Non-linear Dynamics

Every mechanical and physical system is nonlinear. A thorough understanding of the effects of nonlinearities on the dynamics of these systems is therefore often essential, especially when amplitude of speeds are large. Nonlinear dynamics focuses on the measurement and mathematical description of the motion in nonlinear systems. It thus enables the design of new nonlinear functionalities and maximization of operating ranges while avoiding failure. Nonlinearities cause a wide variety of interesting new physical and dynamic phenomena that cannot be seen in linear systems. Examples include resonances with multiple coexisting solutions and chaotic dynamics. The nonlinear differential equations that govern nonlinear systems have implications that reach far beyond the realm of mechanics, including physics, meteorology, fluid dynamics, chemistry and biology. This course focuses on teaching the fundamental principles of nonlinear dynamics, to bring students to a level where they master the essential concepts of nonlinear dynamics and can apply them to solve practical cases.

The course will discuss the following topics:

• Introduction to nonlinear dynamics together with a brief review of linear vibrations

• Local geometric theory, stability of equilibrium points and phase space diagrams

• Limit cycles and self-sustained oscillations

• Static and dynamic bifurcations

• Analytic techniques for nonlinear dynamics and perturbation methods

• Nonlinear resonances: Primary, sub harmonic and super harmonic response, jump phenomena, hysteresis, internal resonance, parametric resonance

• Computational methods for nonlinear dynamics: Pseudo arc length continuation and shooting method

• Physical sources of nonlinearities

• Applications of nonlinear dynamics at the macroscopic and microscopic scale (MEMS/NEMS)

• Brief introduction to chaos