symbolic computation of normal forms for resonant double hopf bifurcations using a perturbation...
TRANSCRIPT
Journal of Sound and <ibration (2001) 247(4), 615}632doi:10.1006/jsvi.2001.3732, available online at http://www.idealibrary.com on
SYMBOLIC COMPUTATION OF NORMAL FORMS FORRESONANT DOUBLE HOPF BIFURCATIONS USING
A PERTURBATION TECHNIQUE
P. YU
Department of Applied Mathematics, ;niversity of=estern Ontario, ¸ondon, Ont., Canada N6A 5B7.E-mail: [email protected]
(Received 28 September 2000, and in ,nal form 2 April 2001)
This paper presents a perturbation method for computing the normal forms of resonantdouble Hopf bifurcations with the aid of a computer algebraic system. This technique, basedon the method of multiple time scales, can be used to deal with general n-dimensionalsystems without the application of center manifold theory. Explicit iterative formulas havebeen derived for uniquely determining the coe$cients of normal forms and associatednon-linear transformations up to an arbitrary order. User-friendly symbolic computerprograms written in Maple have been developed, which can be easily applied for computingthe normal forms of a given system. Examples are presented to show the applicability of themethodology and the convenience of using the computer software.
( 2001 Academic Press
1. INTRODUCTION
Normal form theory has been widely used to simplify the analysis of dynamic behavior ofdi!erential equations near an equilibrium [1}3]. The basic idea of the method of normalforms is to employ successive co-ordinate transformations to systematically constructa simpli"ed form of the original di!erential equations without changing the fundamentaldynamical behavior of the system in the vicinity of the equilibrium. Normal form theory isusually applied along with center manifold theory [4]. Center manifold theory is appliedbefore normal form theory, to obtain a locally invariant small dimensional manifold in thevicinity of an equilibrium, called center manifold. Although it is easy to construct an&&abstract'' normal form for a given singularity, it is di$cult to compute the explicitexpressions of a normal form in terms of the coe$cients of the original di!erentialequations. Therefore, the crucial part in computing the normal form of a system is thecomputation e$ciency. Moreover, algebraic manipulations become very involved and timeconsuming as the order of the normal form increases. The introduction of computer algebrasystems such as Maple, Mathematica, Reduce, etc., becomes necessary at this point.Recently, many researchers have paid attention to the development of e$cientcomputational methods with the aid of computer algebra systems [5}11].
In this paper, a perturbation technique is used to compute the normal forms ofdi!erential equations whose Jacobian, evaluated at an equilibrium, has two pairs of purelyimaginary eigenvalues. The ratio of the two eigenvalues is a rational number, giving rise toresonant oscillations. The perturbation method, combined with multiple time scales, hasbeen successfully developed for computing normal forms of Hopf bifurcation and othersingularities [7, 12, 13]. It has been shown that the approach is very computationally
0022-460X/01/440615#18 $35.00/0 ( 2001 Academic Press
616 P. YU
e$cient and is simple enough to use by a novice to computer algebra. This technique issystematic and can be directly applied to the original n-dimensional di!erential equations,without the application of centre manifold theory. This approach reduces the steps involvedin the application of normal form theory to simultaneously obtain the normal form (on thecenter manifold) as well as the associated non-linear transformation.
This paper focuses on the development of the methodology and symbolic software forcomputing the normal forms of a double Hopf bifurcation associated with various resonantcases, including k
1: k
2(k
1Ok
2, k
1and k
2are positive integers) resonances and 1 : 1 primary
resonance. The separate treatment for the two cases is due to di!erent scaling which isnecessary for the application of perturbation techniques. Uniform scaling is usedfor the k
1: k
2(k
1Ok
2) resonant cases while non-uniform scaling has to be applied for
the 1 : 1 resonance to match the perturbation orders. All the formulas derived in this paperfor computing the coe$cients of the normal forms and the non-linear transformationsare given explicitly in terms of the original system coe$cients. This facilitatesdirect applications to special problems [8, 10, 14]. Based on the explicit recursiveformulas, user-friendly symbolic software using Maple has been developed. Examples arepresented to show that the perturbation technique is computationally e$cient andtherefore, is particularly useful for computing high-dimensional systems and high ordernormal forms.
The k1: k
2(k
1Ok
2) resonant cases are described in the next section, while section 3 deals
with 1 : 1 resonant cases. Section 4 outlines the symbolic computer programs, and examplesare presented in section 5. Conclusions are drawn in section 6.
2. PERTURBATION TECHNIQUE AND FORMULATIONS
2.1. k1: k
2RESONANT CASES
Consider a general n-dimensional system described by the following di!erential equation:
x5 "Jx#f(x), x3Rn, f : RnPRn, (1)
where Jx represents the linear term, and the non-linear function f is assumed to be analytic;and x"0 is an equilibrium of the system, i.e., f (0)"0. Further, assume that the Jacobian ofsystem (1) evaluated at the equilibrium 0 involves two pairs of purely imaginary eigenvalues$iu
1cand $iu
2c. In this section, we assume that
u1c
u2c
"
k1
k2
, where k1
and k2
are integers, (2)
which is called a k1: k
2resonance. Thus, for the convenience of analysis, we may let u
1c"
k1u and u
2c"k
2u, and further, without loss of generality, we may assume that u"1.
