4y±/s
1 ±?1 − 4z/s
?
with
?
allowed.ThereforeU (z)=
s(s − 4z)ss − 2
??
× U
1
2±1
2
1 −2
s
?
1 ±?1 − 4z/s
?
??
. Again, rename z → x to obtain
U±±(x) = s(s − 4x)U±(x±) = s(s − 4x)s
?
s − 2
?
1 ±
1 − 4x/s
??
U
1
2±1
?
1 −2
s
?
1 ±
?
1 − 4x/s
??
, (28)
with any combination of ±s allowed. And to improve the convergence properties of this expression, replace U on the
RHS with U0.
This process may be continued indefinitely, through successive application of the basic substitution U (x) →
s(s − 4x) U (x±).
etc. In general, the nth iteration of the procedure gives
For example, the next set of potentials in the sequence is U±±±(x) = s(s − 4x)U±±(x±),
U± ± ...±
?
???
n times
(x) = s(s − 4x)U± ± ...±
?
???
n-1 times
(x±) .(29)
Finally, at each iteration, we must select appropriate switchback potentials out of the 2ndifferent expressions.
particular, we note that many of the U±±...±(x) will be complex-valued for the x intervals under consideration, and
therefore they are not of immediate interest since they do not govern the particle’s evolution along the real axis. (The
continuation of the particle trajectory into the complex plane is outside the scope of this paper.)
In
IV.NUMERICAL EXAMPLES
A.The potential sequence for 2 < s ≤ 3
These are values of the parameter for which the discrete logistic map converges to a single fixed point: There are
no bifurcations.Nevertheless, there are two sign choices when the functional equation for the potential is applied
once, four choices when it is applied twice, and so on.
V±(x,s) = s(s − 4x)V0
?
1
2±
?1
4−x
s,s
?
(30)
V±±(x,s) = s(s − 4x)s
?
s − 2
?
1 ±
?
1 − 4x/s
??
V0

1
2±
?
?
?
?1
4−1
s
?
1
2±
?1
4−x
s
?
,s

.(31)
As it turns out from numerical studies, for 2 < s ≤ 3 only the positive sign choices are needed to produce the sequence
of switchback potentials.
V1(x,s) = s(s − 4x)V0
?
1
2+
?1
4−x
s,s
?
,V2(x,s) = s(s − 4x)V1
?
1
2+
?1
4−x
s,s
?
,(32)
4

Page 5

etc. In general, there is a recursion relation,
Vn+1(x,s) = s(s − 4x)Vn
?
1
2+
?
1
4−x
s,s
?
.(33)
The evolving particle moves through this sequence of potentials in succession, with the potential index incrementing
up by one each time the particle encounters a turning point.
Consider the specific case s = 5/2. This is representative for 2 < s ≤ 3. For this case, the turning points converge
onto the nontrivial fixed point x∗= 1 − 1/s = 3/5. This is evident in the following graphs.
0.00.10.20.3 0.40.50.6
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
x
V
V0in red, V1in green, and V2in blue, for s = 5/2. The first upper turning point is x = 5/8. Subsequent lower and
upper turning points are obtained just by iterating the s = 5/2 logistic map, starting with x = 5/8, and are shown
on the next magnified graph as colored points.
0.590.600.610.62
-0.010
-0.005
0.000
x
V
V0in red, V1in green, V2in blue, and V3in orange, for s = 5/2. The first upper turning point is x = 5/8, the first
lower turning point is x =
by map iteration. The nontrivial fixed point is at x∗= 3/5, indicated by the black circle on the x axis.
75
128= 0.58594, the second upper turning point is x =19 875
32 768= 0.60654, etc., as obtained
As the zero-energy particle moves through this sequence of increasingly shallow, narrowing potentials, its average
speed decreases, giving the appearance of a dissipative system.
upon convergence into the fixed point at x = 3/5, energy is rigorously conserved through changes in the potential.
Insofar as the turning points are branch points for the corresponding analytic potential function, and the various
switchback potentials are just the values of that analytic function on the various sheets of its Riemann surface, this
is convincing numerical evidence for that function to have an infinite number of such branch points, for generic s.
Nevertheless, even as the particle motion subsides
5

