Schedule

Matlab programs

Requirements

Links

Dynamical Systems and modeling - reading course.

  Fall 2020

Instructor: Vered Rom-Kedar
 

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.

Students are expected to read the material and solve the homework independently, using the zoom meeting for clarifying delicate points.

It will be helpful to send questions regarding the material or homework by Sunday, two days before the meeting.

Collaboration between students in discussion groups is highly recommended.

 

Discussion Date	Topics	Basic Reading	Additional readings and related topics	Homework
27/10/20	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	See also: K. Popper, C1,C2 and S. Strogatz lectures	Homework 1
3/11/20	2. Basic applied math tools: 1) Non-dimensionalization & pi theorem, 2) Scaling. 3) Introductio to perturbation theory: regular vs singular perturbation theory. 4) Simple ODEs and introduction to numerical solutions of odes:	Pi theorem 1 or Pi Theorem 2 (L) Ch. 1.1,1.2 Notes on numerical solutions of ODEs Notes on Perturbation theory	On consistency, stability and convergence of numerical solutions	Homework 2 Homework 3 eul.m quadratic.m rk4.m runeuler.m
11/11/20	3.Mathematical concepts: Complete flows,  existence & uniqueness, conjugacy, equivalence and asymptotics, Stability, Lyapunov Stability (for 1d flows)	(M) Ch 3, 4.1-4.3, 4.4 and 4.5 only the first paragraphs on 1d flows, 4.7 till pg 129		Homework 4
18/11/20	4. 1-d dynamics, implicit  function theorem and local bifurcations. The saddle node bifurcation.	(M) Ch. 8.1,8.2, 8.4	Bacteria--Phagocytes Dynamics, Axiomatic Modelling and Mass-Action Kinetics,	Homework 5
25/11	5.  1 dimensional maps:  intro, continuous interval maps, contraction map principle, circle maps, expanding maps	(D)  1.1-1.5 1-d maps Topological transitivity		Homework 6
2/12	6.  Expanding maps, symbolic dynamics, Chaos	(D) 1.6-1.8		Homework 7
9/12	7. The quadratic map, Local bifurcation of maps, period doubling route to Chaos	(D) 1.17	Sarkovskii's theorem	Homework 8 Homework 8 -summary
16/12	8.Two dimensional systems – linear 2d systems. Lyapunov functions + linear systems in n dimensions	(M) 2.1-2.9		Homework 9
23/12	9. Invariant subspaces, Hartman-Grobman thm, l Center manifolds of n-dim systems.	(M) 4.8, 5		Homework 10
30/12	10. The phase plane.	(M) 6	Index theory	Homework 11
6/1/21	11. Phase plane,     Lyapunov exponents	(M) 6.4, 6.5, 6.6, 7.1,7.2	Limit cycles, Blue sky catastrophe	Homework 12
13/1/21	12.  Local bifurcations,      Stable and unstable manifolds, Global bifurcations	(M) 8.3, 8.8-8.12	Unfolding bifurcations (M) 8.3-8.4, Normal forms (M) 8.5	Homework 13
20/1/21	13.  Forced systems and 2d Poincare maps, Homoclinic tangles, Chaotic dynamics in 2d;  horseshoe map, Dissipative chaos	(M) 8.13, Smale Horseshoe , Smale birkhoff homoclinic theorem	Homoclinic tangles and fluid mixing	Homework 14
27/1/21	14. Strange attractors	(M) 7.3	The Poincare sphere	Homework 15

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

This is a pass/fail reading course.

Do not submit homeworks, rather:

1) try to solve on your own

2) discuss them with friends,

3) write your own solution

4) compare with friends.

If you have doubts/conflicting solutions, please raise this in the discussion session.

 

Bonus questions are for your own curiosity and development. You are welcome to come and discuss them in the discussion session

and/or with me and/or with other students.

Matlab Programs

logistic_map.m 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