Gauss iterated map

{\displaystyle \alpha =4.90} — Cobweb plot of the Gauss map for $\alpha =4.90$ and $\beta =-0.58$ . This shows an 8-cycle.

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).

{\displaystyle \alpha =6.20} — Bifurcation diagram of the Gauss map with $\alpha =6.20$ and $\beta$ in the range −1 to +1.

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