In this paper, we investigate the asymptotic stability of the zero solution and boundedness of all solutions of a certain third order nonlinear ordinary vector differential equation. The results are proved using Lyapunov's second (or direct method). Our results include and improve some well known results existing in the literature.

In this paper, we address the problem of global asymptotic stability and strong passivity analysis of nonlinear time-varying systems controlled by a secondorder vector differential equation. First, we obtain this equation from a nonlinear time varying network of the circuit theory. Then, we construct the Lyapunov candidate function directly from the physical meaning of the given system. By the way, we review a number of previous results from the point view of Lyapunov's direct method. Our system with its real energy function generalize and improve upon some well-known studies. The new concept facilitates the formulation of the energy (Lyapunov) function from the powerenergy relationship of the given system. Then, we also realized that the time derivative of the Lyapunov function for a given dynamical systems is the negative value of the power dissipated in the system. Therefore, with the proposed approach, one can inspect the result of the time derivative of the energy function for a given physical system. Finally, two examples (one with simulations) are used to illustrate the superiority and validity of the obtained results.

This paper deals with the global asymptotic stability (GAS) of certain nonlinear RLC circuit systems using the direct Lyapunov method. For each system a suitable Lyapunov function or energy-like function is constructed and the direct Lyapunov method is applied to the related system. Then the invariant equilibrium point of each system that makes the system solution to the global asymptotic stable is determined. Some new explicit GAS conditions of certain nonlinear RLC circuit systems are derived by Lyapunov's direct method. The presented simulations are compatible with the new results. The results are given with proofs.

This paper deals with the boundedness of solutions to a nonlinear differential equation of fourth order. Using the Cauchy formula for the particular solution of nonhomogeneous differential equations with constant coefficients, we prove that the solution and its derivatives up to order three are bounded.

This paper establishes certain sufficient conditions to guarantee the nonexistence of periodic solutions for a class of nonlinear vector differential equations of fifth order. With this work, we extend and improve two related results in the literature from scalar cases to vectorial cases. An example is given to illustrate the theoretical analysis made in this paper.Abstrak. Makalah ini membahas kondisi cukup untuk menjamin non-eksistensi dari solusi periodek untuk suatu kelas vektor nonlinear dari persamaan differensial orde lima. Dalam pembahasan makalah ini, dua hasil terkait yang ada dalam literatur telah dikembankan dan diperluas dari kasus skalar menjadi kasus vektor. Sebuah contoh diberikan untuk mengilustrasikan analisis teoritis yang dibahas dalam makalah ini.