Revision as of 14:37, 6 October 2023 edit AppliedMathematician (talk \| contribs) 375 edits added an introductory paragraph with some concrete examples to better set up the terminology for "bifurcation parameter" and "asymptotic" behavior. Modified the second paragraph to fit the flow of the first. Tag: Reverted ← Previous edit		Revision as of 19:40, 20 December 2023 edit undo BlueBanana (talk \| contribs) 180 edits Undid revision 1178886211 by AppliedMathematician (talk) Concrete examples of applications before definition make the subject matter more difficult to grasp Tag: Undo Next edit →
Line 1: {{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 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. 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]]. 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 mass and 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. [[Image:diagram bifurkacji anim small.gif\|300px\|thumb\|right\|Animation showing the formation of a bifurcation diagram]]

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