М.Г.Гоман, А.В.Храмцовский (2000) - Численный анализ нелинейной динамики самолёта
М.Г.Гоман (2000) – Динамика нелинейных систем и хаос
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Transcript of М.Г.Гоман (2000) – Динамика нелинейных систем и хаос
19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 1
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