Page 6

While of course there are many well-known functions with this property (for example, the inverses of Bessel functions,
J−1
s, or even for specific s, except in those very special cases where the number of branch points is one or two. Recall the
exact closed-form cases s = 2 and s = 4 have analytic potential functions with one and two branch points, respectively,
as evident in (20) and (21).
We may check numerically the transit times for each potential in the sequence:
expected uncertainties for truncation of the series (11). For example, using Mathematica to compute the initial series
(11) to 200th order and also to evaluate the numerical integrations, we find?5/8
75/128
1281241875/2147483648
5
2x(1 − x)??
of each of the potentials in succession, V0→ V1→ V2→ ···, thereby converging onto the fixed point at x = 3/5. The
numbers confirm that for cases with 2 < s ≤ 3, only positive roots are needed to obtain all the switchback potentials,
as previously indicated in (33).
n (x)), this would seem to account for the historical fact that closed-form solutions have not been found for generic
∆tn= 1, for n ≥ 1, within the
1/2
dx
√−V0= 1.000000,?5/8
2x(1 − x)??
75/128
dx
√−V1=
1.000000,?19875/32 768
etc. These transit times are consistent with the continuously evolving, zero-energy particle moving under the influence
dx
√−V2= 1.000000,?19875/32 768
dx
√−V3= 1.000000, etc., corresponding to the iterations
x=75/128=
x=1/2=
5
8,
5
2x(1 − x)??
x=5/8=
75
128,
5
2x(1 − x)??
19875
32768,
5
x=19 875/32768=
1281241 875
2147483 648,
B.The potential sequence family for 3 < s < 4
These are values of the parameter for which the discrete logistic map produces bifurcations and asymptotes to limit
cycles, rather than unique fixed points, and as s is increased, chaotic behavior erupts. Generally speaking, for these
values of s the analytic potentials driving the trajectories are more complicated than for s ≤ 3, and their branches
are not so easy to enumerate as they are encountered by the evolving particle. The potentials here may be grouped
into a family of sequences, where potentials with common upper and lower turning points constitute a single member
of the family.From one family member to the next, for fixed sequence index, the potentials exhibit a progressive
shallowing (similar to the behavior of the individual potentials for s = 5/2).
member successively deepen as the sequence index increases. Consequently, the locally averaged speed of the particle
moving through the potentials does not necessarily diminish, but often increases, giving the appearance of a driven
system. Nevertheless, energy is still rigorously conserved.
The limit cycle situation may be illustrated by the case s = 10/3, while the chaotic situation is aptly illustrated
by the uppermost value, s = 4.The latter case was fully discussed in [1], and originally led to the notion of the
switchback potentials.On the other hand, the potential structure for the s = 10/3 case is more complicated than
for s = 4, the latter having only one member for its family of potential sequences, and in fact s = 10/3 is much more
representative of the situation for generic parameter values in the range 3 < s < 4. Therefore, we discuss s = 10/3 in
some detail. This case evolves towards a two-cycle, consisting of the points
0.469722. The first few terms in the potential sequence (33) are shown here, for this particular s.
However, the V s for a given family
1
20
√13+13
20= 0.830278and13
20−1
20
√13 =
0.00.10.20.30.4 0.50.6 0.70.8
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
x
V
V0in red, V1in green, V2in blue, V3in orange, and V4in purple, for s = 10/3. The upper turning point is x = 5/6.
The two-cycle points are small black circles on the axis. The small black square is the nontrivial fixed point of the
map, x∗= 7/10, and is revisited after integer time steps, but it is not a fixed point for the continuous evolution.
Deepening of the V s is evident for this “mother sequence.” The turning points for this sequence do not converge in
this s = 10/3 case. Rather, they are fixed at5
6for the upper turning point, and25
54for the lower turning point, the
6

