On the geometric and analytical properties of the anharmonic oscillator

On a Family of Hamilton–Poisson Jerk Systems

“…where W ′ (q) = c − g 2 β(q). Thus, on each level set O c the dynamics of system (10) are given by system (16) or Equation (17), representing a nonlinear oscillator with the kinetic energy T = 1 2 p 2 and the potential energy V(q) = −W(q) = g 1 α(q) + g 2 qβ(q) − cq (for details about the system ẍ = f (x), see, e.g., [10,20]).…”

“…In Section 3, we analyze some dynamical properties of the obtained system, namely, the stability of the equilibrium points, the existence of the periodic orbits around some nonlinearly stable equilibria, and the existence of homoclinic or heteroclinic orbits. In Section 4, we apply these results to a family of anharmonic oscillators [16].…”