Topological invariants and detection of periodic orbits
Let \(M\) be an \(n\)-dimensional smooth manifold without boundary and let \(f\) be the local flow on \(M\) generated by the differential equation \(\dot x = v(x)\). The author introduces the notion of \(p\)-detectability for the pair \((f,C)\) where \(C \subset M \times (0,\infty)\) is an isolated set of periodic orbits of \(f\). If the homology index \(I(f,C) \in H_ 1(M)\) is nonzero then \((f,C)\) is 1-detectable. The author considers also the Fuller index \(I_ F(f,C) \in \mathbb{Q}\) and the detection number \(D_ p(f,C) \in \mathbb{Z}_ p\) which is defined for a positive integer which is relatively prime with the multiplicities of periodic orbits in \(C\). The condition \(I_ F(f,C) \neq 0\) does not imply that \((f,C)\) is 1-detectable but the condition \(D_ p(f,C) \neq 0\) implies that \((f,C)\) is \(p\)- detectable. If \(I_ F(f,C) \neq 0\) then \((f,C)\) is \(p\)-detectable, provided that \(p\) is a sufficiently large prime number. Some applications are given.
scientific article; zbMATH DE number 635886