MPŠ MP&Scaron MP&Scaron MP&Scaron Avtorji

Jožef Stefan
Postgraduate School

Jamova 39
SI-1000 Ljubljana

Phone: +386 1 477 31 00
Fax: +386 1 477 31 10


Course Description

Mathematical Modelling of Complex Systems (ICT2)


Information and Communication Technologies, second-level study programme


prof. dr. Đani Juričić


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.


1) Uvod
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:

• M. Žele, Đ. Juričić, S. Strmčnik, D. Matko, " A probabilistic measure for model purposiveness in identification for control" , Int. J. Syst. Sci., vol. 29, str. 653-662, 1998.
• M. Kinnaert, D. Vrančić, E. Denolin, Đ. Juričić, J. Petrovčič, " Model-based fault detection and isolation for a gas-liquid separation unit" , Control Engineering Practice, vol. 8, str. 1273-1283, 2000.
• S. Gerkšič, Đ. Juričić, S. Strmčnik, D. Matko, " Wiener model based nonlinear predictive control" , Int. J. Syst. Sci., vol. 31, str. 189-202, 2000.
• D. Vrančić, S. Strmčnik, Đ. Juričić " A magnitude optimum multiple integration tuning for filtered PID controller" , Automatica, vol. 37, str. 1473-1479, 2001.
• Đ. Juričić, M. Žele, " Robust detection of sensor faults by means of a satistical test" . Automatica 2002, vol. 38, str. 737-742, 2002.
• A. Rakar, Đ. Juričić, " Diagnostic reasoning under conflicting data : the application of the transferable belief model" . J. process control, vol. 12, str. 55-67, 2002.
• M. Žele, Đ. Juričić, " Estimation of the confidence limits for the quadratic forms in normal variables using a simple Gaussian distribution approximation" . Comput. stat. (Z) 2005, vol. 20, str. 137-150, 2005.


Seminar and oral exam.

Students obligations:

Seminar and oral exam.