department civil and structural engineering normal form

A. Y. T. Leung Department of Civil and Structural

Engineering University of Hong Kong

Hong Kong

Zhang Qichang

Chen Yushu Department of Mechanics

Tianjin University China

Normal Form Analysis of Hopf Bifurcation Exemplified by Duffing's Equation

A method is proposed for calculating the normal form coefficients of the degenerate Hopf bifurcation system and the steady periodic solutions of a nonlinear vibration system. The results obtained by this method are the same as those obtained by the classical one. The present method is much simpler and can easily be implemented: that is, given the coefficients of the governing equations, the response is obtained directly by substitution. © 1994 John Wiley & Sons, Inc.

INTRODUCTION

The idea of the normal transformation came from Poincare (1889). It has been taken up by many authors (Birkhoff, 1966; Arnold, 1978; Moser, 1973; 100s and Joseph, 1980). It consists of carrying out a near identity change of variable allowing one to transform one system of nonlinear ordinary differential equations to a simpler one. The simplest forms are called normal forms. We can study the bifurcation behavior and the characteristics of the solution of the original equations more easily by normal form equations due to the reduced number of essential parameters. Mathematicians are interested in the kinds of normal form different differential equations can have. So many kinds of normal forms have been given at present, such as the Hopf, pitchfork, saddle-node bifurcation for the generate and degenerate cases. Engineers and scientists are more interested in how to get the normal form for a given system of equations, that is, what is the relationship of the coefficients between the origi-

Received July 23, 1993; accepted August 30, 1993.

Shock and Vibration, Vol. 1, No.3, pp. 233-240 (1994) © 1994 John Wiley & Sons, Inc.

nal equations and the normal form equations. To this end, we give a simple method to calculate the coefficients of the normal form of a Hopf bifurcation system for both the nondegenerate and the degenerate case. Furthermore, we use this method to get the steady periodic solution of nonlinear free vibrations. We explain the process by solving Duffing's equation.

NORMAL FORM THEORY

The vector normal form (or Poincare normal form, standard form) is a simplified analytic expression. Such a simplified expression is obtained by a series of successive near identity nonlinear transformations of variables. The qualitative geometric structure can be acquired for every bifurcation of flows through the analysis of dynamical character of normal forms.

Consider the nonlinear differential equations

x = f(X), f E CK(Rn), K large integer. (1)

CCC 1070-9622/94/030233-08

233

234 Leung, Qichang, and Yushu

We expand the vector field f(x) into the Taylor series

f(X) = AX + Fl(X) + ... + F(X) (2) + O(IXlr+I), X E Rn

where A = Dxf(O) can be changed to a diagonal Jordan block form by a linear transformation, Fk(X) E H~ is a homogeneous polynomial of order k with n variables. We may suppose, without loss of generality, that matrix A is in Jordan standard form. Then Eq. (1) can be written as:

x = AX + F2(X) + ... + F(X) + O(IXlr+I), X ERn.

(3)

We introduce a near identity nonlinear transformation of variables to simplify the nonlinear part of Eq. (3). We take

X = Y + Pk(y), Pk(Y) E H~, k 2': 2. (4)

If we substitute (4) into (3), the left side is

and the right side is

f(X) = :t [Y + pk(Y)] = A[Y + Pk(y)] (6)

+ ... + Fk[y + pk(y)] + o(IYlk+l)

where [I + Dypk(y)] is invertible for sufficient small Y. According to the binomial theorem for matrices, we get

[I + DyPk(Y)]-1 = I - Dypk(y) + [D ypk(Y)]2 + ....

From (5) and (6)

Y = AY + F2(Y) + ... + Fk-I(y) + Fk(y) (7) - ad~pk(Y) + 0(1 Ylr+I), Y E Rn

where ad~pk(y) is a linear operator, defined as: ad~: H~~ H~

We find that the term whose order is less than k is not changed, but the terms whose order equal or are greater than k are changed after the trans-

formation. If

Fk(Y) - ad~pk(y) = 0 (8)

the kth order term is the simplest form. Because ad~ is a linear operator, Eq. (8) is solvable if ad~ is invertible or Fk(y) is the range of ad~. Otherwise choose a complement Gk for ad~ (H~) in H~, so that

then Fk(Y) can be written as:

where rk(Y) E R(ad~) and gk(y) E Gk. Then we can find a pk( Y) such that ad~

pk(y) = rk(Y). Equation (7) becomes:

Y = AY + F2(Y) + ... + Fk-I(y) + gk(Y) + O(IYlk+I).

