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Chaos Theory

Chaos Theory. MS Electrical Engineering Department of Engineering GC University Lahore. Course Contents. Introduction Flows on the line Bifurcations Flows on the circle Linear Systems Phase Planes Limit Cycles Lorenz Equation One dimensional Maps Fractals Strange Attractors. Books.

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Chaos Theory

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  1. Chaos Theory MS Electrical Engineering Department of Engineering GC University Lahore

  2. Course Contents • Introduction • Flows on the line • Bifurcations • Flows on the circle • Linear Systems • Phase Planes • Limit Cycles • Lorenz Equation • One dimensional Maps • Fractals • Strange Attractors

  3. Books “NONLINEAR DYNAMICS AND CHAOS With Applications to Physics, Biology, Chemistry, and Engineering”, STEVEN H. STROGATZ “CHAOS AND NONLINEAR DYNAMICS: An Introduction for Scientists and Engineers”, Robert C. Hilborn “The illustrated Dictionary of NONLINEAR DYNAMICS AND CHAOS”, Tomasz Kapitaniak, Steven R. Bishop

  4. Research • Chaos, An Interdisciplinary Journal of Nonlinear Science • International Journal of Bifurcation and Chaos • Chaos, Solitons and Fractals

  5. Chaos – Meaning • Pronunciation:/keɪɒs/noun • complete disorder and confusion • Example: “snow caused chaos in the region” • Physics: • the property of a complex system whose behaviour is so unpredictable as to appear random, owing to great sensitivity to small changes in conditions. • the formless matter supposed to have existed before the creation of the universe. • Reference: Oxford Dictionary

  6. Probabilistic vs Deterministic • Random • Deterministic

  7. Static vs Dynamic • Static Systems • Dynamic Systems • Dynamics: Subject that deals with change • systems that evolve in time, settles down to equilibrium, keeps repeating in cycles, or does something more complicated

  8. Brief History of Dynamics • 17th Century – Newton solving Two-Body Problem (Sun & Earth) using Differential Equations and Law of Gravitation • Three-Body Problem (Sun, Moon and Earth) – N o explicit solution • Late 19th Century – Poincare – Qualitative Solution rather than Quantitative, Geometric Approach • Dynamics restricted to Nonlinear oscillators in radio, radar, phase-locked loops, and lasers • Lorenz's discovery in 1963 of chaotic motion on a strange attractor – Weather Model: Aperiodic, very sensitive to Initial Conditions

  9. Types of Dynamical Systems • Differential equations: the evolution of systems in continuous time, whereas iterated maps arise in problems • Iterated maps (Difference equations): time is discrete

  10. Differential Equations LINEAR

  11. Differential Equations • Exponential Growth of population of organisms

  12. Differential Equations NONLINEAR

  13. Trajectory & Phase-Space Draw the trajectories without actually solving the system

  14. Non-autonomous Systems • Time Dependent Systems

  15. Dimension of the Phase Space • n = 1: Growth, Decay or Equilibrium e.g. RC Circuit (Linear), Logistic Equation (Nonlinear) • n = 2: Oscillations e.g RLC Circuit (Linear), Pendulum (Nonlinear) • n >= 3: Three-Body Problem, Chaos & Fractals (Nonlinear)

  16. Example:

  17. Example (Continued)

  18. Application Areas • Mathematics • Biology • Computer science • Economics • Engineering • Finance • Philosophy • Physics • Politics • Population dynamics • Psychology

  19. Chaos in Electrical Circuits

  20. Chaos in Electrical Circuits • Secure Communication

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