1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2023/2024
4. Mathematics - drugiego stopnia z tyt. magistra
5. Qualitative theory of differential equations

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Qualitative theory of differential equations - opis przedmiotu

Cel przedmiotu

The main aim of this course is to familiarize students with the theory of ordinary differential equations, with particular emphasis on the qualitative theory.

Wymagania wstępne

Zapisz zmiany

Mathematical Analysis 1 and 2, Linear Algebra 1 and 2, Mathematical Software.

Zakres tematyczny

Lecture:

1. First-order ordinary differential equations. Basic concepts. Geometrical interpretation of ODE. ODE integrable by quadratures.
2. Existence and uniqueness of local solutions of the initial problems for ODE. Cauchy problem for ODE. Existential theorems (Picard-Lindelöf theorem, Peano theorem). Extension of solutions of the initial problems for ODE. Dependence of the solution to Cauchy problem on initial conditions and the right-hand side of the equation.
3. High-order ordinary differential equations.Types of equations reducible to first-order ordinary differential equations. Linear second-order differential equations. Sturm-Liouville boundary problem.
4. Dynamical interpretation of systems of ODE. Autonomous systems. Phase trajectories and phase portraits. Flows and orbits. First integrals.
5. Systems of linear ordinary differential equations. Methods of solving of homogeneous and inhomogeneous systems of linear equations. Classification and stability of critical points of systems of linear ODE in the plane. Phase portraits.
6. Systems of nonlinear ordinary differential equations. Local phase portraits. Linearization, Grobman-Hartman theorem. Classification and stability of critical points of systems of nonlinear ODE in the plane. Global phase portraits.
7. Periodic orbits and limits cycles. Limits sets. Poincaré-Bendixson theorem.
8. Elements of the stability theory. Lyapunov stability. Hurwitz theorem. Lyapunov function and fundamental stability theorems.
9. Bifurcations and chaos. Hopf bifurcation. The Lorenz model.
10. Some differential models in physics, biology, medicine and economics. Van der Pol oscillator. Lotka-Volterra systems. Epidemiological models. Maya model. Solow model and economic cycle models.

Classes : Solving of problems related to contents of lectures with particular emphasis on practical applications of learned concepts.
Laboratory : Solving problems related to differential equations using a mathematical package, with particular emphasis on the numerical aspect.

Metody kształcenia

Traditional lectures; classes with the lists of exercises to solve by students; computer lab.

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

A grade from the course is consisted of the grade from laboratory (20%), the grade from classes (30%) and the grade from the final exam (50%). To take a final exam, students must receive a positive grade from classes and laboratory. To attain a pass in the course students are required to pass the final exam.

Class and Laboratory: learning outcomes will be verified through two tests consisted of exercises of different degree of difficulty. A grade determined by the sum of points from these two tests is a basis of assessment.

Lecture: final exam. A grade determined by the sum of points from that exam is a basis of assessment.

Literatura podstawowa

1.  F. Verhust, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag Berlin-Heidelberg, 1990.

2.  E. Kreyszig, Advanced Engineering Mathematics, John Wiley and Sons Inc. New York, 1993

Literatura uzupełniająca

1. D. K. Arrowsmith, C.M. Place, Ordinary differential equations, A qualitative approach with applications, Chapman and Hall, London, 1982.

2. Ph. Hartman, Ordinary Differential Equations, Wiley, New York, 1964.

Uwagi

Zmodyfikowane przez dr Ewa Sylwestrzak-Maślanka (ostatnia modyfikacja: 10-04-2024 16:09)