T riply degenerate interactions in 3-component RD system
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Triply degenerate interactions in 3-component RD system
Toshi OGAWA (Meiji University)Takashi OKUDA (Kwansei Gakuin University)
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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
Non-uniform stationary solution
Turing instability
Wave instability
Obtain the linearization from mode interactions.
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Typical patterns near uniform steady state
Turing instability in 2D Wave instability in 1D
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Any non-trivial secondary bifurcation?
See simulations for the following 3-component RD system
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Neutral Stability Curve(0)
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Mode Interactions(0)
Periodic B.C.
fundamental wave number:
Draw the neutral stability curve for each mode
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Neutral Stability Curve(1)
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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.
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Normal form for n,n+1 modesinteraction with n>1
By using SO(2) invariance we obtain the normal form for two critical modes:
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Neutral Stability Curve(1)
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Neutral Stability Curve(2)
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Mode Interactions(2)
Periodic B.C.
Triple mode interaction 0,1,2 modesappears by adjusting the parmeters.
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{1,2} or {0,1,2} Mode Interaction
D.Armbruster, J.Guckenheimer and P.Holmes, 1988
T.R.Smith, J.Moehlis and P.Holmes, 2005
There are quadratic resonance terms in the case of1,2 mode interaction.
Periodic orbits, Rotating waves, Heteroclinic cycles, …
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No periodic motion under the Neumann setting
If we restrict the problem under the Neumann BC,then the normal form variable in the previous ODEare going to be all real. Moreover it turns out to bethere are NO Hopf bifurcation from the non-trivialequilibriums in this dynamics.
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{0,1,2} mode interactionwith up-down symmetry
By assuming the up-down symmetry quadratic terms do not appearin the normal form:
Notice that this includes the AGH 1-2 normal form as its sub dynamics:
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ODE system with 3-real variables
Under the Neuman boundary conditionthe previous ODEs can be reduced to thefollowing real 3-dim ODEs.
This system is invariant under the mappings:
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Three types of Equilibriums
6 Pure mode equilibriums
4 doubley mixed mode equilibriums
4 triply mixed mode equilibriums
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Hopf Bifurcation around the equilibriums
Pure mode
Doubley mixed mode
Triply mixed mode
Hopf instability
Hopf instability
Chaotic Attractor coming from heteroclinic cycle.
Only have Hopf instability
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Hopf bifurcation criterion around P1
Linearize around pure 1-mode stationary solution:
Let
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Hopf bifurcation criterion
Eigenvalues for the linearized matrix A are:
Hopf Instability
Hopf Instability occurs along this segment.
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{0,1,2}-mode interaction in 3 comp RDand Hopf bifurcation from 1-mode
See simulation
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Chaotic attractor of the NF
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Chaotic attractor of the PDE
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More chaotic patterns in 3-comp RD
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1D behaviors (Wave-Turing mix)tim
e
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1D behaviors (animation)
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1-mode(wave) vs 2-mode(Turing)interaction
See simulation
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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.