Rahul Sharma (Editor)

Abel equation

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The Abel equation, named after Niels Henrik Abel, is a type of functional equation which can be written in the form

Contents

f ( h ( x ) ) = h ( x + 1 )

or, equivalently,

α ( f ( x ) ) = α ( x ) + 1

and controls the iteration of   f.

Equivalence

These equations are equivalent. Assuming that α is an invertible function, the second equation can be written as

α − 1 ( α ( f ( x ) ) ) = α − 1 ( α ( x ) + 1 ) .

Taking x = α−1(y), the equation can be written as

For a function f(x) assumed to be known, the task is to solve the functional equation for the function α−1≡h, possibly satisfying additional requirements, such as α−1(0) = 1.

The change of variables sα(x) = Ψ(x), for a real parameter s, brings Abel's equation into the celebrated Schröder's equation, Ψ(f(x)) = s Ψ(x) .

The further change F(x) = exp(sα(x)) into Böttcher's equation, F(f(x)) = F(x)s.

The Abel equation is a special case of (and easily generalizes to) the translation equation,

ω ( ω ( x , u ) , v ) = ω ( x , u + v )   ,

e.g., for ω ( x , 1 ) = f ( x ) ,

ω ( x , u ) = α − 1 ( α ( x ) + u ) .     (Observe ω(x,0) = x.)

History

Initially, the equation in the more general form was reported. Even in the case of a single variable, the equation is non-trivial, and admits special analysis.

In the case of a linear transfer function, the solution is expressible compactly.

Special cases

The equation of tetration is a special case of Abel's equation, with f = exp.

In the case of an integer argument, the equation encodes a recurrent procedure, e.g.,

α ( f ( f ( x ) ) ) = α ( x ) + 2   ,

and so on,

α ( f n ( x ) ) = α ( x ) + n   .

Solutions

  • formal solution : unique (to a constant)
  • analytic solutions (Fatou coordinates) = approximation by asymptotic expansion of a function defined by power series in the sectors around parabolic fixed point

    • Fatou coordinates describe local dynamics of discrete dynamical system near a parabolic fixed point.

    References

    Abel equation Wikipedia
    (Text) CC BY-SA