Schedule

Text books

Detailed program

Matlab programs

Tutorials

Requirements

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Dynamical Systems and modeling.

  Winter 2008

Lectures: Mondays,  11:00-13:00,  Ziskind 261

Instructor: Vered Rom-Kedar 

Teaching assistants:   Eli Shlizerman  and  Roy Malka
Tutorials:  Thursdays, 10:00-11:00, Zyskind 261.

Announcements:
 Last updated: June 12, 2008

The exam date is set to June 23, 11:00-13:00.

 

The exam is with open notes but no books (or copies of books).

 

Please submit all the homework by June 11.
These will be returned by June 18 so you will have time to study and complete any gaps you had.
Homework submitted after this date and before June 18 will count,
but these will be returned only after the exam.
You can colloaborate on homework - but please submit individual works (i.e. in your own words) and indicate your collaborator.
I will not accept homework after June 18.

 

Good luck to all!

Go to MAAA seminars on Tuesdays @ 11:00
 

The course will introduce the students to some basic mathematical concepts of dynamical system theory and chaos. These concepts will be demonstrated using simple fundamental model systems based on discrete maps and ordinary differential equations. Motivation for the models arising in various fields of physics and biology will be discussed. The aim of this course is to provide the students with analytical methods, concrete approaches and examples, and geometrical intuition so as to provide them with working ability with non-linear systems.

To participate students should have mathematical background in linear algebra, differential equations and some functional analysis. 

Date	Topics	Tutorials	Homework	Reading
22/10/07	1. Introduction: motivation, modeling & examples, existence & uniqueness, stability.	a. Simple odes  b. Introduction to numerical     solutions of odes.	1-tex , 1-pdf  2-tex , 2-pdf 2-programs	C1,C2,
21/1/08	2. Introduction:  Modeling and non-dimensionalization, Pi theorem; The bacteria model	c. regular vs singular     perturbations.  (BO) Sec 7	3-tex , 3-pdf	Pi theorem, (LS) Sec 6.2
27/1/08 Sunday	1. Introduction once more
28/1/08	3. 1 dynamics: 1-d flows and local bifurcations, Implicit function theorem, 1-d maps.	d. 1d bifurcations	4-tex , 4-pdf	C4 (Ott) chap 2 (GH) sec 3
4/2/08	4. 1 dimensional maps:  intro, continuous interval maps
11/2/08	5. Chaos-expanding maps and symbolic dynamics			(D) ch 1.1-1.8
18/2/08	6. Chaos-the quadratic map, symbolic dynamics		5-tex , 5-pdf MChomewrk5	(D) ch 1.5,1.6,1.7,1.10
25/2/08	7. Period doubling route to chaos, Intermittency.			(D) 1.17, (St) 10.2-7
3/3/08	8.  Two dimensional systems - linear systems, invariant subspaces, Grobman-Hartman thm		6-tex , 6-pdf	(GH)1.1-1.3,1.6
10/3/08	9.  Two dimensional systems - Lyapunov Stability (Avi Soffer)		7-tex  7-pdf
17/3/08	10. Two dimensional systems - Poincare-Bendixon Thm, index theory & the GN model (Roy Malka & Eliezer Shochat)
24/03/08	11. Structural stability and two-dimensional flows			(GH) 1.7-1.9
31/03/08	12. Co-dimension 1 bifurcations			Bifurcations
07/04/08	13. Normal forms.		8-tex  8-pdf
14/04/08	14.Hopf bifurcation & limit cycles.			Limit cycles Blue sky catastrophe
28/04/08	15. Horseshoe map and symbolic dynamics, Smale-Birkhoff homoclinic theorem.
05/05/08	16.  3d flows: Lorenz attractor, Shilnikov mechanism.		9-tex  9-pdf
12/05/08	17. Hamiltonian systems; Forced systems : Global bifurcations - Melnikov integral, the separatrix map and transport, resonances and Avergaging.
19/05/08	18. Near integrable Hamiltonian systems - Arnold Liouville theorem, KAM, resonances		10-tex  10-pdf
27/05/08	19. Near integrable Hamiltonian systems - KAM, the resonance web, Energy momentum diagrams and higher dimensional chaotic mechanisms.			Tutorial on KAM
23/06/08	Final exam; open notes, no books.

