A Beginners' Introduction to Elementary Concepts of Dynamics

by

Frederick David Abraham

A Workshop for the SCTPLS 11th Annual
Convention

University of Wisconson, Madison August 2, 2001

This workshop presents many elementary concepts of dynamical systems theory, as well as concepts of design and analysis methods for nonlinear research. It consists of a series of four PowerPoint presentations. There are hyperlinks to two dynamics programs (Berkeley Madonna for model construction; Santis for data analysis), for illustration, exercises, and participant use. This outline shows the topics covered to help potential participants determine the desirability of attending. The presentation will be available on a take-home CD for continued review of the basic principles and for completing exercises for which there was insufficient time during the workshop. The workshop preparations were assisted by Olga Mitina, especially the more technical aspects involved in the use of the computer programs. Time considerations may require some topics be skipped or under-cranked.

PART
I **Introduction
to Basic Concepts** (26
slides)

1. Basic Philosophy, Basic and Essential Features

2. State Space

3. Vectorfield

4. Trajectory

5. Portrait

6. Example systems: Conflict, Prey-Predator (portraits and equations)

7. Features of Limit Sets: Attractors, Repellors, Saddles; Point, Periodic, Chaotic

8. Basins and Separatrices

9. Equations: Ordinary differential for continuous variables; Iterative for discrete variables.

10. The Logistic Equation

11. Bifurcations

12. Summary

PART II **Characteristic Properties of
Features of Portraits** (17 slides)

1. Liapunov Characteristic Exponents

2. Characteristic Exponents

3. Poincaré Characteristic Multipliers

4. Poincaré Section and First Return Map

5. Chaotic Attractors

6. Sensitivity and Insensitivity to Initial Conditions: Attractor Invariants

7. Divergence and Convergence: Loss/Gain of Information

8. Fractal Microstructure

9. Relationship: Portraits, Time Series, Spectra

10. Summary

Part III **Bifurcations, Structural Stability,
Networks & Self-Organization** (12 slides)

1. Catastrophic Bifurcation: Static Fold in 1D; Response Diagram and Characteristic Exponents

2. Static Cusp Catastrophe

3. Subtle Bifurcations: Poincaré-Andronov-Hopf Bifurcation

4. Subtle Bifurcations: Classification

5. Explosive Bifurcations: Blue-Loop in 2D

6. Structural Stability

7. Bifurcation Sequence with Hysteresis

8. Networks & Self-Organization

9. Summary

Part IV **Experimental Design and Analysis
in Nonlinear Research** (24 slides)

1. Traditional Linear Methods: Frequency Spectra (power, co-power, coherence, phase); Canonical and Monte Carlo Methods

2. Comparison of Linear and Nonlinear Methods

3. Reconstruction of State Space and Attractor from Lagged Variables

4. Determining the Lag: Mutual Information

5.
Finding the
Embedding Dimension, d_{E}

6.
Criteria for knowing
when d_{E }is sufficiently large: False nearest neighbors, Liapunov
Exponents, fractal dimension, vectorfield, unfolding of the attractor.

7. Local Dynamics & Dimension: Local Liapunov Exponents & False Nearest Neighbors.

8. Invariant Characteristics of the Dynamics: Fractal Dimension & Liapunov Exponents

9.
D_{0} from
Box Counting

10. D2, Correlation Dimension

11. Global Liapunov Exponents

12. Liapunov Dimension

13. Local Liapunov Exponents

14. Recurrence Plots

15. Design & Analysis Considerations: Monte Carlo (surrogate, randomization for inferential estimation, other data mangling techniques), addition of different amounts of noise, passing known attractors through the analysis routine, etc.

16. Chaophilosophy

fda: june 30, 2001; update jan 29, 2002