Schedule first semester Schedule second semester

Text books

Matlab programs

Requirements

Links

Dynamical Systems and modeling.

  Fall 2014-Spring 2015

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

Instructor: Vered Rom-Kedar 
Tutor: Mary Kloc (email: MaryKloc@gmail.com)

Announcements:

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. 

In the first semester we will study basic topics in dynamical systems theory and modeling.

The second semester will be dedicated to Hamiltonian systems with applications to mechanics, fluid mixing and billiards.

Students may take only the first semester for 2 credit points.

Second semester - advanced topics in Hamiltonian systems and modeling

Hamiltonian systems are fascinating: on one hand these systems model phenomena appearing in nature and on the other, their analysis involves
a rich and elegant mathematical theory.

While the solutions of integrable Hamiltonian systems may be completely described, most Hamiltonian systems that
model real problems are not integrable and typically have a mixed phase space in which both chaotic and regular motions coexist.

In this course we will learn a little bit about integrable systems (the Arnold-Liouville theorem, Energy-momentum diagrams and Fomenko Graphs) and then concentrate on studying near-integrable (various types of Hamiltonian chaotic mechanisms) and far from integrable (near billiards dynamics) Hamiltonian systems.

The course is advanced. Students attending this course should have good knowledge of dynamical systems (preferably attend the first semester or an equivalent course), ordinary differential equations, linear algebra and some functional analysis.  

Date	Topics		Homework	Reading/materials
15/3/15	1.  Introduction to Hamiltonian systems: Definitions, some applications, 1 dof systems, phase space, canonical coordinates,  integrability and non-integrability,		hmwrk 1	(M) 9.1-9.3
22/3/15	2. Loop action, the action principle, billiards			(M) 9.4-9.6
29/3/15	3.Linear Hamiltonian systems, Hamiltonian and Symplectic matrices, the structure of energy surfaces near fixed points.		hmwrk 2	(M) 9.10 (MH) 1,3 (LU)
12/4/15	4. Generating functions, action-angle coordinates			(MH) 6
19/4/15	5. Liouville theorem		hmwrk 3
26/4/15	6. Symmetries, Noether thm     1.5 dof systems:  motion in the extended phase space,  a few words on averaging theory KAM theory and resonances			(M) 6.4, (MH) 8.4 (M) 9.13, (GH), (W)
3/5/15	7. KAM theory, The Arnold resonance web,  Arnold diffusion, Global bifurcations, homoclinic tangles
10/5/15	8. Melnikov integral,			(M) 8.12
17/5/15	9.  The separatrix map, Lobe dynamics and transport., Resonances.		hmwrk 4 henon.m
24/5/15	Shavuot
2/6/15	10. Energy-momentum bifurcation diagrams and the branched surfaces, Resonances: elliptic, hyperbolic and parabolic resonances, degenerate resonances (Zaslavski web maps).		hmwrk 5
7/6/15	11.Fluid mixing - characterization of mixing, coherent structures and dividing surfaces, 2d vs 3d finite time and finite resolution effects
14/6/15	12. Mixing : fluid mixing and DS mixing, Lyapunov exponents
21/6/15	13.Billiards - billiard flows and billiard maps, integrable billiards and chaotic billiards,                       some basic properties of Sinai billiards.			(MC), (KH)
23/6/15	14. Soft billiards and Hamiltonian systems with soft impacts. (11:15-13:15 in room 1)			(MC), (KH)
Exam

 

First semester - basic notions

Date	Topics	Homework	Reading
28/10/14	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) Pg 23 ex 1,2,4 Bonus: integrate the eq.  numerically and draw phase portraits.   Due: 4/11/14	(M) Ch. 1 C1,C2
4/11/14	2. Modeling: non-dimensionalization, pi theorem, scaling,  introto perturbation theory: regular vs singular perturbation theory.	(M) Pg 23 ex 3,7,10 Bonus: integrate the eq.  numerically and draw phase portraits.   Due: 11/11/14	(L) Ch. 1 Pi Theorem K. Popper
11/11/14	3. 1-d dynamics, stability, implicit function theorem, co-dim 1 bifurcations	(M) pg 325 ex 1(a,b,d), 3. Bonus: 1) Read the following papers on Bacteria-Neutrophils dynamics: Motivation and application [1] Mathematical derivation [2] 2) Provide a modification which appears relevant to you and discuss its outcome. Due: 18/11/14	(M) Ch. 8.1,8.2, 8.4
18/11/14	4. Saddle node bifurcations:  flows and maps.  Basic notions of  DS: Complete flows.	Find the bifurcation diagram for the fixed points of the logistic map  for 0<r<3 . Bonus: read about the logistic map. Due: 25/11/14	(M)  Ch 3
25/11/14	5. Existence and uniqeness & contraction map principle.	(M) pg 102 ex 5	(M)  Ch 3,4
2/12/14	6. Global existence, homeomorphisms, diffeomorphism, conjugacy.		(M) Ch 4
9/12/14	7. Numerical integrations and Matlab intro (Mary Kloc)
16/12/14	8.  Equivalence, omega/alpha limit sets		(M) 4.9
23/12/14	9. omega/alpha limit sets, 1d maps	homework 6 homework 6(tex file)	(D), (KH)
30/12/14	10.    1d maps, chaos and symbolic dynamics.		(D), (KH)
6/1/15	11.  Attractors, linear Systems, linear stability, invariant manifolds,	homework 7 homework 7(tex file)	(M) Ch 6  Bifurcations
13/1/15	12.  Lyapunov Stability, Lyapunov functions	homework 8 homework 8(tex file)	(M) Ch 6
20/1/15	13.  Periodic orbits and Poincare maps, Poincare-Bendixon Thm, index theory.	homework 9 homework 9(tex file) [see re discussion on Euler schemes:  http://people.sc.fsu.edu/~jpeterson/IVP.pdf ]	(M) Ch 6
27/1/15	14.  Global bifurcations, Smale horseshoe  and symbolic dynamics. Relation to dissipative chaos - strange attractors ( Lorenz attractor, Shilnikov's chaos).
			Limit cycles Blue sky catastrophe
12/2/15	Exam	See previous exams for the style of questions. Notice - the course changes every time - so some topics in these exams were not covered in this course/some topics were added - the exam will change accordingly, and will definitely be shorter (these exams took people much more than 3 hours): exam1  exam 2 exam 3

Textbooks:

• (MH) Introduction to Hamiltonian dynamical systems and the N-body problem, Meyer, K.,  G.R.  Hall, D. Springer-Verlag, New York, 1992 
• (M) Differential Dynamical Systems, James D. Meiss, SIAM 2007.
• (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.

 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
• (D) An Introduction to Chaotic dynamical systems. R. L. Devaney, 2nd Edition, 1989.
• (KH) Introduction to the Modern Theory of Dynamical Systems,  Anatole Katok, Boris Hasselblatt, Volume 54 of Encyclopedia of Mathematics and its Applications
• (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.
Grade:  60% homework (best 80%) +  40% exam.

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

Matlab Programs

rk2  rk4
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