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

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

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

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

 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.

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

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.

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

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

Types of nonstationary regimes ν ν λ ν ν ν ν λλ λ λ λ

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

Family and profiles

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

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

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

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

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.

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, Yudovich, V. I. On bifurcations under cosymmetry-breaking perturbations. Dokl. Phys., Frischmuth K., Tsybulin V. G.,” Cosymmetry preservation and families of equilibria.In”, Computer Algebra in Scientific Computing--CASC Frischmuth K., Tsybulin V. G., ”Families of equilibria and dynamics in a population kinetics model with cosymmetry”. Physics Letters A, 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.