Investigation of Well-posedness an‎d Dynamic Analysis of Nonlinear Ordinary Differential Equation: insights from Duffing model

12/9/2026 12:00:00 AM

In this thesis, we investigate the dynamical behavior an‎d bifurcation structure of a nonlinear oscillatory system of Duffing type. The Duffing equation, as one of the fundamental models in the theory of dynamical systems, has a strong capability to describe many physical an‎d engineering phenomena such as mechanical oscillators, bridges, elastic beams, an‎d nonlinear electrical circuits. First, by applying the Banach fixed point theorem an‎d the Picard–Lindelöf theorem, we derive sufficient conditions for the existence an‎d uniqueness of local an‎d global solutions. These results show that, under suitable assumptions on the coefficients an‎d the forcing term, the Duffing equation is a well-posed system from the viewpoint of existence an‎d uniqueness. Next, we study Hyers–Ulam an‎d Hyers–Ulam–Rassias stability for second-order differential equations, including the Duffing equation, an‎d then investigate the Lyapunov stability of the Duffing equation together with its numerical simulations. Subsequently, we first consider the unperturbed system an‎d, by analyzing its equilibrium points, their types of stability, an‎d the existence of homoclinic an‎d heteroclinic orbits, we employ qualitative tools such as phase portraits an‎d stable an‎d unstable manifolds. We then examine the effect of introducing small perturbations an‎d periodic forcing into the system an‎d, in order to obtain a more precise analysis of homoclinic an‎d heteroclinic bifurcations, we make use of the Melnikov method an‎d the first- an‎d second-order averaged systems. These approaches provide conditions for the occurrence of various bifurcations, including the emergence of new periodic orbits, period-doubling, an‎d the onset of chaotic behavior. Furthermore, by means of the Poincaré map, bifurcation diagrams, an‎d numerical simulations, the analytical results are validated an‎d the qualitative changes of the solutions in different regions of the parameter space are described. Finally, while pointing to several applications of the Duffing equation in the modeling of real systems, we emphasize the role of homoclinic an‎d heteroclinic bifurcations in the transition from regular to chaotic behavior an‎d their importance for understan‎ding an‎d controlling nonlinear systems

معادله دافينگ , وجود و يكتايي , پايداري هايرز-اولام-راسياس , پايداري لياپانوف , انشعاب , تابع ملنيكوف , آشوب

Duffing equation , existence an‎d uniqueness , Hyers–Ulam–Rassias stability , , Lyapunov stability , bifurcation , Melnikov function , chaos

Pouria Heydari

Mohammad Bagher Ghaemi