Theory of Nonlinear Phenomena (graduate level)
General Information
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Method: 2 hours lecture and 2 hours seminar per week
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Instructor: Dr. Sorinel Adrian Oprisan
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Prerequisite:
- Advanced Calculus (Ordinary and Partial Differential Equations)
- Programming languages (Turbo Pascal, C, etc.) or an integrated package of your choice (Mathematica, Matlab, Maple, etc.)
Course Description:
The goal is to learn the analytical and numerical techniques available for nonlinear models synthesis. While the topic of nonlinear phenomena is very extensive, our focus will be on some very specific concepts. We will study both analytical and numerical methods for one-dimensional nonlinear models. The steady solutions and their stability are investigated to obtain the bifurcation diagram of the model. Two and three-dimensional nonlinear models described by ordinary differential equations are also covered. The steady solution(s) and the stability analysis is based on Hartman-Grobmann theorem. Nonlinear models involving partial differential equations develop spatial and temporal patterns through Turing instability mechanism. We also cover numerical methods for computation of power spectrum, autocorrelation function, Lyapunov exponents, Poincare sections, and fractal dimensions.
Instructional Objectives:
At the end of the course, the students should be able to analytically extract the steady state solution, determine its stability, and extract the bifurcation diagram. Although analytical methods provide a global view of the dynamic it is rarely possible to solve analytically a nonlinear model. The students should be familiar with the available software packages that implement fundamental concepts like: power spectrum, autocorrelation function, Lyapunov exponents, Poincare sections, and fractal dimensions. The students should be able to present their work both as a short (15 minutes) talk and in a written form.
Texts:
1. Murray J.D., Mathematical Biology, Springer-Verlag Berlin, Heidelberg, 1989
2. Parker T. S., Chua L. O., Practical Numerical Algorithms for Chaotic Systems, Springer - Verlag, New York, 1989
Topics:
1. One-dimensional nonlinear modes
1.1. Continuous differential systems. Steady solutions and their stability. Bifurcation diagram
1.2. Delayed differential equations. Limit cycle bifurcations
1.3. Nonlinear maps. Fix points and their stability
1.4. Maps with delay. Chaos transitions
2. Two and three-dimensional differential equation systems
2.1. General rules for nonlinear model synthesis by including elementary processes of
2.1.1. competition
2.1.2. symbiosis
2.1.3. threshold phenomenon
2.2. Two-dimensional maps. Phase space synchronization of coupled maps
3. Nonlinear mechanisms in complex systems
3.1. Autocatalytic effects. Excitation and inhibition processes
3.2. Multiple steady states and hysteresis
3.3. Nonlinear feedback control
3.4. Relaxation mechanisms in excitable media
4. Spatial and temporal patterns
4.1. Reaction-diffusion models
4.2. Waves in excitable media
5. Computational methods of nonlinear dynamics
5.1. Phase space reconstruction. Lyapunov exponents. Poincare sections
5.2. Multifractal analysis
Evaluation and Grading
Homework: One homework every two weeks will be assigned. The purpose is to highlight special techniques presented during the lectures. The assignments may be both analytical and computational. The usual due date is usually after two weeks. Late homework are severely penalized (ten percent of the total grade each day). Academic dishonesty will not be tolerated.
Midterm Examinations: There are two written, in-class, partial examinations. There is one comprehensive final exam. No makeup exams. The exams are tentatively scheduled for ...
Project: A recent paper (less than five years) of your choice must be thoroughly understood and prepared for in-class presentation. The review papers may involve library search, web search, numerical algorithm implementation and testing, and slides preparation. Instead of a review paper you may present your graduate research results if they are related to our topics. However, it is expected that your in-class presentation will also be presentable at a national-level conference.
Grading: Final grade is the weighted average with
Tentative student research projects
    1. Critical self-organization processes (SOC). Scaling laws for avalanches
    2. Neural networks for nonlinear feature extraction
    3. Sandpile model for SOC
    4. Sonoluminescence
    5. N body problem. Self-organizing structures and Universe models
    6. Diffusion limited aggregation (DLA)
    7. Synchronization of chaotic oscillators
    8. Bifurcations and chaos in plasma
    9. Pierce diode
    10. Thermodynamics of crystals with fractal energy spectrum
    11. Nonlinear models for lasers
    12. Experimental analysis of EEG using nonlinear methods
    13. Fractals and physiology
    14. Lotka-Volterra nonlinear models
    15. Nonextensive (Tsallis) statistics applications
    16. Earthquake - a self-organized critically model
    17. Nonlinear oscillations of magnetic domain walls
Supplementary bibliography
  1. Arnold V. I., Geometrical Methods in the Theory of Ordinary Differential Equations, Springer - Verlag, New York, 1988.
  2. Arnold V. I., Ordinary Differential Equations, Springer - Verlag, New York, 1992.
  3. Bergé P., Pomeau Y., Vidal C., Order Within Chaos: Towards a Deterministic Approach to Turbulence, Wiley, New York, 1984.
  4. Bogoliubov N. N., Mitropolsky Y. A., Asymptotic Method in the Theory of Nonlinear Oscillations, Gordon and Breach, New York, 1961.
  5. Coddington E. A., Levinson N., Theory of Ordinary Differential Equation, McGraw - Hill, New York, 1955.
  6. Gaylord R. J., Wellin P. R., Computer Simulations with MATHEMATICA. Explorations in Complex Physical and Biological Systems, Springer - Verlag, New York, 1995.
  7. Glansdoff P., Prigogine I., Thermodynamics Theory of Structures , Stability and Fluctuation, Wiley Interscience, New York, 1971.
  8. Keizer J., Statistical Thermodynamics of Nonequilibrium Processes, Springer - Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, 1987.
  9. Krylov N., Bogoliubov N. H., Introduction to Nonlinear Mechanics, Princeton Univ. Press, Princeton, New Jersey, 1974.
  10. Kubo R., Toda M., Hashitsume N., Statistical Physics, I and II, Springer - Verlag, Berlin, 1985.
  11. Mandelbrot B. B., The Fractal Geometry of Nature, W. H. Freeman, New York, 1983.
  12. Preston K., Duff M., Modern Cellular Automata, Theory and Applications, Plenum Press, 1984.
  13. Prigogine I., From Being to Becoming, Freeman, New York, 1980.
  14. Prigogine I., Stengers J., Order out of chaos, Heinemann, London, 1984.
  15. Ott E., Chaos in dynamical Systems, Cambridge Univ. Press, Cambridge, England, 1993.
  16. Wiggins S., Introduction to Applied Nonlinear Dynamical system and Chaos, Springer - Verlag, New York, 1990
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