In this course the student acquires the essential knowledge and technical skills for analyzing Dynamical Systems, i.e, systems of coupled ordinary differential equations, with emphasis on nonlinear equations. The student becomes acquainted with the analytical tools to correctly interpret the dynamical behavior in phase space, which is one of the central concepts in the field of Dynamical Systems. The course is structured in such a way that the level of complexity is increased gradually: the student is first introduced to Dynamical Systems which have a two-dimensional phase space, and later to systems with three or more dimensions, where the dynamics can become chaotic. The notion of chaos (i.e. sensitive dependence on initial conditions, also called "the butterfly effect") and its ubiquity in the natural world, in economics, in the health sciences and innumerable other applications of practical interest, is one of the major new insights of Mathematics of the past 50 years.
After having successfully followed the course, the student will have acquired a sound working knowledge of Dynamical Systems and will be able to follow the new scientific developments in this active and highly relevant research field.
Course Content (Syllabus)
Autonomous systems of ODEs in the two-dimensional phase plane, equilibrium points and their stability properties, the importance of the nonlinear terms. Population dynamics of two competing species (Lotka-Volterra model) and other applications. Hamiltonian dynamical systems, gradient systems. Local vs. global stability, Lyapunov functions. Periodic solutions, limit cycles and the Poincaré-Bendixson theorem. The Van der Pol oscillator and other applications. The notion of structural stability/instability. Bifurcations of equilibrium points and periodic trajectories: saddle-node, transcritical, pitchfork and Hopf bifurcations. Systems of ODEs with a phase space of three or more dimensions, the appearance of chaotic behavior. The Lorenz attractor and other chaotic ("strange") attractors in phase space. Study of stationary and travelling wave solution of non-linear PDEs, soliton equations (Korteweg-de Vries, Nonlinear Schrodinger & sine-Gordon). We study the existence and stability of solitons with methods of applied dynamical systems.
Additional bibliography for study
1. Jordan D W and P. Smith, Nonlinear Ordinary Differential Equations, 2nd Edition, Clarendon Press, Oxford, 1987.
2. E. Infeld & G. Rowlands, Nonlinear Waves Solitons and Chaos, Cambridge University Press, 2000.
3. T. Kapitula & K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Springer 2013.
4. M.J. Ablowitz, Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons, Cambridge University Press, 2011.
5. R.V.Churchill and J.W.Brown, Μιγαδικές Συναρτήσεις και Εφαρμογές. Πανεπιστημιακές Εκδόσεις Κρήτης.