Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 4: Analysis Based On Continuity 1 This lecture presents several techniques of qualitative systems analysis based on what is frequently called topological arguments, i.e. on the arguments relying on continuity of functions involved. 4.1 Analysis using general topology arguments This section covers results which do not rely speciflcally on the shape of the state space, and thus remain valid for very general classes of systems. We will start by proving gener- alizations of theorems from the previous lecture to the case of discrete-time autonomous systems. 4.1.1 Attractor of an asymptotically stable equilibrium Consider an autonomous time invariant discrete time system governed by equation x(t + 1) = f(x(t)), x(t) ? X, t = 0,1,2,..., (4.1) where X is a given subset of R n , f : X ∞? X is a given function. Remember that f is called continuous if f(x k ) ? f(x ? ) as k ? → whenever x k ,x ? ? X are such that x k ? x ? as k ? →). In particular, this means that every function deflned on a flnite set X is continuous. One important source of discrete time models is discretization of difierential equations. R n Assume that function a : ∞? R n is such that solutions of the ODE x˙(t) = a(x(t)), (4.2) 1 Version of September 17, 2003 2 with x(0) = x exist and are unique on the time interval t ? [0,1] for all ?? x ? R n . Then discrete time system (4.1) with f(?) = x(1, ?) describes the evolution of continuous time x x system (4.2) at discrete time samples. In particular, if a is continuous then so is f. Let us call a point in the closure of X locally attractive for system (4.1) if there exists d > 0 such that x(t) ? ? ?x 0 as t ?→for every x = x(t) satisfying (4.1) with |x(0)?x 0 | < d. Note that locally attractive points are not necessarily equilibria, and, even if they are, they are not necessarily asymptotically stable equilibria. x 0 ? R n the set A = A(? x ? X in (4.1) which deflne a For ? x 0 ) of all initial conditions ? solution x(t) converging to ? xx 0 as t ?→ is called the attractor of ? 0 . Theorem 4.1 If f is continuous and ?x 0 is locally attractive for (4.1) then the attractor A = A(?x 0 ) is a (relatively) open subset of X, and its boundary d(A) (in X) is f-invariant, i.e. f(?) ? d(A) whenever ?x x ? d(A). Remember that a subset Y ‰ X ‰R n is called relatively open in X if for every y ? Y there exists r > 0 such that all x ? X satisfying |x?y| < r belong to Y . A boundary of a subset Y ‰ X ‰R n in X is he set of all x ? X such that for every r > 0 there exist y ? Y and z ? X/Y such that |y ?x| < r and z ?x| < r. For example, the half-open interval Y = (0,1] is a relatively closed subset of X = (0,2), and its boundary in X consists of a single point x = 1. Example 4.1 Assume system (4.1), deflned on X = R n by a continuous function f : R n ∞? R n , is such that all solutions with |x(0)| < 1 converge to zero as t ? →, and all solutions with |x(0)| > 100 converge to inflnity as t ? →. Then, according to Theorem 4.1, the boundary of the attractor A = A(0) is a non-empty f-invariant set. By ? ?assumptions, 1 ? |x| ? 100 for all x ? A(0). Hence we can conclude that there exist solutions of (4.1) which satisfy the constraints 1 ?|x(t)|? 100 for all t. Example 4.2 For system (4.1), deflned on X = R n by a continuous function f : R n ∞? R n , it is possible to have every trajectory to converge to one of two equilibria. However, it is not possible for both equilibria to be locally attractive. Otherwise, according to The- orem 4.1, R n would be represented as a union of two disjoint open sets, which contradicts the notion of connectedness of R n . 4.1.2 Proof of Theorem 4.1 According to the deflnition of local attractiveness, there exists d > 0 such that x(t) ? ?x 0 as t ? → for every x = x(t) satisfying (4.1) with |x(0) ? ?x 0 | < d. Take an arbitrary x 1 ? A(? x 1 . Then x 1 (t) ? ? 0 as? x 0 ). Let x 1 = x 1 (t) be the solution of (4.1) with x(0) = ? x t ?→, and hence |x 1 (t 1 )| < d/2 for a su–ciently large t 1 . Since f is continuous, x(t) is a continuous function of x(0) for every flxed t ?{0,1,2,...}. Hence there exists – > 0 such x xthat |x(t 1 ) ?x 1 (t 1 )| < d/2 whenever |x(0) ? ? 1 | < –. Since this implies |x(t 1 ) ? ? 0 | < d, 3 we have ? x x ? X such that |x? ? 1 | < –, which proves that A = A(? 0 )x ? A(? 0 ) for every ? ? x x is open. To show that d(A) is f-invariant, note flrst that A is itself f-invariant. Now take an arbitrary x ? d(A). By the deflnition of the boundary, there exists a sequence ?? x k ? A x x k ) converges to f(?converging to ?. Hence, by the continuity of f, the sequence f(? x). If f(?) ≤? A, this implies f(?x x) ? d(A). Let us show that the opposite is impossible. Indeed, if f(?) ? A then, since A is proven open, there exists ? > 0 such that z ? A for every x z ? X such that |z ? f(?)| < ?. Since f is continuous, there exists – > 0 such that x |f(y) ? f(? xx)| < ? whenever y ? X is such that |y ? ?