An Algorithm for Higher Order Hopf Normal Forms

Shock and Vibration, Sep 2018

Normal form theory is important for studying the qualitative behavior of nonlinear oscillators. In some cases, higher order normal forms are required to understand the dynamic behavior near an equilibrium or a periodic orbit. However, the computation of high-order normal forms is usually quite complicated. This article provides an explicit formula for the normalization of nonlinear differential equations. The higher order normal form is given explicitly. Illustrative examples include a cubic system, a quadratic system and a Duffing–Van der Pol system. We use exact arithmetic and find that the undamped Duffing equation can be represented by an exact polynomial differential amplitude equation in a finite number of terms.

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An Algorithm for Higher Order Hopf Normal Forms

Shock and Vibration, Vol. An Algorithm for Higher Order Hopf Normal Forms 0 Department of Civil and Structural Engineering University of Hong Kong Hong Kong - INTRODUCTION The invariant manifold of a nonlinear oscillator near an equilibrium or a periodic orbit is deter­ mined by the structure of its vector field. Two often-used mathematical tools to simplify the original system are center manifold and normal forms. The normal form theory is a technique of transforming the original nonlinear differential equation to a simpler standard form by appropri­ ate changes of coordinates, so that the essential features of the manifold become more evident. Basic references on normal forms and their appli­ cations may be found in Poincare (1889) , Birkhoff (1927) , Arnold (1983) , Chow and Hole (1982) , Guckenheimer and Holmes (1983) , Iooss and Joseph (1980), Sethna and Sell (1978) , Van der Beek (1989 ), Vakakis and Rand (1992) , and Leung and Zhang (1994) . In this article a compu­ tational approach based on the classical normal form theory of Poincare and Birkhoff is intro­ duced. The relationship of the coefficients between the original equations and the normal form equations is explicitly constructed. The tech­ nique presented in our approach follows the idea of Takens (1973) . A linear operator and its adja­ cent operator are defined with an inner product on the space of homogeneous polynomials. The resultant normal form keeps only the resonant terms, which cannot be eliminated by a nonlinear polynomial changes of variables, in the kernel of the adjacent linear operator. The normal form simplifies the original systems so that the dy­ namic stability and bifurcation can be studied in a standard manner and the classification of mani­ fold in the neighborhood of an equilibrium or a periodic orbit can be achieved with relatively lit­ tle efforts. This article is a further development of previous work (Leung and Zhang, 1994) . The higher order normal forms of several typical non­ linear oscillators: cubic system, quadratic sys­ tem, and Duffing-Van der Pol system are pro­ vided and the steady-state solutions of the method are compared with existing results. TRANSFORMATION TO NORMAL FORM Substituting Eq. (7) into Eq. (6) gives Consider the nonlinear ordinary differential equations i = (/ + Dyhk(Y))Y = J(y + hk(y)) + ... + Fk(y + hk(y)) + O(lylk+I), (8) wherefis an n-vector function of u, differentia­ ble up to order r. Suppose Eq. (1) has a fixed point at u = Uo. We first perform a few linear transformations to simplify eq. (1). By the vari­ able change v = u - uo, we eliminate the constant terms and shift the fixed point to the origin under which Eq. (1) becomes v = f(v + uo) == H(v), where H(v) is at least linear in v. We next split the linear part of the ordinary differential equa­ tion and write (2) as follows v = D"H(O)v + H(v), where H(v) == H(v) - D"H(O)v and, H(v) = O(lv F) is at least quadratic in v and D" denotes differentiation with respect to v. We further transform D"H(O) into Jordan canonical form by the canonical matrix T, i.e., v = Tx, and obtain i = lID"H(O)Tx + T-IH(Tx). which can be written alternatively as i = Jx + F(x). where J is the Jordan Canonical form of D"H(O) and F(x) is the nonlinear part of the equation. In Poincare's normal form theory, the nonlinear function F(x) is expanded by a series of homoge­ neous polynomials, i = Jx + F 2(x) + F3(X) + ... + F,(x) (6) + O(lxl,+I) where Fk E Hko which is the set of homogeneous polynomials or order k. To transform Eq. (6) into its normal form, Poincare introduces a nearly identity nonlinear coordinate transformation of the form (2) (3) (4) (5) where / denotes the n x n identity matrix and the term (/ + Dyhk(y)) is invertible for sufficient small Y so that Substituting Eq. (9) into Eq. (8) and applying similar nearly identity nonlinear transformations up to the kth order, we obtain Y = Jy + F 2(y) + ... + Fk-I(y) + Fk(y) + (Jhk(y) - Dyhk(y)Jy) + O(lylk+l) = Jy + F 2(y) + ... + Fk-I(y) + Fk(y) + adJhk(y) + O(lylk+l) where the overbar is used to represent the origi­ nal polynomial matrices and adJhk(y) denotes an adjacent operator equivalent to the function of the Lie Bracket, To simplify the terms of order k as much as pos­ sible, we choose a specific form for hk(y) such that (10) (12) If