(Otherwise, one may use a time scaling t@"ut to scale the frequency u to 1.) Consequently,the Jacobian matrix of system (1) evaluated at the equilibrium, x"0, can be assumed(with the aid of linear transformation if necessary) in the Jordan canonical form, to be
J"
0 k1
0 0 0
!k1
0 0 0 0
0 0 0 k2
0
0 0 !k2
0 0
0 0 0 0 A
, A3R(n~4)](n~4), (3)
NORMAL FORMS FOR DOUBLE HOPF BIFURCATION 617
such that k1(k
2. Here A is hyperbolic, i.e., all of its eigenvalues have non-zero real parts,
with either a real or complex conjugate. For physically interesting cases, the unstablemanifold is assumed to be empty, i.e., all the eigenvalues of A have negative real parts. Let usrewrite system (1) in component form as
xR1"k
1x2#f
1(x), xR
2"!k
1x1#f
2(x), (4)
xR3"k
2x4#f
3(x), xR
4"!k
2x3#f
4(x), (5)
xRp"!a
pxp#f
p(x) (p"5, 6,2 , m
1#4), (6)
xRq"!a
qxq#u
qxq`1
#fq(x),
xRq`1
"!uqxq!a
qxq`1
#fq`1
(x) (q"m1#5, m
1#7,2, n!1), (7)
where ap'0, a
q'0, u
q'0, and 4#m
1#2m
2"n. Equations (4)}(7) show that the matrix
A has m1
real eigenvalues, !ap, p"1, 2,2, m
1, and m
2pairs of complex conjugate
eigenvalues, !aq$iu
q, q"1, 2,2, m
2. The functions f
i(x) satisfy f
i(0)"0 and
L fi(0)/Lx
j"0, i, j"1, 2,2, n. That is, the Taylor expansion of f
i(x) about x"0 starts with
quadratic terms.The underlying idea of the method of multiple scales is to consider the expansion of
representing the response as a function of multiple independent variables, or scales, insteadof a single time variable. This can be achieved by introducing new independent variablesaccording to
¹k"ekt, k"0, 1, 2,2 . (8)
It follows that the derivatives with respect to t now become expansions in terms of thepartial derivatives with respect to ¹
k, given by
d
dt"
d¹0
dt
LL¹
0
#
d¹1
dt
LL¹
1
#
d¹2
dt
LL¹
2
#2"D0#eD
1#e2D
2#2, (9)
where the di!erentiation operator Dk"L/L¹
k.
Now suppose that the solution of equation (1) (or equivalently, equations (4)}(7)) in theneighborhood of x"0 is represented by an expansion of the form
xi(t; e)"ex
i1(¹
0, ¹
1,2)#e2x
i2(¹
0, ¹
1,2)#2 (i"1, 2,2, n). (10)
Note that the number of independent time scales needed depends on the order to which theexpansion is carried out. In general, to "nd a normal form up to an order n, ¹
0, ¹
1,2, ¹
nshould be used in solution (10). However, one can employ the idea of the intrinsic harmonicbalance (IHB) technique [15] and let n be open so that the process of the method will takeinto account all essential terms up to the required order, as seen in the following procedure.
Substituting solution (10) into equations (4)} (7) with the aid of equation (9) and balancingthe like powers of e results in the perturbation equations in the order given:
e1 : D0x11"k
1x21
, D0x21"!k
1x11
, (11)
D0x31"k
2x41
, D0x41"!k
2x31
, (12)
D0xp1"!a
pxp1
(p"3, 4,2 , m1#2), (13)
618 P. YU
D0xq1"!a
qxq1#u
qx(q`1)1
,
D0x(q`1)1
"!uqxq1!a
qx(q`1)1
(q"m1#3, m
1#5,2 , n!1). (14)
e2 : D0x12"k
1x22!D
1x11#f
12(x
1) , D
0x22
"!k1x12!D
1x21#f
22(x
1) , (15)
D0x32"k
2x42!D
1x31#f
32(x
1), D
0x42"!k
2x32!D
1x41#f
42(x
1), (16)
D0xp2"!a
pxp2#f
p2(x
1) (p"3, 4,2, m
1#2), (17)
D0xq2"!a
qxq2#u
qx(q`1)2
#fq2
(x1),
D0x(q`1)2
"!uqxq2!a
qx(q`1)2
#f(q`1)2
(x1) (q"m
1#3, m
1#5,2 , n!1), (18)
etc., where fi2"(d2/de2) [ f
i(x
1)]e/0
are functions of xi1
(i"1, 2,2 , n) which have beenobtained from the e1 order perturbation equations (11)}(14). In general, functions f
ikonly
involve the variables which have been solved from the previous 1, 2,2, (k!1) orderperturbation equations.
To "nd the solutions of the e1 order equations (11)} (14), "rst note that these equationscan be divided into two groups, one of which consists of the "rst four equations given inequations (11) and (12), and the other one includes the remaining equations. The formergroup is associated with the critical eigenvalues (having zero real parts), and the lattercorresponds to the non-critical eigenvalues (having non-zero real parts). Second, the "rstorder solutions for the variables of the second group are obtained only from the "rst fourvariables x
1, x
2, x
3and x
4, since the perturbation technique is based on the assumed
asymptotic solution (10). However, it is interesting to note that the normal form obtainedbased on equation (10) not only represents the asymptotic behavior, but also the transientproperty of the system.
