1 Introduction Definitons Markov... 2

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1 Compact course notes Dynamic systems Fall 2011 Professor: Y. Kudryashov transcribed by: J. Lazovskis Independent University of Moscow December 23, 2011 Contents 1 Introduction Definitons Markov Perturbations and structural stability Stability Newhouse phenomenon Anosov diffeomorphisms Attractors Defintions Fixed point classification Note: For a more complete exposition, see the following textbooks, which were used in accompaniment with this course: Introduction to the Modern Theory of Dynamical Systems, Anatole Katok and Boris Hasselblatt An Introduction to Dynamical Systems, D.K. Arrowsmith and C.M. Place

2 1 Introduction 1.1 Definitons Definition Given a map F : X X,a point x X is termed a fixed point of F iff F (x) = x. The orbit of x is the set Fix(x) = {F n (x) n Z0 }. Definition Two maps F : X X and G : Y Y are termed conjugate if there exists a bijection h : X Y such that G h = h F. Proposition If F and G are conjugated by h, then h Fix(F ) : Fix(F ) Fix(G) is a bijection. Proposition Suppose G h = h F and F (x 0 ) = x 0 with h (x 0 ) 0. Then G (h(x 0 )) = F (x 0 ). Definition Let X be a space and F : X X a function. Then the fate of a point x X is the sequence..., ω n ω n+1... ω 1 ω 0 ω 1... ω n ω n+1... with ω i = k F i (x) R k for R k a predefined region of X. Definition A diffeomorphism between manifolds is an isomorphism that is both differentiable and has a differentiable inverse. Definition An Anosov diffeomorphism is a diffeomorphsim f : M M from a C 1 manifold M to itself such that the tangent bundle of M is hyperbolic to f. An example is the set of matrices with unit length determinant. Remark Let f : X f(x) be a map. Then det(f) = Area(f(X)) Area(X). Moreover, if A M n, then the number of fixed points of A k is det(a k I) for k Z 0. Remark Here on in, we use the notation R 2 /Z = T 2 the real torus. 1.2 Markov Definition A Markov chain for k random variables is a k k matrix A such that A ij = P (ω n+1 = j ωn = i) for any n and for all 1 i, j k. Theorem [Perron, Frobenius] Suppose the following hold: A = P ij is a map with p ij 0 for all i, j For all k > 0, the graph corresponding to A k is strongly connected Then there exists a unique set {p 1,..., p s } with p i > 0 and (p 1... p s )A = (p 1... p s ). Moreover, if p ij > 0, then for any initial set of probabilities q 1,..., q s we have (q 1... q s )A = (p 1... p s ). Definition Let f : S S be a bijective map preserving orientation. Then F : R R is termed a lifting of f if and only if π F = f π. Equivalently, when the following diagram commutes. R F R π S f π S = R/Z In this case we also define the rotation number by [ F n ] (x) x ρ(f) = n with the following properties: 1. ρ(f) = 0 f has a fixed point 2. ρ(f k ) = kρ(f) 3. f(x + 1) = f(x) + 1 2

3 Theorem [Denjoy] Let f : S 1 S 1 be a bijective C 2 -map. Suppose that ρ(f) / Q and the second derivative is continuous. Then f is conjugated to R ρ (f) : X to Theorem Let {x} denote the fractional part of x and #S for S a set denote the cardinality of S. If I is an interval on R/Z and α and angle, then for m R, [ ] #{0 m n {mα} I} = I n Moreover, if α / Q, then for any function ϕ on R, [ ] n 1 1 ϕ({x + αk}) n 2 Perturbations and structural stability 2.1 Stability k=0 = 1 0 ϕ(y)dy Definition A map F is termed structurally stable if there exists ɛ > 0 such that for every F with F F C 1 < ɛ, F is conjugated to F. Definition For X a space, F : X X is termed a contracting map if there exists λ (0, 1) such that for all x, y X, d(f (x) F (y)) λd(x, y). Theorem [Contracting map principle] Let X be a complete metric space and F : X X a contracting map. Then F has a unique fixed point. In other words, there exists unique x X such that F (x) = x. Note that the above implies both the inverse and implicit function theorems. 2.2 Newhouse phenomenon Definition Given a fixed point and its phase portrait, a curve with a transversal intersection with another curve is termed a separatrix. Definition Given a phase portrait of a dynamical system, if a separatrix intersects the same curve non-transversally that it separates, then the non-transversal intersection is termed a homoclinic tangency. Theorem [Newhouse phenomenon] Consider a one-parameter family of dynamical systems with a set of fixed points and homoclinic tangencies of F 0 with F ɛ, such that increasing the parameter changes a homoclinic tangency to two transversal intersections. Then there exist many ɛ > 0 such that F ɛ has infinitely many stable periodic points. Definition Let U i be an open, dense set. Then U i is termed a residual set. The set S = {α ɛ > 0, N N, there exists p q Q, q > N with α p q < ɛ q 3 } is a residual set of Lebesgue measure Anosov diffeomorphisms Theorem Let A = ) be a map A : T 2 T 2. Then for each ɛ > 0 there exists a δ > 0 such that there is a sequence {P n } n= with AP n P n+1 < δ and p T 2 such that A n p P n < ɛ. Definition A sequence {P n } n= is termed a δ-pseudo orbit of a torus map A if AP n P n+1 < δ. i=1 3

