آشوب و بررسی آن در سیستم های بیولوژیکی
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Transcript of آشوب و بررسی آن در سیستم های بیولوژیکی
آشوب و بررسی آن در سیستم های بیولوژیکی
ارائهراحله داودی
استاددكتر فرزاد توحيدخواه
1388دی
دانشگاه صنعتي اميركبير
دانشكده مهندسي پزشكي
What is talked in this seminar:Introduction to chaosChaos propertiesHistory FractalsChaos and stochastic processLogistic Map
What is talked in this seminar: (continue)Biological models producing chaos Chaos in heart sign of healthy or disease?Application:
Model of heart rate Applying chaos theory to a cardiac
arrhythmia
What chaos is:
One behavior of nonlinear dynamic systems
Unpredictable for long time but limited to a
specific area (attractor)
Seems to be random while it happens in
deterministic systems
Highly sensitive to initial condition
Chaos Properties:
• Fractal (Self Similarity)
• Liapunove Exponent (Divergence)
• Universality
Henri Poincaré - 1890while studying the three-body problem, he found that there can be orbits which are non-periodic, and yet not forever increasing nor approaching a fixed point.
History
Poincare &Three body problem
The problem is to determine the
possible motions of three point
masses m1,m2,and m3, which
attract each other according to
Newton's law of inverse squares.
In 1977, Mitchell Feigenbaum published the noted
article “ Quantitative Universality for a Class of
Nonlinear Transformations", where he described
logistic maps. Feigenbaum notably discovered the
universality in chaos, permitting an application of
chaos theory to many different phenomena.
History …
Edward Lorenz whose interest in chaos
came about accidentally through his work
on weather prediction in 1961.
small changes in initial conditions
produced large changes in the long-term
outcome. Predictability: Does the Flap of
a Butterfly’s Wings in Brazil set off a
Tornado in Texas?
History …
The flapping of a single butterfly's wing today produces a tiny
change in the state of the atmosphere. Over a period of time,
what the atmosphere actually does diverges from what it would
have done. So, in a month's time, a tornado that would have
devastated the Indonesian coast doesn't happen. Or maybe one
that wasn't going to happen, does.
(Ian Stewart, Does God Play Dice? The Mathematics of Chaos,
pg. 141)
Butterfly Effect
The father of fractals: Gaston Julia. 1900
There were some other works out there, such as
Sierpinski’s triangle and Koch’s curve.
Mandelbrot 1970 :Mandelbrot Set.
History of Fractals
Fractals …
Koch’s curveFractals …
Fractals …
Fractals …
Types of Attractors:
Types of Attractors:
Self Similarity in Chaos
• mean• variance• power
spectrum
Chaos and stochastic process
Similar time series
RANDOMrandomx(n) = RND
CHAOSDeterministicx(n+1) = 3.95 x(n) [1-x(n)]
How to recognize chaos from random
Power spectraStructure in state spaceDimension of dynamicsSensitivity to initial condition Lyapunov ExponentsPredictive Ability
Controllability of Chaos
Structure in state space
Poincare Section
divergence
Divergence
Divergence
Divergence
Predictive Ability
)1(1 nnn xrxx
Logistic Map
0 < r < 3 The sequence approaches to a stable value.
3 < r < 3.570 The sequence jumps among some stable values.
3.570 < r < 4 The sequence shows chaotic behavior.
Feigenbaum Number
Biological models producing chaosNonlinearity
Time delay
Compartment Cascades
Forcing Functions
Chaos in Biology
why chaos is so important in Biology?
Chaotic systems can be used to show :
rhythms of heartbeats
walking strides
Fractals can be used to model:
Structures of nerve networks
circulatory systems
lungs
DNA
why chaos is so important in Biology?
Evidence for chaotic healthy hearts
Applying chaos theory to a
cardiac arrhythmia
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