Now consider equation set (11). Di!erentiating the "rst equation of equation (11) andthen substituting the second equation of equation (11) into the resulting equation yieldsa simple second order ordinary di!erential equation,
D20x11#k2
1x11"0. (19)
The solution of equation (19) can be written in a general form as
x11"r
1(¹
1, ¹
2,2) cos [k
1¹0#/
1(¹
1, ¹
2,2)],r
1cos (k
1¹0#/
1),r
1cos h
1, (20)
where r1
and /1
represent, respectively, the amplitude and phase of motion, andh1"k
1¹0#/
1. Once x
11is determined, x
21can be directly solved from the "rst equation
of equation (11). Note that solution (20) implies that
D0r1"0 and D
0/
1"0, (21)
since r1
and /1
do not contain the variable ¹0. (Remember that D
0denotes di!erentiation
with respect to ¹0.) Similarly, we can obtain solutions from equation (14) as follows:
x31"r
2(¹
1, ¹
2,2) cos [k
2¹0#/
2(¹
1, ¹
2,2)],r
2cos (k
2¹0#/
1),r
2cos h
2(22)
and
D0r2"0 and D
0/
2"0. (23)
NORMAL FORMS FOR DOUBLE HOPF BIFURCATION 619
x41
can be determined from the "rst equation of equation (12) after x31
is solved. Theasymptotic e1 order solutions of the second group, described by equations (13) and (14), areobviously given by
xi1"0, i"5, 6,2 , n, (24)
which actually represent the "rst order steady state solutions of the second group equations.Next, to solve the e2 order perturbation equations (15)}(18), the procedure described
above can be applied. Thus, di!erentiating the "rst equation of equation (15) and thensubstituting the second equation of equation (15) into the resulting equation results ina non-linear homogeneous ordinary di!erential equation:
D20x12#k2
1x12"!D
1D
0x11!D
1x21
#D0f12#f
22. (25)
Then, substituting the solutions x11
and x21
into the right-hand side of equation (25)gives an expression in terms of the trigonometric functions cos k¹
0and sin k¹
0. To
eliminate possible secular terms which may appear in the solution of x12
, it is requiredthat the coe$cients of the two terms cos k
1¹0
and sin k1¹0
be equal to zero, whichin turn yields the explicit solutions for the derivatives D
1r1
and D1/1. Then, the solution
of x12
can be found from the remaining terms of equation (23), and thus, x12
containsonly a particular solution. Having found x
12, x
22is determined from the "rst
equation of equation (15) as x22"(1/k
1) (D
0x12#D
1x11!f
12(x
1)). This procedure
can be carried out to higher order perturbation equations, and thus we can "nd D1r2,
D1/2, x
32and x
42, etc. Finally, the normal forms, given in polar co-ordinates, can be
written as
dr1
dt"
Lr1
L¹0
L¹0
Lt#
Lr1
L¹1
L¹1
Lt#
Lr1
L¹2
L¹2
Lt#2"D
0r1#D
1r1#D
2r1#2, (26)
dh1
dt"k
1#
L/1
L¹0
L¹0
Lt#
L/1
L¹1
L¹1
Lt#
L/1
L¹2
L¹2
Lt#2"k
1#D
0/
1#D
1/1#D
2/1#2,
(27)
dr2
dt"
Lr2
L¹0
L¹0
Lt#
Lr2
L¹1
L¹1
Lt#
Lr2
L¹2
L¹2
Lt#2"D
0r2#D
1r2#D
2r2#2, (28)
dh2
dt"k
2#
L/2
L¹0
L¹0
Lt#
L/2
L¹1
L¹1
Lt#
L/2
L¹2
L¹2
Lt#2"k
2#D
0/
2#D
1/2#D
2/2#2,
(29)
where the back scaling eriPr
i(i.e., ex
iPx
i) has been used (or simply set e"1). It should be
pointed out here that the particular solutions of equation (25), etc., can be found by usingthe IHB method [15] so that the solution is uniquely determined. Thus, D
irj
and Di/j
( j"1, 2) are also uniquely de"ned, which implies that the normal form given in equations(26)}(29) are actually uniquely determined. It is also noted that, unlike the non-resonantcase, in which the derivatives D
irjand D
i/jare functions of r
1and r
2only, here they are
given explicitly in terms of amplitudes r1, r
2and phase k
2h1!k
1h2.
620 P. YU
2.2. 1 : 1 RESONANT CASE
Again, consider system (1), but now the Jacobian evaluated on the equilibrium x"0 isgiven by
J"
0 u 0 1 0
!u 0 0 0 1
0 0 0 u 0
0 0 !u 0 0
0 0 0 0 A
, A3R(n~4)](n~4), (30)
where u can be assumed as u"1 if a time scaling t@"ut is applied. Note that this is calledthe non-semisimple case since two 1's are present in J.
This resonant case is quite similar to the resonant cases discussed in section 2.1. The onlydi!erence for this particular case is that unlike a general k
1: k
2(k
1Ok
2) resonance, we
cannot use a uniform scaling xiPex
ito obtain the ordered perturbation equations. It is
straightforward to show that a uniform scaling leads to the solution of non-lineardi!erential equations at the "rst perturbation order. This di$culty is due to thenon-semisimple resonance, that is, linear terms x
3and x
4are present in the "rst and second
equations (see the Jacobian given by equation (30)). In order for the perturbation techniqueto work on the 1 : 1 non-semisimple resonant case, we must use di!erent scalings betweenx1, x
2and other variables, x
i, i"3, 4, 5,2 , n. A careful consideration shows that the ratio
of the perturbation order for x1, x
2and the remaining variables should be 2 : 3. Thus, let
x1Pe2x
1, x
2Pe2x
2, x
iPe3x
i, i"3, 4,2 , n. (31)
Then, unlike the k1: k
2(k
1Ok
2) resonances, where the highest degree of harmonics at an
order k is consistently equal to k#1, now the kth order terms in the original di!erentialequations may spread into di!erent order ('k) perturbation equations due to the di!erentscalings. However, it can be shown that the highest order of the perturbation equationwhich should be included for the original kth order terms is 3k!2. Therefore, the highestdegree of the harmonics for each kth order must be 3k!2. In other words, if an nth ordernormal form is needed, then the perturbation equations up to 3n!2 order must beconsidered. Once this scaling is applied and the solution form is appropriately set, thena solution procedure similar to that presented in section 2.1 can be obtained and theformulas similar to that given in section 2.1 can be found.
It should be noted, unlike for the k1: k
2(k
1Ok
2) resonant cases, a kth order normal form
and the associated non-linear transformations (solutions) obtained for this 1 : 1 case mayinvolve higher order terms which actually do not belong to the kth order expression. This isdue to the di!erence in scaling. This suggests that the di!erence in scaling has caused thelower order terms to spread into higher order perturbation equations, and has causedhigher order terms to appear in lower order ((3n!2) perturbation equations. This is thefundamental di!erence between the 1 : 1 primary resonance and the k
1: k
2(k
1Ok
2)
resonance. In order to remove the terms which do not belong to the nth order normal form,one may just easily truncate the normal form and solutions up to nth order. (Note: here theorder means the order of the original di!erential equations.) This redundant calculationincreases the complexity and the time required for the computation.