Page 7

latter obtained by acting on the former with the s =10
While these points are not far from the two-cycle points, as evident below, close does not count (except when playing
p´ etanque).
3discrete logistic map :
5
6= 0.833333 ?→25
54= 0.462963.
0.8200.8250.8300.8350.840
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
x
V
V0red, V1green, V2blue, V3orange, & V4purple, for
s = 10/3. The upper turning point is x =5
6.
0.460 0.4650.4700.4750.480
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
x
V
V1green, V2blue, V3orange, & V4purple, for s = 10/3.
The lower turning point is x =25
54.
Other sequences in the family of potentials are needed to achieve convergence of the continuous particle trajectory,
for unit time steps, onto the two-cycle of the map.
So, we define a second sequence in the family of potentials by changing the sign of the root in the arguments of
the potentials in the first sequence, while noting that W0(x,s) = s(s − 4x)V0
More interesting cases of these “offspring” potentials are given by
?
1
2−
?
1
4−x
s,s
?
just reproduces V0.
Wn(x,s) = s(s − 4x)Vn
?
1
2−
?1
4−x
s,s
?
,forn ≥ 1.(34)
These do not reproduce any of the previous potentials. We plot a few for the s = 10/3 case.
0.8290.8300.8310.8320.833
-0.003
-0.002
-0.001
0.000
x
W(x)
W1green, W2blue, W3orange, and W4purple, for s = 10/3.
For this second sequence the lower turning point can be obtained by acting with the discrete map on the lower turning
point for the first sequence, i.e.
4374= 0.828761, while the upper turning point remains the same
as for the first sequence, i.e.
6= 0.833333. Continuing the enumeration of the potential family members, we define
a third sequence of potentials, and we plot a few.
?
25
54= 0.462963 ?→3625
5
Xn(x,s) = s(s − 4x)Wn
1
2+
?
1
4−x
s,s
?
, forn ≥ 1,(35)
7

Page 8

0.4640.4660.4680.4700.472
-0.015
-0.010
-0.005
0.000
x
X(x)
X1green, X2blue, X3orange, and X4purple, for s = 10/3.
For this third sequence the lower turning point remains the same as for the first sequence, i.e.
the upper turning point can be obtained by acting with the discrete map on the lower turning point for the second
sequence, i.e.
28697 814= 0.473054.
This procedure may be continued indefinitely. We define fourth and fifth sequences as†
25
54= 0.462963, while
3625
4374= 0.828761 ?→13575 625
Yn(x,s) = s(s − 4x)Xn
?
?
1
2−
?1
?1
4−x
s,s
?
?
,(36)
Zn(x,s) = s(s − 4x)Yn
1
2+4−x
s,s
,(37)
etc., and we plot the first few of these new sequences.
0.8290.8300.831
-0.0006
-0.0004
-0.0002
0.0000
x
Y(x)
Y1green, Y2blue, and Y3orange, for s = 10/3.
†A more systematic notation would employ two potential indices, to denote an infinite matrix of potentials.
example, we could designate Vn= V(0)
the family number.
For
n , Wn= V(1)
n , Xn= V(2)
n , Yn= V(3)
n , Zn= V(4)
n , etc. The super-index here is
8

Page 9

0.4690.4700.4710.4720.473
-0.003
-0.002
-0.001
0.000
x
Z(x)
Z1green, Z2blue, and Z3orange, for s = 10/3.
The story here is somewhat analogous to, but more complicated than, the situation for 2 < s ≤ 3.
individual potentials in the sequence {Vn} converging onto the fixed point, as would be the case for 2 < s ≤ 3, here
each of the individual Vnhas proliferated into a “sideways” sequence of potentials, {Vn,Wn,Xn,Yn,Zn,···}, which
converge to the points in the 2-cycle. To show what we mean by this, we plot {V1,W1,Y1} and {V1,X1,Z1} near the
lower and upper points, respectively, in the two-cycle.
Instead of
0.464 0.4660.4680.470 0.472
-0.02
-0.01
0.00
x
V1, W1, and Y1near the lower point in the two-cycle for
s = 10/3.
0.8290.8300.8310.8320.833
-0.004
-0.002
0.000
x
V1, X1, and Z1near the upper point in the two-cycle
for s = 10/3.
This type of behavior is repeated by {Vn,Wn,Yn} and {Vn,Xn,Zn} for each n.
To complete the picture for the evolving particle, it is necessary to understand the order in which the potentials
act, i.e. to determine how the switches are made from one potential to another as the turning points are encountered
by the particle. Unlike the situation with s ≤ 3, here this can be a bit tedious.
various transit times (and also from the branch structure of the underlying Schr¨ oder auxiliary function, as exhibited in
Appendix B) we infer the order of the potentials for the s = 10/3 case to produce a chemin des ´ energies potentielles:
From a numerical examination of
V0→ V1
????
∆t=1
→ V2→ W1
?
???
∆t=1
→ W2→ V3→ X1
????
∆t=1
→ X2→ V4→ W3→ Y1
????
∆t=1
→ Y2→ W4→ V5→ X3→ Z1
????
∆t=1
→ ··· ,(38)
etc. Here we have also indicated how the potentials combine into groups with unit total transit time. For example,
again using Mathematica to compute the initial series (11) to 200th order and computing the transit times numerically,
we find:
?5/6
25/54
25/54
It is not difficult to extend (38) by defining additional sequence family members.
match-up for adjacent potentials along the path. Also note the family number plus the sequence number (i.e.
25/54
dx
√−V1= 1.000000;?5/6
25/54
dx
√−X1= 0.164433+ 0.661295+ 0.174272 = 1.000000; etc.
dx
√−V2+?5/6
3625/4374
dx
√−W1= 0.825728+0.174272= 1.000000;?5/6
3625/4374
dx
√−W2+
?5/6
dx
√−V3+?13575625/28697814
Note the turning points must
9

