Schedule

Matlab programs

Requirements

Links

Dynamical Systems and modeling.

  Spring 2019

Second semester: Hamiltonian Systems - Sundays at 10:15-12:15
Zyskind 261

Instructor: Vered Rom-Kedar 
Tutors: Michal Pnueli & Ori Katz

Announcements:

On the following Mondays there will be a class instead of a tutorial. The class starts at 13:00 !

25.3,  6.5, 3.6, 17.6

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 and some background in programming. 

 

Date	Topics	Reading	Tutorial	Homework
25/3	1. Introduction: motivation, modeling & examples, the geometrical point of view (1d, 2d dynamics), definition of flows, connection to maps - time 1 map, extended phase-space	(M) Ch. 1 C1,C2		Homework 1 BonusQ
27/3	2. Modeling: non-dimensionalization, pi theorem, scaling,  introto perturbation theory: regular vs singular perturbation theory.	(M, L) Ch. 1 Pi Theorem K. Popper	1/4 Simple ODEs and introduction to numerical solutions of odes: Notes	Homework 2 eul.m quadratic.m rk4.m runeuler.m
3/4	3.Mathematical concepts: Complete flows,  existence & uniqueness, conjugacy, equivalence and asymptotics, Stability, Lyapunov Stability	(M) Chapter 3-4	8/4 Perturbation theory: Notes	Homework 3
10/4	4. 1-d dynamics, implicit  function theorem and local bifurcations Bacteria--Phagocytes Dynamics, Axiomatic Modelling and Mass-Action Kinetics,	(M) Ch. 8.1,8.2, 8.4	15/4 Local bifurcation of maps: Notes	Homework 4
17/4	5. The saddle node bifurcation. 1 dimensional maps:  intro, continuous interval maps, contraction map principle, circle maps, expanding maps	(D)  1.1-1.5 1-d maps Topological transitivity	29/4 Contraction map principle. Notes
1/5	6.  Expanding maps, symbolic dynamics, Chaos	(D) 1.6-1.8		Homework 5
6/5 (Monday!)	7. The quadratic map, period doubling route to Chaos	(D) 1.17	13/5 Sarkovskii's theorem Notes	Study for midterm: instructions for midterm, to be taken on May 23.
15/5	8.Two dimensional systems – linear 2d systems.	(M) 2-6	20/5 Gronwall ineq Notes	23/5 Mid-term (=Homework 6)
22/5	Linear Systems Notes	(M) 2-6	27/5 Lyapunov functions Notes	Homework 7
29/5	9. Invariant subspaces, Grobman-Hartman thm,linear systems in  n-d, 1d/2d  as center manifolds of n-dim systems.	(M) 2-6
3/6 (Monday)	10. The phase-plane	(M) 6	10/12 Index theory Notes	Homework 8
12/6	11.  The phase-plane	(M) 6  Limit cycles, Blue sky catastrophe
17/6 (Monday)	12.  Global bifurcations	(M) 7, 8
19/6	13.  Forced systems and 2d Poincare maps, Homoclinic tangles, Chaotic dynamics in 2d;  horseshoe map, Dissipative chaos	(M) 7	24/6 Smale Horseshoe Notes	Homework 9 See related paper
26/6	Lyapunov exponents and other chaos indicators Notes		1/7 The Poincare sphere Notes	Homework 10
3/7	14. Billards		8/7 Symmetries Notes
22/7	Take home Exam - due on the same day.

Textbooks:

• (M) Differential Dynamical Systems, James D. Meiss, SIAM 2007.
• (D) An Introduction to Chaotic dynamical systems. R. L. Devaney, 2nd Edition, 1989.

 Additional reading:

• (PR) Introduction to Dynamics. Ian Percival and Derek Richards, 1982 Cambridge University Press. 
• (H)  Chaos near resonance. G. Haller, Applied Mathematical Sciences, 138. Springer-Verlag, New York 1999
• (L)  Applied Mathematics, J. David Logan, 2013, John Wiley and Sons, New York
• (KH) Introduction to the Modern Theory of Dynamical Systems,  Anatole Katok, Boris Hasselblatt, Volume 54 of Encyclopedia of Mathematics and its Applications
• (LU) Four-dimensional integrable Hamiltonian systems with simple singular points (topological aspects). Lerman, L. M.; Umanskiy, Ya. L.  Translated from the Russian manuscript by A. Kononenko and A. Semenovich. Translations of Mathematical 
  Monographs, 176. American Mathematical Society, Providence, RI, 1998.
• (MH) Introduction to Hamiltonian dynamical systems and the N-body problem, Meyer, K.,  G.R.  Hall, D. Springer-Verlag, New York, 1992 
• (MC) Chaotic Billiards Nikolai Chernov and Roberto Markaria, Publication Year: 2006, Mathematical Surveys and Monographs, vol. 127

• (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.

• (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.
• (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  (no late submissions).
There will be an exam+midterm
Grade:  50% homework (best 80%) +  10% mid-term + 40% exam.

Take home midterm and exam – these will be due within the same day it is given (expected length 2-4 hours).

No collaboration or internet use in your exam!

You can discuss homeworks - but please submit separately and indicate collaborations/sources.

Bonus questions are for your own curiosity and development. You are welcome to come and discuss them with us or with other students.

Matlab Programs

eul.m quadratic.m rk4.m runeuler.m

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