It is interesting to note that the normal form obtained using the perturbation techniquevia di!erent scalings actually agree with that found by using other theories and methods
NORMAL FORMS FOR DOUBLE HOPF BIFURCATION 621
such as PoincareH normal form theory and the IHB technique. This will be demonstrated bythe examples presented in section 4.
3. OUTLINE OF THE SYMBOLIC COMPUTATION
All the formulas presented in the previous section are given explicitly in terms of thecoe$cients of the original di!erential equations, and thus can be easily implemented usinga computer algebraic system. Maple has been used to code these explicit formulas. In thissection, we shall outline the Maple programs.s
3.1. MAPLE SOURCE CODE
(1) Read a prepared input "le. The input "le provides the original di!erent equations,with the number of non-zero real eigenvalues, N
1, and the number of the pairs of
complex conjugate eigenvalues, N2, as well as the order for computing the normal
form, norder.(2) Create procedures.* The procedure Findcoef for computing all the coe$cients of harmonics in terms of
cos (mh1#nh
2) and sin (mh
1#nh
2) for a given trigonometric function F.
* The procedure solu1A}B for computing the coe$cients of harmonics in solutions
xi, i"1, 2, 3, 4 (corresponding to the center manifold) using the method of
harmonic balancing.* The procedure solu2A
}B for computing the coe$cients of harmonics in solutions
xi, i"5, 6,2, N
1#4 (corresponding to the non-zero real eigenvalues) using the
method of harmonic balancing.* The procedure solu3A
}B for computing the coe$cients of harmonics in solutions
xi, i"N
i#5, N
1#6,2, N
1#2]N
2(corresponding to the complex conjugate
eigenvalues) using the method of harmonic balancing.(3) Create the main procedure solution for computing the normal form and non-linear
transformation (i.e., periodic solution).* Establish ordered perturbation equations f
iby separating the original di!erential
equations Dxiaccording to the powers of e.
* Create the formal solutions for xi, i"1, 2, 3, 4, given in terms of basic
trigonometric functions.* Call the procedure Findcoef to "nd the harmonic coe$cients of the original
di!erential equations after the formal solutions are used.* Find the normal form terms D
nr1, D
np1, D
nr2, D
np2
by eliminating the secularterms.
* Call the procedure solu1A}B to "nd the solutions for x
i, i"1, 2, 3, 4.
* For the part associated with the non-zero real eigenvalues, call the procedureFindcoef "rst and then the procedure solu2A
}B to obtain the solutions for x
i,
i"5, 6,2 , N1#4.
* For the part associated with the complex conjugate eigenvalues, call theprocedure Findcoef "rst and then the procedure solu3A
}B to solve the solutions
for x , i"N #5, N #6,2 , N #2]N .
sThe Maple source codes and sample computer input "les can be found from the author's website:http://pyu1.apmaths.uwo.ca/pyu/software. The source code name is program5 and the name of the sample input"le is input5.
i 1 1 1 2
TABLE 1
Input ,le to Maple source code
N1 :" 1:N2 :" 1:N :" 4#N1#N2*2:norder :" 4:case :" 10:Dx[1] :" omg1*x[2]#x[1] L 2#x[1]*x[3]#2/3*x[3]*x[4]#x[4]*x[5]:Dx[2] :"!omg1*x[1]#2*x[4] L 2#x[5] ( 2#1/2*x[1]*x[2]#x[2]*x[4]:Dx[3] :" omg2*x[4]#4*x[3] L 2#x[5] ( 2#x[1]*x[2]:Dx[4] :"!omg2*x[3]#2*x[2] L 2#x[3]*x[4]#x[3]*x[5]:Dx[5] :"!x[5]#x[2] ( 2#1/3*x [2] *x[4]:Dx[6] :"!x[6]#x[7]#x[2] ( 2#5*x[1]*x[3]:Dx[7] :"!x[6]!x[7]#2/5*x[1] ( 2:if case"11 then
omg1 :"1:omg2 :"2:Dx[1] :"Dx[1]#x[3]:Dx[2] :"Dx[2]#x[4]:
elseomg1 :"2:omg2 :"3:
":
622 P. YU
(4) Transform the original di!erential equations into the summation of ordered terms byintroducing perturbation parameter e.
(5) Transform the original variables xi
into the summation of ordered terms byintroducing perturbation parameter e.
(6) Set the zero order solutions xi0
and zero order normal form terms D0r1, D
0p1, D
0r2,
D0p2.
(7) Call the main procedure solution to obtain the normal form and solutions for anarbitrary perturbation order n.
(8) Write the normal form dr1, dp1, dr2, dp2 into the output "le &&Nform''.
3.2. CREATE THE INPUT FILE
(1) Set the variables:N1*the number of the non-zero real eigenvalues of the Jacobian.N2*the number of pairs of complex conjugate eigenvalues of the Jacobian.N2the dimension of the system.norder2the order of normal forms to be computed.
(2) Create the vector "eld, i.e., the functions Dxi, i"1, 2,2 , n; n"4#N
1#2N
2, the
dimension of the system.
As an example, the input "le for the third example (2 : 3 resonant case) given in the nextsection is shown in Table 1.
4. EXAMPLES
In this section, three examples are presented to show the applicability of the method andthe e$ciency of the Maple programs. The "rst two examples are chosen from the problems
NORMAL FORMS FOR DOUBLE HOPF BIFURCATION 623
which have been studied before, in order to verify the results obtained using the programsdeveloped in this paper. The third example is used to demonstrate the computationale$ciency of the Maple programs for computing high order normal forms ofa higher-dimensional system. All the results including the normal forms and correspondingnon-linear transformations are obtained by executing the Maple programs on a PC(PENTIUM III-700MMX 256K system).
