Computer Tools for Bifurcation Analysis: General Approach with

Computer Tools for Bifurcation Analysis 1005
The calculation of the coeffcients using the
6.2. Four-wave resonance in
+ U † f uuu UUŨ . (96) c 6, 2 (R 3 )=U † f uR UR 3 .
oscillatory instability
above results for the model of two-level laser shows
that the interaction coefficients are independent of
A more complicated case of the resonance arises the value of the angle α: c 3, 4 = c 3, 5 = c 3, 6 =
for short-scale oscillatory instability. The simplest c 3, 7 = −1/b; the self-interaction coefficient c 3, 8 =
waves configuration contains four waves with equal −1/(2b).
frequencies w and the wavevectors making the following
angles with the x-axis: 0, α,π,π+α.Inthis
case the Ginzburg–Landau equation in addition to 6.3. Algebraic degeneracy —
the terms describing two-modes interaction, a term Hopf–Turing bifurcation
pertaining to interaction of three modes will also
Here we present an example of usage of the function
appear:
BifurcationTheory for analysis of algebraically degenerated
cases. Consider a situation where both
∂ 1 a 1 = c 2, 1, 1
(ik 1 ∇ 1 )a 1 + c 2, 3(R 1 )
a 1 ,
Hopf and Turing bifurcations are permitted simultaneously.
It may occur if the minima of the bi-
c 2, 2 c 2, 2
∂ 2 a 1 = c 3, 4
|a 4 | 2 a 1 + c 3, 6
|a 3 | 2 a 1 + c 3, 7
|a 2 | 2 furcation curves for long-scale oscillatory (k lo =
a 1
c 3, 3 c 3, 3 c 3, 3
0, w = w lo ≠ 0) and for short-scale monotonic
(k = k sm ≠0,w sm = 0) bifurcations are at the
+ c 3, 8
|a 1 | 2 a 1 + c 3, 5
a 2 a 4 a ∗ same level.
3
c 3, 3 c 3, 3 The function is called as follows:
+ c 3, 9(R 1 , R 2 )
a 1 + c 3, 1, 1
BifurcationTheory[
(∇ 1 ∇ 1 )a 1
c 3, 3 c 3, 3
DD.(Nabla[r].Nabla[r])**u + f[u,R] -
Nabla[t]**u == 0,
+ c 3, 1, 2
(ik 1 ∇ 1 ) 2 a 1 + c 3, 2, 1(R 1 )
u, R, t, {r,2}, {r,2}, {{a1,{k,0},0,U,Ut,
(ik 1 ∇ 1 )a 1 .
c 3, 3 c 3, 3 3},
(94) {a2,{0,0},w,V,Vt,2}}, c, eps, 5, {a1,a2}]
Here U,Ut denote the eigenvectors corresponding to
The interaction coefficient c 3, 7 between two modes
the Turing mode (its amplitude a1 is of the thirdorder
of smallness); V,Vt correspond to the Hopf
a 1 and a 2 with corresponding wavevectors k 1 , k 2
making the angle α is given by expression:
mode with second-order amplitude a2. Scaling exponent
c 3, 7 = −U † f uu UR(f u )f uu UŨ
in (58) α = 2. The last argument of the
function shows that the normal forms are produced
− U † f uu UR(f u − 2k 2 forbothmodes.
D(1 − cos α))f uu UŨ
The resulting equations are:
− U † f uu ŨR(f u − 2k 2 D(1 + cos α)
− 2iwI)f uu UU + U † f uuu UUŨ . (95)
∂ 3 a 1 = c 6, 2(R 3 )
a 1 ,
c 6, 1
The expressions for coefficients c 3, 6 ,c 3,4 can be produced
out of the above by the replacements α → π
∂ 4 a 2 = c 6, 4
|a 2 | 2 a 2 + c 6, 5(R 2 , R 4 )
a 2
c 6, 3 c 6, 3
(97)
and α → π + α, respectively. The self-interaction
coefficient c 3, 8 is found as half of the value of c 3, 7
+ c 6, 1, 1
(∇ 1 ∇ 1 )a 2 .
c 6, 3
at α =0.
Finally, we present the expression for the threemodes
The normalization coefficients are given by:
interaction coefficient:
c 6, 1 = U † U ,
c 3, 5 = −U † f uu UR(f u − 2k 2 D(1 + cos α))f uu UŨ
− U † f uu ŨR(f u − 2iwI)f uu UU
− U † f uu UR(f u − 2k 2 D(1 − cos α))f uu UŨ
c 6, 3 = V † V .
The linear term coefficient c 6, 2 (R 3 ) of the Turing
mode is found as: