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.
М.Г.Гоман (2000) – Динамика нелинейных систем и хаос
1. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
1
Nonlinear Systems Dynamics and ChaosNonlinear Systems Dynamics and Chaos
M.G.Goman
Institute of Mathematical and Simulation Sciences
De Montfort University, Leicester LE1 9BH
2. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
2
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
3. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
3
Examples of Chaotic DynamicsExamples of Chaotic Dynamics
The Lorenz System
3-dim continuos system
The Henon Attractor
2-dim invertible discrete map
T ehe Feigenbaum Cascad
1-dim non-invertible discrete map
4. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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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
5. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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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)
6. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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Stability CriteriaStability Criteria
7. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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PoincarePoincare Mapping TechniqueMapping Technique
8. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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Stable and Unstable ManifoldsStable and Unstable Manifolds
W
W
u
2
s
n-1
G
n-1,2
W
W
W
s
n-1
n-1
u
u
1
1
L
9. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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Domains of AttractionDomains of Attraction
10. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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Bifurcations of Equilibrium PointsBifurcations of Equilibrium Points
11. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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Bifurcations of Closed OrbitsBifurcations of Closed Orbits
12. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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HomoclinicHomoclinic BifurcationsBifurcations
Homoclinic intersection
Homoclinic bifurcation
and basin boundary
“metamorphosis”
13. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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Example of Attraction DomainExample of Attraction Domain
Fractal BoundariesFractal Boundaries
14. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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HomoclinicHomoclinic Trajectories and ChaosTrajectories and Chaos
15. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
15
Bifurcation Scenarios Leading to ChaosBifurcation Scenarios Leading to Chaos
Landau-Hopf Sequence
Period-Doubling Cascade
16. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
16
Geometrical Properties of Strange AttractorGeometrical Properties of Strange Attractor
17. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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RayleighRayleigh--BenardBenard Convection ProblemConvection Problem
18. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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The Lorenz SystemThe Lorenz System
19. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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QualitativeQualitative AnalisysAnalisys of the Lorenz Systemof the Lorenz System
20. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
20
Bifurcation Diagram for Lorenz SystemBifurcation Diagram for Lorenz System
21. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
21
Phase Portraits of Lorenz SystemPhase Portraits of Lorenz System
22. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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Lorenz Strange AttractorLorenz Strange Attractor
23. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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Sensitivity to Initial ConditionsSensitivity to Initial Conditions
24. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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TheThe HenonHenon MapMap
25. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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TheThe HenonHenon Strange AttractorStrange Attractor
26. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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Chaotic Trajectory on theChaotic Trajectory on the HenonHenon AttractorAttractor
27. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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The Logistic Map (1)The Logistic Map (1)
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IMA 1st Conference in Fractal
Geometry, DMU
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The Logistic Map (2)The Logistic Map (2)
29. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
29
Period Doubling Bifurcation SequencePeriod Doubling Bifurcation Sequence
in Logistic Mapin Logistic Map
30. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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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
31. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
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Concluding Remarks (II)Concluding Remarks (II)
l “Stretching and folding” generates chaos
l 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 ones
l Statistical methods expand the understanding of Chaos