Behavior of orbits near a critical equilibrium of a two-dimensional system

A mechanical system consisting of two coupled springs moving in a medium of negligible damping is considered and the sensitivity of orbits and their higher derivatives to initial conditions in a neighborhood of the equilibrium state of this system is analyzed. The springs are assumed to obey a nonlinear extension of Hooke's law. The general system describing the motion of the springs is a four-dimensional nonlinear system whose linear part is a conservative system depicting the motion of the uncoupled springs obeying Hooke's law. The system is analyzed in polar coordinates, centering on the two amplitude equation. Each of the amplitudes of the springs is found to decay to zero due to the effects the nonlinearities.