 

Textbooks:

• (GH) Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Guckenheimer, J and P. Holmes, Springer-Verlag, 1983.
• (C) Online lecture notes by M. Cross (including Java numerical simulations)
• (Cv) Chaos: classical and quantum by P. Cvitanović et al., Online.

Additional reading:

• (W) Introduction to Applied Nonlinear Dynamical Systems and Chaos. Stephen Wiggins, 1990. (Texts in Applied Mathematics, Vol 2).
• (PR) Introduction to Dynamics. Ian Percival and Derek Richards 1982 Cambridge University Press.
• (Ott) Chaos in dynamical systems. Edward Ott. 1993 Cambridge University Press.
• (St) Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering by Steven H. Strogatz
• (Sc) Deterministic Chaos: An Introduction; Heinz Georg Schuster, [VCH, 2nd edition, 1989]
• (JS) Classical Dynamics, a contemporary approach. Jorge V. Jose and Eugene J. Saletan. 1993 Cambridge University Press.
• (H)  Chaos near resonance. G. Haller, Applied Mathematical Sciences, 138. Springer-Verlag, New York 1999
• (D) An Introduction to Chaotic dynamical systems. R. L. Devaney, 2nd Edition, 1989.
• (LS) Mathematics applied to deterministic problems in the natural sciences, C.C. Lin and L.A. Segel, McMillan Publishing Co., 1974.
• (BO) Advanced mathematical methods for scientists and engineers,  C.M. Bender and S.A. Orszag, McGraw-Hill book company, 1978

Grades etc

Homework assignments will be given every week and will be discussed in the tutorials (no late submissions).
There will be an exam.
Grade:  60% homework (best 80%) +  40% exam.

Matlab Programs

logistic map  Euler scheme circle map  lorenz lorenz w manifolds  henon pendulum standard map  pendulum w forcing

Links

Online "Labs":

• The Physics Lab by E. Neumann
• M. Cross Lab
• B. Fraser lab
• The pendulum lab of F-J Elmer.
• Billiards, standard map and others by V. Donnay et al.
• Double pendulum by Wolfram

Online courses

• Eli Tziperman
  
• M. Cross

Local activities:

 Mathematical Analysis and Applications Seminar

Detailed Program:

Introduction:

• A bit of history (Poincare and the 3 body problem, Lorentz and the ``butterfly effect'')
  
• Modeling - defining phase space, variables, parameters.
  
• Classes of  Dynamical systems (Examples: population dynamics, pendulum, Lorenz eq):
• continuous vs discrete time/space (ODEs vs maps; St 348)
  
• conservative vs dissipative (St 312)
  
• linear vs non-linear
  
• autonomous vs non-autonomous systems.
  
• initial value vs boundary value
  
• deterministic versus stochastic modeling
  
• finite vs infinite dimensional models (PDE's, integral eq.)
• Existence, uniqueness and smooth dependence of solutions of ODE's on initial conditions and parameters.
• The geometric approach to dynamical systems.
  
• Fixed points, linearization, and stability.
• Non-dimensionalization, the Buckingham Pi theorem (see MathWorld or here ),
  
• small parameters, scales.
  
• Perturbation theory - regular vs singular perturbations
  

 

Bifurcations in one dimensional systems

• What is a bifurcation, local vs global bifurcations (GH § 3.1).
  
• Implicit function theorem
  
• classification of bifurcations by number and type (real/ complex) eigenvalues that cross the imaginary axis.
  
• saddle-node bifurcation (St § 3.1; GH § 3.4)
  
• Transcritical bifurcation, super critical and sub critical (St § 3.2; GH § 3.4).
  
• Pitchfork, super-critical and sub-critical. bead on a rotating hoop, higher order nonlinear terms and hysteresis (St § 3.4; GH § 3.4)
  
• Some generalities: center manifold and normal form. (GH § 3.2-3.3).
  
