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آشوب و بررسی آن در سیستم های بیولوژیکی

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دانشگاه صنعتي اميركبير دانشكده مهندسي پزشكي. آشوب و بررسی آن در سیستم های بیولوژیکی. ارائه راحله داودی استاد دكتر فرزاد توحيدخواه دی 1388. What is talked in this seminar: Introduction to chaos Chaos properties History Fractals Chaos and stochastic process Logistic Map.

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slide1
دانشگاه صنعتي اميركبير

دانشكده مهندسي پزشكي

آشوب و بررسی آن در سیستم های بیولوژیکی

ارائه

راحله داودی

استاد

دكتر فرزاد توحيدخواه

دی 1388

slide2
What is talked in this seminar:
  • Introduction to chaos
  • Chaos properties
  • History
  • Fractals
  • Chaos and stochastic process
  • Logistic Map
slide3
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
slide4
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
slide5
Chaos Properties:
  • Fractal (Self Similarity)
  • Liapunove Exponent (Divergence)
  • Universality
slide6
History
  • Henri Poincaré - 1890
  • while 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.
slide7
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.

slide8
History …

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.

slide9
History …

Edward Lorenzwhose 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?
slide10
Butterfly Effect

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)

slide11
History of Fractals

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.

slide13
Fractals …

Koch’s curve

slide21
Chaos and stochastic process
  • mean
  • variance
  • power spectrum
slide23
RANDOM

random

x(n) = RND

slide24
CHAOS

Deterministic

x(n+1) = 3.95 x(n) [1-x(n)]

slide28
How to recognize chaos from random
  • Power spectra
  • Structure in state space
  • Dimension of dynamics
  • Sensitivity to initial condition
    • Lyapunov Exponents
    • Predictive Ability
  • Controllability of Chaos
slide29
Structure in state space

Poincare Section

slide41
Biological models producing chaos
  • Nonlinearity
  • Time delay
  • Compartment Cascades
  • Forcing Functions
slide43
why chaos is so important in Biology?
  • Chaotic systems can be used to show :
  • rhythms of heartbeats
  • walking strides
slide44
why chaos is so important in Biology?
  • Fractals can be used to model:
  • Structures of nerve networks
  • circulatory systems
  • lungs
  • DNA
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