Spatio-temporal Oscillations by Wave BifurcationToshi OGAWA
Graduate School of Engineering ScienceOsaka University
E-mail: [email protected]
Abstract
We shall describe the mechanism for the various type of spatio-temporal os-cillations by finding a sort of organizing center in the sense of bifurcation.Wave bifurcation is one of the key idea to understand such a structure. Weshall show that the wave instability can be seen in 3-component reaction-diffusion system under certain conditions. The interaction between the waveand Turing bifurcations can be also seen universally. We shall introduce thebifurcation analysis about these critical points and discuss how much spatio-temporal oscillations we can explain by the wave bifurcation.
Temporal oscillations are commonly ob-served in a variety of phenomena such as BZchemical reaction. Consider now a coupledsystem of these oscillatory dynamics so thatit exhibits spatio-temporal oscillations. Moti-vated by chemical reactions we consider spa-tial diffusion effects as the coupling and studyhow these systems exhibit spatio-temporal os-cillations. Actually we study the reaction dif-fusion(RD) system with oscillatory dynamics.One can observe a variety of complex activi-ties there, compared to the RD systems withexcitable dynamics or the so-called Turing dy-namics. We shall concentrate on the bifurca-tions from the rest state first and then try tounderstand more global view of behaviors.
Oscillating motion appears as the Hopf bi-furcation for a certain parameter. Now, weneed to distinguish the following two situa-tions. One possibility is that the Hopf bifur-cation to a spatially uniform oscillation takesplace earlier than that to spatially non-trivialoscillations. There could be another possibil-ity, namely, the Hopf bifurcation to the non-zero wave number mode might take place ear-lier than the uniform mode. We call the lattercase ”wave bifurcation”. (See 2), 4) and 6).)
It is easy to show that there is no wave bi-furcation in 2-component RD systems as weshall see later. Then, how we can under-stand the various spatio-temporal oscillationswhich we can actually observe in both actualand numerical experiments? One may imaginethe case that the branch of the spatially non-trivial oscillation may recover its stability farfrom an equilibrium even if the correspondingHopf bifurcation takes place later than that ofthe uniform mode. Or branches of the spatio-temporal oscillations do not necessary connectto the rest state. However, we shall not studythese spacial cases here since we think thewave bifurcation is the first key to understandthe spatio-temporal oscillations. In fact we
will show that 3-component RD systems canproduce the wave bifurcation. Moreover wewill introduce the bifurcation analysis for thesolutions resulting from this wave bifurcation.
Let us start with the 2-dimensional ODEwhich has a limit cycle coursed by the Hopfbifurcation:
u = f(u, v) := p(u − v) − u3,v = g(u, v) := qu − v.
Here, we fix a constant q > 1 and consider pas the bifurcation parameter. In fact, p = 1 isthe supercritical Hopf bifurcation point. Now,consider the RD system:
ut = D1∆u + f(u, v),vt = D2∆v + g(u, v).
The linearized stability about its trivial uni-form solution is given by the following matrixfor each wave number k:
Ak =
„
p − D1k2 −p
q −1 − D2k2
«
.
Since TraceAk = p − 1 − (D1 + D2)k2 the
critical set for the Hopf bifurcation is given by
CH := {(k, p); TraceAk = 0} = {p = 1+(D1+D2)k2}
if the imaginary part of the eigenvalues is notzero. The critical mode for k = 0 may bedestabilized at p = 1 earlier than the othercritical mode for k = 0. Therefore, spa-tially non-uniform oscillation may bifurcatelater than the uniform oscillation and, as aresult, only the uniform oscillation can be ob-servable. This means that we do not have wavebifurcation in 2-component RD systems. Itshould be mentioned here that we did not sayanything about static bifurcations. In fact, astatic bifurcation can take place earlier thanthe uniform Hopf mode by taking the ratio oftwo diffusion coefficients appropriately.
2007 International Symposium on Nonlinear Theory and itsApplicationsNOLTA'07, Vancouver, Canada, September 16-19, 2007
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Next consider the following 3-componentRD system where the third component hasnegative feedback effect to the oscillator.
ut = D1∆u + f(u, v) − swvt = D2∆v + g(u, v)
τwt = D3∆w + u − w
If the time constant τ for the third compo-nent w is sufficiently small the wave bifurca-tion occurs by taking the feedback constant sappropriately. In fact take τ = 0 and we haveformally
D3∆w + u − w = 0.
By solving this limit equation as
w =1
2π
Z +∞
−∞e−|x−η|/D3u(η)dη
the system can be reduced to the following 2-component RD system with a global coupling.
ut = D1∆u + f(u, v) −sR
e− |x−η|
D3 u(η)dηvt = D2∆v + g(u, v)
The linearized eigenvalue problem is now givenby
Ak =
„
p − D1k2 − s
1+D3k2 −p
q −1 − D2k2
«
.
By calculating TraceAk we can conclude thatthe wave bifurcation occurs when
D3 >D1 + D2
s.
We shall introduce the bifurcation analysisresults about this wave bifurcation and more-over the mixed bifurcation between the waveand Turing bifurcations.
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Figure 1: Rotating and standing oscillationsof 3-component RDs with different nonlinearterms. Vertical and horizontal axis correspond tospace(periodic) and time (from left to right).
Figure 2: Mixed mode oscillations be-tween standing and uniform oscillations in a 3-component RD.
Figure 3: Mixed mode oscillations between waveand Turing bifurcations in 3-component RDs.
Figure 4: Wave-Turing bifurcation can producea variety of 2D Turing patterns.
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