  Determination of stability and instability of singular points in nonlinear
dynamical systems is an important issue that has attracted considerable
attention in different fields of engineering and science. So far, different
well-defined theories have been presented to study the stability of singular
points among which the Lyapunov theory is well-known. However, the instability
problem of singular points has been neglected to some extent in spite of its
application in oscillator design. Besides, it is often difficult to achieve a
proper Lyapunov function for a given complex system. This work presents a novel
theorem based on defining two distinct functionals and some straightforward
criteria to study motion stability that significantly facilitate the
determination of equilibrium status at singular points without the requirement
to analytical solution. Indeed, this method is applicable to both stability and
instability problems of linear and nonlinear dynamical systems. In addition,
the presented theorem is further extended to achieve a new linearization
approach of dynamical systems based on averaging technique, which is superior
to the Jacobian approach. Lastly, the proposed linearization method is
generalized to study the stability/instability of higher-order liner/nonlinear
systems. The obtained results clearly show the effectiveness of the proposed
theorem to assess the motion stability/instability.
