Visualization of sudden behavior changes caused by continuous parameter changes
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In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the system.[citation needed] 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.
A bifurcation diagram of the logistic mapBifurcation diagram of the circle map. Black regions correspond to Arnold tongues.
and 22 Related for: Bifurcation diagram information
omitted. Bifurcationdiagrams enable the visualization of bifurcation theory. In the context of discrete-time dynamical systems, the diagram is also called...
specifically bifurcation theory, the Feigenbaum constants /ˈfaɪɡənˌbaʊm/ are two mathematical constants which both express ratios in a bifurcationdiagram for...
{\displaystyle r=3.678...,x=0.728...} , two chaotic bands of the bifurcationdiagram intersect in the first Misiurewicz point for the logistic map. It...
In bifurcation theory, a field within mathematics, a pitchfork bifurcation is a particular type of local bifurcation where the system transitions from...
{p}{q}}} is known as a bifurcation: the attracting fixed point "collides" with a repelling period-q cycle. As we pass through the bifurcation parameter into the...
and the Julia set is the whole interval from μ − μ2/2 to μ/2 (see bifurcationdiagram). If μ is between 1 and 2 the interval [μ − μ2/2, μ/2] contains both...
Aspen by Pierre Hohenberg, presented his experimental observation of the bifurcation cascade that leads to chaos and turbulence in Rayleigh–Bénard convection...
can be chaotic. The map is also called the mouse map because its bifurcationdiagram resembles a mouse (see Figures). Chaos and nonlinear dynamics: an...
context of a biological system, bifurcation theory describes how small changes in an input parameter can cause a bifurcation or qualitative change in the...
Bifurcation memory is a generalized name for some specific features of the behaviour of the dynamical system near the bifurcation. An example is the recurrent...
values of x {\displaystyle x} are visited in turn (a period-doubling bifurcation), or, as a result of further doubling, any number k × 2 n {\displaystyle...
bifurcations called a Hopf bifurcation and an infinite period bifurcation.[citation needed] Biological applications of bifurcation theory Biophysics Biostatistics...
{\displaystyle K} , the bifurcationdiagram around this paragraph is obtained, where we can observe periodic orbits, period-doubling bifurcations as well as possible...
region, the mapping undergoes bifurcations at precise values of the parameter. At first, one stable point occurs, then bifurcates to an oscillation between...
constant δ is the limiting ratio of each bifurcation interval to the next between every period-doubling bifurcation. The logistic map is a polynomial mapping...
saddle-node bifurcation or limit point; the supercritical or stable-symmetric bifurcation; the subcritical or unstable-symmetric bifurcation; and the transcritical...
species or taxa during a specific time. In other words, it is a branching diagram or a tree showing the evolutionary relationships among various biological...
one value. These long-term values can be plotted against r in a bifurcationdiagram to show the different behaviours of the function for different values...
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