Dynamical Systems
| Type of study | Follow-up Master |
|---|---|
| Language of instruction | English |
| Code | 470-4120/02 |
| Abbreviation | DS |
| Course title | Dynamical Systems |
| Credits | 4 |
| Coordinating department | Department of Applied Mathematics |
| Course coordinator | prof. RNDr. Marek Lampart, Ph.D. |
Osnova předmětu
Talks:
One-dimensional and two-dimensional population, economical and infection discrete models. General definition of dynamic system and its stability. The system of quadratic functions and its bifurcation diagram. Symbolic dynamics, topological conjugacy, transitivity and sensitivity on initial conditions. Introduction of chaos. Lyapunov exponents.
Difference equations of the first order (continuous logistic model, Poincaré map). Planar continuous linear systems. Phase portraits of the planar systems (classification of dynamic properties). Nonlinear continuous systems (continuous dependence on initial conditions). Equilibria of nonlinear systems (saddles, stability, bifurcation). Closed orbits and limit sets (Poncaré-Bewndrixon Theorem)
Practice:
Solving problems on the topic: modeling of discrete dynamic systems.
Solving problems on the topic: analysis of properties of discrete dynamic systems.
Solving problems on the topic: classification of chaotic behavior of discrete dynamic systems.
Solving problems on the topic: modeling of continuous dynamic systems.
Solving problems on the topic: analysis of properties of continuous dynamic systems
Solving problems on the topic: classification of chaotic behavior of continuous dynamic systems.
Individual projects:
Two individual projects on the topic:
Discrete dynamic systems.
Continuous dynamic systems.
One-dimensional and two-dimensional population, economical and infection discrete models. General definition of dynamic system and its stability. The system of quadratic functions and its bifurcation diagram. Symbolic dynamics, topological conjugacy, transitivity and sensitivity on initial conditions. Introduction of chaos. Lyapunov exponents.
Difference equations of the first order (continuous logistic model, Poincaré map). Planar continuous linear systems. Phase portraits of the planar systems (classification of dynamic properties). Nonlinear continuous systems (continuous dependence on initial conditions). Equilibria of nonlinear systems (saddles, stability, bifurcation). Closed orbits and limit sets (Poncaré-Bewndrixon Theorem)
Practice:
Solving problems on the topic: modeling of discrete dynamic systems.
Solving problems on the topic: analysis of properties of discrete dynamic systems.
Solving problems on the topic: classification of chaotic behavior of discrete dynamic systems.
Solving problems on the topic: modeling of continuous dynamic systems.
Solving problems on the topic: analysis of properties of continuous dynamic systems
Solving problems on the topic: classification of chaotic behavior of continuous dynamic systems.
Individual projects:
Two individual projects on the topic:
Discrete dynamic systems.
Continuous dynamic systems.
E-learning
Consultation through MS Teams.
Povinná literatura
[1] Hirsch MW, Smale S, Devaney RL. Differential Equations, Dynamical systems, and an Introduction to Chaos. Elsevier – Acadenic Press USA; 2012. ISBN 978-0-12-382010-5
[2] Devaney RL. An Introduction to Chatic Dynamical Systems. Westview Press, USA; 2003. ISBN 978-0-8133-4085-2
[2] Devaney RL. An Introduction to Chatic Dynamical Systems. Westview Press, USA; 2003. ISBN 978-0-8133-4085-2
Doporučená literatura
[1] Hirsch MW, Smale S, Devaney RL. Differential Equations, Dynamical systems, and an Introduction to Chaos. Elsevier – Acadenic Press USA; 2012. ISBN 978-0-12-382010-5
[2] Devaney RL. An Introduction to Chatic Dynamical Systems. Westview Press, USA; 2003. ISBN 978-0-8133-4085-2
[3] Marotto FR. Introduction to Mathematical Modeling Using Discrete Dynamical Systems. Thomson, USA; 2006. ISBN 0-495-01417-6
[4] Sandefur JT. Discrete Dynamical Systems Theory and Applications. Clarendon Press, England; 1990. ISBN 0-19-853384-5
[2] Devaney RL. An Introduction to Chatic Dynamical Systems. Westview Press, USA; 2003. ISBN 978-0-8133-4085-2
[3] Marotto FR. Introduction to Mathematical Modeling Using Discrete Dynamical Systems. Thomson, USA; 2006. ISBN 0-495-01417-6
[4] Sandefur JT. Discrete Dynamical Systems Theory and Applications. Clarendon Press, England; 1990. ISBN 0-19-853384-5