On Application of Lyapunov and Yoshizawaâ€™s Theorems on Stability, Asymptotic Stability, Boundaries and Periodicity of Solutions of Duffingâ€™s Equation
Keywords:Lyapunov theorems, Yoshizawaâ€™s theorem, stability, asymptotic stability, boundedness, periodicity Duffingâ€™s equation
Stability is one of the properties of solutions of any differential systems. A dynamical system in a state of equilibrium is said to be stable. In other words, a system has to be in a stable state before it can be asymptotically stable which means that stability does not necessarily imply asymptotic stability but asymptotic stability implies stability. For a system to be stable depends on the form and the space for which the system is formulated. Results are available for boundedness and periodicity of solutions of second order non-linear ordinary differential equation. However, the issue of stability, asymptotic stability, with boundedness and periodicity of solutions of Duffingâ€™s equation is rare in literature. In this paper, our objective is to investigate the stability, asymptotic stability, boundedness and periodicity of solutions of Duffings equation. We employed the Lyapunov theorems with some peculiarities and some exploits on the first order equivalent systems of a scalar differential equation to achieve asymptotic stability and hence stability of Duffings equation and again using Yoshizawas theorem we proved boundedness and periodicity of solutions of a Duffings equation. Furthermore, we use fixed point technique and integrated equation as the mode to confirm apriori-bounds in achieving periodicity and boundedness of the solution.
The results obtained showed the consequences of the cyclic relationship between different properties of solutions because the asymptotic stability converges uniformly to a point and limit of the supremum of the absolute value of the difference between the distances existed and are unique and it is this uniqueness that implies the existence of stability. The space where this existed is the space which confirmed continuous closed and bounded nature of the solution and hence the existence of optimal solution and opened the window for application of abstract implicit function theorem in Banachâ€™sSpace to guarantee uniqueness and asymptotic stability, ultimate boundedness and periodicity of solutions of Duffings equation. We concluded that the objectives for the paper were achieved based on our deductions.
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