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^{MATH 614: Dynamical Systems and Chaos (Nataliya Goncharuk, fall 2024)}

Homeworks:

HW1, HW1 Solutions, HW2, HW2 Solutions, HW3, HW3 Solutions, HW4, HW4 Solutions, HW5, HW5 Solutions, HW6, HW6 Solutions, HW7, HW7 Solutions,

 

Midterm exam with Solutions and Final exam

Textbooks

[1] Robert L. Devaney, An Introduction to Chaotic Dynamical Systems, Third edition.

[2] John Guckenheimer and Philip Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields.

List of topics for presentations

 Schedule, reading, past assignments and solutions.

 

 

Week 1. Sec 1-5 of [1]. Examples of DS with continuous and discrete time. Fixed points and periodic orbits, stability and asymptotic stability. Examples of aperiodic behavior. Notion of attractor and basin of attraction.

On Monday, I gave definitions of dynamical systems (DS) with discrete and continuous time. For DS with discrete time, I gave definitions of fixed and periodic points, and discussed one-dimensional examples. On Wednesday, I showed that the orbits of irrational rotation are dense on the circle. Also, I introduced hyperbolic fixed points (in dimension 1) and proved criteria for attraction and repulsion; we looked together at the graphs of ax(1-x) and its iterates and discussed stability of fixed and periodic orbits. On Friday, I proved contraction mapping principle, and briefly explained how it implies Existence and Uniqueness for solutions of differential equations. Lec3

Week 2. Sec 6 - 8 of [1]. Notion and philosophy of (deterministic) chaos. Encoding and itineraries of points. Examples: circle expansion x->2x, logistic map x->ax(1-x) for a>4.

On Monday, we discussed the circle expanding map x->2x. I showed that it has a dense set of periodic orbits, demonstrated that numerical experiments do not work for this map, and introduced the itinerary of a point (which happens to coincide with the binary expansion). On Wednesday, I discussed itinerary map for the map x->5x(1-x) for points that never leave [0,1] -- this set is Cantor-like (and explained what is a standard Cantor set). Also, I introduced the metric in the space of sequences and proved that Bernoulli shift is continuous with respect to it. On Friday, I proved that the itinerary map for x->2x on the circle is almost 1-to-1, not continuous, but its inverse is continuous. Also, the itinerary map for x->5x(1-x) is 1-to-1, continuous, and its inverse is continuous. I used this to show that x->2x on the circle has a dense orbit, and x->5x(1-x) has an orbit that is dense in its non-escaping Cantor-like set. Lec4 Lec5 Lec6

Week 3. Sec 28, 30 of [1], or Sec.5.1 of [2]. Notion and philosophy of (deterministic) chaos. Encoding and itineraries of points. Examples: Smale's horseshoe, Baker's map.

On Wednesday, I introduced the Smale's horseshoe (a version given by particular linear maps). We found its fixed points, and I explained why the itinerary map is one-to-one and continuous with its inverse. On Friday, I discussed the Devaney's definition of the chaotic map (see Sec.8 in [1]), including sensitivity on initial conditions, and explained why the Bernoulli shift on bi-infinite sequences --- and thus the Smale's horseshoe map --- has this property. Also, I introduced the sets W^s(y) and W^u(y) and described them for the Bernoulli shift and for the Smale's horseshoe. Lec7 Lec8

Week 4. Sec 5.8 of [2]. Optional reading: Katok&Hasselblat, Modern theory of dynamical systems, sec. 4.1, 4.2. Baker's map. Measure and measure-preserving maps. Ergodicity of irrational rotation. Ergodicity of circle doubling and the general notion of ergodicity; Birkhoff's ergodic theorem.

On Monday, I introduced the Baker's map (given by (x,y)-> (2x, y/2) on the left half on the unit square and by (x, y) -> (2x-1, y/2+1/2) on the right half), and compared it to the Smale's horseshoe map. Also, I introduced the notion of measure and Lebesgue measure, and gave examples. On Wednesday, I introduced measure-preserving maps, defined ergodicity, and formulated a version of Birkhoff ergodic theory that involves average time spent in a given set. On Friday, I proved Birkhoff ergodic theorem for the Bernoulli shift using the Law of Large numbers; this implies ergodicity of x->2x on the circle and the Baker's map (wrt Lebesgue measure) and of the Smale's horseshoe, x->5x(1-x), and a version of the tent map from the home assignment (wrt some other measures supported on the respective Cantor sets). I briefly discussed what does it mean for a map to have several ergodic measures. Lec9 Lec10 Lec11

Week 5, Sec 5.8 of [2]. Optional reading: Katok-Hasselblat, Modern theory of dynamical systems, sec. 4.1, 4.2. Birkhoff ergodic theorem; comments about the proof. Definition of mixing. Mixing of circle doubling and non-mixing of rotation. Mixing for Baker's map.