(9)

THEOREM 1. Let X = f(X) be a cr system of differential equations with f(O) = 0 and Df(O) = A. Choose a complement Gkfor ad~(H~) in H~, so that H~ = ad~(H~) + Gk. Then there is an analytic change of coordinates in a neighborhood of the origin that transforms the system

X = f(X) to Y = g(Y) = AY + g(2)(y) + ... + g(r)(y) + Rr

with g(k)(y) E Gk for 2 2': k 2': rand Rr =

0(1 Ylr).

Refer to Wang (1990) for the proof of this theorem. DEFINITION 1. Intercept system

Y = A Y + g2( Y) + . . . + gr( Y) (10)

is called an A-normal form up to order r.

HOPF BIFURCATION

Consider the system:

x=f(x,/L), xER2, /LER (11)

Suppose

Let

{~ = x, + ~XZ Z = x, - lXZ

and substitute these transformations into Eq. (11). We have:

[Z] [z] [g(Z' z) ] (12) z = Ao z + g(z, z, IL)

where

[WOi

g = o(\z\Z), Ao = 0

The eigenvalues corresponding to Ao are lq = woi and Az = -woi. According to the Poincare resonance condition: (m, A) - As = 0, where <., .) is the scalar product, we know that it holds if m, - mz = 1 and m, + mz = m, that is, m is an odd number, otherwise it fails. So there are only odd order terms in its normal form. The normal form of Eq. (12) in complex form is:

where h.o.t. denotes higher order terms. The real normal form in rectangular coordi

nates is:

if = AU + W3(U) + W5(U) + h.o.t. (14)

where

U = [::l Ws(U) = (uj + u~)Z [az -bz] U.

bz az

In polar coordinates, it is:

{r = (dlL + a,rz + azr4)r + h.o.t. . (15)

iJ = Wo + CIL + b, rZ + bzr4 + h.o. t.

According to the following theorem of Hopf bifurcation, we know that Eq. (12) is a Hopf bifurcation system.

Normal Form Analysis of Hopf Bifurcation 235

THEOREM 2. (Hopf, 1942). Suppose that the system x = f(x, IL), x ERn, IL E R has an equilibrium point (xo, ILo) at which the following properties are satisfied: (H1) Dxf(xo, ILo) has a simple pair of pure imaginary eigenvalues and no other eigenvalues with zero real parts.

Then (H1) implies that there is a smooth curve of equilibrium [X(IL) , IL] with X(ILo) = Xo. The eigenvalues A(IL) , ~(IL) ofDxf(xo, ILo) that are imaginary at IL = ILo vary smoothly with IL. If, moreover, (H2)

then there is a unique three-dimensional center manifold passing through (xo, ILo) in Rn x Rand there is a smooth system of coordinates (preserving the planes IL = const.) for which the TayLor expansion of degree 3 on the center manifoLd is given by Eq. (14). If a, =1= 0, there is a surface of periodic solutions in the center manifold that has quadratic tangency with the eigenspace of MILo), ~(ILO) agreeing to the second order with the paraboloid IL = -(aI/d) (X Z + yZ). If a, < 0, then these periodic soLutions are stable limit cycles, but if a, > 0, the periodic solutions are repelling.

Note 1. It is called the first kind of degenerate case if d = O. Note 2. It is called the second kind of degenerate case if a, = o. The stability of the system is determined by the coefficient of high order terms of its A-normal form.

Method of Calculating Coefficients of Normal Form

Consider the system [Eq. (2)]:

f(X) = AX + FZ(X) + ... + P(X) + o(\x\r+'), X ERn

where

[Fk(l)(X)]

Fk(X) - -Fk(Z)(X) k

2: bijx\x~ j=O

k = 2, ... ,5; i = k - j.


We have given the expressions of al and b l of Eq. (15) in Chen (1990) for the nondegenerate case:

al = (b2lWo + 3a30wO + 3b03Wo + a12WO - b02b ll + 2a02bo2 + alla02 + alla20 - b20bll - 2b20a20)/(8wo)

bl = (3b12wo - 9a03wO - 3a21wO + 9b30Wo - 4bfi2 + 5bo2all - lObo2 b2o - aTI + allb20 - 10b~o - lOafi2 + a02bll - 10ao2a2o - bll + 5blla20 - 4a~o)/ (24wo)

(16)

N ow we extend the method to calculate a2 and b2 for the second kind of degenerate case.

