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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
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Dynamics of nonlinear parabolic equations with cosymmetry Vyacheslav G. Tsybulin Southern Federal University Russia Joint work with: Kurt Frischmuth Department.

Dec 18, 2015

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Page 1: Dynamics of nonlinear parabolic equations with cosymmetry Vyacheslav G. Tsybulin Southern Federal University Russia Joint work with: Kurt Frischmuth Department.

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

Page 2: Dynamics of nonlinear parabolic equations with cosymmetry Vyacheslav G. Tsybulin Southern Federal University Russia Joint work with: Kurt Frischmuth Department.

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

Agenda

Page 3: Dynamics of nonlinear parabolic equations with cosymmetry Vyacheslav G. Tsybulin Southern Federal University Russia Joint work with: Kurt Frischmuth Department.

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

Page 4: Dynamics of nonlinear parabolic equations with cosymmetry Vyacheslav G. Tsybulin Southern Federal University Russia Joint work with: Kurt Frischmuth Department.

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

Page 5: Dynamics of nonlinear parabolic equations with cosymmetry Vyacheslav G. Tsybulin Southern Federal University Russia Joint work with: Kurt Frischmuth Department.

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

Page 6: Dynamics of nonlinear parabolic equations with cosymmetry Vyacheslav G. Tsybulin Southern Federal University Russia Joint work with: Kurt Frischmuth Department.

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

Page 7: Dynamics of nonlinear parabolic equations with cosymmetry Vyacheslav G. Tsybulin Southern Federal University Russia Joint work with: Kurt Frischmuth Department.

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

Page 8: Dynamics of nonlinear parabolic equations with cosymmetry Vyacheslav G. Tsybulin Southern Federal University Russia Joint work with: Kurt Frischmuth Department.

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

Page 9: Dynamics of nonlinear parabolic equations with cosymmetry Vyacheslav G. Tsybulin Southern Federal University Russia Joint work with: Kurt Frischmuth Department.

Regions of the different limit cycles- chaotic regimes

- tori

- limit cycles

Page 10: Dynamics of nonlinear parabolic equations with cosymmetry Vyacheslav G. Tsybulin Southern Federal University Russia Joint work with: Kurt Frischmuth Department.

Types of nonstationary regimes νν

λ

ν

ν

νν

λ λ

λ λ λ

Page 11: Dynamics of nonlinear parabolic equations with cosymmetry Vyacheslav G. Tsybulin Southern Federal University Russia Joint work with: Kurt Frischmuth Department.

Families and spectrum; λ=15

Cosymmetry effect: variability of stability spectra along the family

Page 12: Dynamics of nonlinear parabolic equations with cosymmetry Vyacheslav G. Tsybulin Southern Federal University Russia Joint work with: Kurt Frischmuth Department.

Family and profiles

Page 13: Dynamics of nonlinear parabolic equations with cosymmetry Vyacheslav G. Tsybulin Southern Federal University Russia Joint work with: Kurt Frischmuth Department.

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..

Page 14: Dynamics of nonlinear parabolic equations with cosymmetry Vyacheslav G. Tsybulin Southern Federal University Russia Joint work with: Kurt Frischmuth Department.

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

Page 15: Dynamics of nonlinear parabolic equations with cosymmetry Vyacheslav G. Tsybulin Southern Federal University Russia Joint work with: Kurt Frischmuth Department.

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

Page 16: Dynamics of nonlinear parabolic equations with cosymmetry Vyacheslav G. Tsybulin Southern Federal University Russia Joint work with: Kurt Frischmuth Department.

Destruction of the family of equilibrium

- - family;

limit cycle.

* Yudovich V.I., Dokl. Phys., 2004.

Page 17: Dynamics of nonlinear parabolic equations with cosymmetry Vyacheslav G. Tsybulin Southern Federal University Russia Joint work with: Kurt Frischmuth Department.

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.

Page 18: Dynamics of nonlinear parabolic equations with cosymmetry Vyacheslav G. Tsybulin Southern Federal University Russia Joint work with: Kurt Frischmuth Department.

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.