Continuous iterates continue to be interesting, after 150 years
As a first illustration, we display the continuous iterates of the
sine function, sin[t](x). Note that the
maximum values at x = π/2 are approximately given by
Evolution surfaces and Schröder functional methods
Including time-sliced progressions for selected examples.
This page (© Thomas Curtright
2010, 2011) is based on concepts developed in T. Curtright and C. Zachos, J. Phys.
A: Math. Theor. 42
(2009) 485208 and J. Phys. A: Math. Theor. 43 (2010) 445101 (Cf. arXiv:0909.2424 [math-ph] and arXiv:1002.0104 [nlin.CD])
. Also see T. Curtright and A.
Veitia, Phys. Lett. A 375
(2011) 276-282 (arXiv:1005.5030 [math-ph]), T. Curtright and C.
Zachos, Phys. Rev. D 83
(2011) 065019 (arXiv:1010.5174 [hep-th]), T. Curtright, SIGMA 7 (2011) 042 (arXiv:1011.6056
[math-ph]), and T. Curtright, X. Jin, and C. Zachos, J. Phys. A:
Math. Theor. 44 (2011)
harmonic oscillator interpolates, for fixed energy, 1 = v2
+ x2 .
potential: V(x) = x4 - x2,
zero energy continuous interpolates.
Goldstone potential: Right-moving (solid green) and
left-moving (dashed green) zero-energy profiles for V(x) = - ( 1 - x2
)2 (shown in orange).
Schröder example: Continuous interpolates of
x → 2x(1+x), and its inverse.1,3
Schröder example: Continuous interpolates of x →
4x(1-x) (shown in orange) evolving under the influence of the
indexed potentials VP(x) = (ln4)2 x(x-1) (
(-1)P Floor[(1+P)/2] π +(arcsin(√x ) )2 with P
= 0 plotted in green.2,3 The switchback effect4
is evident. As a function of the distance traveled by a
particle moving through the sequence of switchbacks, the VP potentials
patch together to give a progressively deepening, single-valued but
cusped function, V(X), on the real half-line X ≥ 0 cover of the unit
interval, as shown
here. Also, energy is conserved for a particle moving
through the switchbacks only if the potentials are indeed switched,
as is clear from plotting E(t) = v2(x(t)) + VP(x(t)),
here for P = 0, 1, 2, 3, and 4.
More generally, for the map x → s x(1-x) , the Schröder
auxiliary function, Ψ(x), and its inverse, Φ(x) ≡ Ψ-1(x),
may be constructed as formal series in x for any s ≠ 1. A
simple Mathematica® program to do this is shown here (in PDF, the
actual code is here).
These formal series become rather unwieldly after the first ten or
so terms, i.e. beyond O(x10). However, for specific
numerical choices of s, the series can be constructed easily to
include several hundred terms. This can again be done using
Mathematica® as shown here (in PDF, the code
is here) for s =
2, one of the cases solvable in closed-form, with the result used to
plot the effective potential V(x) = - ( ln(s) Ψ(x)/Ψ'(x) )2
and compare it to the exact answer. By modifying the latter
code, one can acquire a sense of what happens for intermediate
values, 2 < s < 4. (But one must keep in mind the
default accuracy for decimal real numbers in Mathematica®
is just machine precision, say 16 digits, while arbitrarily high
precision arithmetic will be invoked if values of s are taken as
fractions of integers. So, the choice s = 3.5 will produce erratic results
for the coefficients after about 80 or 90 terms in the series, but
the choice s = 7/2 gives smooth
behavior for several hundred coefficients, or beyond, thereby
permitting computation of the effective potential as well as
allowing the radius of convergence of the series to be accurately
established by the ratio test.)
Alternatively, the potential may be constructed directly as a
solution of the functional equation which it inherits from Ψ. The
theory behind this is developed here.
example: Continuous interpolates of x → x exp(x)
and its inverse, LambertW(x). The splinter of this example is
also known as the Ricker model5
introduced in the 1950s to understand fish populations.
Mathematica® routines to compute Ψ(x), Ψ'(x), and
V(x) for the map x → s x exp(x), with parameter s, are given here for generic s and here for specific
numerical choice of s (in PDF, with the codes here and here, respectively).
Actually, a more standard parameterization of the Ricker model is x
→ k x exp(-x)
for parameter k, where
x is now the population, and therefore x > 0 is the region of
interest. This exhibits period doubling for large k, as is evident from iterating the
map as well as from the behavior of the interpolating Ricker
trajectories, a few of which are shown here.
An animation of the tent
map interpolation (for μ = 2) is given here.
And to close with an image "reflective" of the sine map above,
consider the s = 1 logistic map surface.
1 After change of
variables, this is familiar as the s = 2 special case of the logistic map,
namely, xn+1 = 2 xn(1-xn), a
well-known case that can be solved exactly for xn.
2 This is familiar as the s = 4 logistic map, xn+1
= 4 xn(1-xn), another well-known case that can
be solved exactly.
3 For a thorough discussion of the s = 4 logistic map,
among other things, see J.V. Whittaker, "An Analytical Description
of Some Simple Cases of Chaotic Behaviour" in The American Mathematical Monthly,
Vol. 98, No. 6 (Jun. - Jul., 1991), pp. 489-504. However, this
paper does not cite E. Schröder who found and
published the exact solutions of the s = 2 and s = 4 maps in the
4 T. Curtright and C. Zachos, J. Phys. A: Math. Theor. 43 (2010) 445101. ANL-HEP-PR-10-3 and
5 W. E. Ricker, Computation
interpretation of biological statistics of fish populations.
Ottawa: Department of the Environment, Fisheries and Marine Service