When a particle of mass m moves on the x -axis in a potential of the form V ( x ) = kx 2…

When a particle of mass m moves on the x-axis in a potential of the form V(x) = kx², it performs simple harmonic motion. The corresponding time period is proportional to as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of x = 0 in a way different from kx² 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) = ax⁴ (a > 0) for |x| near the origin and becomes a constant equal to V₀ for |x| ≥ X₀ (see figure).


The acceleration of this particle for |x| > X₀ is