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 Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand ...

: given the function , find the function such that Schröder's equation is an eigenvalue equation for the

composition operator In mathematics, the composition operator C_\phi with symbol \phi is a linear operator defined by the rule C_\phi (f) = f \circ \phi where f \circ \phi denotes function composition. The study of composition operators is covered bAMS category 47B33 ...

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 of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

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 holomorphic ma ...

. Equations such as Schröder's are suitable to encoding

self-similarity __NOTOC__ 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 se ...

, and have thus been extensively utilized in studies of

nonlinear dynamics In mathematics and science, a nonlinear 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, mathematicians, and many other ...

(often referred to colloquially as

chaos theory Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have co ...

). 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 a 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 term renormalization group (RG) refers to 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 ...

. An equivalent transpose form of Schröder's equation for the inverse of Schröder's conjugacy function is . The change of variables (the

Abel function The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form :f(h(x)) = h(x + 1) or :\alpha(f(x)) = \alpha(x)+1. The forms are equivalent when is invertible. or control the iteration of . Equivalence The se ...

) further converts Schröder's equation to the older

Abel equation The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form :f(h(x)) = h(x + 1) or :\alpha(f(x)) = \alpha(x)+1. The forms are equivalent when is invertible. or control the iteration of . Equivalence The seco ...

, . Similarly, the change of variables converts Schröder's equation to

Böttcher's equation Böttcher's equation, named after Lucjan Böttcher, is the functional equation ::F(h(z)) = (F(z))^n where * is a given analytic function with a superattracting fixed point of order at , (that is, h(z)=a+c(z-a)^n+O((z-a)^) ~, in a neighbour ...

, . Moreover, for the velocity, , ''

Julia Julia is usually a feminine given name. It is a Latinate feminine form of the name Julio and Julius. (For further details on etymology, see the Wiktionary entry "Julius".) The given name ''Julia'' had been in use throughout Late Antiquity (e.g ...

'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

. 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 to ...

, cf.

Carleman matrix In mathematics, a Carleman matrix is a matrix used to convert function composition into matrix multiplication. It is often used in iteration theory to find the continuous iteration of functions which cannot be iterated by pattern recognition alone ...

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 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 object or position in space such as a p ...

(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 ide ...

'', see

iterated function In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function with itself a certain number of times. The process of repeatedly applying the same function is ...

) are provided by the orbit for real — not necessarily positive or integer. (Thus a full

continuous group In mathematics, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continui ...

.) 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: ''x''·''y'', or simply ''xy'', ...

) 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

. 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 ...

is , so that , and so on. For example, special cases of the

logistic map The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popular ...

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. :

h_t(x)= \Psi^\big(2^ \Psi(x)\big) = \frac.

References

{{DEFAULTSORT:Schroder's equation Functional equations Mathematical physics

Functional significance

Solutions

Applications

See also

References