Page 10

N = m + n for the potential matrix element V(m)
group with total ∆t = 1, and this N increments by one as the particle moves from one potential group to the next.
n
of the last footnote) is the same for each potential belonging to a
V.CONCLUSION
The point of view supported in this paper is that the logistic map, and other discrete time-stepped dynamical
models, may be regarded as continuously evolving Hamiltonian systems sampled at integer times. For this view to be
valid, the continuous system must be allowed to undergo a series of switchbacks whereupon the potential affecting the
dynamics changes when the evolving particle encounters a turning point. From a perspective of configuration space
covering manifolds, in the case of simple one-dimensional motion, the particle moves from one sheet of a Riemann
surface to another, to experience a different branch of the underlying analytic potential.
The methods of this paper may be used directly to determine such branches of the potential for the logistic map,
for any value of s, as well as other one-dimensional maps.
[9].While peculiar behavior is possible for exceptional maps (say, for special values‡of s), so far as we are aware,
any such behavior can always be analyzed using the potential framework presented here.
In total, for all parameter values governing a particular map, the collection of potential sequence families constitute
what we may call a “potential fractal” with self-similarities qualitatively evident in the various graphs, as visible
in the above.Perhaps such potential fractals have a significant role to play in continuum physics.
might involve any of the usual systems exhibiting chaotic behavior [7], including accelerator beams [10], or perhaps
cosmological models [11].
A more extensive study of other examples is underway
Applications
Acknowledgments
We thank David Fairlie, Xiang Jin, Luca Mezincescu, and especially Cosmas Zachos, for sharing their thoughts
about functional evolution methods.One of us (TC) thanks the CERN Theoretical Physics Group for its gracious
hospitality and generous support. The numerical calculations and graphics in this paper were made using MapleR ?,
MathematicaR ?, and MuPADR ?. This work was also supported by NSF Award 0855386.
[1] T. Curtright and C. Zachos, J. Phys. A: Math. Theor. 42, 485208 (2009), arXiv:0909.2424 [math-ph];
T. L. Curtright and C. K. Zachos, “Chaotic Maps, Hamiltonian Flows, and Holographic Methods” to appear in a well-known
journal, eventually, arXiv:1002.0104 [nlin.CD].
[2] E. Schr¨ oder, Math. Ann. 3, 296-322 (1871).
[3] R. M. May, Nature 261, 459-467 (1976).
[4] P. Collet and J. P. Eckmann, Iterated Maps On The Interval As Dynamical Systems, Birkh¨ auser, Boston (1980).
[5] M. J. Feigenbaum, Physica D 7, 16-39 (1983).
[6] L. Kadanoff, Physics Today, 46-53 (December 1983).
Also see http://en.wikipedia.org/wiki/Logistic map .
[7] P. Cvitanovi´ c, et al., Chaos: Classical and Quantum, http://chaosbook.org/ .
[8] For example, see http://mathworld.wolfram.com/RadiusofConvergence.html .
[9] T. Curtright and A. Veitia, in preparation.
[10] S. Y. Lee, Accelerator Physics, World Scientific (2004).
[11] G. C. Corrˆ ea, T. J. Stuchi, and S. E. Jor´ as, Phys. Rev. D81, 083531 (2010), arXiv:1005.3273 [gr-qc].
‡A peculiar example is provided by s = 1. For this case the appropriate limit of (11) is not convergent but rather
an asymptotic series. (See Appendix C.)
10

