Stability in nonlinear systems

Stability in nonlinear systems

Members of the research team from the DAMS

  • doc. Mgr. Richard Kollár, PhD.

The stability of solutions of nonlinear differential equations has an important role in many applications in geophysics, astrophysics, materials engineering, fluid flow mechanics, condensed matter physics, evolutionary biology and in many other fields. In addition to identifying stable states, questions about the changes in stability under the influence of system parameter's changes are also important. In the linear approximation, the stability is characterized by the spectrum of the corresponding operator. In our research we focuses on the study of qualitative spectrum changes, especially via the Krein signature, which identifies potentially unstable modes of Hamiltonian systems.

Selected papers

  1. R Kollár, Homotopy method for nonlinear eigenvalue pencils with applications, SIAM Journal on Mathematical Analysis 43 (2011), No. 2, 612-633.
  2. R. Kollár, R. L. Pego: Spectral stability of vortices in two-dimensional Bose–Einstein condensates via the Evans function and Krein signature, Applied Mathematics Research eXpress 2012 (2012), No. 1, 1-46.
  3. R. Kollár, P. D. Miller: Graphical Krein Signature Theory and Evans-Krein Functions, SIAM Review 56 (2014), No. 1, 73-123.
  4. R. Kollár, R. Bosák: Index Theorems for Polynomial Pencils, in Nonlinear Physical Systems. Spectral Analysis, Stability and Bifurcations, ISTE London, 2014, p. 177-202.