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 d2 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 The￾orem 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