COURSE GOALS: The aim of the course is to familiarize students with recent developments in the theory of nonlinear conservative and dissipative systems in various fields of science (physics, biology, engineering, social sciences) with reference to the phenomenological, experimental and mathematical aspects of the analysis of such systems and their solutions.
LEARNING OUTCOMES AT THE LEVEL OF THE PROGRAMME:
Upon completing the degree, students will be able to:
1. KNOWLEDGE AND UNDERSTANDING
1.1 formulate, discuss and explain the basic laws of physics including mechanics, electromagnetism and thermodynamics
1.2 demonstrate a thorough knowledge of advanced methods of theoretical physics including classical mechanics, classical electrodynamics, statistical physics and quantum physics
2. APPLYING KNOWLEDGE AND UNDERSTANDING
2.1 identify the essentials of a process/situation and set up a working model of the same or recognize and use the existing models
2.3 apply standard methods of mathematical physics, in particular mathematical analysis and linear algebra and corresponding numerical methods
4. COMMUNICATION SKILLS
4.2 present one's own research or literature search results to professional as well as to lay audiences
5. LEARNING SKILLS
5.1 search for and use physical and other technical literature, as well as any other sources of information relevant to research work and technical project development (good knowledge of technical English is required)
LEARNING OUTCOMES SPECIFIC FOR THE COURSE:
Upon completing the course Nonlinear science, students will be able:
- to analyze the behavior of the first-order autonomous dynamical systems by using analytical and numerical methods;
- to analyze the behavior of the second-order autonomous dynamical systems by using analytical and numerical methods;
- to discuss the concept of bifurcation and list bifurcation types;
- to describe standard scenarios of transition to chaos for dissipative systems;
- to analyze the Feigenbaum scenario of transition to chaos by using analytical and numerical methods;
- to discuss the basic concepts of fractal objects and calculate the fractal dimension for a few selected fractal objects;
- to use the Kolmogorov-Arnold-Moser (KAM) and Poincare-Birkhoff theorems in order to describe the transition to chaos for Hamiltonian systems;
- to analyze the dynamics of the standard map by using numerical methods;
- to solve the Fermi-Pasta-Ulam problem;
- to solve the Korteweg-de Vries equation as a typical example of a nonlinear partial differential equation and discuss the concept of soliton;
COURSE DESCRIPTION:
- Phase space, Lie's derivative, Liouville's theorem, types of stability.
- First-order autonomous systems: fixed points, structural stability, examples of the first-order autonomous systems.
- Second-order autonomous systems: stability of the fixed points, limiting cycle, examples of the second-order autonomous systems.
- Poincare's sections.
- Bifurcations.
- Feigenbaum's scenario and logistic map.
- Standard scenarios of transition to chaos for dissipative systems.
- Lorentz model.
- KAM theorem and Poincare-Birkhoff theorem, standard map.
- Fermi-Pasta-Ulam problem.
- Analytical and numerical solution of the KdV equation.
REQUIREMENTS FOR STUDENTS:
Students are required to regularly attend classes, participate actively in solving problems and write a seminar paper.
GRADING AND ASSESSING THE WORK OF STUDENTS:
At the end of the course an oral examination is held for students who have successfully completed the requirements of the course.
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