On Monday, I formulated and discussed the integral version of the Birkhoff ergodic theorem (time average equals space average). On Wednesday, I gave a couple of examples ((r,phi) -> (r, phi+r) and (r, phi)-> (1+0.5(r-1), phi) in polar coordinates) and outlined the proof of the Birkhoff theorem. On Friday, I introduced the notion of mixing, explained why Baker's map and circle doubling are mixing, and why circle rotation is not. Also, I showed that mixing implies ergodicity but ergodicity does not imply mixing. Lec12 Lec13 Lec14

Week 6, Sec 7- 9, 14 of [1], also 6.2 of [2]. Conjugacy. Classification of generic fixed points in dimension 1 up to continuous conjugacy. Structural stability; structural stability of circle doubling. Bifurcations.

On Monday, I introduced notions of smooth and topological conjugacy, showed that conjugacy maps periodic orbits to periodic orbits and preserves convergence, and explained why hyperbolic fixed points fall into four classes up to topological conjugacy. On Wednesday, I introduced structural stability and proved that in dimension 1, hyperbolic fixed points are structurally stable in their neighborhoods. Also, I defined bifurcations and discussed saddlenode bifurcations and period-doubling bifurcations in dimension 1. On Friday, I explained why the circle rotation is not structurally stable and proved that the circle doubling is structurally stable. Also, I defined bifurcation diagrams, plotted a bifurcation diagram for an overfishing model x-> 2x(1-0.5 x)-c, and discussed the relation of the bifurcation diagram of the map x->ax(1-x) and the famous Mandelbrot set, which is the bifurcation diagram of the complex map z->z^2+c for complex c. Lec15 Lec16 Lec17

Week 7, Sec 14, 15 of [1], also 6.2 of [2]. Rotation numbers and classification of circle diffeomorphisms up to continuous conjugacy. Denjoy theorem (statement) and Arnold-Herman-Yoccoz theorem (statement). Denjoy examples.

On Monday, I discussed two possible modes of behavior for circle homeomorphisms: some have periodic orbits (whether or not two such maps are topologically conjugate will then depend on the type and order of these periodic orbits) and some have not (those are conjugate to irrational rotations if smooth enough -- Denjoy's theorem). The angle of this irrational rotation is the rotation number of the circle map. I introduced and discussed the definition of the rotation number. On Wednesday, gave a definition of rotation number in terms of lifts to a real line, and I proved properties of rotation numbers: rho(f) does not depend on a point, is continuous on f, and is rational if and only if f has a periodic orbit. On Friday, I proved existence of limit in the definition of the rotation number, defined Arnold tongues, and started discussing the Denjoy theorem. Lec18 Lec19 Lec20

Week 8-10, Sec 27, 29 - 31 of [1]. Hyperbolic fixed points. Statement of Grobman-Hartman theorem. Statement of Hadamard-Perron theorem: existence of stable and unstable manifolds. Hyperbolic sets and cone condition. Examples: Smale's horseshoe, Baker's map, Arnold's cat. Markov partition and encoding for Arnold's cat; its structural stability (sketch of the proof). Homoclinic and heteroclinic intersections.

On Wednesday 10/9, I discussed Denjoy examples and corollaries of the Denjoy theorem, and formulated Arnold-Herman-Yoccoz result on linearization of circle diffeomorphisms. Also, I gave definitions of hyperbolic attractors, repellors, and saddle fixed points in dimension n. On Friday, Prof.Nekrashevych introduced and discussed Arnold's cat map, proved that periodic orbits for this map are dense, and proved density of irrational lines on the torus. Lec21 Lec22

On Monday 10/14, I proved that Arnold's cat map is sensitive to initial conditions, and proved mixing with respect to the Lebesgue measure. Mixing implies ergodicity and transitivity of the map. On Wednesday, I formulated Grobman-Hartman theorem and Hadamard-Perron theorem on hyperbolic fixed points, explained the effects of the homoclinic intersection and the creation of the Smale horseshoe, and started to define an itinerary map for the Arnold's cat map. On Friday, I defined the Markov partition of the Arnold's cat map and showed that the itinerary map is (almost) one-to-one onto the set of sequences with prohibited subwords. Also, I proved transitivity of the Arnold's cat map.