Page 11

VI.APPENDIX A: FUNCTIONAL CONJUGATION
Functional conjugacy for the logistic map may be expressed in terms of a linear function. Namely,
g (sx(1 − x)) = (2 − s)g (x)(1 − g(x)) , (39)
where
g (z) =
1
2 − s(1 − s + sz) ,g−1(z) =1
s(s − 1 + (2 − s)z) .(40)
It is often useful to refer to these maps as g(z) = gs(z) and g−1(z) = g2−s(z). Pursuing the conjugacy a bit farther,
let fs(x) = sx(1 − x), then the conjugacy equation for the map can be written in various ways:
g ◦ fs= f2−s◦ g ,
g ◦ fs◦ g−1= f2−s,
Moreover, Schr¨ oder’s equation is the functional composition
fs◦ g−1= g−1◦ f2−s,
g−1◦ f2−s◦ g = fs.
(41)
(42)
s ◦ Ψ = Ψ ◦ fs,(43)
where s : x → sx is the simple multiplicative map. Under functional conjugacy by g, the RHS of Schr¨ oder’s equation
becomes
g ◦ Ψ ◦ fs◦ g−1= g ◦ Ψ ◦ g−1◦ f2−s= Ψg◦ f2−s
Ψg≡ g ◦ Ψ ◦ g−1
(44)
(45)
A. Alternate fixed point expansions
Consider again the logistic map, x ?→ sx(1 − x). On the one hand, with a subscript to distinguish other construc-
tions to follow, expansions about the trivial fixed point at x = 0 stem from:
sΨ0(x,s) = Ψ0(sx(1 − x),s) ,
Φ0(sx,s) = sΦ0(x,s)(1 − Φ0(x,s)) .
(46)
(47)
For the inverse Schr¨ oder functions, we consistently use the notation Φ = Ψ−1. On the other hand, expansions about
the non-trivial fixed point at x = 1 − 1/s stem from:
λΨ∗(z) = Ψ∗
Φ∗(λz) = λΦ∗(z) + (λ − 2)Φ2
where
?λz + (λ − 2)z2?
,(48)
∗(z) ,(49)
λ = 2 − s ,s = 2 − λ ,x = 1 −1
s+ z ,z = x +1 − s
s
.(50)
The two expansions produce the same functions, only slightly disguised. Here is the detailed relation between them,
to be proved in the next subsection.
Ψ∗(z) =
λ
2 − λΨ0
?2 − λ
λ
z,λ
?
,i.e.Ψ∗
?
x +1 − s
s
?
=2 − s
s
Ψ0
?
s
2 − s
?
x +1 − s
s
?
,2 − s
?
.(51)
Φ∗(z) =
λ
2 − λΦ0
?2 − λ
λ
z,λ
?
,i.e.Φ∗
?
x +1 − s
s
?
=2 − s
s
Φ0
?
s
2 − s
?
x +1 − s
s
?
,2 − s
?
.(52)
These relations are an extension of the previously known functional conjugacy that relates s = 4 and s = −2.
It is better notation to call the alternate series solutions Ψ∗(x,s) and Φ∗(x,s), instead of Ψ∗(z) and Φ∗(z). Then
the “dual” parameter is
s∗= 2 − s ,(53)
11

