Triply 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.

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.

Typical patterns near uniform steady state

Turing instability in 2D Wave instability in 1D

Any non-trivial secondary bifurcation?

See simulations for the following 3-component RD system

Neutral Stability Curve(0)

Mode Interactions(0)

Periodic B.C.

fundamental wave number:

Draw the neutral stability curve for each mode

Neutral Stability Curve(1)

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.

Normal form for n,n+1 modesinteraction with n>1

By using SO(2) invariance we obtain the normal form for two critical modes:

Neutral Stability Curve(1)

Neutral Stability Curve(2)

Mode Interactions(2)

Periodic B.C.

Triple mode interaction 0,1,2 modesappears by adjusting the parmeters.

{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, …

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.

{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:

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:

Three types of Equilibriums

6 Pure mode equilibriums

4 doubley mixed mode equilibriums

4 triply mixed mode equilibriums

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

Hopf bifurcation criterion around P1

Linearize around pure 1-mode stationary solution:

Let

Hopf bifurcation criterion

Eigenvalues for the linearized matrix A are:

Hopf Instability

Hopf Instability occurs along this segment.

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

See simulation

Chaotic attractor of the NF

Chaotic attractor of the PDE

More chaotic patterns in 3-comp RD

1D behaviors (Wave-Turing mix)tim

e

1D behaviors (animation)

1-mode(wave) vs 2-mode(Turing)interaction

See simulation

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.