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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.
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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
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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
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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
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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
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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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is shown on the horizontal axis of the plot and the vertical axis shows the set of values of the
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Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering
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May, Robert M. (1976). "Simple mathematical models with very complicated dynamics".
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Visualization of sudden behavior changes caused by continuous parameter changes
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shows the values visited or approached asymptotically (fixed points,
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571:. Bifurcation diagrams are used to visualize these changes.
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visited asymptotically from almost all initial conditions.
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as the bifurcation parameter, but for different values of
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Wikiversity: Discrete-time dynamical system orbit diagram
399:{\displaystyle {\ddot {x}}+f(x;\mu )+\varepsilon g(x)=0,}
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quantifies these changes by showing how fixed points,
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532:This is illustrated in the animation on the right.
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132:A bifurcation diagram of the logistic map
77:Learn how and when to remove this message
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262:{\displaystyle x_{n+1}=rx_{n}(1-x_{n}).}
183:is shown on the vertical line at that
305:Symmetry breaking in bifurcation sets
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567:of a system change as a function of
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610:Stability, Instability and Chaos
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591:Skeleton of bifurcation diagram
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329:In a dynamical system such as
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144:. Black regions correspond to
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290:for which bifurcation occurs
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469:{\displaystyle \varepsilon }
284:period-doubling bifurcation
171:Bifurcation diagram of the
140:Bifurcation diagram of the
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722:The Logistic Map and Chaos
614:Cambridge University Press
608:Glendinning, Paul (1994).
429:{\displaystyle \mu \neq 0}
271:The bifurcation parameter
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688:Strogatz, Steven (2000).
552:branch of mathematics, a
296:first Feigenbaum constant
32:This article includes a
61:more precise citations.
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449:{\displaystyle \mu }
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163:List of chaotic maps
647:1976Natur.261..459M
554:bifurcation diagram
408:structurally stable
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810:Chaos theory stubs
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641:(5560): 459–467.
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317:as the parameter
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159:Dynamical systems
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53:Please help
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476:, the case
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91:mathematics
59:introducing
794:Categories
565:attractors
157:See also:
142:circle map
110:attractors
67:March 2013
513:≠
510:ε
484:ε
464:ε
444:μ
421:≠
418:μ
406:which is
373:ε
364:μ
343:¨
292:converges
241:−
177:attractor
575:See also
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643:Bibcode
562:chaotic
294:to the
107:chaotic
55:improve
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673:934280
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634:Nature
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323:ε
319:ε
175:. The
745:This
677:S2CID
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702:ISBN
669:PMID
618:ISBN
161:and
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659:hdl
651:doi
639:261
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