Lec23 Lec24 Lec25 On Monday 10/21, I explained the relation of the Markov partition for the Arnold's cat map and the Markov process (aka Markov chain), and sketched an alternative proof of ergodicity for Arnold's cat map. On Wednesday, I gave a definition of a hyperbolic maps and formulated a theorem on existence of invariant stable/unstable foliations. Also, I introduced the Smale-Williams solenoid map. On Friday, I sketched a proof of the Hadamard-Perron theorem on existence of invariant stable and unstable manifolds of hyperbolic fixed points. Lec26 Lec27 Lec28

Week 11, Sec 27, 29 - 31 of [1]; Sec. 1.0-1.4 of [2]. Hyperbolic maps; Cone condition. Plykin attractor, Henon map. Dynamical systems with continuous time: singular points, periodic orbits, hyperbolicity and invariant manifolds.

On Monday, I formulated and discussed the cone condition for hyperbolic maps, and formulated the invariant manifold theorem; also, I explained why the Smale-Williams solenoid map is hyperbolic, and discussed its stable/unstable foliations. On Wednesday, I described stable/unstable manifolds of the Smale-Williams solenoid using itineraries; also, I introduced and discussed Plykin attractor and the Henon map. On Friday, I started dynamical systems with continuous time: formulated Existence and Uniqueness theorem for differential equations, defined flow maps, discussed examples in dimension 1. Approximations for stable and unstable manifolds, Henon map: https://www.desmos.com/calculator/vig5gevsny Lec29 Lec30 Lec31

Week 12, Chapter 1 of [2]. Vector fields on the plane. Grbman-Hartman and Hadamard-Perron theorems; Poincare-Bendixson theorem and Peixoto theorem on structural stability.

On Monday, I discussed solutions of linear systems, defined hyperbolic fixed points, discussed their stability, and formulated Grobman-Hartman and Hadamard-Perron theorems for vector fields. On Wednesday, I gave examples of applications of the Grobman-Hartman theorem and Lyapunov functions, provided a definition of a limit cycle, and formulated Poincare-Bendixson theorem on limit behavior of trajectories of vector fields on R^2. On Friday, I defined structural stability and formulated Andronov-Pontriagin/Peixoto's criterion for structural stability on the plane/sphere. Lec32 Lec33 Lec34

Week 13, Sec. 3.4, 3.5; 2.3, 5.7 of [2]. Examples of bifurcations: Saddlenode bifurcation, Andronov-Hopf bifurcation, Separatrix Loop bifurcation. Lorenz Attractor and its relation to the horseshoe. Introduction to Complex Dynamics.

On Monday, I discussed a Saddlenode, an Andronov-Hopf, and a Separatrix Loop bifurcation. On Wednesday, I described the Lorenz attractor. On Friday, prof. Nekrashevych was covering for me; he gave a definition of the Julia set and stated the classification of Fatou components for rational functions. For quadratic polynomials z^2+c, he explained why the Julia set is disconnected if and only if the orbit of the critical point tends to infinity (and is a Cantor-like set in this case). Also, he gave a defintion of the Mandelbrot set. Lec35 Lec36 Lec37

Week 14-15, Sec. 18 - 26; 12-15 of [1] Introduction to Complex Dynamics. On the family z^2+c: types of fixed points and periodic points; Fatou and Julia sets; classification of components of Fatou sets; role of the critical orbit; Mandelbrot set; hyperbolic components of the Mandelbrot set. Newton's method and Newton's fractal. Returning to ax(1-x) and period-doubling bifurcations. Renormalization and Feigenbaum universality.

On Monday, I described the dynamics on the Fatou set for quadratic polynomials in the case when the critical point escapes to infinity, introduced Bottcher coordinate at infinity, and defined external rays. Also, I showed some pictures. On Wednesday, I defined hyperbolic components of the Mandelbrot set, explained why they are open. Also, I introduced linearizing coordinates of attracting periodic orbits and explained why each attracting basin must contain a critical point. Finally, I described the structure of the attracting basin if (1) c is in the main cardioid of the Mandelbrot set and (2) if c is in the period-3 component attached to the main cardioid from above (Douady rabbit). On Friday, I described the dynamics of the Douady rabbit from the inside (explained which ears are mapped to which ears and found the second repelling fixed point) and from the outside via external rays. Lec38 Lec39 Lec40

On Monday 11/25, I explained the structure of the parabolic basin of a parabolic fixed point, and described the parabolic bifurcation that occurs at c=1/4 (cauliflower, fold point of the cardioid), and the bifurcation of the fat rabbit. On Monday 12/02, I explained the creation of Siegel discs on the boundary of the main cardioid, showed how period-doubling bifurcations of the family ax(1-x) relate to components of the Mandelbrot set that intersect the real axis, and demonstrated the Newton's fractal.

Lec41 Lec42