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Nonlinear Systems Dynamics and ChaosNonlinear Systems Dynamics and Chaos

M.G.GomanInstitute of Mathematical and Simulation Sciences

De Montfort University, Leicester LE1 9BH

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Chaos in Deterministic Systems:Chaos in Deterministic Systems:What is chaos, Why and When it appears?What is chaos, Why and When it appears?

l Nonlinear dynamic systems and qualitative methods of analysis - equilibria, closed orbits, complex attractors, domains of attraction,

bifurcations,etc.

l Examples of chaotic dynamics- Lorenz system, Henon map, Feigenbaum cascade

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Examples of Chaotic DynamicsExamples of Chaotic Dynamics

The Lorenz System3-dim continuos system

The Henon Attractor2-dim invertible discrete map

T ehe Feigenbaum Cascad1-dim non-invertible discrete map

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What is Chaos?What is Chaos?

l “…it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the future. Prediction becomes impossible…”

Henri Poincare, 1897

l Chaos: Steady behavior of dynamical system , when all trajectories converge to the strange attractor and exponentially diverge their from each other

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Different Types of AttractorsDifferent Types of Attractors

Stable equilibrium (D=0) Stable closed orbit (D=1)

Stable toroidal manifold (D 2) Strange attractor (D=fractional, fractal geometry)

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Stability CriteriaStability Criteria

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PoincarePoincare Mapping TechniqueMapping Technique

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Stable and Unstable ManifoldsStable and Unstable Manifolds

W

W

u2

sn-1

Gn-1,2

W

W

W

sn-1

n-1

u

u

1

1

L

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Domains of AttractionDomains of Attraction

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Bifurcations of Equilibrium PointsBifurcations of Equilibrium Points

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Bifurcations of Closed OrbitsBifurcations of Closed Orbits

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HomoclinicHomoclinic BifurcationsBifurcations

Homoclinic intersection

Homoclinic bifurcation and basin boundary “metamorphosis”

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Example of Attraction DomainExample of Attraction Domain

Fractal BoundariesFractal Boundaries

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Homoclinic Homoclinic Trajectories and ChaosTrajectories and Chaos

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Bifurcation Scenarios Leading to ChaosBifurcation Scenarios Leading to Chaos

Landau-Hopf Sequence

Period-Doubling Cascade

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Geometrical Properties of Strange AttractorGeometrical Properties of Strange Attractor

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RayleighRayleigh--Benard Benard Convection ProblemConvection Problem

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The Lorenz SystemThe Lorenz System

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Qualitative Qualitative Analisys Analisys of the Lorenz Systemof the Lorenz System

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Bifurcation Diagram for Lorenz SystemBifurcation Diagram for Lorenz System

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Phase Portraits of Lorenz SystemPhase Portraits of Lorenz System

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Lorenz Strange AttractorLorenz Strange Attractor

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Sensitivity to Initial ConditionsSensitivity to Initial Conditions

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The The Henon Henon MapMap

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The The Henon Henon Strange AttractorStrange Attractor

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Chaotic Trajectory on the Chaotic Trajectory on the Henon Henon AttractorAttractor

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The Logistic Map (1)The Logistic Map (1)

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The Logistic Map (2)The Logistic Map (2)

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Period Doubling Bifurcation Sequence Period Doubling Bifurcation Sequence in Logistic Mapin Logistic Map

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Concluding Remarks (I)Concluding Remarks (I)

l Regular dynamics (linear or nonlinear) is governed by normal, classical geometry

l Irregular or chaotic dynamics is linked with fractal geometry

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Concluding Remarks (II)Concluding Remarks (II)

l “Stretching and folding” generates chaosl Essence of Chaos is the “sensitive dependence on initial

conditions”, so that even unmeasurable differences can lead to enormously differing results

l Qualitative methods are powerful but not unique onesl Statistical methods expand the understanding of Chaos