4.1. EXAMPLE 1 FOR 1 : 2 RESONANCE
Consider the following four-dimensional general system, described by
x5 "Jx#f (x, l), x3R4, l3R2, (32)
where l is a two-dimensional vector parameter called perturbation parameter (unfolding),and the Jacobian is given by
J"
0 u1c
0 0
!u1c
0 0 0
0 0 0 u2c
0 0 !u2c
0 (x"0, l"0),
(33)
satisfying u1c
/u2c"1 : 2. Thus, we may assume that u
1c"1 and u
2c"2. This example was
studied before using the IHB technique [16]. Since our interest here is focused on thecomputation of the normal form, we may ignore the perturbation parameter l (or simply setl"0). The general normal form up to second order was obtained using the IHB method inreference [16] as
dr1
dq1
"r1r2(A cosU#B sinU), r
1
d/1
dq1
"r1r2(B cosU!A sinU),
dr2
dq2
"r21(C cosU#D sinU), r
2
d/2
dq2
"r21(!D cosU#C sinU), (34)
where U"2/1!/
2, and
A"14
[( f113
!f223
)#( f124
#f214
)], C"14
[( f114
!f224
)!( f123
#f213
)],
C"14
[( f311
!f322
)#( f412
#f421
)], D"14
[( f411
!f422
)!( f312
#f321
)], (35)
in which fijk"L f
i(0, 0)/Lx
jLx
kfor f"( f
1, f
2, f
3, f
4)T. Note that the scaled times
q2"2q
1,2q, and q
k"u
kt.
Now rewrite equation (32) in the expanded form
x5 "Jx#Q(x)#2, (36)
where Q(x)"(q1, q
2, q
3, q
4)T represents the quadratic terms with
qi"1
2f111
x21#1
2f122
x21#1
2f133
x23#1
2f144
x24#f
112x1x2#f
113x1x3
#f114
x1x4#f
123x2x3#f
124x2x4#f
134x3x4
(i"1, 2, 3, 4). (37)
624 P. YU
Then executing the Maple program developed in this paper yields the following normalform:
dr1
dt"!r
1r2(A cosW#B sinW ), r
1
dh1
dt"r
1!r
1r2(B cosW!A sin W),
dr2
dt"!
1
2r21(C cosW#D sinW ), r
2
dh2
dt"2r
2!
1
2r21(!D cosW#C sinW), (38)
where A, B, C and D are given in equation (35), and W"h2!2h
1. One can show that
equation (38) is actually identical to equation (34) by noting the di!erent time scales, and thephase di!erence between system (38) and (34) (due to the di!erent notations used in theMaple program), which is !n/2. This indicates that the results obtained in this paper usingthe multiple scales are the same as those given by a di!erent method such as the IHBtechnique [16].
Here, it should be pointed out that the Maple program developed in this paper can beemployed to "nd normal forms up to an arbitrary order as long as the machine memory isallowed. However, for the IHB method, only the form up to the leading order terms iscorrect. For example, for the 1 : 2 resonant case, it can be shown that, in general, the normalform obtained using the IHB approach is correct only up to the second order.
4.2. EXAMPLE 2 FOR 1 : 1 RESONANCE
The double pendulum system shown in Figure 1 consists of two rigid weigthless links ofequal length l which carry two concentrated masses 2m and m respectively. A follower forceP1
and a constant directional force (vertical) P2
are applied to this system. This system hasbeen studied by many authors for di!erent types of bifurcations (e.g., see references[14, 17]).
Figure 1. A double pendulum system.
NORMAL FORMS FOR DOUBLE HOPF BIFURCATION 625
The system energy for the three linear springs h1, h
2and h
3is assumed to be given by [17]
<"12[(h
1#h
2#h
3l2 )h2
1#2(h
3l2!h
2)h
1h2#(h
2#h
3l2 )h2
2]!1
6h3l2(h
1#h
2) (h3
1#h3
2),
(39)
where h1
and h2
are the generalized co-ordinates which specify the con"guration of thesystem completely.
The kinetic energy ¹ of the system is expressed by
¹"
ml2
2X2[3hQ 2
1#hQ 2
2#2hQ
1hQ2cos (h
1!h
2)], (40)
where X is an arbitrary value rendering the time variable non-dimensional, and the overdotdenotes di!erentiation with respect to the non-dimensional time variable q and q"Xt.
