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19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
1
Nonlinear Systems Dynamics and ChaosNonlinear Systems Dynamics ...
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
2
Chaos in Deterministic Systems:Chaos in Deterministic Systems:
...
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
3
Examples of Chaotic DynamicsExamples of Chaotic Dynamics
The Lo...
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
4
What is Chaos?What is Chaos?
l “…it may happen that small diffe...
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
5
Different Types of AttractorsDifferent Types of Attractors
Stab...
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
6
Stability CriteriaStability Criteria
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
7
PoincarePoincare Mapping TechniqueMapping Technique
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
8
Stable and Unstable ManifoldsStable and Unstable Manifolds
W
W
...
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
9
Domains of AttractionDomains of Attraction
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
10
Bifurcations of Equilibrium PointsBifurcations of Equilibrium ...
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
11
Bifurcations of Closed OrbitsBifurcations of Closed Orbits
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
12
HomoclinicHomoclinic BifurcationsBifurcations
Homoclinic inter...
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
13
Example of Attraction DomainExample of Attraction Domain
Fract...
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
14
HomoclinicHomoclinic Trajectories and ChaosTrajectories and Ch...
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
15
Bifurcation Scenarios Leading to ChaosBifurcation Scenarios Le...
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
16
Geometrical Properties of Strange AttractorGeometrical Propert...
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
17
RayleighRayleigh--BenardBenard Convection ProblemConvection Pr...
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
18
The Lorenz SystemThe Lorenz System
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
19
QualitativeQualitative AnalisysAnalisys of the Lorenz Systemof...
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
20
Bifurcation Diagram for Lorenz SystemBifurcation Diagram for L...
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
21
Phase Portraits of Lorenz SystemPhase Portraits of Lorenz Syst...
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
22
Lorenz Strange AttractorLorenz Strange Attractor
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
23
Sensitivity to Initial ConditionsSensitivity to Initial Condit...
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
24
TheThe HenonHenon MapMap
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
25
TheThe HenonHenon Strange AttractorStrange Attractor
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
26
Chaotic Trajectory on theChaotic Trajectory on the HenonHenon ...
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
27
The Logistic Map (1)The Logistic Map (1)
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
28
The Logistic Map (2)The Logistic Map (2)
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
29
Period Doubling Bifurcation SequencePeriod Doubling Bifurcatio...
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
30
Concluding Remarks (I)Concluding Remarks (I)
l Regular dynamic...
19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
31
Concluding Remarks (II)Concluding Remarks (II)
l “Stretching a...
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М.Г.Гоман (2000) – Динамика нелинейных систем и хаос

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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.

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М.Г.Гоман (2000) – Динамика нелинейных систем и хаос

  1. 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. 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. 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. 4. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 4 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. 5. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 5 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. 6. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 6 Stability CriteriaStability Criteria
  7. 7. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 7 PoincarePoincare Mapping TechniqueMapping Technique
  8. 8. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 8 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. 9. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 9 Domains of AttractionDomains of Attraction
  10. 10. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 10 Bifurcations of Equilibrium PointsBifurcations of Equilibrium Points
  11. 11. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 11 Bifurcations of Closed OrbitsBifurcations of Closed Orbits
  12. 12. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 12 HomoclinicHomoclinic BifurcationsBifurcations Homoclinic intersection Homoclinic bifurcation and basin boundary “metamorphosis”
  13. 13. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 13 Example of Attraction DomainExample of Attraction Domain Fractal BoundariesFractal Boundaries
  14. 14. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 14 HomoclinicHomoclinic Trajectories and ChaosTrajectories and Chaos
  15. 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. 16. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 16 Geometrical Properties of Strange AttractorGeometrical Properties of Strange Attractor
  17. 17. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 17 RayleighRayleigh--BenardBenard Convection ProblemConvection Problem
  18. 18. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 18 The Lorenz SystemThe Lorenz System
  19. 19. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 19 QualitativeQualitative AnalisysAnalisys of the Lorenz Systemof the Lorenz System
  20. 20. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 20 Bifurcation Diagram for Lorenz SystemBifurcation Diagram for Lorenz System
  21. 21. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 21 Phase Portraits of Lorenz SystemPhase Portraits of Lorenz System
  22. 22. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 22 Lorenz Strange AttractorLorenz Strange Attractor
  23. 23. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 23 Sensitivity to Initial ConditionsSensitivity to Initial Conditions
  24. 24. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 24 TheThe HenonHenon MapMap
  25. 25. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 25 TheThe HenonHenon Strange AttractorStrange Attractor
  26. 26. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 26 Chaotic Trajectory on theChaotic Trajectory on the HenonHenon AttractorAttractor
  27. 27. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 27 The Logistic Map (1)The Logistic Map (1)
  28. 28. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 28 The Logistic Map (2)The Logistic Map (2)
  29. 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. 30. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 30 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. 31. 19 Sept 2000 IMA 1st Conference in Fractal Geometry, DMU 31 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

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