1MA217 Homepage
A collection of links to various Dynamical Systems resources
Homework
Timetable


Course Info:
Course: Dynamical Systems 1MA217
Lecturer: Denis Gaidashev, Ång 14231, gaidash at math.uu.se
First Day Handout
Textbooks:
Primary text: M. Brin, G. Stuck, Introduction to Dynamical Systems,
Cambridge University Press 2002.
Secondary text: M. W. Hirsch, S. Smale, R. L. Devaney, Differential Equations, Dynamical Systems and an Introduction to Chaos,
Academic Press (Elsevier) 2004.
Bits and pieces from: A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press 1995.
TAKEHOME, DUE JAN 1, 2017, FINAL VERSION, 9 problems
Things on the current agenda:
My Lecture Notes, based on BS and KH, a work in progress, updated 2016/11/28.
Period three implies chaos, J. Yorke ans T.Y. Li.
A little bit about the exponential map and geodesics on Riemannian manifolds.
A horseshoe and stable/unstable manifolds for Henon maps, an applet .
O. Lanford's computer assisted proof of the Feigenbaum conjecture and the renormalization hyperbolicity .
Universal properties of the maps of the interval, P. Collet, J.P. Eckmann and O. Lanford .
C. Liverani's notes on the stableunstable manifold theorem .
John Milnor's lectures on dynamics in one complex variable .
A paper about the recurrence property of the Arnold cat map .
Jaume Alonso's slides on the Siegel theorem .
Iterations and the bifurcation diagram for the quadratic map
Interactive cobweb plot for the quadratic map by Andre Burbanks
Paper by J. Banks et al, "On Devaney's Difinition of Chaos".
Course Objectives/Outcomes:
1) Understand the fundamental concepts of the dynamical systems, specifically:
 dynamics of the quadratic family,
 conjugacies, specifically as relates to circle maps,
 topological dynamics; recurence, mixing and transitivity,
 symbolic dynamics, shifts
 hyperbolicity and structural stability,
 stable and unstable manifolds, homoclinic and heteroclinic intersections, horseshoes,
 fundamental concepts of continuous dynamics, PoincareBendixson theorem, Poincare return maps, etc
 fundamental concepts of Hamiltonian dynamics and KolmogorovArnoldMoser theory,
 fundamental concepts of ergodic theory,
 some concepts from complex dynamics, such as Julia and Mandelbrot sets.
2) Solve representative problems in the abovementioned topics.
3) Carry out numerical studies of dynamical systems.
4) Understand and be able to explain/present some applications of the theory.
Grading:
There will be a takehome final, 100% of the final grade.
Schedule
Below: BS stands for BrinStuck, HSD for Hirsch, Smale and Devaney, KH for Katok Hasselblat.
 20160830
 Introduction, course information.
 What is "dynamics"?
 The quadrtic family as the fundamental example of a chaotic dynamical system.
Reading: 1.1 BS, 1.5 BS, 15.1 HSD, 15.3 HSD, 15,4 HSD
 20160901
 The quadratic family continued.
Reading: 1.1 BS, 1.5 BS, 15.1 HSD, 15.3 HSD, 15,4 HSD
 20160907
 The quadratic family continued.
 Expanding endomorphisms of the circle
Reading: 1.3 BS, 1.4 BS. 1.9 a) and b) in KH
 20160908
 Symbolic dynamics.
 Topological dynamics: transitivity and mixing.
Reading: 2.2 and 2.3 BS, Lemma 1.4.2 in KH
 20160914
 Topological dynamics continued: transitivity and mixing.
 Discussion of the Devaney's definition of chaos.
 Topological dynamics continued: limit sets, reccurent sets.
Reading: Paper by J. Banks under "Current Agenda", 2.1 BS, 3.3 KH, 10.1 HSD
 20160914
 Topological dynamics continued: transitivity and mixing.
 Discussion of the Devaney's definition of chaos.
 Topological dynamics continued: limit sets, reccurent sets.
Reading: Paper by J. Banks under "Current Agenda", 2.1 BS, 3.3 KH, 10.1 HSD
 20160916
 Irrational rotations of the circle.
 Topological dynamics continued: topological entropy.
Reading: 1.3 BS, 3.3 KH, 2.5 BS, 2.6 BS, 1.9 c) and d) in KH
 20160920
 Topological dynamics continued: topological entropy.
Reading: 2.5 BS, 2.6 BS, 1.9 c) and d) in KH
 20160923
 Subshifts.
 PerronFrobenius Theorem.
Reading: 3.1 BS, 3.2, 3.3 BS, 1.9 in KH
 20160926
 PerronFrobenius Theorem.
 Circle homeomorphisms.
Reading: 3.3, 7.17.2 BS
 20160930
 Circle homeo and diffeomorphisms.
Reading: 7.17.2 BS
 20161005
 Circle homeomorphismsm, rotation number.
 Denjoy's example of a nontransitive circle diffeomorphism
Reading: 7.2 BS.
 20121011
 Poincare's classification of irrational circle homeos, possible limit sets for an irrational circle homeo.
Reading: 7.17.2 BS, 11.1 KH.
 20121014
 Ordered families of orientation preserving homeomorphisms, monotonicity of the rotation number.
Reading: my notes, 11.1 KH.
 20121024
 Devils staircase, Arnold's tongues
 Rational rotations
 Circle diffeos, Denjoy's theorem
Reading: my notes, 11.1 KH, 11.2 KH.
 20161031
 Introduction to KolmogorovArnoldMoser theory.
Reading: Jaume Alonso's slides on the Siegel theorem in Current Agenda, a full proof in CarlesonGamelin, 12.3 KH, 12.5 KH,
Another proof of a version of the theorem, page 21 here
 2012113
Reading: 7.3 BS.
 20121108
Reading: 7.3 BS.
 20121115
 Period doubling and renormalization
Reading: my lecture notes
 20121117
 The linearization problem and the HartmanGrobman theorem
 The HadamardPerron Theorem.
Reading: my lecture notes, Liverani's notes in Current Agenda
 20121128
 The HadamardPerron Theorem.
 Hyperbolic sets and shadowing.
Reading: my lecture notes, Liverani's notes in Current Agenda
 20121130
 Hyperbolic sets and shadowing.
 The horseshoe.
Reading: my lecture notes
 2012127
 Persistence of hyperbolic sets.
Reading: my lecture notes
