THE BEST BOUND ON THE ROTATIONS IN THE STABILITY OF PERIODIC SOLUTIONS OF A NEWTONIAN EQUATION

In most cases, the third order approximation of a scalar Newtonian equation can lead to the Lyapunov stability of a periodic solution through the obtaining of a nonzero twist coefficient. Recently, Ortega obtained the twist property of a periodic solution when the second order coefficient does not change sign and the third one is negative under a crucial limitation to the rotation of the linearization equation. The paper finds that the best bound on the limitation of the rotations is $\theta^*_0=\arccos(-1/4)$ .