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Topic 1- Perturbation Method Applied to PDE Problems With Application in Mechanical Engineering
Abstract
In this course initial-boundary value problems for weakly nonlinear string, beam, membrane, and plate equations will be studied. Asymptotic approximations of the solutions for these problems will be constructed by using perturbation methods such as averaging methods and multiple time-scales methods. The validity of these approximations will be proved for large times. Also, bifurcation methods will be used to detect resonances and to reduce undesired oscillations in such systems. Recently published journal papers will be incorporated in the course to show the direct applications of the (in this course presented) advanced mathematical techniques in the field of mechanical engineering.
Topic 2 – The Application of Averaging Method to Autoparametric Systems
Abstract
This is an introductory course on the method of averaging, an asymptotic approximation method for solutions of perturbed systems. We present the theory and demonstrate the methodology through examples on finding the approximate solutions of autoparametric systems, coupled oscillator systems that have many applications in mechanical engineering.
Topic 3 – Averaging and Perturbation from Geometric Viewpoint
Abstract
In these lectures, we will focus on using the averaging method to construct an approximation for Poincare’ map for weakly nonlinear oscillations. We will show that under some conditions, global information which is valid for a semi-infinite time interval can be obtained. Next, we will study perturbations of the Hamiltonian system possessing a homoclinic orbit. This perturbation method is known as the Melnikov method. Finally, we will explain the relation with the celebrated Kolmogorov-Arnold-Moser Theorem and the Moser Twist Theorem for the Hamiltonian system.
Topic 4 – Elementary Bifurcation Theory with Applications to Medical Sciences
Abstract
In these lectures, we will introduce some basic concepts of Bifurcation and their applications in Medical Sciences. The lecture will start with the definition of Dynamical Systems, the concept of topological equivalent, and one parameter bifurcation of equilibria in a continuous time dynamical system. Then we will show some examples from Medical Science where bifurcation occurs. We also introduce a numerical continuation software that can be used to compute the bifurcation diagram.
Topic 5 – Perturbation Methods for Delay Differential Equations with Small Delays
Abstract
This talk introduces the application of perturbation methods to delay differential equations (DDEs), focusing on systems with small time delays. Delay effects are common in various physical, biological, and engineering systems, often leading to complex dynamics even when the delays are small. By treating the delay as a small parameter, we demonstrate how Taylor expansions can be used to approximate DDEs by ordinary differential equations, enabling the use of classical perturbation techniques. Several illustrative examples will be discussed to show how small delays influence the system’s stability and long-term behavior. The session aims to provide both theoretical insights and practical tools for analyzing weakly delayed systems, and highlights connections to bifurcation phenomena and nonlinear dynamics.
