A stability criterion for periodic systems with first integrals
Searching stability criteria first integrals can be useful for systems with \(m+ k\) degrees of freedom -- where \(k\) coordinates are ignorable. In this paper system \(\dot x= f(t, x)\) is investigated supposing the origin is an equilibrium point admitting a first integral, and assuming certain properties of \(f\), between them that \(f\) is \(T\)-periodic in \(t\). Under suitable hypotheses for this first integral \(x= 0\) is uniformly stable if \(x= 0\) is uniformly asymptotically stable. The first example is a spherical pendulum in a gravity force field with a viscous friction, connecting point of the pendulum moves along the vertical axis according to \(z= a\) cost. The second example is the system \(\ddot x= - h(t) x(x\dot x+ y^3 \dot y^3)- kx\), \(\ddot y= - h(t)y(x\dot x+ y^3 \dot y^3)- ky\), where \(k> 0\), \(h(t)> 0\), \(h\) is continuous and periodic. In the last example the system \(\ddot x+ g(x, \dot x) \dot x + (a+ \varepsilon\cos t) x= 0\), \(\varepsilon'= 0\), with \(a> 0\), \(g(x, \dot x)\geq 0\) is considered.
scientific article; zbMATH DE number 763299