1. SylabUZ
2. Faculty of Physics and Astronomy
3. winter term 2019/2020
4. Physics - Second-cycle studies leading to MS degree
5. Dynamics of nonlinear systems

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Dynamics of nonlinear systems - course description

Aim of the course

The aim of the course is to familiarize students with methods and fundamental facts of the theory of nonlinear dynamical systems with continuous and discrete time.

Prerequisites

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Mathematical Analysis I and II, algebraic and geometric methods in physics, mathematical methods of physics, classical mechanics

Scope

1. Basic concepts of the theory of ordinary differential equations.

2. Linear differential equations.

3. Singular points, invariant sets and attractors of differential equations.

4. Stability in the Lyapunov sense and linearization.

5. Stability of linear systems with constant coefficients.

6. Periodic solutions and their stability.

7. Methods of numerical investigations of differential equations: Poincare cross-section, Lyapunov exponents.

8. Examples of systems with chaotic behavior.

9. Invariant sets and bifurcations

10. Basic notions of dynamical systems with dicrete time: orbits, periodic points, limit sets.

11. Examples of dynamical systems with discrete time: logistic and baker maps, Bernoulli shifts, Henon map.

12. Investigation of the logistic map.

14. Basic notions of fractals geometry. The Mandelbrota and Julia sets

Teaching methods

Conventional, conversational and problem lecture

Learning outcomes and methods of theirs verification

Outcome description	Outcome symbols	Methods of verification	The class form

Assignment conditions

The exam consists of theoretical questions and short tasks to solve and verifies the learning outcomes in terms of knowledge and skills. Obtaining 50% of points guarantees a positive grade.

Recommended reading

1. W. I. Arnold „Równania Różniczkowe Zwyczajne”, PWN, 1975, eng. V.I. Arnold, Ordinary Differential Equations, Springer, 1992
2. J. Ombach „Wykłady z Równań Różniczkowych”, Wyd. UJ Kraków, (Wyd II) 1999
3. E. Ott, Chaos w układach dynamicznych, WNT, 1997, eng. E. Ott, Chaos in Dynamical Systems, Cambridge University Press, 2nd ed., 2000
4. H. G. Schuster, Chaos deterministyczny, PWN, Warszawa 1993, eng. H. G. Schuster, Deterministic Chaos, 4th ed., Wiley, 2005
5. Materials provided by a lecturer

Further reading

1. R. L. Devaney, An Introduction to Chaotic Dynamical Systems, 2003

2. L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag New York 2001

Notes

Modified by dr hab. Maria Przybylska, prof. UZ (last modification: 05-03-2020 17:12)