4 Remark Any map close enough to ) is topologically conjugated to it, and therefore is an Anosov diffeomorphism. Theorem Let F be a function on vectors with v 1, v 2 vector fields on R n. If df, df dx dx the angle between v 1 (F (x)) and v 2 (F (x)) v2(x) v1(x) for all x R n, then F is an Anosov diffeomorphism. Definition A map A : T 2 T 2 is termed ergodic if for any continuous function ϕ, almost everywhere 3 Attractors 3.1 Defintions [ n 1 k=0 ϕ(an (x, y)) n ] = ϕ dx dy T 2 Definition An attractor, very generally, is a subset of the phase space such that all points except a set of measure 0 tend to the subset. Definition A maximal attractor for a map F : U U for U open with F (U) U and F (U) compact in U is the set A max := F n (U). Essentially, we have n=0 U F (U) F 2 (U) F (U) F 2 (U) F 3 (U) Definition Let F : R n R n be a map. A Milnor attractor is the minimal closed set A M such that [d(f n (x), A M )] = 0 for all x R n except possibly a set of measure 0. Definition Let F : M M be a map with attractor A M. Then F is termed Lyapunov stable if for fixed ɛ > 0, there exists δ > 0 such that for all x M with d(x, A) < δ, d(f n (x), A) < ɛ for all n. In other words, if we start in an ɛ-neighborhood of x, then we never leave a δ-neighborhood of x. Open problem How can it be determined for a generic dynamical system that A M stable? is Lyapunov Theorem Let F : M M be a map with attractor A M and x M. Then x A M for all open U x, measure(s) > 0 for S = {y for all N N, there exists n > N such that F n (y) U}. Definition Let F : R n R n be a map with x R n a Lyapunov stable point. Then x is termed asymptotically stable if there exists ɛ > 0 such that for all y R n, d(x, y) < ɛ = [F n (y)] = x. Theorem Let F : R n R n be a map in C 1 with x 0 R n such that F (x 0 ) = x 0. Let A = F (x 0 ) with λ 1,..., λ n be the eigenvalues of A. Then λ i < 1 for all i = x 0 is asymptotically stable λ i > 1 for at least one i = x 0 is not asymptotically stable Proposition The Milnor attractor is invariant under forward and backward applications of F. 4

5 3.2 Fixed point classification A map F : R 2 R 2 may have several types of fixed points. To determine the type of fixed point at (x, y) R 2 for F (x, y) = (u(x), v(y)), let ) J = ( du dx dv dx be the Jacobian of F. Evaluate J(x, y) and identify it with one of the matrix types below. du dy dv dy Saddle point Node Axis of fixed points Center ( 1 ( 0 ) Stable if ad > 0 Unstable if ad < 0 Stable if a = d < 0 Unstable if a = d > 0 Stable if d < 0 Unstable if d > 0 Focus Jordan cell ( 0 ) 2 ) Stable if bc > 0 Unstable if bc < 0 Stable if a, d < 0 Unstable if a, d > 0 ( ) a b The variables above refer to the general matrix. c d 5