Dynamics of nonlinear parabolic equations with cosymmetry

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

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

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 theequilibrium 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. Solution scheme 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). Рis a positive-definite matrix; QandSare skew-symmetric matrix; F(Y) - a nonlinear term.

8. Numerical results(k1=1; k2=0.3; k3=1) --- neutral curve; m – monotonic instability; o – oscillator instability. nonstationary regimes nonstationary regimes Families of equilibria coexistence Stable zero equilibrium nonstationary regimes nonstationary regimes coexistence Families of equilibria

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

10. Types of nonstationary regimes ν ν ν λ λ λ ν ν ν 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=12.5 λ=13 λ=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.