The components of the generalized forces corresponding to the generalized co-ordinatesh1
and h2
may be written as
Q1"P
1l sin (h
1!h
2)#2P
2l sin h
2, Q
2"P
2l sin h
2(41)
and the damping can be expressed by
D"12[d
1hQ 21#d
2(hQ
1!hQ
2)2]#1
4[d
3hQ 41#d
4(hQ
1!hQ
2)4], (42)
where d1
and d2
represent the linear parts, while d3
and d4
describe the non-linear parts.With the aid of the ¸agrangian equations, in addition, choosing the state variables
z1"h
1, z
2"hQ
1, z
3"h
2and z
4"hQ
2(43)
and by rescaling the coe$cients to be dimensionless coe$cients as
f1"
h1X2
ml2, f
2"
h2X2
ml2, f
3"
h3X2
m, f
4"
P1X2
ml, f
5"
P2X2
ml,
f6"
d3X4
ml2, f
7"
d4X4
ml2, g
1"
d1X2
ml2, g
2"
d2X2
ml2, (44)
one can derive a set of "rst order di!erential equations up to third order terms as follows:
dz1
dq"z
2,
dz2
dq"!(1
2f1#f
2!1
2f4!f
5) z
1!(1
2g1#g
2) z
2#( f
2!1
2f4!1
2f5) z
3#g
2z4
#(14
f1#3
4f2!1
3f4!2
3f5) z3
1!(3
4f2#1
2f3!1
3f4! 7
12f5) z3
3#f
7z34
!(12
f6#f
7) z3
2#(1
4g1#3
4g2) z2
1z2!(1
2f1#9
4f2!1
2f3!f
4!3
2f5) z2
1z3
!34
g2z21z4!1
2z1z22#1
2z22z3#3 f
7z22z4#(1
4f1#9
4f2!f
4!3
2f5)z
1z23
#(14
g1#3
4n2) z
2z23!3
4g2z23z4!1
2z1z24!3 f
7z2z24#1
2z3z24
!( 12
g1#3
2g2) z
1z2z3#3
2g2z1z3z4,
626 P. YU
dz3
dq"z
4,
dz4
dq"(1
2f1#2 f
2!f
3!1
2f4!f
5) z
1#(1
2g1#2g
2)z
2!( 2 f
2#f
3!1
2f4!3
2f5) z
3
!2 g2z4!(1
2f1#5
4f2!1
6f3! 7
12f4!7
6f5) z3
1#(5
4f2#7
6f3! 7
12f4!f
5) z3
3
#(12
f6#2 f
7) z3
2!2f
7z34!(1
2g1#5
4g2) z2
1z2#5
4g2z21z4#3
2z1z22
#( f1#15
4f2!1
2f3!7
4f4!11
4f5)z2
1z3!(1
2f1#15
4f2!1
2f3!7
4f4!5
2f5)z
1z23
!32z22z3!6 f
7z22z4!(1
2g1#5
4g2) z
2z23#5
4g2z23z4#1
2z1z24
#6 f7z2z24!1
2z3z24#(g
1#5
2g2) z
1z2z3!5
2g2z1z3z4, (45)
where fi*0 (for i"1, 2, 3) due to physical conditions, and g
1and g
2are used to indicate
the system parameters. Note that g1
and g2
represent the linear parts of the damping,assumed to be non-negative.
We choose a critical point de"ned by
g1c"g
2c"0 (46)
and the parameter values
f1"10, f
2"2, f
3"5, f
4"!6, f
5"8, f
6"!14, f
7"30, (47)
at which the Jacobian of equation (45) has non-semisimple double imaginary eigenvalues:
j1"I, j
2"!I, j
3"I, j
4"!I. (48)
This 1 : 1 resonant case has been studied in reference [14], where the normal form wasobtained using PoincareH normal form theory. In the following, we list the two normal formsfor a comparison. The Maple program developed in reference [9] based on PoincareHnormal form theory can be applied to "nd the following normal form for system (45) up tothird order, given in polar co-ordinates:
rR1"r
2cos (t), r
1hQ1"r
1#r
2sin (t),
rR2"21 r2
1r2!5r3
1cos (t)#
21
2r31sin (t)!58 r
1r22cos (t)#63 r
1r22sin (t)
!21 r21r2cos (2t)#r2
1r2sin (2t),
r2hQ2"r
2!22r2
1r2#
21
2r31cos (t)#5 r3
1sin (t)
#63 r1r22cos (t)#58 r
1r22sin (t)#r2
1r2cos (2t)#
21
2r21r2sin (2t), (49)
where the phase di!erence t"h2!h
1. Note that these four di!erential equations are not
independent, and can be reduced to three independent equations by combining the second
NORMAL FORMS FOR DOUBLE HOPF BIFURCATION 627
and fourth equations to form an equation for tQ . Note that since normal forms are generallynot unique, equation (49) only represents one of the normal forms.
Similarly, we can execute the Maple program developed in this paper based on themethod of multiple scales to obtain the normal form:
rR1"r
2cos (t)!
21
2r31!
105
4r1r22#
83
4r21r2cos (t)!
315
8r21r2sin (t)
#
131
16r32cos (t)#
22347
32r32sin (t)#
21
2r1r22cos (2t)!31 r
1r22sin (2t),
r1hQ1"r
1#r
2sin (t)!9r3
1!12 r
1r22!
63
8r21r2cos (t)#
129
4r21r2sin (t)
!
22347
32r32cos (t)#
131
16r32sin (t)#31 r
1r22cos (2t)#
21
2r1r22sin (2t),
rR2"41 r2
1r2!
1971
4r32!5r3
1cos (t)#
21
2r31sin(t)
#
21
2r1r22cos (t)!
53
4r1r22sin (t)!
21
2r21r2cos (2t)#10 r2
1r2sin (2t),
r2hQ2"r
2!4r2
1r2!
63
4r32#
21
2r31cos (t)#5 r3
1sin (t)
#
189
8r1r22cos (t)#
71
4r1r22sin (t)#10r2
1r2cos (2t)#
21
2r21r2sin (2t). (50)
It is obvious from equations (49) and (50) that the normal form obtained using the multiplescales has more terms than that obtained using PoincareH normal form theory. However, thenormal forms obtained using the two di!erent methods for the non-resonance and allresonant cases (except for the 1 : 1 resonance) have been found to actually have the samenumber of terms up to a "xed order. The di!erence that appeared in the 1 : 1 resonance isdue to the fact that the approach of multiple scales chooses a "xed basis and thus the form isuniquely determined at each order, while in applying PoincareH normal form theory, theform is not uniquely determined and one may select a di!erent basis which may result indi!erent number of terms. However, we can show that normal form (50) is equivalent tonormal form (49). In other words, we can "nd a third order near identity non-lineartransform which transforms equation (50) into equation (49). The proof is given in AppendixA. Further, it can be shown that we may use additional undetermined coe$cients, theso-called relaxing parameter approach which has been applied to compute the simplestnormal forms [18], in the process of calculating the normal form to directly "nd the normalform for the 1:1 resonance given in the form of equation (49). This will be discussed in moredetail in a di!erent paper.
