The number of linking of periodic solutions of nondissipative systems
The author considers the system of ordinary differential equations, \(x'=f(t,x)\), \(t\in {\mathbb{R}}\), \(x\in {\mathbb{R}}^ 2\), where f is a \(C^ 1\)- map frm \({\mathbb{R}}^ 3\) to \({\mathbb{R}}^ 2\) and is periodic in t of period 1. It is assumed that every solution of this equation is defined on an interval (a,\(\infty)\), where \(a\geq -\infty\) may depend on the solution. The author uses the theory of braids as a tool of his investigations and he formulates his main results in terms of this theory. Let \(C=\{C_ 1,...,C_ n\}\) be a set of n distinct periodic solutions of the above equation such that the condition \(\{C_ 1(0),...,C_ n(0)\}-\{C_ 1(t),...,C_ n(t)\}\) is satisfied. Then the set of functions \(\{C_ 1(t),...,C_ n(t)\}\), for \(0\leq t\leq 1\) define a braid with n strings denoted by \(\sigma\) (C). It is known that the group \(B_ n\) of braids with n strings has a representation by (n-1)\(\times (n-1)\)-matrices with coefficients in the ring of integer polynomials in the variables a and \(a^{-1}\), which is called the Buran representation. Given a braid \(\sigma \in B_ n\) and an integer k, let \(r_ k(\sigma)\) be the coefficient of \(a^ k\) in the polynomial trace of the matrix corresponding to \(\sigma\) in this representation. For a braid \(\sigma \in B_ n\) and a positive integer p let M(\(\sigma\),p) be the number of integers k satisfying the following: \(r_ k(\sigma^ p)\neq 0\), k and p is relatively prime and k is not divisible by n. for a positive integer p let N(C,p) be the number of p-periodic solutions of the above equation not contained in \(C=\{C_ 1(t),...,C_ n(t)\}\), where C as mentioned above defines the braid \(\sigma\) (C). As his main result the author proves that \(N(C,p)\geq p(M(\sigma (C),p)-n)\) for all p and \(N(C,p)\geq pM(\sigma (C),p)\) for sufficiently large p.
scientific article; zbMATH DE number 4069358