Page 12

and the previous relations are more succinctly written as:
Ψ∗(x,s) =s∗
s
Ψ0
?sx
s∗,s∗
?
,Φ∗(x,s) =s∗
s
Φ0
?sx
s∗,s∗
?
.(54)
That is to say, the series solution of Schr¨ oder’s equation for map parameter 2 − s, about the trivial fixed point at
x = 0, is conjugate to the series solution of Schr¨ oder’s equation for map parameter s, about the nontrivial fixed
point at x∗ = 1 − 1/s.
g ◦ s ◦ g−1(u) = g(s − 1 + (2 − s)u) =(1−s)2
g ◦ s ◦ g−1◦ Ψg(w,s) =(1 − s)2
Now, as previously noted, functional conjugation by g gives (44) and (45).
+ su , so
But then,
2−s
2 − s
+ sΨg(w,s) .(55)
That is to say, the g-conjugated Schr¨ oder function obeys the inhomogeneous functional equation,
(1 − s)2
2 − s
+ sΨg(x,s) = Ψg((2 − s)x(1 − x),s) , (56)
whereas Ψ∗obeys a homogeneous equation, but with shifted position-dependent arguments.
?
To summarize the relations between the various functions:
(2 − s)Ψ∗
x +1 − s
s
,s
?
= Ψ∗
?
sx(1 − x) +1 − s
s
,s
?
. (57)
Ψ0(x,s) =2 − s
s
Ψ∗
?
?
sx
2 − s,2 − s
sx
2 − s,2 − s
?
?
s
2 − sΨ0
+2 − s
s
?
?
,(58a)
Ψ∗(x,s) =2 − s
s
Ψ0
, (58b)
Ψg(x,s) =1 − s
2 − s+ Ψ∗
x +s − 1
2 − s,2 − s
x +s − 1
s
?s − 1 + (2 − s)x
?1 − s + sx
?
?
(58c)
Ψ∗(x,s) =1 − s
s
+ Ψg
,2 − s(58d)
Ψg(x,s) =1 − s
2 − s+
s
,s
?
, (58e)
Ψ0(x,s) =s − 1
s
Ψg
2 − s
,s
?
,(58f)
For the potential itself, functional conjugation leads to
V (x,s) =
?2 − s
?(2 − s)ln(2 − s)
s
?2
V
?
s
2 − s
?
x −
?
1 −1
s
?
??
,2 − s
?
?
,(59)
U (x,s) =
slns
?2
U
?
s
2 − s
x −1 −1
s
??
,2 − s
?
.(60)
Perhaps it is worthwhile to work out the various functions explicitly for the cases which can be solved in closed-
form, with Ψ0and Φ0= Ψ−1
0
as given in (17). Consider the first and the last of these. The corresponding dual and
conjugated functions are given by:
Ψ∗(x,4) = −1
2Ψ0(−2x,−2) = −
√3
12
?
?
2π − 3arccos
?
?
√3
12(2π − 3arccos(2x − 1)) .
?
2x +1
2
??
, (61a)
Ψ∗(x,−2) = −2Ψ0
?
−1
2x,4
?
= −2arcsin
−1
2x
?2
?
,(61b)
Ψg(x,4) =3
2+ Ψ∗
?
?
x −3
2,−2
?
=3
=3
2− 2arcsin
1
2
?3
2− x
??2
,(61c)
Ψg(x,−2) =3
4+ Ψ∗
x −3
4,4
?
4−(61d)
We plot these, say for x such that the arguments of the inverse trigonometric functions are real, and the functions are
principal-valued.
12

Page 13

1
-2
2
x
Psi(4)
Ψ0(x,4) red, Ψ∗(x,4) blue, and Ψg(x,4) green.
-2-11
-4
-2
2
x
Psi(-2)
Ψ0(x,−2) red, Ψ∗(x,−2) blue, and Ψg(x,−2) green.
B.Alternate series solution theorem
Consider the Poincar´ e functional equation for the Schr¨ oder function inverse, rendered to facilitate expansion about
the nontrivial fixed point “∗” corresponding to x∗= 1 − 1/s,
Φ∗((2 − s)z,s) = (2 − s)Φ∗(z,s) − sΦ2
where
∗(z,s) withΦ∗(z,s) = z + O?z2?
, (62)
x = 1 −1
s+ z ,z = x +1
s− 1 .(63)
Let
Φ∗(z,s) =2 − s
s
φ(z,s) , (64)
to change the functional equation to
2 − s
s
φ((2 − s)z,s) =(2 − s)2
s
φ(z,λ)(1 − φ(z,λ)) .(65)
That is to say,
φ((2 − s)z,s) = (2 − s)φ(z,s)(1 − φ(z,s)) withφ(z,s) =
sz
2 − s+ O?z2?
. (66)
Comparing this last relation to the original Poincar´ e functional equation tailored to yield the expansion about the
trivial fixed point “0”,
Φ0(sx,s) = sΦ0(x,s)(1 − Φ0(x,s))withΦ0(x,s) = x + O?x2?
,(67)
it follows that
φ(z,s) = Φ0
?
sz
2 − s,2 − s
?
.(68)
So then
Φ∗(z,s) =2 − s
s
Φ0
?
sz
2 − s,2 − s
?
,Φ0(z,s) =2 − s
s
Φ∗
?
sz
2 − s,2 − s
?
,(69)
and therefore the Schr¨ oder function, i.e. the inverse of the inverse function Ψ = Φ−1, is
Ψ∗(z,s) =2 − s
s
Ψ0
?
sz
2 − s,2 − s
?
,Ψ0(z,s) =2 − s
s
Ψ∗
?
sz
2 − s,2 − s
?
.(70)
Hence Φ∗(Ψ∗(z,s),s) = z = Ψ∗(Φ∗(z,s),s). This proves the relationships between Φ∗,Ψ∗and Φ0,Ψ0. ?
13

