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 I 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 specifically 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 4.1.1 Attractor of an asymptotically stable equilibrium Consider an autonomous time invariant discrete time system governed by equation r(t+1)=f(x(t),r(t)∈X,t=0.,1,2, (4.1) where X is a given subset of R",f: X H X is a given function. Remember that f is called continuous if f(xk)→f(x)ask→ o whenever k,x∈ X are such that k-Doo as k-oo). In particular, this means that every function defined on a finite set X is continuous One important source of discrete time models is discretization of differential equations Assume that function a: RhR is such that solutions of the ODe I Version of September 17, 2003
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 specifically on the shape of the state space, and thus remain valid for very general classes of systems. We will start by proving generalizations 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 Rn, f : X ∞� X is a given function. Remember that f is called continuous if f(xk) � f(x�) as k � → whenever xk, x� ⊂ X are such that xk � x� as k � →). In particular, this means that every function defined on a finite set X is continuous. One important source of discrete time models is discretization of differential equations. Rn Assume that function a : ∞� Rn is such that solutions of the ODE x˙ (t) = a(x(t)), (4.2) 1Version of September 17, 2003
with x(0)=I exist and are unique on the time interval t E [ 0, 1] for allTER".Then discrete time system(4. 1)with f(5)=r(, i)describes the evolution of continuous time system(4.)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 r(t)-io as t-o for every a= a(t) satisfying(4.1) with z(0)-iol <d Note that locally attractive points are not necessarily equilibria, and, even if they are they are not necessarily asymptotically stable equilibria For Io E R the set A=A(Co) of all initial conditions i E X in(4.1) which define a solution (t) converging to io as t-o is called the attractor of To Theorem 4.1 If f is continuous and io is locally attractive for(4.1) then the attractor A=A(Co)is a(relatively) open subset of X, and its boundary d(a)(in X)is f-invariant, ie.f()∈d(A) whenever at∈d() Remember that a subset y CxCr is called relatively open in X if for every yEY there exists r>0 such that all E X satisfying lar-yl <r belong to Y. a boundary of a subset y C XCr in X is he set of all E X such that for every r>0 there exist yEY and zE X/Y such that ly-a <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 ngle point x= 1 Example 4.1 Assume system(4.1), defined on X= R by a continuous function f R" R", is such that all solutions with r(0)< 1 converge to zero as t-0o and all solutions with (0)>100 converge to infinity as t-o0. Then, according to Theorem 4.1, the boundary of the attractor A= A(0) is a non-empty f-invariant set. By assumptions, 1 <i <100 for all I E A(0). Hence we can conclude that there exist solutions of(4.1)which satisfy the constraints 1 <r(t)l< 100 for all t Example 4.2 For system(4. 1), defined on X=R" by a continuous function f: R"H R", 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" would be represented as a union of two disjoint open sets, which contradicts the notion of connectedness of r 4.1.2 Proof of Theorem 4.1 According to the definition of local attractiveness, there exists d >0 such that c(t)To as t=o0 for every a a (t) satisfying(4.)with z(0)-iol< d. Take an arbitrary I1 EA(o). Let 21=TI(t)be the solution of(4.)with a(0)=T1. Then 1(t)-+io as t-00, and hence a1(t1) d/2 for a sufficiently large t1. Since f is continuous, a(t)is a continuous function of r(0)for every fixed t E 10, 1, 2,.... Hence there exists>0 suck that x(t1)-T1(ti)l< d/2 whenever la(0)-i1l <8. Since this implies a(t1)-iol d
2 with x(0) = x¯ exist and are unique on the time interval t ⊂ [0, 1] for all x¯ ⊂ Rn. Then discrete time system (4.1) with f(¯) x x = x(1, ¯) describes the evolution of continuous time 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. x0 ⊂ Rn For ¯ the set A = A(¯x0) of all initial conditions x¯ ⊂ X in (4.1) which define a solution x(t) converging to x¯0 as t � → is called the attractor of x¯0. Theorem 4.1 If f is continuous and x¯0 is locally attractive for (4.1) then the attractor A = A(¯x0) is a (relatively) open subset of X, and its boundary d(A) (in X) is f-invariant, i.e. f(¯) x ⊂ d(A) whenever x¯ ⊂ d(A). Remember that a subset Y � X � Rn 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 � Rn 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), defined on X = Rn by a continuous function f : Rn ∞� Rn, is such that all solutions with |x(0)| < 1 converge to zero as t � →, and all solutions with |x(0)| > 100 converge to infinity 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), defined on X = Rn by a continuous function f : Rn ∞� Rn, 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 Theorem 4.1, Rn would be represented as a union of two disjoint open sets, which contradicts the notion of connectedness of Rn. 4.1.2 Proof of Theorem 4.1 According to the definition 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(¯x0). Let x1 = x1(t) be the solution of (4.1) with x(0) = x¯1. Then x1(t) � x¯0 as t � →, and hence |x1(t1)| < d/2 for a sufficiently large t1. Since f is continuous, x(t) is a continuous function of x(0) for every fixed t ⊂ {0, 1, 2, . . . }. Hence there exists � > 0 such that |x(t1) − x1(t1)| < d/2 whenever |x(0) − x¯1| < �. Since this implies |x(t1) − x¯0| < d
