Stability of nonlinear systems described by a second-order vector differential equation

The stability of a class of nonlinear dynamical systems described by a second-order vector differential equation Md² x/dt²+Ddx/dt+f( x)=0 is considered. It is shown that for such systems all the equilibrium points are hyperbolic. Moreover, that the number of right half plane eigenvalues of the system Jacobian matrix depends only on f(x), independent of the elements of M and D. The asymptotic behavior of the trajectories of the system is studied, showing that every bounded trajectory (x(t), dx(t)/dt) of the system converges to one of the equilibrium points as t approaches ∞. It is also shown that without the transversality condition, the stability boundary of the second-order system is contained in the union of the stable manifolds of the equilibrium points on the stability boundary and that the stability region of the second-order system is unbounded