1. Dynamics of nonlinear parabolic equations with cosymmetry Vyacheslav G. Tsybulin Southern Federal University Russia Joint work with: Kurt Frischmuth Department of Mathematics University of Rostock Germany Ekaterina S. Kovaleva Department of Computational Mathematics Southern Federal University Russia

2.  Population kinetics model  Cosymmetry  Solution scheme  Numerical results  Cosymmetry breakdown  Summary Agenda

3. Population kinetics model Initial value problem for a system of nonlinear parabolic equations: (1) where - the density deviation; - the matrix of diffusive coefficients;

4. Cosymmetry Yudovich (1991) introduced a notion cosymmetry to explain continuous family of equilibria with variable spectra in mathematical physics. L is called a cosymmetry of the system (1) when Let w * - equilibrium of the system (1): If it means that w* belongs to a cosymmetric family of equilibria. Linear cosymmetry is equal to zero only for w= 0. Fricshmuth & Tsybulin (2005): cosymmetry of (1) is

5.  The system of equations (1) is invariant with respect to the transformations:  The system (1) is globally stable when λ=0 and any ν.  When ν=0 and the equilibrium w=0 is unstable.

6. Solution scheme Method of lines, uniform grid on Ω = [0,a]: Centered difference operators: Special approximation of nonlinear terms

7. The vector form of the system: Where Technique for computation of family of equilibria was realized firstly Govorukhin (1998) based on cosymmetric version of implicit function theorem (Yudovich, 1991). Solution scheme Р is a positive-definite matrix; Q and S are skew-symmetric matrix; F(Y) - a nonlinear term.

8. Numerical results ( k 1 =1; k 2 =0.3; k 3 =1 ) Stable zero equilibrium nonstationary regimes Families of equilibria --- neutral curve; m – monotonic instability; o – oscillator instability. coexistence

9. Regions of the different limit cycles - chaotic regimes - tori - limit cycles

10. Types of nonstationary regimes ν ν λ ν ν ν ν λλ λ λ λ

11. Families and spectrum; λ=15 Cosymmetry effect: variability of stability spectra along the family

12. Family and profiles

13. Coexistence of limit cycle and family of equilibria; ν=6 λ=12.5λ=13λ=13.3 –-- trajectory of limit cycle; - - - family of equilibria; *, equilibrium.

14. Cosymmetry breakdown Consider a system (1) with boundary conditions Due to change of variables w=v+  we obtain a problem where

15. Neutral curves for equilibrium w= (  1, 0,0)

16. Destruction of the family of equilibrium - - family; limit cycle. * Yudovich V.I., Dokl. Phys., 2004.

17. Summary A rich behavior of the system: - families of equilibria with variable spectrum; - limit cycles, tori, chaotic dynamics; - coexistence of regimes. Future plans: - cosymmetry breakdown; - selection of equilibria.

18. Some references Yudovich V.I., “Cosymmetry, degeneration of solutions of operator equations, and the onset of filtration convection”, Mat. Zametki, 1991 Yudovich V.I., “Secondary cycle of equilibria in a system with cosymmetry, its creation by bifurcation and impossibility of symmetric treatment of it ”, Chaos, 1995. Yudovich, V. I. On bifurcations under cosymmetry-breaking perturbations. Dokl. Phys., 2004. Frischmuth K., Tsybulin V. G.,” Cosymmetry preservation and families of equilibria.In”, Computer Algebra in Scientific Computing--CASC 2004. Frischmuth K., Tsybulin V. G., ”Families of equilibria and dynamics in a population kinetics model with cosymmetry”. Physics Letters A, 2005. Govorukhin V.N., “Calculation of one-parameter families of stationary regimes in a cosymmetric case and analysis of plane filtrational convection problem”. Continuation methods in fluid dynamics, 2000.