Some problems in stability of the equilibrium related to coexistence of solutions to Hill’s equations. (English) Zbl 0685.34060

Differential equations, Proc. EQUADIFF Conf., Xanthi/Greece 1987, Lect. Notes Pure Appl. Math. 118, 773-780 (1989).
[For the entire collection see Zbl 0675.00011.]
The paper deals with the system \[ (*)\quad \ddot x+xf(x)=0,\quad \ddot y+yw(x)=0,\quad x,y\in {\mathbb{R}},\quad f(0)>0. \] If there exists \(s(x,\dot x)\) such that \(\dot ys-y\dot s\) is a first integral, and some smoothness and nondegeneracy conditions hold, then the stability of the origin for (*) is equivalent to “coexistence” of periodic solutions of every Hill’s equation in a certain family. Given the functions s and f, there exists at most one function w such that the system (*) admits \(\dot ys- y\dot s\) as first integral, but generally no such w exists. The paper determines all the functions s which have the property that w can be found in connection with each f so that (*) has the first integral \(\dot ys-y\dot s\) (an example is \(s(x,\dot x)=x\) where we can choose \(w=f)\). Each of these special functions s generates the following problem: determine all the functions f such that the origin is a stable equilibrium for (*) with w defined by s and f. We call such problems free coexistence-like.
Reviewer: G.Zampieri


34D20 Stability of solutions to ordinary differential equations


Zbl 0675.00011