In order to simplify the second order terms, we take the transformation:

x = X + P2(X).

Note. The same variable value X is used for the purpose of computer implementation. The following second order transformation is to be determined,

2

2: dijxilx~ j~O

Substituting into Eq. (2), we get

i = 2 - j.

+ ~ ~o Dm~~(x) [p2(x)]m} (17)

= AX + Pr(X) + Pj(X) + Pj(X) + pi(X)

where

k = 2, ... ,5;

k

2: I .. b··x'lx!, IJ ~

j~O

i = k - j.

Note. The explicit forms of Dmp(x) are given in the Appendix.

By the normal form equation of Hopf bifurcation (14), we know that

PI = p2 + Ap2 - Dp2AX = O. (18)

This means that the second order normal form is required to be identically zero because the Hopf bifurcation has only odd order terms. We obtain P2(X) by the above equation with results given below:

d20 a20 - b ll + 2aQ2

d ll 2b20 - all - 2bo2

d02 2a20 + bll + a02

3wo -b20 - all - 2bo2 (19)

e20

ell 2a20 + bll - 2a02

CO2 -2b20 + all - b02

pj = p3 + Dp2p2 - Dp2 . PI

pi = p4 + 1. D2p2(P2)2 + Dp3p2 - Dp2 . Pi 2

(20)

In order to simplify the third order term in Eq. (17), we take the transformation, X = X + P3(X). Then we have

+ ~ ~o Dm~\(x) [p3(x)]m} (21)

= AX + P~(X) + pi(X) + pi(X)

where

3

2: eijX~X~ j~O

3

2: dijx~x~ j~O

k '" 2 . . L.J aijx'l x~

[P~(l)(X)] pk(X) = = 2 P~(2)(X)

j=O

k

2: b~x~x~ j~O

k = 3, 4, 5; i = k - j.

i = 3 - j

By the normal form equation of Hopf bifurcation (14), it is required that:

F~ = F1 + A . p3 - Dp3 . A . X = W3(X). (22)

We obtain P3(X) by the above equation with results given below (C30, C03 can be any constant):

C2l = (8 . Wo . C03 + bll - bA3 + 3 . alo - 3 . al~/(8 . wo)

Cl2 = (8 . Wo . C30 - blo + bb + 3 . all - 3 . aA3)/(8 . wo)

d30 = (-4 . Wo . C03 - bll - bA3 + alo + ab)1 (4 . wo)

d2l = (8 . Wo . C30 + 5 . blo - bb + all + 3 . aA3)/(8 . wo)

(23)

d l2 = (-8 . Wo . C03 + bll - 5 . bA3 + 3 . alo + ab)/(8 . wo)

d03 = (4 . Wo . C30 + blo + bb + all + a&3)1 (4 . wo)

F~ = F1 (24)

F~ = F1 + DF{p3 - Dp3 . F~. (25)

In order to simplify the fourth order term in Eq. (20), we take the transformation, X = X + P4(X). Then we have

x = ~o (-l)N[Dp4(X)]N. {AX + Ap3X

where

+ ~ ~o Dm~~(x) [p3(x)]m}

= AX + F~(X) + Pi(X) + Fj(X)

4

2: CijX~X~ j=O

(26)

i = 4 - j 4

2: dijx~x~ j=O

Fk(X) = = [F~(1)(X)] 3 Fr2)(X) k

" 3 .. L..J bijxix~ j=O

k = 4,5; i = k - j.


By the normal form equation of Hopf bifurcation (14), it is required that:

Pi = F~ + A . p4 - Dp4 . A . X = o. (27)

We obtain P4(X) by the above equation with results given below:

d40 = (-3 . b~1 - 2 . b13 + 3 . a~o + 2 . a~2 + 8 · a~)/(15 . wo)

d31 = (12 . b~o - 2 . bh - 8 . b~ - 3 . a~1 - 2 . a13)/(15 . wo)

d22 = (bjl - hi3 + 4 . a~o + a~2 + 4 . a~)/(5 . wo)

dl3 = (8 . b~o + 2 . b~2 - 12 . b~ - 2 . a~1 - 3 . a13)(1(15 . wo)

d04 = (2 . bjl + 3 . b13 + 8 . a~o + 2 . a~2 + 3 . a~)/(15 . wo)

