An Introduction to Chaos An Introduction to Chaos Theory WorkshopTheory Workshop

AmirAmir Reza Reza SaffariSaffari AzarAzarElectrical Engineering Department, Electrical Engineering Department, SahandSahand University of TechnologyUniversity of Technology

ContentsContents

Philosophical AspectsPhilosophical AspectsConcepts of Order and DisorderConcepts of Order and DisorderCausality, Determinism, and RandomnessCausality, Determinism, and RandomnessIntelligence and ComplexityIntelligence and Complexity

Mathematical AspectsMathematical AspectsIntroduction to StateIntroduction to State--SpaceSpaceDefinition of ChaosDefinition of ChaosChaotic Systems PropertiesChaotic Systems PropertiesFractal Geometry and ChaosFractal Geometry and Chaos

Concepts of Order and DisorderConcepts of Order and Disorder

What is Order ?!!!What is Order ?!!!… and Disorder… and DisorderWhy Order ?!!!Why Order ?!!!… and Disorder… and DisorderHybrid SystemsHybrid Systems

Causality, Determinism, and Causality, Determinism, and RandomnessRandomness

CausalityCausalityDeterminismDeterminismFreewillFreewillRandomnessRandomnessGoal Directed ActionGoal Directed Action

Intelligence and ComplexityIntelligence and Complexity

What is Intelligence?What is Intelligence?Artificial Intelligence ?!!!Artificial Intelligence ?!!!Simple BehaviorsSimple BehaviorsComplex BehaviorsComplex BehaviorsConclusionConclusion

Introduction to StateIntroduction to State--SpaceSpace

System ModelingSystem ModelingDynamical SystemsDynamical Systems

Example: Logistic MapExample: Logistic Map

m00 )t(x,x)t(x,)x(fx ℜ∈==&

mn00n1n x,xx,)x(fx ℜ∈==+

)x1(axx nn1n −=+

SteadySteady--State Behavior State Behavior –– IIFixed PointFixed Point

SteadySteady--State Behavior State Behavior –– IIIILimit CycleLimit Cycle

SteadySteady--State Behavior State Behavior –– IIII… … Limit CycleLimit Cycle

SteadySteady--State Behavior State Behavior –– IVIVQuasiQuasi--periodicperiodic

SteadySteady--State Behavior State Behavior –– VVChaosChaos

SteadySteady--State Behavior State Behavior –– VV… … ChaosChaos

SteadySteady--State Behavior State Behavior –– VV… … ChaosChaos

Definition of Chaos Definition of Chaos -- II

Historical PerspectiveHistorical PerspectiveLaplaceLaplace , 1776, 1776PoincarePoincare, 1903, 1903BirkhoffBirkhoff, 1920, 1920KolmogorovKolmogorov, 1960, 1960Lorenz, 1963Lorenz, 1963……

Definition of Chaos Definition of Chaos -- IIII

None of the rest !!!None of the rest !!!

Deterministic RandomnessDeterministic Randomness

Unpredictable CausalityUnpredictable Causality

Chaotic Systems Properties Chaotic Systems Properties -- II

NonNon--periodic Bounded Motion with periodic Bounded Motion with Initial Condition SensitivityInitial Condition Sensitivity

Butterfly Effect !!!Butterfly Effect !!!Restless Motion !!!Restless Motion !!!Heisenberg’s Uncertainty Principle Heisenberg’s Uncertainty Principle Naturally Unpredictable !!!Naturally Unpredictable !!!

Continuous, BroadContinuous, Broad--band, Noiseband, Noise--like like Power SpectrumPower Spectrum

Chaotic Systems Properties Chaotic Systems Properties -- IIII

Stretching and FoldingStretching and FoldingLyapunovLyapunov ExponentsExponentsGrowing UncertaintyGrowing Uncertainty

Information Creation and DestructionInformation Creation and Destruction22ndnd Law of Thermodynamics SuspectedLaw of Thermodynamics Suspected

Chaotic Systems Properties Chaotic Systems Properties -- IIIIII

Bifurcation DiagramBifurcation Diagram

Chaotic Systems Properties Chaotic Systems Properties -- IIIIII

… … Bifurcation DiagramBifurcation Diagram

Chaotic Systems Properties Chaotic Systems Properties -- IIIIII

… … Bifurcation DiagramBifurcation Diagram

Chaotic Systems Properties Chaotic Systems Properties -- IVIV

Strange AttractorStrange Attractor

Chaotic Systems Properties Chaotic Systems Properties -- IVIV

… … Strange AttractorStrange Attractor

More Strange AttractorsMore Strange Attractors

More Strange AttractorsMore Strange Attractors

More Strange AttractorsMore Strange Attractors

Fractal DimensionFractal Dimension

What is Dimension?What is Dimension?Why Euclidian?Why Euclidian?What is Fractal?What is Fractal?What is Fractal DimensionWhat is Fractal Dimension

More FractalsMore Fractals

SierpinskySierpinsky TriangleTriangle

More FractalsMore Fractals

FernFern

More FractalsMore Fractals

Even CompleterEven Completer

More FractalsMore Fractals

And Natural OnesAnd Natural Ones