М.Г.Гоман (2000) – Динамика нелинейных систем и хаос
DESCRIPTION
М.Г.Гоман «Динамика нелинейных систем и хаос», доклад на 1-й конференции Института математики и приложений (IMA) по фрактальной геометрии, г.Лейстер (Великобритания), 19 сентября 2000 года. M.G.Goman "Nonlinear Systems Dynamics and Chaos", presentation at the IMA (Institute of Mathematics and its Applications) 1st Conference in Fractal Geometry, De Montfort University, Leicester, the UK, 19 September 2000.TRANSCRIPT
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