Continuous Dynamical Systems 360-MS2-2CDSa
Wykład (WYK) Rok akademicki 2022/23

Concept of system. Mathematical model of a continuous dynamical system. Autonomous systems. Formal definition of autonomous system. About solutions of autonomous systems. Trajectories of autonomous systems. Three types of trajectories. Limit points and limit sets, periodic solutions, existence of limit cycles, their applications. Qualitative structure of phase space. Basic concepts of stability of solutions of dynamical systems. Stability, asymptotic, uniform, global asymptotic stability, Lyapunov theorems for dynamical systems. First approximation of stability. The use of the first integrals of the system of differential equations in the study of the stability of solutions. The nature of the equilibrium points of a non-linear planar system of differential equations. Three-dimensional dynamical systems and chaos. Applications in economics and natural sciences.

6 stages of uncertainty. Financial predictions. Utility and mean-variance analysis. Dynamic Models. Pricing contingent conditions. Lack of arbitrage in one-period models; hedging portfolios; martingale probabilities and pricing claims in the binomial model. Extension to the multi-period binomial model. Axiomatic derivation. Application of uncertainty.