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.

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
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. 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. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
6
Stability CriteriaStability Criteria

7. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
7
PoincarePoincare Mapping TechniqueMapping Technique

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. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
9
Domains of AttractionDomains of Attraction

10. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
10
Bifurcations of Equilibrium PointsBifurcations of Equilibrium Points

11. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
11
Bifurcations of Closed OrbitsBifurcations of Closed Orbits

12. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
12
HomoclinicHomoclinic BifurcationsBifurcations
Homoclinic intersection
Homoclinic bifurcation
and basin boundary
“metamorphosis”

13. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
13
Example of Attraction DomainExample of Attraction Domain
Fractal BoundariesFractal Boundaries

14. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
14
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
17
RayleighRayleigh--BenardBenard Convection ProblemConvection Problem

18. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
18
The Lorenz SystemThe Lorenz System

19. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
19
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
22
Lorenz Strange AttractorLorenz Strange Attractor

23. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
23
Sensitivity to Initial ConditionsSensitivity to Initial Conditions

24. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
24
TheThe HenonHenon MapMap

25. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
25
TheThe HenonHenon Strange AttractorStrange Attractor

26. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
26
Chaotic Trajectory on theChaotic Trajectory on the HenonHenon AttractorAttractor

27. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
27
The Logistic Map (1)The Logistic Map (1)

28. 19 Sept 2000
IMA 1st Conference in Fractal
Geometry, DMU
28
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
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. 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