Introduction to Stability Theory of Dynamical Systems Michael Seligman April 25, 2015 Chapter 1 Dynamical Systems 1.1 Definitions and Related Notation Definition 1.1.1. A dynamical system on X is the triplet (X; R;'), where ' is a map from the product space X × R into the space X satisfying the following axioms: '(x; 0) = x; 8x 2 X (1.1) '('(x; t1); t2) = '(x; t1 + t2); 8x 2 X and t1; t2 2 R (1.2) ' is continuous (1.3) Given a dynamical system acting on X, the space X is called the phase space and the map ' is called the phase map. We will delete the symbol ' and denote the image '(x,t) of a point (x,t) in X×R as t.x. The above identities will then read 0:x = x; 8x 2 X (1.4) t2:(t1:x) = (t1 + t2):x; 8x 2 X and t1; t2 2 R (1.5) In accordance to this notation, if M ⊂ X and A ⊂ R, then A:M is the set ft:x : x 2 M and t 2 Ag. If either M or A is a singleton, i.e., M = fxg or A = ftg, then we simply write A:x or t:M for fxgA and Mftg, respectively. For any x 2 X, the set R:x is called the trajectory through x. The phase map determines two other maps when one of the variables x or t is fixed. Thus for fixed t 2 R, the map 't : X ! X defined by 't(x) = t:x is called a transition, and for a fixed x 2 X, the map 'x : R ! X defined by 'x(t) = t:x is called a motion (through x). Note that 'x maps R onto R:x. 1 CHAPTER 1. DYNAMICAL SYSTEMS 2 1.2 Examples of Dynamical Systems Definition 1.2.1. Ordinary Autonomous Differential Systems. Consider the autonomous differential system dx =x _ = f(x); (1.6) dt where f : Rn ! Rn(Rn is the real n-dimensional euclidean space) is con- tinuous and moreover assume that for each x 2 Rn a unique solution t:x exists which is defined on R and satisfies 0:x = x. Then it is well known that the uniqueness of solutions implies t1:(t2:x) = (t1 + t2):x; 8t1; t2 2 R; (1.7) and considered as a function from R × Rn into Rn;' is continuous in its ar- guments. We remark that the conditions on solutions of 1.6, as required above, are obtained, for example, if the function f satisfies a global Lipschitz condition, i.e., there is a positive number k such that n kf(x) − f(y)k 5 kkx − yk; 8x; y 2 R (1.8) Definition 1.2.2. Ordinary Non-autonomous Differential Systems. Consider the differential system n x_ = f(t; x); x 2 R ; t 2 R (1.9) n n n where f : R×R ! R is continuous. Assume that for each (t0; x0) 2 R×R , 1:9 possesses a unique solution '(t; t0; x0);'(t0; t0; x0) = x0, defined 8t 2 R. y_ = g(y); where y = (t; x), and g(y) = (1; f(y)): (1.10) Chapter 2 Elementary Concepts 2.1 Invariant Sets and Trajectories Definition 2.1.1. A set M ⊂ X is called invariant whenever t:x 2 M; 8x 2 M and t 2 R: (2.1) It is called positively invariant whenever 2.1 holds with R replaced by R+, and is called negatively invariant if the same holds with R replaced by R−. Note that 2.1 is equivalent to R:M = M. Theorem 2.1.2. Let fMig be a collection of positively invariant, negatively invariant, or invariant subsets of X. Then their intersection and their union have the same property. Theorem 2.1.3. Let M ⊂ X be positively invariant, negatively invariant, or invariant. Then the closure M has the same property. Theorem 2.1.4. A set M ⊂ X is positively invariant if and only if the set X n M is negatively invariant. M is invariant if and only if X n M is invariant. Theorem 2.1.5. A set M ⊂ X is invariant if and only if it is both positively and negatively invariant. Definition 2.1.6. We introduce the maps γ; γ+, and γ− from X into 2X by defining 8x 2 X, γ(x) = ft:x : t 2 Rg; (2.2) + + γ (x) = ft:x : t 2 R g; (2.3) − − γ (x) = ft:x : t 2 R g; (2.4) 3 CHAPTER 2. ELEMENTARY CONCEPTS 4 8x 2 X, the sets γ(x); γ+(x), and γ−(x) are, respectively called the tra- jectory, the positive semi-trajectory, and the negative semi-trajectory through x. Note that 