By Anthony Michel, Kaining Wang, Bo Hu
Illuminates crucial result of the Lyapunov and Lagrange balance concept for a normal category of dynamical platforms by way of constructing subject matters in a metric house independantly of equations, inequalities, or inclusions. Applies the overall thought to express sessions of equations. provides new and improved fabric at the balance research of hybrid dynamical structures and dynamical platforms with discontinuous dynamics.
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Extra info for Qualitative theory of dynamical systems : the role of stability preserving mappings
Sample text
Suppose n > 1 and f:D n+ Sn-' is a map such that f o i = identity, where i is the inclusion of Sn-' in D". 7 28 1 SINGULAR HOMOLOGY. THEORY homomorphisms is commutative: However, this gives a factorization of the identity on an infinite cyclic group through zero which is impossible. Therefore, no such retraction f exists. 18 COROLLARY (Brouwer $xed-point theorem) Given a map f: D" -+ Dn,there exists an x in D nwith f ( x ) = x . - PROOF Suppose f: Dn D" without fixed points. E Dn there is a well-defined ray starting at f ( x ) and passing through x .
If X is a topological space define - - - - - D = {(x,x)~ X E X }C X X X - the diagonal in X x X . Recall that X is Hausdorff if and only if the diagonal is a closed subset of X x X . Now let be an equivalence relation on X and denote by L3 the diagonal in ( X / - ) x (X/-). Note that the continuous function x x n: X x X + (X/-) x (X/-) has (nx n>-'(d) = ((x, Y ) 1 x Y } . - This subset of X x X is the graph of rhe relation. closed if and only if its graph is a closed subset of the above that if X / - is a Hausdorff space, then We now show that the converse is true whenever - - The relation on X is X x X .
B is closed, so S” - B is open and hence locally pathwise connected. This implies that the path components are components. Let C, and C, be the components of S” - B. Since C, u B is closed, the boundary of C, is contained in B. (Here we mean by the boundary of C , , the set dC, = - C,”). The proof will be complete when we show that B c d C , . Let x E B and U be a neighborhood of x in S”. 13). 28 H*(Sn - ( B - K ) ) = 2 with generator in dimension zero. Thus, S” - ( B - K ) has one path component.