Editing Bifurcation diagram

{{short description|Visualization of sudden behavior changes caused by continuous parameter changes}}
{{no footnotes|date=March 2013}}
In [[mathematics]], particularly in [[dynamical systems]], a '''bifurcation diagram''' shows the values visited or approached asymptotically ([[fixed point (mathematics)|fixed points]], [[periodic orbit]]s, or [[chaos (mathematics)|chaotic]] [[attractor]]s) of a system as a function of a [[Bifurcation theory|bifurcation parameter]] in the system.{{Fact|reason=Is the parameter of the function to be iterated really called "bifurcation parameter"? What sources substantiate this statement?|date=February 2024}} It is usual to represent stable values with a solid line and unstable values with a dotted line, although often the unstable points are omitted. Bifurcation diagrams enable the visualization of [[bifurcation theory]]. In the context of discrete-time dynamical systems, the diagram is also called '''orbit diagram'''.

[[File:LogisticMapOrbitDiag InclNegR.png|300px|thumb|right|A bifurcation diagram of the logistic map]]
[[Image:Circle map bifurcation.jpeg|thumb|upright|right|Bifurcation diagram of the [[circle map]]. Black regions correspond to [[Arnold tongues]].]]

==Logistic map==
{{See also|Dynamical systems|List of chaotic maps}}
[[File:Logistic Map Bifurcation Diagram, Matplotlib.svg|300px|thumb|left|Bifurcation diagram of the [[logistic map]]. The [[attractor]] for any value of the parameter ''r'' is shown on the vertical line at that ''r''.]]
An example is the bifurcation diagram of the [[logistic map]]:
<math display="block"> x_{n+1}=rx_n(1-x_n). </math>

The bifurcation parameter ''r'' is shown on the horizontal axis of the plot and the vertical axis shows the set of values of the [[logistic function]] visited asymptotically from almost all initial conditions.

The bifurcation diagram shows the forking of the periods of stable orbits from 1 to 2 to 4 to 8 etc. Each of these bifurcation points is a [[period-doubling bifurcation]].
The ratio of the lengths of successive intervals between values of ''r'' for which bifurcation occurs [[convergent series|converges]] to the [[first Feigenbaum constant]].

The diagram also shows period doublings from 3 to 6 to 12 etc., from 5 to 10 to 20 etc., and so forth.
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==Symmetry breaking in bifurcation sets==

[[Image:Asymbif.gif|300px|thumb|Symmetry breaking in [[pitchfork bifurcation]] as the parameter ''&epsilon;'' is varied. ''&epsilon;''&nbsp;=&nbsp;0 is the case of symmetric pitchfork bifurcation.]]
In a dynamical system such as
<math display="block"> \ddot {x} + f(x;\mu) + \varepsilon g(x) = 0,</math>
which is [[structurally stable]] when <math> \mu \neq 0 </math>, if a bifurcation diagram is plotted, treating <math> \mu </math> as the bifurcation parameter, but for different values of <math> \varepsilon </math>, the case <math> \varepsilon = 0</math> is the symmetric pitchfork bifurcation. When <math> \varepsilon \neq 0 </math>, we say we have a pitchfork with ''broken symmetry.'' This is illustrated in the animation on the right.
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== Applications ==
Consider a system of [[differential equation|differential equations]] that describes some physical quantity, that for concreteness could represent one of three examples: 1. the position and velocity of an undamped and frictionless pendulum, 2. a neuron's membrane potential over time, and 3. the average concentration of a virus in a patient's bloodstream. The differential equations for these examples include *parameters* that may affect the output of the equations. Changing the pendulum's length will affect its oscillation frequency, changing the magnitude of injected current into a neuron may transition the membrane potential from resting to spiking, and the long-term viral load in the bloodstream may decrease with carefully timed treatments. 

In general, researchers may seek to quantify how the long-term (asymptotic) behavior of a system of differential equations changes if a parameter is changed. In the [[dynamical systems]] branch of mathematics, a '''bifurcation diagram''' quantifies these changes by showing how fixed points, [[periodic orbit]]s, or [[chaos (mathematics)|chaotic]] [[attractor]]s of a system change as a function of [[Bifurcation theory|bifurcation parameter]]. Bifurcation diagrams are used to visualize these changes.

== See also ==
* [[Bifurcation memory]]
* [[Chaos theory]]
* [[Complex quadratic polynomial#Critical curves|Skeleton of bifurcation diagram]]
* [[Feigenbaum constants]]
* [[Sharkovskii's theorem]]

== Further reading ==
* {{cite book |first=Paul |last=Glendinning |title=Stability, Instability and Chaos |publisher=[[Cambridge University Press]] |year=1994 |isbn=0-521-41553-5 }}
* {{cite journal |last=May |first=Robert M. |year=1976 |title=Simple mathematical models with very complicated dynamics |journal=[[Nature (journal)|Nature]] |volume=261 |issue=5560 |pages=459–467 |doi=10.1038/261459a0 |bibcode=1976Natur.261..459M |pmid=934280 |hdl=10338.dmlcz/104555 |s2cid=2243371 |hdl-access=free }}
* {{cite book |first=Steven |last=Strogatz |title=Non-linear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering |publisher=[[Perseus Books]] |year=2000 |isbn=0-7382-0453-6 |url-access=registration |url=https://archive.org/details/nonlineardynamic00stro }}

==External links==
* [http://www.egwald.ca/nonlineardynamics/logisticsmapchaos.php The Logistic Map and Chaos] by Elmer G. Wiens, egwald.ca
* [[Wikiversity: Discrete-time dynamical system orbit diagram]]

[[Category:Chaos theory]]
[[Category:Bifurcation theory]]

[[de:Bifurkationsdiagramm]]

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