8x 2 X; γ(x) = R:x, etc. Proposition 2.1.7. A set M ⊂ X is invariant, positively invariant, or neg- atively invariant if and only if, respectively, γ(M) = M; γ+(M) = M, or γ−(M) = M. Another characterization of invariance is the following: Proposition 2.1.8. A set M ⊂ X is invariant, positively invariant, or nega- tively invariant if and only if for each x 2 M, respectively, γ(x) ⊂ M; γ+(x) ⊂ M; γ−(x) ⊂ M. 2.2 Critical Points and Periodic Points Definition 2.2.1. A point x 2 X is said to be a critical point if x = t:x; 8t 2 R. Theorem 2.2.2. Let x 2 X. Then the following are equivalent. x is critical; (2.5) fxg = γ(x); (2.6) fxg = γ+(x); (2.7) fxg = γ−(x); (2.8) fxg = [a; b]:x for some a<b: (2.9) There is a sequence ftng; tn > 0; tn ! 0 with x = tn:x for each n: (2.10) Theorem 2.2.3. A point x 2 X is critical if and only if every neighborhood of x contains a semi-trajectory. Definition 2.2.4. A point x 2 X is said to be periodic if there is a T 6= 0 such that t:x = (t + T ):x; 8t 2 R: (2.11) A number T 2 R for which 2.11 holds will be called a period of x. If a point x 2 X is periodic then both the motion and the trajectory are said to be periodic. Note that every x 2 X has the period T = 0, but it may not be periodic. Further, if x 2 X is critical, then every T 2 R is a period of x, and indeed x is periodic. The following characterization of a periodic point is very useful. CHAPTER 2. ELEMENTARY CONCEPTS 5 Proposition 2.2.5. x 2 X is periodic if and only if there is a T 6= 0 with x = T:x. Theorem 2.2.6. If x 2 X is periodic, but not critical, then there is T > 0 such that T is the smallest positive period of x. Further if τ is any other period of x, then τ = T · n for some integer n. Definition 2.2.7. If a point x is periodic but not critical, then the smallest positive period of x is called its fundamental or primitive period. 2.3 Trajectory Closures and Limit Sets Definition 2.3.1. Define maps Λ+; Λ− from X into 2X by setting for each x 2 X, + Λ (x) = fy 2 X : 9 ftng 2 R with tn ! +1 and tn:x ! yg; (2.12) − Λ (x) = fy 2 X : 9 ftng 2 R with tn ! −∞ and tn:x ! yg; (2.13) For any x 2 X, the set Λ+(x) is called its positive (or omega) limit set, and the set Λ−(x) is called its negative (or alpha) limit set. Definition 2.3.2. If x 2 X is periodic, then Λ+(x) = Λ−(x) = γ(x). Example 2.3.3. Examples of Limit Sets. Consider the differential system defined in R2 by the differential equations in Polar coordinates: dr dθ = r(1 − r); = 1 (2.14) dt dt It can easily be verified that the solutions are unique and all solutions are defined on R. Thus 2:14 defines a dynamical system. The trajectories are shown in F igure 3:1. These consist of: (i) a critical point, namely the origin 0, (ii) a periodic trajectory γ coinciding with the unit circle, (iii) spiraling trajectories through each point P = (r; θ) with r 6= 0; r 6= 1. For points P with 0 < r < 1, Λ+(P ) is the unit circle and Λ−(P ) is the singleton f0g. For points P with r > 1; Λ+(P ) is the unit circle and Λ−(P ) = ;. CHAPTER 2. ELEMENTARY CONCEPTS 6 Figure 2.1: The differential system can be redefined in Cartesian coordinates: dx d dr dθ = (rcosθ) = cosθ − rsinθ (2.15) dt dt dt dt dy d dr dθ = (rsinθ) = sinθ + rcosθ (2.16) dt dt dt dt dx = r(1 − r)cosθ − rsinθ = r(1 − r)cosθ − rsinθ (2.17) dt dy = r(1 − r)sinθ + rcosθ = r(1 − r)sinθ + rcosθ (2.18) dt dx p = x − x x2 + y2 − y (2.19) dt dy p = y − y x2 + y2 + x (2.20) dt dy y − ypx2 + y2 + x = (2.21) dx x − xpx2 + y2 − y Chapter 3 Stability Theory 3.1 Stability and Attraction for Compact Sets Definition 3.1.1. With a given M we associate the sets + A!(M) = fx 2 X :Λ (x) \ M 6= ;g; (3.1) A(M) = fx 2 X :Λ+(x) 6= ; and Λ+(x) ⊂ Mg; (3.2) + + Au(M) = fx 2 X : J (x) 6= ; and J (x) ⊂ Mg; (3.3) + + J (x) = fy 2 X : 9fxng 2 X and ftng 2 R s.t.