By: Bahareh Taghizadeh

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Presentation transcript:

By: Bahareh Taghizadeh Chaos JAN 2007 By: Bahareh Taghizadeh

Non-linear Dynamics Orbit Attractor

Basin of attraction Bifurcation diagram

Chaos in mathematical models Discrete Population growth The Logistic map Continiouse The Duffing oscilator The Lorenz model

Population growth

The logistic map The effect of control parameter a= 0.9 a= 2.6 a= 3.2

Logistic curve : Fixed Point

(Period 1) Fixed point=0.5 Fixed point attractor (Period 1) (Period 1)

(Period 2) (Period 2) (Period 2)

(Period 4)

(Chaos)

Bifurcation Diagram for Logistic Map

Stable Nuturally Stable Unstable

Feigenbaum number

Lorenz model

Duffing oscilator

Frequency Spectrum Structure in Phase Space Signatures of Chaos

Dimensions of Dynamics Correlation Dimension Fractal Dimension Information Dimension etc. Signatures of Chaos

The Butterfly Effect Sensitivity to Initial Conditions Lyapunov Exponents Signatures of Chaos The Butterfly Effect

Lyapunov Exponents Signatures of Chaos

Fractal Fractal Geometry, proposed by Benoit Mandelbrot

Biological systems chaos & fractal Parkinson Human Heart DNA Lung Blood Vessels Protein Ion Channels