Schröder's equation,
named after
Ernst Schröder, is a
functional equation
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
with one
independent variable
A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical function ...
: given the function , find the function such that
Schröder's equation is an eigenvalue equation for the
composition operator that sends a function to .
If is a
fixed point of , meaning , then either (or ) or . Thus, provided that is finite and does not vanish or diverge, the
eigenvalue
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
is given by .
Functional significance
For , if is analytic on the unit disk, fixes , and , then
Gabriel Koenigs showed in 1884 that there is an analytic (non-trivial) 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 In mathematics, the Koenigs function is a function arising in complex analysis and dynamical systems. Introduced in 1884 by the French mathematician Gabriel Koenigs, it gives a canonical representation as dilations of a univalent function, univalent ...
.
Equations such as Schröder's are suitable to encoding
self-similarity
In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar ...
, and have thus been extensively utilized in studies of
nonlinear dynamics
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathem ...
(often referred to colloquially as
chaos theory
Chaos theory is an interdisciplinary area of Scientific method, scientific study and branch of mathematics. It focuses on underlying patterns and Deterministic system, deterministic Scientific law, laws of dynamical systems that are highly sens ...
). It is also used in studies of
turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers with no disruption between ...
, as well as the
renormalization group
In theoretical physics, the renormalization group (RG) is a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the underlying p ...
.
An equivalent transpose form of Schröder's equation for the inverse of Schröder's conjugacy function is . The change of variables (the
Abel function) further converts Schröder's equation to the older
Abel equation, . Similarly, the change of variables converts Schröder's equation to
Böttcher's equation, .
Moreover, for the velocity,
, ''
Julia's equation'', , holds.
The -th power of a solution of Schröder's equation provides a solution of Schröder's equation with eigenvalue , instead. In the same vein, for an invertible solution of Schröder's equation, the (non-invertible) function is also a solution, for ''any'' periodic function with period . All solutions of Schröder's equation are related in this manner.
Solutions
Schröder's equation was solved analytically if is an attracting (but not superattracting)
fixed point, that is by
Gabriel Koenigs (1884).
In the case of a superattracting fixed point, , 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
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation t ...
, cf.
Carleman matrix.
Applications
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 , can have its smooth
orbit
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
(or
flow) reconstructed from the solution of the above Schröder's equation, its
conjugacy
equation.
That is, .
In general, ''all of its functional iterates'' (its ''regular iteration
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
'', see
iterated function
In mathematics, an iterated function is a function that is obtained by composing another function with itself two or several times. The process of repeatedly applying the same function is called iteration. In this process, starting from some ...
) are provided by the orbit
for real — not necessarily positive or integer. (Thus a full
continuous group
In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...
.)
The set of , i.e., of all positive integer iterates of (
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
) is called the ''splinter'' (or Picard sequence) of .
However, ''all iterates'' (fractional, infinitesimal, or negative) of are likewise specified through the coordinate transformation determined to solve Schröder's equation: a holographic continuous interpolation of the initial discrete recursion has been constructed;
in effect, the entire
orbit
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
.
For instance, the
functional square root
In mathematics, a functional square root (sometimes called a half iterate) is a square root of a function with respect to the operation of function composition. In other words, a functional square root of a function is a function satisfying fo ...
is , so that , and so on.
For example, special cases of the
logistic map such as the chaotic case were already worked out by Schröder in his original article
(p. 306),
: , , and hence .
In fact, this solution is seen to result as motion dictated by a sequence of switchback potentials, , a generic feature of continuous iterates effected by Schröder's equation.
A nonchaotic case he also illustrated with his method, , yields
: , and hence .
Likewise, for the
Beverton–Holt model, , one readily finds
, so that
[ See equations 41, 42.]
:
See also
*
Böttcher's equation
References
{{DEFAULTSORT:Schroder's equation
Functional equations
Mathematical physics