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T riply degenerate interactions in 3-component RD system

T riply degenerate interactions in 3-component RD system. Toshi OGAWA (Meiji University) Takashi OKUDA ( Kwansei Gakuin University). Pattern dynamics near equilibriums. Consider R-D systems on a finite interval [ 0,L] with Neumann or periodic boundary conditions.

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T riply degenerate interactions in 3-component RD system

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  1. Triply degenerate interactions in 3-component RD system Toshi OGAWA (Meiji University) Takashi OKUDA (KwanseiGakuin University)

  2. Pattern dynamics near equilibriums Consider R-D systems on a finite interval [0,L] with Neumann or periodic boundary conditions. If we have equilibriums we linearise around it … Uniform stationary solution Turing instability Wave instability Non-uniform stationary solution Obtain the linearization from mode interactions.

  3. Typical patterns near uniform steady state Turing instability in 2D Wave instability in 1D

  4. Any non-trivial secondary bifurcation? See simulations for the following 3-component RD system

  5. Neutral Stability Curve(0)

  6. Mode Interactions(0) Periodic B.C. fundamental wave number: Draw the neutral stability curve for each mode

  7. Neutral Stability Curve(1)

  8. Mode Interactions(1) Periodic B.C. Neutral stability curve for each mode Steady (Pitchfork) bifurcations to pure mode solutions occur. Number of mode depends on the system size L. Moreover there are degenerate bifurcation point for n and n+1 modes.

  9. Normal form for n,n+1 modesinteraction with n>1 By using SO(2) invariance we obtain the normal form for two critical modes:

  10. Neutral Stability Curve(1)

  11. Neutral Stability Curve(2)

  12. Mode Interactions(2) Periodic B.C. Triple mode interaction 0,1,2 modes appears by adjusting the parmeters.

  13. {1,2} or {0,1,2} Mode Interaction There are quadratic resonance terms in the case of 1,2 mode interaction. D.Armbruster, J.Guckenheimer and P.Holmes, 1988 T.R.Smith, J.Moehlis and P.Holmes, 2005 Periodic orbits, Rotating waves, Heteroclinic cycles, …

  14. No periodic motion under the Neumann setting If we restrict the problem under the Neumann BC, then the normal form variable in the previous ODE are going to be all real. Moreover it turns out to be there are NO Hopf bifurcation from the non-trivial equilibriums in this dynamics.

  15. {0,1,2} mode interactionwith up-down symmetry By assuming the up-down symmetry quadratic terms do not appear in the normal form: Notice that this includes the AGH 1-2 normal form as its sub dynamics:

  16. ODE system with 3-real variables Under the Neuman boundary condition the previous ODEs can be reduced to the following real 3-dim ODEs. This system is invariant under the mappings:

  17. Three types of Equilibriums 6 Pure mode equilibriums 4doubley mixed mode equilibriums 4triply mixed mode equilibriums

  18. Hopf Bifurcation around the equilibriums Pure mode Only have Hopf instability Hopf instability Doubleymixed mode Triply mixed mode Hopf instability Chaotic Attractor coming from heteroclinic cycle.

  19. Hopf bifurcation criterion around P1 Linearize around pure 1-mode stationary solution: Let

  20. Hopf bifurcation criterion Eigenvalues for the linearized matrix A are: Hopf Instability HopfInstability occurs along this segment.

  21. {0,1,2}-mode interaction in 3 comp RDand Hopf bifurcation from 1-mode See simulation

  22. Chaotic attractor of the NF

  23. Chaotic attractor of the PDE

  24. More chaotic patterns in 3-comp RD

  25. 1D behaviors (Wave-Turing mix) time

  26. 1D behaviors (animation)

  27. 1-mode(wave) vs 2-mode(Turing)interaction See simulation

  28. Summary ・We introduce a 3-component RD system which have (0,1,2)-mode interaction. ・We can obtain all the possibility of Hopf bifurcation from the equilibrium in the (0,1,2)-normal form with Neumann boundary condition. Moreover we can construct RD systems which have these periodic motion. ・(0,1,2)-normal form may have chaotic solution. The corresponding RD system seems to have such “chaotic” motion.

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