Dynamics of nonlinear parabolic equations

with cosymmetryVyacheslav G. Tsybulin

Southern Federal University Russia

Joint work with:Kurt Frischmuth

Department of Mathematics University of Rostock

Germany

Ekaterina S. KovalevaDepartment of Computational Mathematics

Southern Federal University Russia

Population kinetics modelPopulation kinetics model CosymmetryCosymmetry Solution schemeSolution scheme Numerical resultsNumerical results Cosymmetry breakdown Cosymmetry breakdown SummarySummary

Agenda

Population kinetics modelInitial value problem for a system of nonlinear parabolic equations:

(1)

where

xtxw

axxwxw

wwwFwMwKw

,0),(

],0[),()0,(

)(),(0

,

00

00

0

M

),,( 321 kkkdiagK ),,( 321 wwww - the density deviation;

- the matrix of diffusive coefficients;

.

2

2

3

1331

1221

11

wwww

wwww

ww

KF

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

),,1(,)( 321 diagBMwBKwL )2(

.0),( L

.0* w0* Lw

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.

},,,,{},,,,{:

},,,,{},,,,{:

321321

321321

wwwwwwR

wwwwwwR

y

x

akkcrit /2 31

Solution scheme

).1/(,1,...,0, nahnjjhx j

.2

)()(

,2

)()(

2

112

111

h

uuuuDu

h

uuuDu

jjjjj

jjjj

Method of lines, uniform grid on Ω = [0,a]:

Centered difference operators:

).()'(23

1

2

)(

3

1

2

)(

3

2),( 211111111 hOvu

h

vuvuu

h

vvv

h

uuvuD j

jjjjj

jjj

jjj

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.

),...,,,...,,...,( ,31,3,21,2,11,1 nnn wwwwwwY

)()( YFYSQPY

Numerical results (k1 =1; k2=0.3; k3=1)

Stable zero equilibrium

nonstationary regimes

nonstationary regimes

nonstationary regimes

nonstationary regimes

Families of equilibria

Families of equilibria

--- neutral curve;

m – monotonic instability;

o – oscillator instability.coexistence

coexistence

Regions of the different limit cycles- chaotic regimes

- tori

- limit cycles

Types of nonstationary regimes νν

λ

ν

ν

νν

λ λ

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=12.5 λ=13 λ=13.3

–-- trajectory of limit cycle;

- - - family of equilibria;

*, equilibrium..

Cosymmetry breakdownConsider a system (1) with boundary conditions

Due to change of variables w=v+ we obtain a problem

where

.),(),0( tawtw

,,0),(

,)()()0,(

),,,(00

xtxv

xxwxvxv

vvvMvKv

.

'2'

'2'

'3

'2'

'2'

'3

1331

1221

11

1331

1221

11

vv

vv

v

K

vvvv

vvvv

vv

K

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 referencesSome 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.