Gauss iterated map

In today's article we will explore Gauss iterated map, a topic that has captured the attention of people around the world. Since its emergence, Gauss iterated map has generated a wide spectrum of opinions and emotions, becoming a central point of discussion in different areas. Over the years, Gauss iterated map has proven its relevance in society, triggering intense debates and generating a significant impact on people's lives. Through this article, we will delve into the different facets of Gauss iterated map, exploring its origin, evolution and its influence on various aspects of daily life. Get ready to enter the fascinating world of Gauss iterated map and discover everything this theme has to offer.

In mathematics, the Gauss map (also known as Gaussian map^[1] or mouse map), is a nonlinear iterated map of the reals into a real interval given by the Gaussian function:

x_{n+1}=\exp(-\alpha x_{n}^{2})+\beta ,\,

where α and β are real parameters.

Named after Johann Carl Friedrich Gauss, the function maps the bell shaped Gaussian function similar to the logistic map.

Properties

In the parameter real space $x_{n}$ can be chaotic. The map is also called the mouse map because its bifurcation diagram resembles a mouse (see Figures).

References

^ Chaos and nonlinear dynamics: an introduction for scientists and engineers, by Robert C. Hilborn, 2nd Ed., Oxford, Univ. Press, New York, 2004.

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

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Gauss iterated map

Properties

References

Wikious

Boobota

Sagapedia