we have i E A(o) for every T E X such that Ii-i1l <8, which proves that A= A(Lo) Is open. To show that d (a) is f-invariant, note first that A is itself f-invariant. Now take an arbitrary i E d(A). By the definition of the boundary, there exists a sequence Tk E A converging to i. Hence, by the continuity of f, the sequence f(Ik) converges to f(i). If f(I) A, this implies f(r)Ed(A). Let us show that the opposite is impossible. Indeed if f(a)E A then, since A is proven open, there exists E>0 such that z E A for every z E X such that z-f(iI< E. Since f is continuous, there exists 8>0 such that f(y)-∫()< e whenever y∈ X is such that|y-到<6. Hence j(y)∈ A whenever ly-i <8. Since, by the definition of attractor, f(y) E A imlies y E A, y E A whenever ly-i < 8, which contradicts the assumption that i E d(a) 4.1.3 Limit points of planar trajectories For a given solution a =a(t)of(4.2), the set lim(a)CR" of all possible limits z(tk)-i as k-o0, where tk converges to infinity, is called the limit set of c Theorem 4.2 Assume that a: R"HR" is a locally Lipschitz function. If r: [0, oo)H r is a solution of (4. 2)then the set lim(a)of its limit points is a closed subset ofR" and every solution of (4.2 )with initial conditions in lim(a) lies completely in lim(a) Proof First, if tk. g→ oo and a(tkq,x(0)→五ask→∞ for every q, and tg→oas q→∞ then one can select q=q(k) such that tk.a(k)→andx(tk.q),x(0)→as k-00. This proves the closedness(continuity of solutions was not used yet) Second, by assumption (tk,x(0) Hence, by the continuous dependence of solutions on initial conditions r(t, io)= lim a(t, r(tk, (0))= lim x(t +tk, (0)) In general, limit sets of ODE solutions can be very complicated. However, in the case when n=2, a relatively simple classification exists Theorem 4.3 Assume that a: R HR is a locally Lipschitz function. Let To [0, oo)HR be a solution of (4.2). Then one of the following is true (a)|xo(t)→∞ast→ th ere er stsT>0 and a non- constant solution x:(-∞,+∞)→R2 such that p(t+r)=lp(t)for all t, and the set of limit points of a is the trajectory(the age)of
3 we have x¯ ⊂ A(¯x0) for every x¯ ⊂ X such that |x¯ − x¯1| < �, which proves that A = A(¯x0) is open. To show that d(A) is f-invariant, note first that A is itself f-invariant. Now take an arbitrary x¯ ⊂ d(A). By the definition of the boundary, there exists a sequence x¯k ⊂ A converging to x¯. Hence, by the continuity of f, the sequence f(¯xk) converges to f(¯x). If f(¯) x ≤⊂ A, this implies f(¯x) ⊂ d(A). Let us show that the opposite is impossible. Indeed, if f(¯) x ⊂ A then, since A is proven open, there exists � > 0 such that z ⊂ A for every z ⊂ X such that |z − f(¯) x | < �. Since f is continuous, there exists � > 0 such that |f(y) − f(¯x)| < � whenever y ⊂ X is such that |y − x¯| < �. Hence f(y) ⊂ A whenever |y − x¯| < �. Since, by the definition of attractor, f(y) ⊂ A imlies y ⊂ A, y ⊂ A whenever |y − x¯| < �, which contradicts the assumption that 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) � Rn of all possible limits x(tk) � x˜ as k � →, where {tk} converges to infinity, is called the limit set of x. Theorem 4.2 Assume that a : Rn ∞� Rn is a locally Lipschitz function. If x : [0,→) ∞� Rn is a solution of (4.2) then the set lim(x) of its limit points is a closed subset of Rn, and every solution of (4.2) with initial conditions in lim(x) lies completely in lim(x). Proof First, if tk,q � → and x(tk,q, x(0)) � x¯q as k � → for every q, and x¯q � x¯ as � q � → then one can select q = q(k) such that tk,q(k) � → and x(tk,q(k), x(0) � x¯ as � k � →. This proves the closedness (continuity of solutions was not used yet). Second, by assumption x¯0 = lim x(tk, x(0)). k�� Hence, by the continuous dependence of solutions on initial conditions, x(t, x¯0) = lim x(t, x(tk, x(0))) = lim x(t + tk, 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 classification exists. Theorem 4.3 Assume that a : R2 ∞� R2 is a locally Lipschitz function. Let x0 : [0,→) ∞� R2 be a solution of (4.2). Then one of the following is true: (a) |x0(t)| � → as t � →; (b) there exists T > 0 and a non-constant solution xp : (−→, +→) ∞� R2 such that xp(t + T) = xp(t) for all t, and the set of limit points of x is the trajectory (the range) of xp;
