Admission Requirements
Mathematical analysis, differential equations, numerical methods

Learning Outcomes

Following the successful completion of the course the following main learning outcomes are expected:

Knowledge:

• Describe the basic concepts, definitions and proofs in the area of attractors, basin boundaries, and stability theory
• Explain a range of mathematical methods used to develop and analyze mathematical models of nonlinear systems.
• Describe the concept of stochastic and chaotic dynamics and principles for chaos control.

Skills:

• Develop and analyze mathematical models of nonlinear systems rising from various interdisciplinary application areas.
• Use basic numerical methods and tools of mathematical computing for the analysis of nonlinear systems and their bifurcations.
• Apply mathematical methods for the analysis of relationships between various nonlinear systems, their bifurcations, and Puancare diagrams.

Competencies:

• Identify nonlinear effects (phenomena, situations, processes) at a mathematical modelling context, characterize them quantifiably and qualitatively.
• Choose and apply relevant mathematical models and algorithms for the synchronization and control of nonlinear systems.
• Interpret the results of nonlinear system model numerical analysis under various conditions.
• Study individually, make progress in the field of nonlinear systems theory and plan the study process.

Programme
Classification of linear systems. Liapunov and structural stability. Attractors, their classification, basin boundaries. Construction of mathematical models of deterministic discrete and continuous nonlinear systems. Perturbation methods. Lindsted method. Duffing equation. Limit cycles. Asymptotic stability and its evaluation. Van der Pol equation. Bifurcations and their classification. Hopf bifurcation. Relaxation oscillations. Subharmonic resonance. Skeleton curves. Holoclinic orbits. Puancare diagrams. Cusp catastrophe, bifurcations and stability alternations. Nonlinear parametric resonance. Van der Pol equation with external excitation. Nonlinear non-autonomous systems. Floke theory for nonlinear systems. Matje equation. Harmonic balance methods. Influence of damping to nonlinear dynamics. Hill equation. Nonlinear eigenforms. Modal equations. Invariant manifolds. Central manifolds and their existence conditions. Chaos in conservative and dissipative dynamical systems. Stochastic and chaotic dynamics. Measures of chaos. Routes leading to chaos. Chaos control.

References

• Jan Awrejcewicz, Rajasekar Shanmuganathan, Minvydas Ragulskis. Recent Trends in Chaotic, Nonlinear and Complex Dynamics. World Scientific Series on Nonlinear Science Series B. World Scientific. ISBN: 978-981-122-189-7. August 2021, Singapore
• D.W. Jordan, P. Smith. Nonlinear Ordinary Differential Equations. Introduction to Dynamical Systems. Oxford University Press. 1999, p.550. 1

Teaching Methodology
Lecture, laboratory classes, discussion

ECTS Credits
7.5 ECTS

Period
II semester (STEM)

Examination methodology
Laboratory examinations, written examination

Relevance
Students learn to construct mathematical models of nonlinear systems, analyse and classify bifurcations. They acquire knowledge and experience on dynamical chaos, chaos evolution, and chaos control. The obtained knowledge, skills, and competences enable to model and analyse sustainable development problems such as coordination of sustainable development of the energy sector and social economy with respect to energy policy and environmental protection.

Status
Existing Course but revised