Exponential map (discrete dynamical systems)

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Parameter plane of the complex exponential family f(z)=exp(z)+c with 8 external ( parameter) rays

In the theory of dynamical systems, the exponential map can be used as the evolution function of the discrete nonlinear dynamical system.[1]

Family[edit]

The family of exponential functions is called the exponential family.

Forms[edit]

There are many forms of these maps,[2] many of which are equivalent under a coordinate transformation. For example two of the most common ones are:

The second one can be mapped to the first using the fact that , so is the same under the transformation . The only difference is that, due to multi-valued properties of exponentiation, there may be a few select cases that can only be found in one version. Similar arguments can be made for many other formulas.

References[edit]

  1. ^ Dynamics of exponential maps by Lasse Rempe
  2. ^ Lasse Rempe, Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity