Chaos TheoryChaos Theory

MS Electrical EngineeringMS Electrical Engineering

Department of EngineeringDepartment of Engineering

GC University LahoreGC University Lahore

Course ContentsCourse Contents

IntroductionIntroduction Flows on the lineFlows on the line BifurcationsBifurcations Flows on the circleFlows on the circle Linear SystemsLinear Systems Phase PlanesPhase Planes Limit CyclesLimit Cycles Lorenz EquationLorenz Equation One dimensional MapsOne dimensional Maps FractalsFractals Strange AttractorsStrange Attractors

BooksBooks

““NONLINEAR DYNAMICS AND CHAOS With NONLINEAR DYNAMICS AND CHAOS With Applications to Physics, Biology, Chemistry, Applications to Physics, Biology, Chemistry, and Engineering”, STEVEN H. STROGATZand Engineering”, STEVEN H. STROGATZ

““CHAOS AND NONLINEAR DYNAMICS: An CHAOS AND NONLINEAR DYNAMICS: An Introduction for Scientists and Engineers”, Introduction for Scientists and Engineers”, Robert C. HilbornRobert C. Hilborn

““The illustrated Dictionary of NONLINEAR The illustrated Dictionary of NONLINEAR DYNAMICS AND CHAOS”, DYNAMICS AND CHAOS”, Tomasz Tomasz Kapitaniak, Steven R. BishopKapitaniak, Steven R. Bishop

ResearchResearch

Chaos, An Interdisciplinary Journal of Chaos, An Interdisciplinary Journal of Nonlinear ScienceNonlinear Science

International Journal of Bifurcation and International Journal of Bifurcation and ChaosChaos

Chaos, Solitons and Fractals Chaos, Solitons and Fractals

Chaos – Meaning Chaos – Meaning

Pronunciation:/keɪɒs/nounPronunciation:/keɪɒs/noun complete disorder and confusioncomplete disorder and confusion Example: “snow caused chaos in the region”Example: “snow caused chaos in the region” Physics: Physics: the property of a complex system whose behaviour is the property of a complex system whose behaviour is

so unpredictable as to appear random, owing to great so unpredictable as to appear random, owing to great sensitivity to small changes in conditions. sensitivity to small changes in conditions.

the formless matter supposed to have existed before the formless matter supposed to have existed before the creation of the universe. the creation of the universe.

Reference: Oxford DictionaryReference: Oxford Dictionary

Probabilistic vs Probabilistic vs DeterministicDeterministic

RandomRandom DeterministicDeterministic

Static vs DynamicStatic vs Dynamic

Static SystemsStatic Systems

Dynamic SystemsDynamic 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 DynamicsBrief History of Dynamics

1717thth Century – Newton solving Two-Body Problem (Sun Century – Newton solving Two-Body Problem (Sun & Earth) using Differential Equations and Law of & Earth) using Differential Equations and Law of GravitationGravitation

Three-Body Problem (Sun, Moon and Earth) – N o Three-Body Problem (Sun, Moon and Earth) – N o explicit solutionexplicit solution

Late 19Late 19thth Century – Poincare – Qualitative Solution Century – Poincare – Qualitative Solution rather than Quantitative, Geometric Approachrather than Quantitative, Geometric Approach

Dynamics restricted to Nonlinear oscillators in radio, Dynamics restricted to Nonlinear oscillators in radio, radar, phase-locked loops, and lasersradar, phase-locked loops, and lasers

Lorenz's discovery in 1963 of chaotic motion on a Lorenz's discovery in 1963 of chaotic motion on a strange attractor – Weather Model: Aperiodic, very strange attractor – Weather Model: Aperiodic, very sensitive to Initial Conditionssensitive to Initial Conditions

Types of Dynamical Types of Dynamical SystemsSystems

Differential equations: the evolution of Differential equations: the evolution of systems in continuous time, whereas systems in continuous time, whereas iterated maps arise in problemsiterated maps arise in problems

Iterated maps (Difference equations): Iterated maps (Difference equations): time is discretetime is discrete

Differential EquationsDifferential Equations

LINEAR

Differential EquationsDifferential Equations

Exponential Growth of population Exponential Growth of population of organismsof organisms

Differential EquationsDifferential Equations

NONLINEAR

Trajectory & Phase-SpaceTrajectory & Phase-Space

Draw the trajectories without actually solving the system

Non-autonomous Non-autonomous SystemsSystems

Time Dependent SystemsTime Dependent Systems

Dimension of the Phase Dimension of the Phase SpaceSpace

n = 1: Growth, Decay or Equilibrium e.g. n = 1: Growth, Decay or Equilibrium e.g. RC Circuit (Linear), Logistic Equation RC Circuit (Linear), Logistic Equation (Nonlinear)(Nonlinear)

n = 2: Oscillations e.g RLC Circuit n = 2: Oscillations e.g RLC Circuit (Linear), Pendulum (Nonlinear)(Linear), Pendulum (Nonlinear)

n >= 3: Three-Body Problem, Chaos & n >= 3: Three-Body Problem, Chaos & Fractals (Nonlinear)Fractals (Nonlinear)

Example:Example:

Example (Continued)Example (Continued)

Application AreasApplication Areas

MathematicsMathematics BiologyBiology Computer scienceComputer science EconomicsEconomics EngineeringEngineering FinanceFinance PhilosophyPhilosophy PhysicsPhysics PoliticsPolitics Population dynamicsPopulation dynamics PsychologyPsychology

Chaos in Electrical Chaos in Electrical CircuitsCircuits

Chaos in Electrical Chaos in Electrical CircuitsCircuits Secure Secure

CommunicationCommunication