Entry
Math: Chaos theory: What is a bifurcation?
Nov 29th, 2003 09:24
Knud van Eeden,
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--- Knud van Eeden --- 29 November 2003 - 06:14 pm -------------------
Math: Chaos theory: What is a bifurcation?
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A bifurcation means a doubling, quadrupling, octupling, ... of
the period.
It represents the sudden appearance of a qualitatively different
solution
for a non-linear system as some parameter is varied.
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[book: source: Weisstein, Eric W. - CRC concise encyclopedia of
mathematics - ISBN 0-8493-9640-9 - p. 131 'Bifurcation']
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Bifurcation comes from Latin, with 'bi' meaning 'two, or double' and
'furcation' from 'furca' or thus 'forc']
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[book: see also: Prisma dictionary]
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Period doubling
One important feature of the logistic map is the passage to chaos
through a sequence of period doublings. The bifurcation where this
doubling occurs is called a 'pitchfork' bifurcation, because the local
shape of the bifurcation diagram resembles a pitchfork.
The period doubling route is particularly interesting, because it may
be
characterized by certain universal numbers that do NOT depend (within
certain limits) on the nature of the map (or ordinary differential
equation).
For example, the RATIO of the spacings between consecutive values of
the constant parameter, at this bifurcations, approaches a UNIVERSAL
CONSTANT, called the FEIGENBAUM constant, after its discoverer Mitchell
Feigenbaum.
If the first bifurcation occurs at parametervalue1,
the second bifurcation occurs at parametervalue2,
the third bifurcation occurs at parametervalue3,
and so forth, then this universal constant is defined as:
limit of k approaching infinity of:
( parametervalue[k] - parametervalue[k-1] )/( parameterervalue[k+1] -
parametervalue[k] )
And this limit, called the Feigenbaum constant, turns out to be equal
to
4.6692016091029909...
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[book: source: Baker, Gregory L. / Gollub, Jerry P. - chaotic
dynamics, an introduction - ISBN 0-521-38897-X - p. 81 'period
doubling']
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Bifurcation was introduced in chaos theory, by Robert May
(Australian physicist and biologist),
when trying to describe the behavior of the following logistic
equation:
xnew = constant . xold . ( 1 - xold )
as a function of the constant.
When you choose this constant between 0 and about 3, you will see that
your xnew will after a while settle to one specific value.
Between 3 and about 3.57 the values of xnew will or cycle through
2, 4, 8, 16, 2N values.
Between 3.57 and 4 the values of xnew will or be periodic, or be
chaotic.
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Bifurcations come in four basic varieties:
- flip bifurcation
- fold bifurcation
- pitchfork bifurcation
- transcritical bifurcation
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[book: see also: Gleick, James - chaos - publisher: Sphere books,
1988 - ISBN: 0-7474-0413-5 - p. 69 'bifurcation']
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Internet: see also:
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Math: Chaos theory: Overview: Can you give me an overview of chaos
theory?
http://www.faqts.com/knowledge_base/view.phtml/aid/26929/fid/867
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