Bogdanov map

Chaotic 2D map related to the Bogdanov–Takens bifurcation

In dynamical systems theory, the Bogdanov map is a chaotic 2D map related to the Bogdanov–Takens bifurcation. It is given by the transformation:

{\begin{cases}x_{n+1}=x_{n}+y_{n+1}\\y_{n+1}=y_{n}+\epsilon y_{n}+kx_{n}(x_{n}-1)+\mu x_{n}y_{n}\end{cases}}

The Bogdanov map is named after Rifkat Bogdanov.

References

DK Arrowsmith, CM Place, An introduction to dynamical systems, Cambridge University Press, 1990.
Arrowsmith, D. K.; Cartwright, J. H. E.; Lansbury, A. N.; and Place, C. M. "The Bogdanov Map: Bifurcations, Mode Locking, and Chaos in a Dissipative System." Int. J. Bifurcation Chaos 3, 803–842, 1993.
Bogdanov, R. "Bifurcations of a Limit Cycle for a Family of Vector Fields on the Plane." Selecta Math. Soviet 1, 373–388, 1981.

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