Chaos Theory MS Electrical Engineering Department of Engineering

Chaos Theory MS Electrical Engineering Department of Engineering

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 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 Research Chaos, An Interdisciplinary Journal of Nonlinear Science International Journal of Bifurcation and

Chaos Chaos, Solitons and Fractals Chaos Meaning Pronunciation:/kes/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 Probabilistic vs Deterministic Random Deterministic

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 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

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

Differential Equations LINEAR Differential Equations Exponential Growth of population of organisms

Differential Equations NONLINEAR Trajectory & Phase-Space Draw the trajectories without actually solving the system Non-autonomous

Systems Time Dependent Systems 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) Example: Example (Continued) Application Areas

Mathematics Biology Computer science Economics Engineering Finance

Philosophy Physics Politics Population dynamics Psychology Chaos in Electrical Circuits

Chaos in Electrical Circuits Secure Communication

Recently Viewed Presentations

  • The strength of our commitment to the health

    The strength of our commitment to the health

    UNM Plan for 9200 Bradford Pear Lane: 0.5 acres . 1. Get Expert Lawn Advice 2. Maintain Dense Cover on Turf 3. Choose NOT to fertilize 4. Recycle Lawn Clippings and Compost Fallen Leaves 5. Correct Fertilizer Timing . N/A....
  • Cigna Case Study - The Goldenson Center for Actuarial Research

    Cigna Case Study - The Goldenson Center for Actuarial Research

    Cigna Case Study. By Bliss Joseph, Marney Lariviere, Pari Patel, and Serena Shah. The Patient Protection and Affordable Care Act. Also known as Affordable Care Act or Obamacare . Allowed insurance to be more available to individuals .
  • Missouri Housing Development Commission

    Missouri Housing Development Commission

    WHO WE ARE. MHDC is committed to strengthening communities through the financing, development and preservation of affordable housing. The commission administers the federal and Missouri Low Income Housing Tax Credit (LIHTC) programs, federal HOME funds, the U.S. Department of Housing...
  • e-handout To have these notes without taking notes

    e-handout To have these notes without taking notes

    David Stern on MJ 8.13 No, for the author could hardly have been unaware that the Mosaic Covenant presents itself as eternal; also the context shows that he is speaking only of its system of priests and sacrifices, Mtt. 26.26-29.New...
  • National Quality Strategy - Agency for Healthcare Research ...

    National Quality Strategy - Agency for Healthcare Research ...

    The Relationship Between the Institute for Healthcare Improvement Triple Aim and the National Quality Strategy Three Aims. Healthy People/Healthy Communities: Improve the health of the U.S. population by supporting proven interventions to address behavioral, social, and environmental determinants of health.
  • Bell Ringer: Please Copy in your IAN Sedimentary Rock: Ch4.3

    Bell Ringer: Please Copy in your IAN Sedimentary Rock: Ch4.3

    Calcium Carbonate Critters (Crystalize!) Both compact together and compress to form . Sandstone. Arches National Park in Utah . ... Which of these 2 properties would be more important for classifying each type of sedimentary rocks? Use our notes from...
  • Slope & Y-intercept

    Slope & Y-intercept

    What do you need to do? Identify the slope on a graph First find where the line goes through (intersects the grid lines) Draw a triangle and calculate the slope What do you need to do? Decide if it's a...
  • Educational Advocacy - Illinois Legal Aid

    Educational Advocacy - Illinois Legal Aid

    Educational Advocacy: K-12. Title IX only applies to nearly every elementary and secondary school in the country. Does the school district have a Title IX coordinator? Title IX now requires parents of minor students to be notified. Mandatory Reporting Requirements...