Computer Tools for Bifurcation Analysis: General Approach with

1004 L. M. Pismen & B. Y. Rubinstein
The normal forms for the amplitude a 1 is given by (54) with additional resonant terms:
∂ 1 a 1 = c 2, 1, 1
c 2, 2
(ik 1 ∇ 1 )a 1 + c 2, 4(R 1 )
c 2, 2
a 1 + c 2, 3
c 2, 2
a ∗ 2 a∗ 3 ,
∂ 2 a 1 = c 3, 7
c 3, 5
|a 3 | 2 a 1 + c 3, 9
c 3, 5
|a 2 | 2 a 1 + c 3, 11
c 3, 5
|a 1 | 2 a 1 + c 3, 6(R 1 )
c 3, 5
a ∗ 2 a∗ 3 + c 3, 12(R 1 , R 2 )
c 3, 5
a 1
+ c 3, 1, 1
(∇ 1 ∇ 1 )a 1 + c 3, 1, 2
(ik 1 ∇ 1 ) 2 a 1 + c 3, 4, 1(R 1 )
(ik 1 ∇ 1 )a 1 + c (88)
3, 2, 1
a ∗ 2
c 3, 5 c 3,5 c 3, 5 c (ik 1∇ 1 )a ∗ 3
3, 5
− c 3, 2, 2
a ∗ 2
c (ik 3∇ 1 )a ∗ 3 + c 3, 3, 1
a ∗ 3 3, 5 c (ik 1∇ 1 )a ∗ 2 − c 3, 3, 2
a ∗ 3 3, 5 c (ik 2∇ 1 )a ∗ 2 . 3, 5
Below there are formulae for the coefficients of
the resonant terms. The quadratic term a ∗ 2 a∗ 3 appearing
in the equation at the first time scale has
the coefficient given by:
c 2, 3 = U † f uu ŨŨ . (89)
The Ginzburg–Landau equation contains three
cubic terms — the self-interaction term does not
change upon the addition of the third wave; two
terms describing interaction of the principal harmonic
with the two others have equal coeffcients
c 3, 7 = c 3, 9 given by (66) with an additional term:
2(Ũ† f uu UU)(U † f uu R(f u − k 2 D)ŨŨ)/(Ũ† Ũ) .
(90)
There is a quadratic term a ∗ 2 a∗ 3 in the Ginzburg–
Landau equation for which the coefficient is given
by:
c 3, 6 (R 1 )=U † f uuR ŨŨR 1 +(U † f uu ŨŨ)(U† R(f u − k 2 D)f uR UR 1 )/(U † U)
+4(Ũ† f uR ŨR 1 )(U † R(f u − k 2 D)f uu ŨŨ)/(Ũ† Ũ) − 3U † f uu ŨR(f u − k 2 D)f uR ŨR 1 . (91)
There are also four differential second-order terms in the equation; their coefficients are given below.
c 3, 2, 1 = c 3, 3, 1 =2(U † f uu ŨŨ)(U† R(f u − k 2 D)DU)/(U † U) − U † DR(f u − k 2 D)f uu ŨŨ , (92)
c 3, 2, 2 = c 3, 3, 2 =4(Ũ† DŨ)(U† R(f u − k 2 D)f uu ŨŨ)/(Ũ† Ũ) − 2U † f uu ŨR(f u − k 2 D)f uR DŨ . (93)
6.1.2. Calculations for brusselator model
The calculation of the above coefficients made with
the help of the function CalculateCoefficient
produces the following results. The coefficient of the
resonant quadratic term in the lower-order equation
is found as:
c 2, 3 = 2d√ d( √ d − a)
√
d + a
.
The modes interaction coefficients are equal to:
d 2
c 3, 7 =c 3, 9 =
a( √ d + a) 3 (d − 1) 2
×(−3a 4 +8a 2 d+6a 4 d+d 2 −20a 2 d 2
−3a 4 d 2 +6d 3 +16a 2 d 3 −3d 4 +2a √ d
×(2a 2 +d−2a 2 d−6d 2 +d 3 )) .
The coefficients of the second-order differential
terms are:
4d 2
c 3, 2, 1 = c 3, 3, 1 =
a( √ d + a) 2 (d − 1)
× (d + a 2 d − a 2 − ad √ d) ,
4d 3
c 3, 2, 2 = c 3, 3, 2 =
a( √ d + a) 2 (1 − d)
× (−1+ad √ d) .
Finally the quadratic term coefficient in the
Ginzburg–Landau equation takes the form:
c 3, 6 (R 1 )=
2d 2√ db 1
a( √ d + a) 3 (d − 1) 2 (2a2 − a 2 d − a 2 d 2
− 3d + √ d(ad 2 +3ad − a)) .