C40 = (-3 . b~o - 2 . M2 - 8 . bij4 - 3 . a~1 - 2 · a13)/(15 . wo)

C31 = (3 . b~1 + 2 . b13 + 12 . a~o - 2 . a~2 - 8 . a~)/(15 . wo)

C22 = (-4 . b~o - b~2 - 4 . b~ + a~1 - a13)/(5 . wo)

CI3 = (2 . b~1 + 3 . b13 + 8 . a~o + 2 . a~2 - 12 . a~)/(15 . wo)

C04 = (-8 . b~o - 2 . b~2 - 3 . bij4 + 2 . a~1 + 3 · a13)/(15 . wo) (28)

(29)

In order to simplify the fifth order term in Eq. (23), we take the transformation: X = X + P5(X). Then we have

x = ~o (-l)N[Dp5(X)]N. {AX + Ap5X

+ ~ ~o Dm~~(x) [p5(x)]m} (30)

= AX + F~(X) + F~(X)

where

i = 5 - j 5

2: dijx~x~ j=O


k = 5; i = k - j.

By the normal form equation of the Hopfbifurcation (14), we know that:

We calculate a2, b2, and PS(X) by the above equation with results given below:

This is the coefficients of normal form of the second kind degenerate case. Cso, Cos can be any constant.

C41 = (48 . Wo • Cos + 5 . b~1 + b~3 - 7 . bbs + 25 . a~o - 7 . aj2 - 11 . a~4)/(48 . wo)

C32 = (32 . Wo • Cso - 3 . b~o + bj2 + b~4 + 7 . a~1 - a~3 - 5 . abs)/(16 . wo)

C23 = (32 . Wo • Cos + b~1 + b~3 - 3 . bbs + 5 . a~o + aj2 - 7 . a~4)/(16 . wo)

CI4 = (48 . Wo • Cso - 7 . b~o + bj2 + 5 . b~4 + 11 . a~1 + 7 . a~3 - 25 . abs)/ (48 . wo)

dso = (-12 . Wo • Cos - 2 . b~1 - b~3 - 2 . bbs + 2 . aj2 + 2 . a~4)/(12 . wo)

d41 = (16 . Wo • Cso + 11 . b~o - bj2 - hf4 + a~1 + a~3 + 5 . abs)/(16 . wo)

d32 = (-96 . Wo • Cos + 5 . b~1 - 11 . b~3 - 31 . bbs + 25 . a~o + 5 . aj2 + 13 . a~4)/ (48 . wo)

d23 = (96 . Wo • Cso + 31 . b~o + 11 . b~2 - 5 . b~4 + 13 . a~1 + 5 . a~3 + 25 . abs)/ (48 . wo)

dl4 = (-16 . Wo • Cos + b~1 + b~3 - 11 . bbs + 5 . a~o + a~2 + a14)/(16 . wo)

dos = (12 . Wo • Cso + 2 . b~o + bj2 + 2 . b~4 + 2 . a~1 + a~3 + 2 . a~s)/(12 . wo) (33)

In conclusion, if the coefficients of the governing Eq. (2) are given, the normal form (15) is calculated by Eqs. (16) and (32).

Nonlinear Free Vibration Problems

Consider the Duffing oscillator with no damping term for illustration:

d 2x 2 _ 3 dt2 + WI - CX • (34)

then X2 = WIXI - ~ x~ (35) WI

To solve the equations, we can either repeat the process step by step or calculate the normal form (15) directly by Eqs. (16) and (32) using the given coefficients. We proceed the former.

Substituting the above transformation into Eq. (31), we have:

or simply write it as,

{X} = {X} + F3(X). (37)

Comparing this equation with Eq. (17), we find that b30 = -(c1wl) and all the other coefficients are zero. So if we substitute these coefficients into Eq. (16), we get al = 0, b l = -(3c18wl ). Because there are no second order terms in this system, we directly take the third order transformation of coordinates:

{X} = [A]{Y} + P3(Y).

