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Jožef Stefan
International
Postgraduate School

Jamova 39
SI-1000 Ljubljana
Slovenia

Phone: +386 1 477 31 00
Fax: +386 1 477 31 10
Email: info@mps.si

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Course Description

Mathematical Modeling of Complex Systems

Program

Information and Communication Technologies, second-level study programme

Lecturers:

prof. dr. Đani Juričić

Goals:

Mathematical modelling is a distinctly generic and interdisciplinary branch of science which is applied in almost all branches of natural, technical and, last but not least, e-science. The purpose of this course is to present selected topics related to the issues of the synthesis of models for complex dynamic processes, their simulations and calibration. The presentation will include basic techniques and ideas, available modelling and simulation tools and examples of their practical use.

Content:

1) Introduction
Basic steps of the model synthesis process, basics of nonlinear dynamics

2) Complex dynamics
Models of complex and self-organizing systems; determinism, predictability and causality in (complex) dynamic systems; stochastic processes, Fokker- Planck equation; synchronization.

3) Analysis of complex dynamic systems Spectral methods (Fourier and vawelet analysis), Lyapunov exponent, correlation dimension.

4) Advanced simulation
Basics of numerical integration. Simulation of differential-algebraic equations. Simulation of models with distributed parameters; method of final elements, offline methods (with examples from ecology, heat conduction, Black-Scholes financial model).
Simulation of stochastic systems (Monte Carlo approaches, Markov chains). Simulation tools: Matlab, Simulink, Femlab.

5) Data driven modelling of complex dynamic systems
Basics of linear regression and instrumental variables method. Nonparametric model identification (neural networks, Gaussian processes). Bayesian approach to the identification of complex dynamic systems.
Applications.

Course literature:

• D. Matko, B. Zupančič, R. Karba (1992). Simulation and Modelling of Continuous Systems : A Case Study Approach. Prentice Hall, New York.
• E. Zauderer (2006). Partial Differential Equations of Applied Mathematics. Willey&Sons, New Jersey.
• H. Kantz and Th. Schreiber (2004). Nonlinear Time Series Analysis, University Press, Cambridge.
• E. Cumberbatch and A. Fitt (2001). Mathematical Modelling: Case Studies from Industry. University Press, Cambridge.
• Hangos, K.M. and I.T. Cameron (2001). Process Modelling and Model Analysis. Academic Press, London.
• D. Kaplan and L. Glass (1995). Understanding Nonlinear Dynamics. Springer-Verlag, New York.
• A. Pikovsky, M. Rosenblum and J. Kurths (2003). Synchronization: A Universal Concept in Nonlinear Science. Univesrsity Press, Cambridge.
• A. Scott (2005). Encyclopedia of Nonlinear Science. Routledge, New York.
• R.J. Carroll, D. Ruppert and D.A. Stefanski (2006). Measurement Error in Nonlinear Models: A Modern Perspective. CRC Press.

Significant publications and references:

• Mileva-Boshkoska, B., Boškoski, P., Debenjak, A., Juričić, Đ. Dependence among complex random variables as a fuel cell condition indicator. Journal of Power Sources, [in press], 26 str., 2015.
• Debenjak, A., Boškoski, P., Musizza, B., Petrovčič, J., Juričić, Đ. Fast measurement of proton exchange membrane fuel cell impedance based on pseudo-random binary sequence perturbation signals and continuous wavelet transform. Journal of Power Sources, 254, 112-118, 2014.
• Boškoski, P., Gašperin, M., Petelin, D., Juričić, Đ. Bearing fault prognostics using Rényi entropy based features and Gaussian process models. Mechanical Ssystems and Signal Processing, 11 str.,2014
• Mileva-Boshkoska, B., Boškoski, P., Debenjak, A., Juričić, Đ. Dependence among complex random variables as a fuel cell condition indicator. Journal of Power Sources, [in press], 26 str., 2015.
• Debenjak, A., Boškoski, P., Musizza, B., Petrovčič, J., Juričić, Đ. Fast measurement of proton exchange membrane fuel cell impedance based on pseudo-random binary sequence perturbation signals and continuous wavelet transform. Journal of Power Sources, 254, 112-118, 2014.
• Boškoski, P., Gašperin, M., Petelin, D., Juričić, Đ. Bearing fault prognostics using Rényi entropy based features and Gaussian process models. Mechanical Ssystems and Signal Processing, 11 str.,2014

Examination:

Seminar and oral exam (100%)

Students obligations:

Seminar and oral exam.

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