Schröder's equation

Ernst Schröder (1841–1902) in 1870 formulated his eponymous equation.

Schröder's equation, named after Ernst Schröder, is a functional equation with one independent variable: given the function Template:Math, find the function Template:Math such that

$\mathrm{\forall }x\mathrm{\Psi }(h(x))=s\mathrm{\Psi }(x).$

Schröder's equation is an eigenvalue equation for the composition operator Template:Math that sends a function Template:Math to Template:Math.

If Template:Mvar is a fixed point of Template:Mvar, meaning Template:Math, then either Template:Math (or Template:Mvar) or Template:Math. Thus, provided that Template:Math is finite and Template:Math does not vanish or diverge, the eigenvalue Template:Mvar is given by Template:Math.

Functional significance

For Template:Math, if Template:Mvar is analytic on the unit disk, fixes Template:Math, and Template:Math, then Gabriel Koenigs showed in 1884 that there is an analytic (non-trivial) Template:Math satisfying Schröder's equation. This is one of the first steps in a long line of theorems fruitful for understanding composition operators on analytic function spaces, cf. Koenigs function.

Equations such as Schröder's are suitable to encoding self-similarity, and have thus been extensively utilized in studies of nonlinear dynamics (often referred to colloquially as chaos theory). It is also used in studies of turbulence, as well as the renormalization group.

An equivalent transpose form of Schröder's equation for the inverse Template:Math of Schröder's conjugacy function is Template:Math. The change of variables Template:Math (the Abel function) further converts Schröder's equation to the older Abel equation, Template:Math. Similarly, the change of variables Template:Math converts Schröder's equation to Böttcher's equation, Template:Math.

Moreover, for the velocity, Template:Math,   Julia's equation,   Template:Math, holds.

The Template:Math-th power of a solution of Schröder's equation provides a solution of Schröder's equation with eigenvalue Template:Math, instead. In the same vein, for an invertible solution Template:Math of Schröder's equation, the (non-invertible) function Template:Math is also a solution, for any periodic function Template:Math with period Template:Math. All solutions of Schröder's equation are related in this manner.

Solutions

Schröder's equation was solved analytically if Template:Mvar is an attracting (but not superattracting) fixed point, that is Template:Math by Gabriel Koenigs (1884).

In the case of a superattracting fixed point, Template:Math, Schröder's equation is unwieldy, and had best be transformed to Böttcher's equation.

There are a good number of particular solutions dating back to Schröder's original 1870 paper.

The series expansion around a fixed point and the relevant convergence properties of the solution for the resulting orbit and its analyticity properties are cogently summarized by Szekeres. Several of the solutions are furnished in terms of asymptotic series, cf. Carleman matrix.

Applications

First five half periods of the phase-space orbit of the s = 4 chaotic logistic map Template:Math, interpolated holographically through Schröder's equation. The velocity Template:Math plotted against Template:Math. Chaos is evident in the orbit sweeping all Template:Mvars at all times.

It is used to analyse discrete dynamical systems by finding a new coordinate system in which the system (orbit) generated by h(x) looks simpler, a mere dilation.

More specifically, a system for which a discrete unit time step amounts to Template:Math, can have its smooth orbit (or flow) reconstructed from the solution of the above Schröder's equation, its conjugacy equation.

That is, Template:Math.

In general, all of its functional iterates (its regular iteration group, see iterated function) are provided by the orbit

$h_t(x)={\mathrm{\Psi }}^{-1}(s^t\mathrm{\Psi }(x)),$

for Template:Mvar real — not necessarily positive or integer. (Thus a full continuous group.) The set of Template:Math, i.e., of all positive integer iterates of Template:Math (semigroup) is called the splinter (or Picard sequence) of Template:Math.

However, all iterates (fractional, infinitesimal, or negative) of Template:Math are likewise specified through the coordinate transformation Template:Math determined to solve Schröder's equation: a holographic continuous interpolation of the initial discrete recursion Template:Math has been constructed; in effect, the entire orbit.

For instance, the functional square root is Template:Math, so that Template:Math, and so on.

For example, special cases of the logistic map such as the chaotic case Template:Math were already worked out by Schröder in his original article (p. 306),

Template:Math, Template:Math, and hence Template:Math.

In fact, this solution is seen to result as motion dictated by a sequence of switchback potentials, Template:Math, a generic feature of continuous iterates effected by Schröder's equation.

A nonchaotic case he also illustrated with his method, Template:Math, yields

Template:Math, and hence Template:Math.

Likewise, for the Beverton–Holt model, Template:Math, one readily finds Template:Math, so that

$h_t(x)={\mathrm{\Psi }}^{-1}(2^{-t}\mathrm{\Psi }(x))=\frac{x}{2^t+x(1-2^t)}.$

See also

• Böttcher's equation