Dynamics of nonlinear parabolic equations with cosymmetry

1
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

2
Agenda

• Population kinetics model
• Cosymmetry
• Solution scheme
• Numerical results
• Cosymmetry breakdown
• Summary

3
Population kinetics model

• Initial value problem for a system of nonlinear
  parabolic equations
  
  
  
• 
  
  
  (1)
• where

- the matrix of diffusive coefficients
- the density deviation
4
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

5

• The system of equations (1) is invariant with
  respect to the transformations
• The system (1) is globally stable when ?0 and
  any ?.
• When ?0 and
  the equilibrium
• w0 is unstable.

6
Solution scheme
Method of lines, uniform grid on O 0,a
Centered difference operators
Special approximation of nonlinear terms
7
Solution scheme
The vector form of the system Where Techniqu
e for computation of family of equilibria was
realized firstly Govorukhin (1998) based on
cosymmetric version of implicit function
theorem (Yudovich, 1991).

• ? is a positive-definite matrix
• Q and S are skew-symmetric matrix
• F(Y) - a nonlinear term.

8
Numerical results (k1 1 k20.3 k31)
--- neutral curve m monotonic
instability o oscillator instability.
nonstationary regimes
nonstationary regimes
Families of equilibria
coexistence
Stable zero equilibrium
nonstationary regimes
nonstationary regimes
coexistence
Families of equilibria
9
Regions of the different limit cycles
- chaotic regimes
- tori
- limit cycles
10
Types of nonstationary regimes
?
?
?
?
?
?
?
?
?
?
?
?
11
Families and spectrum ?15
Cosymmetry effect variability of stability
spectra along the family
12
Family and profiles
13
Coexistence of limit cycle and family of
equilibria ?6
?12.5
?13
?13.3

• -- trajectory of limit cycle
• - - family of equilibria
• , equilibrium.

14
Cosymmetry breakdown

• Consider a system (1) with boundary conditions
• Due to change of variables wv? we obtain a
  problem
• where

15
Neutral curves for equilibrium w (?1, 0,0)
16
Destruction of the family of equilibrium
- - family limit cycle.
Yudovich V.I., Dokl. Phys., 2004.
17
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.

18
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,
  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.