• Wednesday 12-14
  
• Rm 304, Gebäude KP/TP
  
• start on April 3, 2019
  

The class deals with aspects of dynamical systems that are (due to time restrictions) usually not taught in the introductory class on NLD  (TNLP I) and also focusses on some important and rather recent aspects and developments in NLD: The tentative outline of the class is as follows:

• very brief overview of the basics of NLD (Recap of TNLP I)
  Here we review briefly the main concepts of (smooth) dynamical systems, their state space description, the main theorems of NLD, organization and classification of orbits in state space, as well as their stability, bifurcation behavior and (if applicable)  routes to chaotic behavior. 
  
• minimal chaotic systems
  Here we address the question how to identify in a systematic way minimal chaotic systems that are much simpler than the standard examples, the Lorenz model and the Rössler model. The key is to consider a specific subset of ODEs, so-called jerky dynamics and, in particular Newtonian jerky dynamics.  Using this approach, we give a classification scheme for the most elementary chaotic systems, explain how chaotic dynamics appears and why it is hard to find analytically closed solutions for chaotic systems and we tell you how to construct your very own chaotic system.
  
• coupled dynamical systems
  Isolated dynamical systems are rare in nature. If two or more (not necessarily) identical dynamical systems interact by  appropriate coupling functions, a plethora of new and interesting dynamical phenomena can appear.  Paradigmatic examples are various types of synchronization, amplitude and oscillation death. All that depends on the kind of the couplings (direct, diffusive, conjugate, environmental, global, attractive, repulsive etc.) and their strength.  Using representative examples and - to some extent - systematic analyses,  the interrelation between coupling and corresponding dynamics is investigated.
  
• non-smooth dynamical systems
  In contrast to smooth dynamical systems, non-smooth dynamical systems, i.e. dynamical systems have a state space that is  partitioned by  switching boundaries into different subspaces with different associated  smooth vector fields, are far less studied. In these systems a plethora of new properties such as grazing bifurcations, period adding bifurcations, sliding and chattering can occur. Starting with examples such as impact, friction and stick-slip oscillators, bouncing ball dynamics, avalanching in granular matter, and rocking block dynamics, basics insights into the dynamics and the bifurcation phenomena of non-smooth systems will be presented.
  
• Hamiltonian systems and Hamiltonian chaos
  In contrast to dissipative dynamical systems, Hamiltonian systems such as the N-body problem governing the evolution of solar systems behave rather differently. In this part we focus on the following aspects: symplectic property, action-angle representation of Hamiltonian systems, integrability vs. non-integrability, Kolmogorov-Arnol'd-Moser theorem,  Poincare-Birkhoff theorem, Hamiltonian chaos, Chirikov-Taylor map, Smale's horseshoe, numerical methods, in particular  symplectic integration.