Since the bifurcation analysis results obtained from equation (49) have been veri"ed bya numerical approach performed based on the original system (45) [14], we will not repeatthe numerical simulation here to verify equation (50).
628 P. YU
4.3. EXAMPLE 3 FOR 2 : 3 RESONANCE
In order to further demonstrate the e$ciency of the Maple program for computing highorder normal form of higher-dimensional systems without applying center manifold theory,consider the following seven-dimensional system:
xR1"u
1x2#x2
1#x
1x3#2
3x3x4#x
4x5,
xR2"u
1x1#2x2
4#x2
5#1
2x1x2#x
2x4,
xR3"u
2x4#4x2
3#x2
5#x
1x2,
xR4"!u
2x3#2x2
2#x
3x4#x
3x5,
xR5"!x
5#x2
2#1
3x2x4,
xR6"!x
6#x
7#x2
2#5x
1x3,
xR7"!x
6!x
7#2
5x21. (51)
Executing the Maple program on a PC (PENTIUM III-700MMX 256K system) takesabout 27 min to obtain the normal form up to 10th order. The normal form up to sixthorder is given below
rR1"
29
84r31#
29
104r1r22#
12713
78336r21r22cos (t)#
24143
19584r21r22sin (t)
#
23630620163
22368648960r51#
138177606761
681135436800r31r22#
177068030053
49695085440r1r42
!
3174158149507271
10440611639377920r41r22cos(t)#
2404182693451369
1305076454922240r41r22sin (t)
#
79586782468307866859
55648460037884313600r41r21cos (t)!
8438641581728237917
55648460037884313600r21r42sin (t),
r1hQ1"2r
1!
313
1344r31!
263
2184r1r22!
24143
19584r21r22cos(t)#
12713
78336r21r22sin(t)
!
4053892009
58432389120r51#
7248108167809
2043406310400r31r22#
12602271904219
12986982328320r1r42
!
373702778230261
522030581968896r41r22
cos (t)#23211401849611067
10440611639377920r41r22sin (t)
#
8438641581728237917
55648460037884313600r21r42cos (t)#
79586782468307866859
55648460037884313600r21r42sin (t),
NORMAL FORMS FOR DOUBLE HOPF BIFURCATION 629
rR2"
41
42r21r2#
1258879
1268064r31r2cos (t)#
868757
2536128r31r2sin (t)
!
2000235383
1091563200r41r2#
2427640143403
1260100558080r21r32!
39909823
49230720r52
#
629839075170792271
278996344363376640r51r2
cos (t)!3982767058488934319
1673978066180259840r51r2sin (t)
#
1000025370085244051
176767944831878400r31r32cos (t)!
221984821019707783
282828711731005440r31r32sin (t),
r2hQ2"3r
2!
5
84r21r2!
5
8r32#
868757
2536128r31r2cos (t)!
1258879
1268064r31r2sin (t)
#
19758721
30321200r41r2!
23095337868209
6930553069440r21r32#
8592833
16410240r52
!
3982767058488934319
1673978066180259840r51r2cos (t)!
629839075170792271
278996344363376640r51r2sin (t)
!
29817963307132463
68743089656841600r31r32cos (t)!
2692641827024565277
1583840785693630464r31r32sin (t), (52)
where t"2h2!3h
1.
5. CONCLUSIONS
A perturbation technique and Maple computer programs have been developed forcomputing the explicit normal forms of resonant double Hopf bifurcations. For a givenarbitrary n-dimensional system, this approach does not require the application of centermanifold theory, which has been included in the perturbation technique so that the normalform on the center manifold and the associated non-linear transformations can be obtainedsimultaneously. The formulas are given in an explicit iterative procedure, and thus areimplemented using computer algebra. Maple programs have been developed forautomating the computations. Examples are presented to verify the method and to showthat the method is computationally e$cient, which is particularly useful forhigh-dimensional systems and higher order normal forms.
ACKNOWLEDGMENT
This work was supported by the Natural Sciences and Engineering Research Council of Canada.
REFERENCES
1. J. GUCKENHEIMER and P. HOLMES 1983 Nonlinear Oscillations, Dynamical Systems, andBifurcations of <ector Fields. New York: Springer-Verlag.
2. A. H. NAYFEH 1993 Methods of Normal Forms. New York: Springer-Verlag.
630 P. YU
3. S. N. CHOW, C. LI and D. WANG 1994 Normal Forms and Bifurcation of Planar <ector Fields.Cambridge: Cambridge University Press.
4. J. CARR 1981 Applications of Center Manifold ¹heory. New York: Springer-Verlag.5. R. H. RAND and W. L. KEITH 1986 in Applications of Computer Algebra (R. Pavelle, editor)
309}328. Boston: Kluwer Academic Press. Normal forms and center manifold calculations onMACSYMA.
6. S. N. CHOW, B. DRACHMAN and D. WANG 1991 Journal of Computation and Applied Mathematics29, 129}143. Computation of normal forms.
7. P. YU 1988 Journal of Sound and <ibration 211, 19}38. Computation of normal forms viaa perturbation technique.
8. Q. BI and P. YU 1998 International Journal of Bifurcation and Chaos 8, 2279}2319. Computationof normal forms of di!erential equations associated with non-semi-simple zero eigenvalues.
9. Q. BI and P. YU 1999 Mathematical and Computer Modelling 29, 49}70. Symbolic softwaredevelopment for computing the normal forms of double Hopf bifurcation.
10. P. YU, W. ZHANG and Q. BI 2001 International Journal of Non-¸inear Mechanics 36, 597}627.Vibration analysis on a thin plate with the aid of computation of normal forms.
11. W. ZHANG and K. HUSEYIN 2000 Applied Mathematical Modelling 24, 279}295. On the relationbetween the methods of averaging and normal forms.
