Continuous iterates continue to be interesting, after 150 years of study.

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 exp[(1-√t) ln(π/2)].

sine iterates

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) 405205 (arXiv:1105.3664 [math-ph]).

Simple harmonic oscillator interpolates, for fixed energy, 1 = v2 + x2 .


Goldstone potential:  V(x) = x4 - x2,  zero energy continuous interpolates.

Inverted Goldstone potential:  Right-moving (solid green) and left-moving (dashed green) zero-energy profiles for V(x) = - ( 1 - x2 )2  (shown in orange).

First Schröder example:  Continuous interpolates of  x → 2x(1+x), and its inverse.1,3

Second 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)), shown 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.

Lambert 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 early 1870s.
4 T. Curtright and C. Zachos, J. Phys. A: Math. Theor. 43 (2010) 445101. ANL-HEP-PR-10-3 and UMTG-13, arXiv:1002.0104 [nlin.CD].
5 W. E. Ricker, Computation and interpretation of biological statistics of fish populations. Ottawa: Department of the Environment, Fisheries and Marine Service (1975).