### Wieslaw Z. Krawcewicz, PhD

#### Education

PhD, Université de Montréal

MSc, Gdánsk University

#### Overview

Topological methods in nonlinear analysis, symmetric differential equations and bifurcation problems.

#### Research Interest

Topological Equivariant Methods in Nonlinear Analysis (TEMNA), with special interests in applications to differential equations (ODEs/FDEs/PDEs/FPDEs) and variational problems with symmetries including symmetric models in mathematical biology (e.g. population dynamics, symmetric configuration of neural networks, pattern formation), fluid dynamics (e.g. Hopf bifurcation in Navier-Stokes equation), electrical engineering (e.g. fluctuations in transmission lines), to mention a few. The Equivariant Degree theory for non-abelian groups, that was developed in recent years, is applied to investigate impact of symmetries on dynamical systems, local and global bifurcation problems, global continuation, existence of multiple periodic solutions and their symmetric properties. Our methods were used, for example, to investigate symmetric configurations of electrical, chemical or biological oscillators, in population models, or electrical transmission lines.

#### Significant Research Contributions

- Substantial contribution to a development of the equivariant degree method for non-abelian compact Lie groups { a new topological method for studying nonlinear equations with symmetries in functional spaces.
- Establishing a standard setting for studying symmetric Hopf bifurcation in ODEs, FDEs, PDEs and FPDEs, using the equivariant degree method outside its topological context (axiomatic approach).
- Application of the equivariant degree method to various models in mathematical biology, engineering and fluid dynamic, to classify and describe the symmetric Hopf bifurcation phenomena.
- Creation of computational methods for degree theoretical calculations and establishing a computational database for standardized usage of the equivariant degree.
- Setting up an equivariant degree approach to detect periodic solutions in symmetric autonomous ODEs and FDEs.
- Applications to symmetric variational problems.

#### Publications

#### A. Monographs

- Z. Balanov, W. Krawcewicz, H. Steinlein, "Applied Equivariant Degree," in AIMS Series in Differential Equations and Dynamical Systems. 2006, American Institute for Mathematical Sciences.
- W. Krawcewicz and J. Wu, "Theory of Degrees with Applications to Bifurcations and Differential Equations", CMS Series of Monographs, John Wiley and Sons, 1997, New York.

#### B. Refereed Book Chapter

- Z. Balanov and W. Krawcewicz, Symmetric Hopf Bifurcation: Twisted Degree Approach, Handbook of Differential Equations, Ordinary Differential Equations, Vol. IV, Edited by F.Battelli and M. Feckan, 2008 Elsvier, 1-131.

#### C. Refereed Journal Publications

- B. Rai and W. Krawcewicz, Symmetric configuration of predator-prey-mutualist systems. Accepted in Nonlinear Analysis, TMA (2008).
- Z. Balanov, W. Krawcewicz and H. Ruan, Periodic solutions to O(2) × S1-symmetric variational problems: Equivariant gradient degree approach. Accepted in Contemporary Mathematics (2008)
- N. Hirano, W. Krawcewicz and H. Ruan, Existence of nonstationary periodic solutions for Γ-symmetric Lotka-Volterra type systems. Accepted in Discrete and Continuous Dynamical Systems, series A (2008)
- Z. Balanov, W. Krawcewicz, and H. Ruan, G.E. Hutchinson's Delay Logistic System with Symmetries and Spatial Diffusion, Nonlinear Analysis, Real World Applications, 8 (2007), 1144-1170.
- Z. Balanov, W. Krawcewicz, and H. Ruan, Hopf Bifurcation in a Symmetric Configuration of Transmission Lines, Nonlinear Analysis, Real World Applications, 8 (2007), 1144–1170.
- Z. Balanov, M. Farzamirad, W. Krawcewicz, and H. Ruan "Applied Equivariant Degree. Part II. Symmetric Hopf Bifurcation for Functional Differential Equations", Discrete Contin. Dyn. Syst. 16 (2006), no. 4, 923–960.
- Z. Balanov, M. Farzamirad and W. Krawcewicz Symmetric Systems of Van der Pol Equations", Topol. Methods Nonlinear Anal. 27 (2006), no. 1, 29-90.
- Balanov, W. Krawcewicz and S. Zur , On Orbital Equivalence of Cubic ODEs in Two-Dimensional Algebras - Topological Methods in Nonlinear Analysis, 25 (2005), 205-234.
- Z. Balanov, W. Krawcewicz, and H. Ruan, Applied Equivariant Degree. Part I. An Axiomatic Approach to Primary Degree, Discrete Contin. Dyn. Syst. 15 (2006), no. 3, 983–1016.

- Updated: May 28, 2013