12. S. ZHU and P. YU 1999 Proceedings of the 17th CANCAM, Hamilton, Canada, 69}70, May30}June 3. Computing normal forms for vibration analysis of nonlinear oscillators via multipletime scales.
13. P. YU Nonlinear Dynamics (in press). Analysis on double Hopf bifurcation using computer algebrawith the aid of multiple scales.
14. P. YU and Q. BI 1998 Journal of Sound and <ibration 217, 691}736. Analysis of non-lineardynamics and bifurcations of a double pendulum.
15. K. HUSEYIN 1986 Multiple-Parameter Stability ¹heory and its Applications. Oxford: OxfordUniversity Press.
16. P. YU and K. HUSEYIN 1993 Quarterly Applied Mathematics 51, 91}100. On phase-locked motionsassociated with strong resonance.
17. V. MANDADI and K. HUSEYIN 1980 International Journal of Non-linear Mechanics 15, 159}172.18. P. YU 2000 International Conference on Di+erential Equations (Berlin 1999), Vol. 2, 832}837.
Singapore: World Scienti"c. A method for computing center manifold and normal forms.
APPENDIX A: THE PROOF FOR EXAMPLE 2
To prove that normal form (50) obtained using the multiple scales is equivalent to normalform (49) obtained using PoincareH normal form theory, we "rst transform the two normalforms, which are given in polar co-ordinates, into Cartesian co-ordinates. To achieve this,applying the transformation
x1"r
1cos (h
1), x
2"r
1sin (h
1), x
3"r
2cos (h
2), x
4"r
2sin (h
2), (A.1)
to equation (50) yields
xR1"x
2#x
3!
21
2x31!9x3
2#
131
16x33!
22347
32x34!9x2
1x2#
83
4x21x3#
315
8x21x4
!
21
2x1x22#
129
4x22x3!
63
8x22x4!
63
4x1x23!43x
2x23!
22347
32x23x4!
147
4x1x24
#19x2x24#
131
16x3x24!
189
4x1x2x3!
23
2x1x2x4#62x
1x3x4#21x
2x3x4,
NORMAL FORMS FOR DOUBLE HOPF BIFURCATION 631
xR2"!x
1#x
4#9x3
1!
21
2x32#
22347
32x33#
131
16x34!
21
2x21x2#
63
8x21x3#
129
4x21x4
#9x1x22!
315
8x22x3#
83
4x22x4!19 x
1x23!
147
4x2x23#
131
16x23x4#43x
1x24
!
63
4x2x24#
22347
32x3x24!
23
2x1x2x3#
189
4x1x2x4#21x
1x3x4!62x
2x3x4,
xR3"x
4!5x3
1#
21
2x32!
1971
4x33!
63
4x34#
21
2x21x2#
63
2x21x3!14x2
1x4
!5x1x22#
105
2x22x3#6x2
2x4!
53
4x1x23#
735
8x2x23!
22347
32x23x4!
71
4x1x24
#
189
8x2x24!
1971
4x3x24#20x
1x2x3!21x
1x2x4!
273
4x1x3x4#
9
2x2x3x4,
xR4"!x
3!
21
2x31!5x3
2#
63
4x33!
1971
4x34!5x2
1x2!6x2
1x3#
105
2x21x4
!
21
2x1x22#14x2
2x3#
63
2x22x4!
189
8x1x23!
71
4x2x23!
1971
4x23x4!
735
8x1x24
!
53
4x4x24#
63
4x3x24!21x
1x2x3!20x
1x2x4#
9
2x1x3x4#
273
4x2x3x4. (A.2)
A similar transformation can be used for equation (49) to obtain
y51"y
2#y
3, yR
2"!y
1#y
4,
yR3"y
4!5y3
1#
21
2y32#
21
2y21y2!23y2
1y4!5y
1y22#42y2
2y3!21y2
2y4!58y
1y23
#63y2y23!58y
1y24#63y
2y24#2y
1y2y3!42y
1y2y4,
y54"!y
3!
21
2y31!5y3
2!5y2
1y2#21y2
1y3#42y2
1y4!
21
2y1y22#23y2
2y3
!63y1y23!58y
2y23!63y
1y24!58y
2y24!42y
1y2y3!2y
1y2y4. (A.3)
Then, with the aid of Maple, we can "nd the following near identity non-lineartransformation, given by
x1"y
1#
115
8y31#
147
16y32#
147
16y21y2!
1017
4y21y3!
483
8y21y4#
115
8y1y22#
1101
4y22y3!
13
8y22y4
#
131
16y1y23#
22347
32y2y23#
131
16y1y24#
22347
32y2y24#
235
4y1y2y3!
1059
2y1y2y4,
632 P. YU
x2"y
2!
147
16y31#
115
8y32#
115
8y21y2!
13
8y21y3!
1101
4y21y4!
147
16y1y22#
483
8y22y3!
1017
4y22y4
!
22347
32y1y23#
131
16y2y23!
22347
32y1y24#
131
16y2y24!
1059
2y1y2y3!
235
4y1y2y4,
x3"y
3#
21
2y31!9y3
2!9y2
1y2#
179
8y21y3#
777
16y21y4#
21
2y1y22!
143
8y22y3#
315
16y22y4
!
1971
4y1y23#
63
4y2y23!
1971
4y1y24#
63
4y2y24!
231
8y1y2y3#
161
4y1y2y4,
x4"y
4#9y3
1#
21
2y32#
21
2y21y2!
315
16y21y3!
143
8y21y4#9y
1y22!
777
16y22y3#
179
8y22y4
!
63
4y1y23!
1971
4y2y23!
63
4y1y24!
1971
4y2y24#
161
4y1y2y3#
231
8y1y2y4, (A.4)
to reduce equations (A.2) to equations (A.3). This proves that equations (A.2) and equations(A.3) (i.e., equations (49) and equations (50)) are indeed equivalent.