Perturbations near zero of the leading coefficient of solutions to a nonlinear differential equation

Abstract:

The equation \[ \frac {d}{{dt}}\left ( {a\left ( t \right ) \cdot \psi \left ( x \right )\frac {{dx}}{{dt}}} \right ) + f\left ( x \right ) \cdot \frac {{dx}}{{dt}} + c\left ( t \right )x = g\left ( t \right )\] that generalizes the more commonly studied selfadjoint second order equation \[ \frac {d}{{dt}}\left ( {a\left ( t \right )\frac {{dx}}{{dt}}} \right ) + c\left ( t \right )x = g\left ( t \right )\] occurs in celestial dynamics, in the study of gyroscopic systems, and in other problems of nonlinear mechanics. Using techniques of nonstandard analysis we derive some properties of the "duck"-type cycles for the solutions of this equation.