We can obtain P3(Y) by Eq. (23). Let C30 = Cm = 0, then

5c d21 = --8 2'

WI

(38)

(39)

Then we can obtain Fi by equation (24) and F~ by equation (25). For Fi = F1 = {O}, i.e., the fourth order terms are already in its simplest form, we directly take the change of variable:

{Y} = {U} + PS(U). (40)

We can obtain P5(U) by Eq. (33). Let C50 = C05 = 0, then

C32 = (21 * c2)/(256 * wi)

C23 = C41 = d50 = dl4 = d32 = 0

CI4 = (25 * c2)/(256 * wi)

d41 = (-39 * c2)/(256 * wi)

d23 = (-67 * c2)/(256 * wi)

d05 = -c2/(32 * wi).

(41)

The coefficients of normal form a2 and b2 can be obtained by Eq. (32).

{a2 = 0 21c2

b2 = - 256w~· (42)

The normal form of Eq. (36) in polar coordinate system is:

{~ : 0 _ 3c 2 _ 2lc2 4 (43) (J - WI 8wI r 256w~ r .

The first order asymptotic solution of this normal form equation is:

UI = r cos (J

dr = 0 dt (44)

The steady period solution of equation (44) is:

In order to get the steady periodic solution of Duffing equation (34), we should utilize the transformation equation (38) and (40).

{X} = {Y} + P3(y)

= {U} + P5(U) + P3[U + P5(U)] (46)

= {U} + P3(U) + P5(U).

The steady periodic solution of the Duffing equation is:


_ ( C 3 23c2 5) X - Xo + 32wI Xo + 1024wi Xo COS (J

( c 3 3c2 5) - 32wT Xo + 128w1 Xo COS 3(J

c2

+ 1024wt xij COS 5(J

(47)

( 3c 2 21c2 4) (J = wI - 8wI Xo - 256w~ Xo t.

In order to prove this new result, we have studied this equation by the Lindstedt-Poincare method and found that they are the same.

CONCLUSION

Applying near identity change of variable through matrix representation, a simple method for calculating the coefficients of normal form of nondegenerate and degenerate Hopf bifurcation systems has been introduced. We use this method to study the steady-state period solution of nonlinear free vibration systems. This method is simple and easy for computer implementation.

REFERENCES

Arnold, V. I., 1978, Mathematical Methods of Class ical Mechanics, Springer-Verlag, Berlin.

Birkhoff, G. D., 1966, Dynamical Systems, Vol. 9, AMS Collection Publications, 1927, reprinted.

Chen, Y., and Zhang, Q., 1990, "A New Method of Calculating the Asymptotic Solution of Nonlinear Vibration Systems," ACTA Mechanic Sinica, Vol. 22, pp. 413-419 (in Chinese).

Guckenheimer, J., and Holmes, P., 1983, Nonlinear Oscillation, Dynamical Systems and Bifurcation of Vector Fields, Springer-Verlag, New York.

Hassards, B. D., 1981, Theory and Application of Hopf Bifurcation, Society Lecture Notes series 41, London.

Hopf, X., 1942, Howland, R. A., and Richardson, D. L., 1986, "Com

puter Implementation of an Algorithm for the Quadratic Analytical Solution of Hamiltonian Systems," Journal of Computational Physics, Vol. 67, pp. 19-27.

looss, G., and Joseph, D. D., 1980, Elementary Bifurcation and Stability Theory, Springer-Verlag, Berlin.

Jezequel, L., and Lamarque, C. H., 1991, "Analysis of Nonlinear Dynamical Systems By the Normal form Theory," Journal of Sound and Vibration, Vol. 149, pp. 429-459.


Moser, J., 1973, Stable and Random Motion in Dynamical Systems, Hermann Weyl Lectures, Princeton, NJ.

Poincare, H., 1889, Les Methods Nouvelles de la Mecanique Celeste, Gauthier-Villars, Paris.

Wang, D., 1990, "An Introduction to the Normal Form Theory of Ordinary Differential Equations," Advances in Mathematics (China), Vol. 19, p. 38-71.

APPENDIX

The definitions of DmF(X). Suppose

l'F' iixi DF(X) =

iiF2 iixi

l"F' iiX2 D2F(X) = I

ii2F2 iiX2

I

l"F' iix 3 D3F(X) = I

ii3F2 iixj

'F] iiX2

iiF2 iiX2

ii2FI

"F'J iiXliix2 iix~

ii2F2 ii2F2 iiXliix2 iiX2 2

ii3FI ii3FI

~F'J iiXIiix2 iixliix~ iiX3

ii3;2 . ii3F2 ii3F2 iiXI iix2 iixliix~ iix~

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