| < –. Hence f(y) ? A whenever x|y ? ?| < –. Since, by the deflnition of attractor, f(y) ? A imlies y ? A, y ? A whenever x| < –, which contradicts the assumption that ?|y ? ? x ? d(A). 4.1.3 Limit points of planar trajectories For a given solution x = x(t) of (4.2), the set lim(x) ‰R n of all possible limits x(t k ) ? ?x as k ?→, where {t k } converges to inflnity, is called the limit set of x. Theorem 4.2 Assume that a : R n ∞?R n is a locally Lipschitz function. If x : [0,→) ∞? R n is a solution of (4.2) then the set lim(x) of its limit points is a closed subset of R n , and every solution of (4.2) with initial conditions in lim(x) lies completely in lim(x). x x q ? ? asProof First, if t k,q ?→ and x(t k,q ,x(0)) ? ? q as k ?→ for every q, and ? x ? q ? → then one can select q = q(k) such that t k,q(k) ? → and x(t k,q(k) ,x(0) ? ? asx ? k ?→. This proves the closedness (continuity of solutions was not used yet). Second, by assumption ?x 0 = lim x(t k ,x(0)). k?? Hence, by the continuous dependence of solutions on initial conditions, x(t, ?x 0 ) = lim x(t,x(t k ,x(0))) = lim x(t + t k ,x(0)). k?? k?? In general, limit sets of ODE solutions can be very complicated. However, in the case when n = 2, a relatively simple classiflcation exists. Theorem 4.3 Assume that a : R 2 ∞? R 2 is a locally Lipschitz function. Let x 0 : [0,→) ∞?R 2 be a solution of (4.2). Then one of the following is true: (a) |x 0 (t)|?→ as t ?→; (b) there exists T > 0 and a non-constant solution x p : (?→,+→) ∞? R 2 such that x p (t + T) = x p (t) for all t, and the set of limit points of x is the trajectory (the range) of x p ; ? 4 (c) the limit set is a union of trajectories of maximal solutions x : (t 1 ,t 2 ) ∞? R 2 of (4.2), each of which has a limit (possibly inflnite) as t ? t 1 or t ? t 2 . The proof of Theorem 4.3 is based on the more speciflc topological arguments, to be discussed in the next section. 4.2 Map index in system analysis The notion of index of a continuous function is a remarkably powerful tool for proving existence of mathematical objects with certain properties, and, as such, is very useful in qualitative system analysis. 4.2.1 Deflnition and fundamental properties of index For n = 1,2,... let S n = {x ? R n+1 : |x| = 1} denote the unit sphere in R n+1 . Note the use of n, not n + 1, in the S-notation: it indicates that locally the sphere in R n+1 looks like R n . There exists a way to deflne the index ind(F) of every continuous map F : S n ∞? S n in such a way that the following conditions will be satisfled: (a) if H : S n ￡ [0,1] ∞? S n is continuous then ind(H(￠,0)) = ind(H(￠,1)) (such maps H is called a homotopy between H(￠,0) and H(￠,1)); (b) if the map F ? : R n+1 ∞? R n+1 deflned by F ? (z) = |z|F(z/|z|) is continuously difierentiable in a neigborhood of S n then ind(F) = det(J x (F ? ))dm(x), x?S n where J x (F ? ) is the Jacobian of F ? at x, and m(x) is the normalized Lebesque measure on S n (i.e. m is invariant with respect to unitary coordinate transformations, and the total measure of S n equals 1). Once it is proven that the integral in (b) is always an integer (uses standard vol- ume/surface integration relations), it is easy to see that conditions (a),(b) deflne ind(F) correctly and uniquelly. For n = 1, the index of a continuous map F : S 1 ∞? S 1 turns out to be simply the winding number of F, i.e. the number of rotations around zero the trajectory of F makes. It is also easy to see that ind(F I ) = 1 for the identity map F I (x) = x,and ind(F c ) = 0 for every constant map F c (x) = x 0 = const. 5 4.2.2 The Brower’s flxed point theorem One of the classical mathematical results that follow from the very existence of the index function is the famous Brower’s flxed point theorem, which states that for every continuous function G : B n ∞? B n , where B n = {x ? R n+1 : |x| ? 1}, equation F(x) = x has at least one solution. The statement is obvious (though still very useful) when n = 1. Let us prove it for ? n > 1, starting with assume the contrary. Then the map G : B n ∞? B n which maps x ? B n to the point of S n?1 which is the (unique) intersection of the open ray starting from G(x) and passing through x with S n?1 . Then H : S n?1 ￡ [0,1] ∞? S n?1 deflned by H(x,t) = ? G(tx) is a homotopy between the identity map H(￠,1) and the constant map H(￠,0). Due to existence of the index function, such a homotopy does not exist, which proves the theorem. 4.2.3 Existence of periodic solutions Let a : R n ￡ R ∞? R n be locally Lipschitz and T-periodic with respect to the second argument, i.e. x,t + T) = a(?a(? x,t) ? x,t where T > 0 is a given number. Assume that solutions of the ODE x˙(t) = a(x(t),t) (4.3) with initial conditions x(0) ? B n remain in B n for all times. Then (4.3) has a T-periodic solution x = x(t) = x(t + T) for all t ? R. x ∞? x(T,0, ?Indeed, the map ? x) is a continuous function G : B n ∞? B n . The solution x = G(?of ? x) deflnes the initial conditions for the periodic trajectory.