In the end of the lectures, the students
1) should have become familiar with the basic theory and methods used for the study of nonlinear dynamical systems in Physics and in other sciences.
2) learn to recognize the usefulness of analytical mathematical methods and their limits and how to manage computational methods.
3) under a systematic way, they come in contact with the modern theory of chaos.
4) since software of symbolic mathematics and numerical computations is widely used, they get this particular mastery too, which is very useful for their scientific field.
Course Content (Syllabus)
Introduction to Dynamical systems, analytic and numerical approach - The programming tool "Mathematica"
· Analytic and Numerical solution of Differential equations with Mathematica
· Basic notions of the Dynamical systems - Phase space - Classification of systems and trajectories.
· Conservative systems of one degree of freedom - oscillations
· Autonomous linear systems 2x2
· Autonomous nonlinear systems - Stability of equilibrium points and phase space diagrams. Applications (Lotka-Voltera models)
· Limit cycles. Application to electrical circuit oscillators (Van der Pol)
· Linear perturbed oscillators – Periodic and quasi-periodic trajectories, limit cycles and Poincare maps.
· Conservative Oscillators – Poincare maps - Homoclinic chaos.
· Limit cycles and strange attractor in dissipative Duffing equation
· Discrete dynamical systems
· Summary and Discussion