• Role of symmetry and symmetry breaking (imperfect bifurcations), relation to catastrophes and sudden transitions. (St § 3.6)
  
• Flows on a circle - oscillators, synchronization (fireflies flashing, Josephson junctions) (St § 4)

 Two-dimensional systems and some more basics

•  Linear systems: classifications, fixed points, stable and unstable spaces (St § 5)
  
•  Non-linear systems: phase portrait (St §6.1), fixed points and linearization(St §6.3), stable and unstable manifolds (St §6.4)
•  conservative systems (St §6.5)
•  reversible systems (St §6.6)
•  Solution of the (fully non-linear) damped pendulum equation (St §6.7)
•  index theory (St §6.8).
  
•  Limit cycles: Ruling out and finding out closed orbits (Lyapunov functions , Poincare Bendixon theorem) (St §7.2 and §7.3)
  
•  relaxation oscillations (relation to "stick-slip", friction) (St §7.5)
•  Non-linear oscillators (Duffing eq) (St §7.6)
• Averaging method and two time-scales (St §7.6)
  
• Hopf bifurcation and oscillating chemical reactions (St §8.2)

Forced two-dimensional systems

•  Global bifurcations of cycles: saddle-node bifurcations, homoclinic bifurcations, examples in Josephson Junction and driven pendulum in 2D (St §8.4 and§ 8.5)
• Quasi periodicity, coupled oscillators, nonlinear resonance/ frequency locking (Frequency locking of glacial cycles to earth orbital variations), (St §8.6)
• Poincare maps (St §8.7) 

Chaos, transitions to chaos 

• The Lorentz model as an introduction to chaotic systems (examples briefly motivating it from atmospheric dynamics and as a model of Magnetic field reversals of the Earth); and then a more systematic characterization of chaotic systems (examples from fluid dynamics and mantle convection) (St § 9).
  
• Some preliminaries: Poincare maps, Lyapunov exponents, delay coordinates, fractal dimensions, Kolmogorov entropy. (St § 8.7, 10.5, 11.4-5).
  
•  Possibility of short-time predictability (Gronwall's Lemma).
•  The problem in relying on simulations, the shadowing lemma.
•  Universal routes to chaos:  
• Period doubling: logistic map, renormalization, universality. (Sc § 3)
  
• Intermittency (types I,II,III Sc § 4)
  
• Quasi-periodicity (1D circle map: El Nino's chaos) (Sc § 6)
  
•  The horseshoe map and homoclinic tangles (GH 230-255, W 420-443, 482-483, 519-552)
  
•  Shilnikov's model and revisiting the Lorenz attractor. (GH 318-340 & 312-318, W 552-591)

 Hamiltonian systems

• Examples (Pendulum, Duffing, The n-body problem, point vortices, Lagrangian advection) (GH, W etc)
  
• Basics (canonical coordinates, Linear systems, generating functions,1 d.o.f. systems) (e.g. PR 42-60, 84-116)
  
• On near integrable (chaotic) systems: (GH 166-193, W 106-175,483-513) 
• 1.5 d.o.f. + Averaging and KAM theory.
  
• Melnikov theory and resonances.
  
•  Transport theory and applications to fluid flows
  
• Influence of small dissipation.
  
•  On n d.o.f. systems: (Encyclopedia of mathematical sciences, vol. 3, papers)
  
• Integrable systems (Liouville-Arnold theorem)
  
• Near integrable systems in Higher dimensions
  
•  The resonance web and instabilities in phase space

Tutorials:

Tutorial 1: Solving simple initial value problems of ODEs analytically: 1d equations, linear systems with constant coefficients, special equations, special systems, linearizations.

Tutorial 2: Solving with matlab: what is a discretization, what are the different ODE packages, errors, global errors, test problems.
The role of computers in nonlinear dynamics, a simple example of a numerical solution method for ODEs (improved Euler scheme). 

Tutorial 3:  Local bifurcations in 1-d maps