COURSE GOALS: The main goal of this course is to teach student the qualitative and quantitative (analytical and numerical) methods of solving of ordinary differential equations and their systems, as well as to demonstrate how to model dynamical systems in various fields of science (physics, biology, chemistry, engineering...) by differential equations. Furthermore, student is taught the key methods of analysis of nonlinear dynamical systems of the 1st and higher orders (bifurcations, mappings, Ly-exponents, Poincare sections, attractors, fractal dimensions...) and is introduced into basics of the theory of chaos.
LEARNING OUTCOMES AT THE LEVEL OF THE PROGRAMME:
1. KNOWLEDGE AND UNDERSTANDING
1.1. demonstrate a thorough knowledge and understanding of the fundamental laws of classical and modern physics;
1.2. demonstrate a thorough knowledge and understanding of the most important physics theories (logical and mathematical structure, experimental support, described physical phenomena);
1.5. describe the purpose and use of common software packages;
1.6. list and describe the methods for manipulating data, basic principles of databases and fundamental algorithms in programming;
2. APPLYING KNOWLEDGE AND UNDERSTANDING
2.1. identify and describe important aspects of a particular physical phenomenon or problem;
2.2. recognize and follow the logic of arguments, evaluate the adequacy of arguments and construct well supported arguments;
2.3. use mathematical methods to solve standard physics problems;
2.6. apply fundamental algorithms in programming;
2.7. use computing technology to solve scientific and technological problems
3. MAKING JUDGMENTS
3.1. develop a critical scientific attitude towards research in general, and in particular by learning to critically evaluate arguments, assumptions, abstract concepts and data;
5. LEARNING SKILLS
5.1. search for and use professional literature as well as any other sources of relevant information;
LEARNING OUTCOMES SPECIFIC FOR THE COURSE:
After finishing the course, the student will be capable to:
* demonstrate qualitative knowledge of basic concepts of dynamical systems specified in the Course description as well as the basic physical models related to them
* demonstrate knowledge of qualitative, quantitative analytical and numerical methods of solving of ordinary differential equations and their systems
* demonstrate ability of quantitative modelling of simple dynamical systems in the field of physics, biology, chemistry and engineering by differential equations
COURSE DESCRIPTION:
Lectures per weeks (15 weeks in total):
week 1-2: (1) Introduction into dynamical systems, (2) Autonomous systems of the 1st order: fixed points, logistic population model, solving of ordinary differential equations of the 1st order by separation of variables, tangent fields, numerical methods of solving of differential equations of the 1st order (Euler, Runge-Kutta, Wolfram Mathematica package), bifurcations - types and examples (model of fishing quotas, model of LASER...)
week 3-4: Linear ordinary differential equations: linear differential equations of the 1st order, exmple of model of polluted lake and Drude model of electrical conductivity, linear differential equations of the 2nd order, forced harmonic oscillator
week 5: Dynamical systems of the higher order: systems of differential equations and related fixed points, Liouville theorem and Lie derivative, stability of trajectories in phase space, quantitative methods of determining of stability and Ly-exponent
week 6-10: Autonomous systems of the 2nd order: classification of fixed points, eigenvectors and solution of the system, example of predator-prey population model, numerical methods of solving of system of differential equations, examples of catastrophes in construction and engineering, phase portraits of damped oscillator (linear and nonlinear), concept and examples of soliton, limit cycle, van der Pol model of clock, two-timescale method, LRC circuit with negative differential conductance, Hopf bifurcation
week 11-14: Systems of the order higher than 2 - entering into chaos: Duffing oscillator, Poincare sections, attractors and scenarios of entering into chaos, fractal objects and Hausdorff-Besichovic dimension, one-dimensional mapping (logistic and tent map), example of kicked rotator - Henon map and strange attractor, Lorenz model
REQUIREMENTS FOR STUDENTS:
Students should attend lectures and exercises and solve the weekly homework. Signature requirements are attendance at at least 40% lectures/exercises, submission and successful elaboration of at least 40% of homework.
GRADING AND ASSESSING THE WORK OF STUDENTS:
Students that have attended at least 75% lectures/exercises, have successfully passed at least 75% homework elaborations, take just a final oral exam. Those that have not fulfilled these requirements, but have fulfilled the signature requirements, take a written exam before the final oral exam. The final grade is formed from: grades from homework elaborations / written exam (1 ECTS) and the final oral exam (2 ECTS).
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- S. T. Strogatz, Nonlinear Dynamics and Chaos with Applications to Physis, Biology, Chemistry and Engineering, Perseus Books, Reading 1994
- nastavna skripta autora programa: http://www.phy.pmf.unizg.hr/~dradic/
- H. G. Schuster, Deterministic Chaos, an Introduction, VCH Verlagsgesellschaft, Weinheim 1995
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