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.
