When a particle of mass m moves on the x-axis in a potential of the form $$V(x) = kx^2$$, it performs simple harmonic motion. The corresponding time period is proportional to $$\sqrt{\frac{m}{k}}$$, as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even whenits potential energy increases on both sides of x = O in a way different from $$kx^2$$ and its total energy is such that the particle does not escape to infinity. Consider a particle of mass m moving on the x-axis. Its potential energy is $$V(x) = \alpha x^4 (\alpha > 0)$$ for $$\mid x \mid$$ near the origin and becomes a constant equal to $$V_0$$ for $$\mid x \mid \geq X_0$$ (see figure).