Unbounded solutions of a class of planar systems
The author obtains some sufficient conditions for the existence of unbounded solutions for the following planar system \[ x'= a^+ y^+- a^- y^-+ f(t),\quad y'= -b^+ x^++ b^- x^-+ g(t),\tag{1} \] where \(a^\pm\), \(b^\pm\) are positive constants satisfying \[ {1\over\sqrt{a^+ b^+}}+ {1\over\sqrt{a^+ b^-}}+ {1\over\sqrt{a^- b^+}}+ {1\over\sqrt{a^- b^-}}= {4\over\omega}, \] and \(x^\pm= \max\{\pm x,0\}\), \(y^\pm= \max\{\pm y,0\}\), \(\omega\in \mathbb{R}^+/Q\), and \(f,g\in L^\infty[0, 2\pi]\) are \(2\pi\)-periodic functions. It is shown, by using some auxiliary functions, that there exists \(R_0> 0\) such that every solutions \(x(t)\) of (1) with the initial value \((x(0),x'(0))\) satisfying \[ x^2(0)+ (x'(0))^2\geq R^2_0 \] goes to infinity in the future or in the past. This result is proved by applying the well-known Birkhoff ergodic theorem. A suggestive example is given.
scientific article; zbMATH DE number 2092958