EXPONENTIAL MAP (DISCRETE DYNAMICAL SYSTEMS)

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

Family

The family of exponential functions is called the exponential family.

Forms

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:

$E_{c}:z\to e^{z}+c$
$E_{\lambda }:z\to \lambda *e^{z}$

The second one can be mapped to the first using the fact that $\lambda *e^{z}.=e^{z+ln(\lambda )}$ , so $E_{\lambda }:z\to e^{z}+ln(\lambda )$ is the same under the transformation $z=z+ln(\lambda )$ . 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

Wikimedia Commons has media related to Exponential maps.

Wikibooks has a book on the topic of: Fractals/exponential

Chaos theory

Concepts

Core	Attractor Bifurcation Fractal Limit set Lyapunov exponent Orbit Periodic point Phase space
Anosov diffeomorphism Arnold tongue axiom A dynamical system Bifurcation diagram Box-counting dimension Correlation dimension Conservative system Ergodicity False nearest neighbors Hausdorff dimension Invariant measure Lyapunov stability Measure-preserving dynamical system Mixing Poincaré section Recurrence plot SRB measure Stable manifold Topological conjugacy
Theorems	Ergodic theorem Liouville's theorem Krylov–Bogolyubov theorem Poincaré–Bendixson theorem Poincaré recurrence theorem Stable manifold theorem Takens's theorem

Theoretical
branches

Chaotic
maps ( list)

Discrete	Arnold's cat map Baker's map Complex quadratic map Coupled map lattice Duffing map Dyadic transformation Dynamical billiards outer Exponential map Gauss map Gingerbreadman map Hénon map Horseshoe map Ikeda map Interval exchange map Irrational rotation Kaplan–Yorke map Langton's ant Logistic map Standard map Tent map Tinkerbell map Zaslavskii map
Continuous	Double scroll attractor Duffing equation Lorenz system Lotka–Volterra equations Mackey–Glass equations Rabinovich–Fabrikant equations Rössler attractor Three-body problem Van der Pol oscillator

Physical
systems

Chaos
theorists

This fractal–related article is a stub. You can help Wikipedia by expanding it.

Forms

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:

E_{c}:z\to e^{z}+c

E_{\lambda }:z\to \lambda *e^{z}

The second one can be mapped to the first using the fact that

\lambda *e^{z}.=e^{z+ln(\lambda )}

, so

E_{\lambda }:z\to e^{z}+ln(\lambda )

is the same under the transformation

z=z+ln(\lambda )

. 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

Wikimedia Commons has media related to Exponential maps.

Wikibooks has a book on the topic of: Fractals/exponential

This fractal–related article is a stub. You can help Wikipedia by expanding it.

Family

Forms

References

Family

Forms

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