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{{Short description|Chaotic 2D map related to the Bogdanov–Takens bifurcation}} In [[dynamical systems theory]], the '''Bogdanov map''' is a [[chaos theory|chaotic]] 2D [[map (mathematics)|map]] related to the [[Bogdanov–Takens bifurcation ...

...d after [[Carl Friedrich Gauss|Johann Carl Friedrich Gauss]], the function maps the bell shaped Gaussian function similar to the [[logistic map]]. In the parameter real space <math>x_n</math> can be chaotic. The map is also called the ''mouse map'' because its [[bifurcation diagram ...

...cau.uni-kiel.de/receive/diss_mods_00000781?lang=en Dynamics of exponential maps by Lasse Rempe]</ref> ...maps,<ref>[http://arxiv.org/abs/0805.1658 "Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and De ...

== Definition for maps == ...ox is subjected to small horizontal oscillations, the pendulum may exhibit chaotic behavior. ...

...ity of the steps. It can be shown that the sequence will be [[Chaos theory|chaotic]], i.e. it is sensitive to the detailed choice of starting angle. The informal reason why the iteration is chaotic is that the angle doubles on every iteration and doubling grows very quickl ...

...nitial conditions that do not have the third integral of motion are called chaotic orbits. |title = Chaotic Behaviour of Deterministic Systems ...

...a [[strange attractor]] observed from a physical electronic [[Chaos theory|chaotic]] circuit (generally, [[Chua's circuit]]) with a single [[Nonlinear system| ...an oscilloscope using the X, Y, and Z output signals of the circuit. This chaotic attractor is known as the double scroll because of its shape in three-dimen ...

...goes a [[Period-doubling bifurcation|period doubling cascade]] and becomes chaotic at <math>b\approx 0.208186</math>. Beyond this the attractor expands, under [[Category:Chaotic maps]] ...

...> is a forcing constant. <math>F=8</math> is a common value known to cause chaotic behavior. [[Category:Chaotic maps]] ...

..., including unpredictable behavior, which is referred to as [[chaos theory|chaotic dynamics]]. This makes the Hindmarsh–Rose model relatively simple and provi [[Category:Chaotic maps]] ...

...otable in [[dynamical systems]] since it can result in [[chaotic attractor|chaotic attractors]] with various dimensions.<ref name="kantz">{{cite book|author1= ...n this delay can be caused by a [[pathology]], which in turn can result in chaotic solutions for the Mackey–Glass equations, especially Equation ({{EquationNo ...

...t_variable.png|400x400px|Activation variable as a function of time for the chaotic regime|thumb]] ...haotic_regime.png|400x400px|Solution for the Chialvo map equations for the chaotic regime|thumb]] ...

...cite journal |last1=Doi |first1= S. |s2cid= 122930351 |date= 1989 |title=A chaotic map with a flat segment can produce a noise-induced order |journal= J Stat ...irst6= F.T. |date= 2003 |title=Constructive effects of noise in homoclinic chaotic systems |journal= Phys. Rev. E |volume=67 |issue= 6 |page= 066220 |doi=10.1 ...

...the characteristics of chaotic motion in the system, cf. [[Double pendulum#Chaotic motion]]|date=October 2019}} ...{Cite book |last=Leah |first=Ganis |title=The Swinging Spring: Regular and Chaotic Motion}}</ref> The motion of an elastic pendulum is governed by a set of co ...

...ww.worldcat.org/oclc/670430585 |title=Elegant chaos : algebraically simple chaotic flows |date=2010 |publisher=World Scientific |isbn=978-981-283-882-7 |locat [[Category:Chaotic maps]] ...

[[Category:Chaotic maps]] ...

[[Image:Chaotic mixing.png|thumb|right|300px|alt=Chaotic_mixing|An example of chaotic mixing]] In [[chaos theory]] and [[fluid dynamics]], '''chaotic mixing''' is a process ...

...]]). They are predominantly used to qualitatively study the [[Chaos theory|chaotic dynamics]] of spatially extended systems. This includes the dynamics of [[w ...monstrate chaotic behavior. Such maps can be found here: [[List of chaotic maps]]. ...

...ith D. K., Place C. M. Dynamical Systems. Differential equations, maps and chaotic behavior. Chapman & Hall, (1992)</ref> ...

...C\left( (X, \tau); (Z, \sigma) \right)</math> denote the set of continuous maps <math>f : (X, \tau) \to (Z, \sigma).</math> The {{em|power with respect to ...ard|2016|pp=55-77}} by: <math>\lim {}_{o_{X}} \mathcal{F} := X.</math> The chaotic preconvergence on <math>X</math> is equal to the canonical convergence indu ...