(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 vol￾ume/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