dynamics of nonlinear parabolic equations with cosymmetry
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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 - PowerPoint PPT PresentationTRANSCRIPT
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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
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Population kinetics modelPopulation kinetics model CosymmetryCosymmetry Solution schemeSolution scheme Numerical resultsNumerical results Cosymmetry breakdown Cosymmetry breakdown SummarySummary
Agenda
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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
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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
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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
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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
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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
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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
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Regions of the different limit cycles- chaotic regimes
- tori
- limit cycles
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Types of nonstationary regimes νν
λ
ν
ν
νν
λ λ
λ λ λ
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Families and spectrum; λ=15
Cosymmetry effect: variability of stability spectra along the family
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Family and profiles
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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..
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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
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Neutral curves for equilibrium w= (1, 0,0)
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Destruction of the family of equilibrium
- - family;
limit cycle.
* Yudovich V.I., Dokl. Phys., 2004.
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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.
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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.