Schedule 
Matlab programs  Requirements  Links 
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 phasespace  (M)
Ch.
1 
See also: K.
Popper, C1,C2 and S. Strogatz lectures 

3/11/20  2. Basic applied math tools: 1) Nondimensionalization & 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 

11/11/20  3.Mathematical concepts: Complete flows, existence & uniqueness, conjugacy, equivalence and asymptotics, Stability, Lyapunov Stability (for 1d flows)  (M) Ch 3, 4.14.3, 4.4 and 4.5 only the first paragraphs on 1d flows, 4.7 till pg 129 

Homework 4 
18/11/20 
4. 1d dynamics, implicit function theorem and local bifurcations. The saddle node bifurcation. 
(M)
Ch. 8.1,8.2, 8.4

BacteriaPhagocytes Dynamics, Axiomatic Modelling and MassAction Kinetics,  
25/11  5. 1 dimensional maps: intro, continuous interval maps, contraction map principle, circle maps, expanding maps  (D) 1.11.5 


2/12  6. Expanding maps, symbolic dynamics, Chaos  (D)
1.61.8 
Homework 7  
9/12  7. The quadratic map, Local bifurcation of maps, period doubling route to Chaos  (D) 1.17  Sarkovskii's theorem  
16/12  8.Two dimensional systems – linear 2d systems. Lyapunov functions + linear systems in n dimensions  (M) 2.12.9 


23/12  9. Invariant subspaces, HartmanGrobman thm, l Center manifolds of ndim 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 
Homework 12  
13/1/21  12. Local bifurcations, Stable and unstable manifolds, Global bifurcations 
(M) 8.3, 8.88.12  Unfolding bifurcations (M) 8.38.4, 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  
27/1/21  14. Strange attractors  (M) 7.3  The Poincare sphere  Homework 15 
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.
logistic_map.m eul.m quadratic.m rk4.m runeuler.m