(c) the limit set is a union of trajectories of maximal solutions a :(t1, t2)HR of (4.2), each of which has a limit(possibly in finite) as t-ti ort-t The proof of Theorem 4.3 is based on the more specific 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 Definition and fundamental properties of index 1.2...le ∈R+:|z|=1} denote the unit sphere in R"+. Note the use of n, not n 1, in the S-notation: it ndicates that locally the sphere in R"f looks like R". There exists a way to define the inder ind (F) of every continuous map F Sn in such a way that the following conditions will be satisfied (a)ifH:S"×0,1→ Sn is continuous then ind(H( O))=ind(H(, 1)) (such maps H is called a homotopy between H(, 0)and H(, 1)) (b)if the map F: R n+I defined by (2)=||F(2/12) is continuously differentiable in a neigborhood of sm then d(F) det(J(F))dm(a) where J_(F)is the Jacobian of F at r, and m(a) is the normalized Lebesque measure on Sn (i.e. m is invariant with respect to unitary coordinate transformations, and the total measure of Sm 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)define ind(F) correctly and uniquely. For n= 1, the index of a continuous map F: SHSturns out to be simply the winding number of F, i.e. the number of rotations around zero the trajectory of F makes 6. It is also easy to see that ind(F1)=1 for the identity map Fr(c)=a, and ind(Fc)=0 every constant map Fc(a)=ro=const
� 4 (c) the limit set is a union of trajectories of maximal solutions x : (t1, t2) ∞� R2 of (4.2), each of which has a limit (possibly infinite) as t � t1 or t � t2. The proof of Theorem 4.3 is based on the more specific 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 Definition and fundamental properties of index For n = 1, 2, . . . let Sn = {x ⊂ Rn+1 : |x| = 1} denote the unit sphere in Rn+1. Note the use of n, not n + 1, in the S-notation: it indicates that locally the sphere in Rn+1 looks like Rn. There exists a way to define the index ind(F) of every continuous map F : Sn ∞� Sn in such a way that the following conditions will be satisfied: (a) if H : Sn × [0, 1] ∞� Sn 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 Fˆ : Rn+1 ∞� Rn+1 defined by Fˆ(z) = |z|F(z/|z|) is continuously differentiable in a neigborhood of Sn then ind(F) = det(Jx(Fˆ))dm(x), x�Sn where Jx(Fˆ) is the Jacobian of Fˆ at x, and m(x) is the normalized Lebesque measure on Sn (i.e. m is invariant with respect to unitary coordinate transformations, and the total measure of Sn equals 1). Once it is proven that the integral in (b) is always an integer (uses standard volume/surface integration relations), it is easy to see that conditions (a),(b) define ind(F) correctly and uniquelly. For n = 1, the index of a continuous map F : S1 ∞� S1 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(FI ) = 1 for the identity map FI (x) = x,and ind(Fc) = 0 for every constant map Fc(x) = x0 = const
4.2.2 The Browers fixed point theorem One of the classical mathematical results that follow from the very existence of the index function is the famous Brower's fixed point theorem, which states that for every continuous function G: B ={x∈Rn+1:||≤1} The statement is obvious(though still very useful) when n= 1. Let us prove it for n> l, starting with assume the contrary. Then the map g: Bn bn which maps E bn to the point of Sm- which is the(unique) intersection of the open ray starting from G(a)and passing through z with Sn-. Then H: Sn-x0, 1]H Sm- defined by H(, t)=G(tr) 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"+R be locally Lipschitz and T-periodic with respect to the second a(a, t+T)=a(i,t)va,t where t>0 is a given number. Assume that solutions of the Ode i(t)=a(r(t), t) (4.3) with initial conditions r(O)E Bn remain in Bn for all times. Then(4. 3)has a T-periodic solution a=r(t)=r(t+T) for all tE R Indeed, the map I H.(T, 0, I)is a continuous function G: Bn+B". The solution
5 4.2.2 The Brower’s fixed point theorem One of the classical mathematical results that follow from the very existence of the index function is the famous Brower’s fixed point theorem, which states that for every continuous function G : Bn ∞� Bn, where Bn = {x ⊂ Rn+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 > ˆ 1, starting with assume the contrary. Then the map G : Bn ∞� Bn which maps x ⊂ Bn to the point of Sn−1 which is the (unique) intersection of the open ray starting from G(x) and passing through x with Sn−1. Then H : Sn−1 × [0, 1] ∞� Sn−1 defined by H(x, t) = Gˆ(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 : Rn × R ∞� Rn be locally Lipschitz and T-periodic with respect to the second argument, i.e. a(¯x, t + T) = 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) ⊂ Bn remain in Bn for all times. Then (4.3) has a T-periodic solution x = x(t) = x(t + T) for all t ⊂ R. Indeed, the map x¯ ∞� x(T, 0, x¯) is a continuous function G : Bn ∞� Bn. The solution of x¯ = G(¯x) defines the initial conditions for the periodic trajectory