Research Seminar " Theory of Bifurcations of Multidimensional Systems"
- To lay down the basis in dynamical systems theory, theory of bifurcations and the necessary understanding of the approaches, methods, results, and terminology used in the modern applied mathematics literature
- Formation of the knowledge and skills applied to the study of main bifurcations using qualitative methods of dynamical systems
- Formation of sufficient knowledge to perform rather complex bifurcation analysis of dynamical systems arising in applications
- Be able to determine the equivalence of dynamic systems, conduct linearization, topological classification of General equilibria and fixed points. Know the Grobman-Hartman theorem. Be able to bring local bifurcations to topological normal forms
- Be able to find the fundamental matrix, the monodromy matrix. Know Floquet's theorem on the form of the fundamental matrix, Lyapunov's theorem on reducibility. existence of a matrix logarithm.
- Have an idea of the birth of the limit cycle from the saddle homoclinic loop on the plane, from the saddle-node homoclinic loop on the plane, from the saddle homoclinic loop in Rn, about the saddle value and stability of the cycle, orientable and non-orientable loops.
- Know the Central manifold theorems. To be able to compute the Central manifold.
- Know the definition of the dynamic system, orbits, and phase portraits. Be able to find invariant sets and limit sets. Be able to solve periodic differential equations, build Poincare maps and iterations of diffeomorphisms
- Know the simplest bifurcation conditions, the normal shape of the fold bifurcation. Have an idea of the General folded bifurcation, the normal form of flip bifurcation.
- Know the simplest bifurcation conditions, the normal shape of the fold bifurcation. Have an idea of the General folded bifurcation, the normal form of the Hopf bifurcation.
- To know the list of bifurcations of codim 2 equilibria. The bifurcation threshold. Bautin bifurcation (generalized Hopf). Bogdanov-Takens (double zero) bifurcation. Bifurcation of fold-Hopf (zero-pair). Hopf-Hopf Bifurcation.
- Introduction to Dynamical Systems
- Linear Periodic Differential Systems
- Codimension One Semi-Local Bifurcations
- Topological Equivalence, Bifurcations, and Structural Stability of Dynamical Systems
- One-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems
- One-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems
- Bifurcations of Equilibria and Periodic Orbits in n-Dimensional Dynamical Systems
- A Review of Two-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems
- Ma, Tian. Bifurcation Theory and Applications [Электронный ресурс ] / Tian Ma, Shouhong Wang; БД ebrary. - Koln: World Scientific Publishing Co, 2005.
- Shilnikov L.P., Shilnikov A.L., Turaev D.V., Chua L.O. Methods Of Qualitative Theory In Nonlinear Dynamics (Part II). World Sci //Singapore, New Jersey, London, Hong Kong. – 2001.