Bifurcation diagram

Bifurcation diagram

In mathematics, particularly in dynamical systems, a bifurcation diagram shows the possible long-term values (equilibria/fixed points or periodic orbits) of a system as a function of a bifurcation parameter in the system. It is usual to represent stable solutions with a solid line and unstable solutions with a dotted line.

Bifurcations in the 1D discrete dynamical systems ( maps )

Logistic map

An example is the bifurcation diagram of the logistic map:

: x_{n+1}=rx_n(1-x_n). ,

The bifurcation parameter "r" is shown on the horizontal axis of the plot and the vertical axis shows the possible long-term population values of the logistic function. Only the stable solutions are shown here, there are many other unstable solutions which are not shown in this diagram.

The bifurcation diagram nicely shows the forking of the possible 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 converges to the first Feigenbaum constant.


= Real quadratic

For x_{n+1}=x_n^2-c; the code in Matlab can be written as:

close all;clear all;c=0; y=0.0;

hold on while c < 4 for i=1:100; y = y.^2 -c; %converge the iteration end for i=1:20 y = y.^2 - c; plot(c,y,'.'); % plot the converged points end c=c+0.01;end

ymmetry breaking in bifurcation sets

[


thumb|right|300px|Symmetry_breaking_in_pitchfork bifurcation as the parameter epsilon is varied. epsilon = 0 is the case of symmetric pitchfork bifurcation.]

In a dynamical system such as

: ddot {x} + f(x;mu) + epsilon g(x) = 0,

which is structurally stable when mu eq 0 , if a bifurcation diagram is plotted, treating mu as the bifurcation parameter, but for different values of epsilon , the case epsilon = 0 is the symmetric pitchfork bifurcation. When epsilon eq 0 , we say we have a pitchfork with "broken symmetry." This is illustrated in the animation on the right.

See also

* Bifurcation theory
* Phase portrait

References

*Paul Glendinning, "Stability, Instability and Chaos", Cambridge University Press, 1994.
*Steven Strogatz, "Non-linear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering", Perseus Books, 2000.

External links

* [http://www.egwald.com/nonlineardynamics/logisticsmapchaos.php The Logistic Map and Chaos]
* [http://home.scarlet.be/kpm/vb/winattract.html A small application for drawing the Logistic Map]


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Bifurcation Diagram — A graph that shows the critical points where bifurcation occurs, and the possible solutions that exist at that point. Bloomberg Financial Dictionary …   Financial and business terms

  • Bifurcation theory — is the mathematical study of changes in the qualitative or topological structure of a given family. Examples of such families are the integral curves of a family of vector fields or, the solutions of a family of differential equations. Most… …   Wikipedia

  • Pitchfork bifurcation — In bifurcation theory, a field within mathematics, a pitchfork bifurcation is a particular type of local bifurcation. Pitchfork bifurcations, like Hopf bifurcations have two types supercritical or subcritical. In flows, that is, continuous… …   Wikipedia

  • Period-doubling bifurcation — In mathematics, a Period doubling bifurcation in a dynamical system is a bifurcation in which the system switches to a new behavior with twice the period of the original system. The hallmark of this is a Floquet multiplier of 1. ExampleConsider… …   Wikipedia

  • Rössler attractor — The Rössler attractor (pronEng|ˈrɒslɚFact|date=December 2007) is the attractor for the Rössler system, a system of three non linear ordinary differential equations. These differential equations define a continuous time dynamical system that… …   Wikipedia

  • Mathematical and theoretical biology — is an interdisciplinary scientific research field with a range of applications in biology, medicine and biotechnology.[1] The field may be referred to as mathematical biology or biomathematics to stress the mathematical side, or as theoretical… …   Wikipedia

  • Cellular model — Part of the Cell Cycle Creating a cellular model has been a particularly challenging task of systems biology and mathematical biology. It involves developing efficient algorithms, data structures, visualization and communication tools to… …   Wikipedia

  • Logistic map — The logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non linear dynamical equations. The map was popularized in a seminal 1976 paper by the… …   Wikipedia

  • Mathematical biology — For use of basic artimethics in Biology, see relevant topic, such as Serial dilution. Mathematical biology, biological mathematical modeling, biomathematics or computational biomodeling is an interdisciplinary field of academic study which aims… …   Wikipedia

  • List of mathematics articles (B) — NOTOC B B spline B* algebra B* search algorithm B,C,K,W system BA model Ba space Babuška Lax Milgram theorem Baby Monster group Baby step giant step Babylonian mathematics Babylonian numerals Bach tensor Bach s algorithm Bachmann–Howard ordinal… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”