Page 14

VII.APPENDIX B: CONSTRUCTING THE BRANCHES OF SCHR¨ ODER’S Ψ FUNCTION
We wish to imitate for Ψ(x,s) what we did to find the branches of the potential in Section III of the text. We start
with the functional equation, (8). Next we write y = sx(1 − x), with solutions x±=1
rename y → x, so the functional relation becomes
?1
One of these (Ψ−) reproduces the original auxiliary function expanded about x = 0, only it does so more accurately
for x → s/4 as may be seen by numerical evaluation, while the other (Ψ+) gives the auxiliary on another sheet of
the function’s Riemann surface. This is useful for determining the first switchback potential. The process may be
repeated to get the other branches of the auxiliary as a sequence of functions, Ψn.
Here we are especially interested in s = 10/3, as we wish to confirm the order in which the switchback potentials
are encountered by the evolving particle for this case. We find the following numerical results.
2
?
1 ±?1 − 4y/s
?
. Now, we
Ψ±(x,s) = sΨ
2
?
1 ±
?
1 − 4x/s
?
,s
?
.(71)
0.0 0.10.2 0.30.40.50.60.70.8
0
10
20
30
40
50
60
70
80
90
100
x
Psi
Eight branches of Ψ(x,10/3).
It follows that the order of the potentials is as given in the text: V0→ V1→ V2→ W1→ W2→ V3→ X1→ X2→ ···.
As a practical matter, it is somewhat easier to compute the inverse function Φ ≡ Ψ−1by combining series solution
14

Page 15

methods with functional extensions. The functional equation for the inverse of Schr¨ oder’s function is
Φ(sx,s) = sΦ(x,s)(1 − Φ(x,s)) .(72)
For s < 1 this quadratic equation may be solved for Φ(x) in terms of Φ(sx),
Φ±(x,s) =1
2
?
1 ±
?
1 −4
sΦ(sx,s)
?
, (73)
and, upon iteration, all values of Φ may be obtained from those values given accurately by the series about x = 0.
For s > 1, on the other hand, we first rescale x in (72) to write
Φ(x,s) = sΦ
?x
s,s
??
1 − Φ
?x
s,s
??
,(74)
and from this equation, upon iteration, all values of Φ may be obtained directly from those values given accurately
by the series about x = 0.
As an example, consider again the case s = 10/3. We find:
0102030 4050 6070 80 90 100
x
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Phi
Seventh order series approximation to Φ(x,10/3), in black, compared to results from using this series in the 1st
(red), 2nd (blue), and 3rd (green) iterates of the functional extension (74).
Flipping this graph about the SW-NE diagonal, we obtain a graph for the multi-valued Ψ = Φ−1.
iterate of (74) (as given in the graph by the green curve) the resulting curve for Ψ(x,10/3) agrees with the previous
graph obtained by solving Schr¨ oder’s equation through use of the series about x = 0, combined with functional
methods.
From the third
VIII.APPENDIX C: THE s = 1 SERIES FOR THE POTENTIAL
For this case the appropriate limit of (11) gives
V (x,1) ≡ − lim
s→1
??ln2s?U (x,s)?
= −x4− 2x5− 4x6−25
3x7−215
12x8−589
15x9−7813
90
x10−60481
315
x11−11821
28
x12+ O?x13?
.(75)
15