On the periods of periodic motions in autonomous systems (Q2768824)

Here, the author establishes lower bounds on the periods of periodic solutions to the autonomous system \(\dot x=f(x)\) determined in a partially ordered Banach space \( \langle X,\|\cdot\|,\preceq\rangle\). It is assumed that the mapping \(f:X\mapsto X\) satisfies the condition \({\mathbf m}(f(x_1)-f(x_2))\preceq_E L{\mathbf m}(x_1-x_2)\) for all \(x_1, x_2\in X\), where \({\mathbf m}:X\mapsto E_+\) is a mapping with norm-like properties, \(E_+\) is a cone of nonnegative vectors in a partially ordered Banach space \(\langle E,\|\cdot\|_E,\preceq_E\rangle\), and the linear operator \(L:E_+\mapsto E_+\) is additive, positively homogeneous and has a nonzero partial spectral radius \(r_{E_+}(L)\). NEWLINENEWLINENEWLINEOne of the main results consists in the assertion that for the period of any \(\omega\)-periodic solution which is not an equilibrium the following inequality holds: \(\omega \geq 6/r_{E_+}(L)\). Once \(L:E\mapsto E\) is a linear operator with the invariant cone \(E_+\) and the spectral radius \(r(L)\), then \(\omega\geq \kappa /r(L)\) where \(\kappa\) is the least positive root of the equation \(1/\kappa = \int_{0}^{1/2}\exp(\tau (\tau -1)\kappa) d\tau\). A similar assertion is true for the \(k\)th-order equation \(x^{(k)}=f(x)\). The results obtained are used for the analysis of convergence conditions on the Samojlenko numerical analytic method [\textit{N. I. Ronto, A. M. Samojlenko} and \textit{S. I. Trofimchuk}, Ukr. Math. J. 51, 1079-1094 (1999; Zbl 0940.34010)].