Fk(y) is the range of L J(Hk)' then all terms of order k can be eliminated completely from Eq. (10). Otherwise we must find a complementary space Gk to LJ(Hk) and let Hk = LAHk) E9 Gk so that only terms of order k which are in Gk remain in the resultant expression. It is interesting to note that simplifying terms at order k would not affect the coefficients of any lower terms. How­ ever, terms of order higher than k will be changed. Therefore, it is only necessary to keep track of the way that the higher order terms are modified by the successive coordinate transfor­ mations. This will be discussed in the next sec­ tion. (7) NORMAL FORMS OF OSCILLATING SYSTEMS where hk(y) is kth order in y, hk(y) E Hb 2 :S k:s r. We now specialize the normal form formulation in two-dimensions for the Hopf bifurcation below. Consider a system with a small perturbed parameter IL and an equilibrium point with eigen­ values ±iwo, wo > O. where i = f(x, IL), x E R2, IL E Rl (13) where we suppose, after shifting of origin and canonical transformation, Perform the Lie bracket operation to each basis element on H 2, thus, LJ (Y01) = [:0 -;0][Y01] - [2~1 ~][:o (15) L J (Y10Y2) = w0 (y~Y-1YYl2)' LJ (y0~) -_ Wo -;0][~:J = Wo [2~r2J and ( -Y1Y2) ,LJ ( 0) 2 2 = Wo Y2 Y 1 (-Yi) , 2Y1Y2 Furthermore, the Taylor expansion of Eq. (13) gives where A ==' Dxf(O, 0) + D2x,J(0, O)IL = [f31L (aIL + wo) -(aIL + wo)], f31L F 2(Y) = [F21(Y)] = [a2o a11 a02]{ Yl }. (16) Fn(Y) b20b110b2 Y1Y2 We would like to find a coordinate transforma­ tion (7) so that the nonlinear terms of order k + 1 in the new system of function Y vanish rather than of order k in the original system of function x. Inserting Y = (Yt. Y2Y E R2 and hk(y) = (hk1(y), hk2(y)V E Hk into (13), we have the Lie bracket, y~ Second-Order Normal Form If the smallest order of nonlinear terms appearing in (13) is two, we try to find a transformation h2 of the form x = Y + hz(y). Because the det[Aj] = 8wo> 0, which does not vanish for Wo > 0, we obtain the following null complementary space for the homogeneous equations Aj g = {O}, Therefore, we can eliminate the quadratic term completely from Eq. (10) by means of the sec­ ond-order coordinate transformation (19) and get the second-order normal form {AY + Ah2(y) + ~ ~o Dr;'~!(y) [h2(y )]m} M Y = ~ (-l)N[Dyh2(y)]N N=O M = Ay + ~ F~l) (y) n=2 where F(1)(y) = [ n F",{(y)] F,,2.J.(y) = [~ aU) Yh~] . ~ bU)Yh~ , (24) in which, the number in the superscript parenthe­ ses refers to the index of coordinate transforma­ tion. From Eq. (22) we know that the comple­ mentary space of second-order normal form, G2, is null so that we have the following equation F~l)(y) = F2(y) + Jh2(y) - Dyh2(Y) . J . Y = O. (25) Then, we find coefficients Cij, dij in Eq. (19) by solving the linear algebraic equation, Eq. (25), C20 c", CO2 d20 d" d02 -1 3wO We further perform the nearly identity transfor­ mation (7) to third order to obtain the more accu­ rate nonlinear description of Eq. (14) in the neighborhood of the original singular point. where y represents the new coordinate and x is its old one. (26) (27) (29) (30) (22) (23) where AJ = Wo 0 -1 3 0 0 1 0 0 H3 = span {(~), (Y~2), (Y~~), (~), (~i)' (yfY2)' (y~y~), (~~)}. Similar to Eq. (20), the matrix representation of LAH3) is given by LJ (H3 = H3 . A} Finally, through a proper coordinate transforma­ tion, we determine the third-order normal form of the original system, where ai and bi are to be determined. We now want to develop a systematic proce­ dure to evaluate the coefficients of normal forms and the normal transformations of third order. We see from Eq. (11) i = M L (-1)N[Dh3(Y )]N N=O M = Ay + F~I)(y) + L F~2)(y) where (31) (32) (33) (34) (35) (37) (38) (39) (40) We observe that the homogenous equations A 3Jg = {O}, g E R8 have two zero eigenvectors for the matrix A} corresponding to its two zero ei­ genvalues, eT = {(1, 0, 1,0,0, 1,0, IV (0, -1,0, -1, 1,0, 1, OV}. Thus, A3 has a complementary space spanned by the monomial basis (29), The resultant linear equation is A3· g = 'Y} where g = {C30, C2)' C\2, C03, d30 , d2), dlb d03Y 'Y} = {am - a), aW + b), alY - a), abY + b), b~~ - b), bW - a), blY - b), abY - alY n = 3, .. " M, i = 2 - j. We know from Eq. (24) that the third-order nor­ mal form is equal to its complement space so that F~2)(y) = F~I)(y) + J. h3(Y) - Dh3(Y) . J . Y = G3(y). (36) where T is the canonical matrix consisting of the eigenvectors of B2 + I. The second equation in (39) becomes in which or B= d= 6= -1 ° 2 0 ° -2 0 1 a= 0 3 0 0 d30 d21 d l2 d03 b~~ - bl bW - al bIY - bl b&Y - al c= 0 0 0 -3 a~~ - al aW + bl a\Y - al abY + bl C30 C21 CI2 C03 Dividing Eq. (38) into two parts, we obtain { WO(B2 + I) . c = B . ii. - 6 wO(B2 + I) . a= B . 6 + ii.' Note that there are two more unknowns existing in Eq. (39) than the number of equations. How­ ever, the coefficients of the normal form a), bl can be evaluated independently by performing an orthogonal linear transformation on one of the equations in Eq. (39), T= 0 -3 LHS = T-I (B2 + l) T . a= Wo RHS = 11 (B . b + a) = 0 -6 -Sal + alY + 3am + 3b&Y + bW By comparing the terms in the last two rows of Eq. (41), we obtain L Y = A Y + 2: W2i+I(Y) + O(jyj2L+3), (44) i=l { al = ~ (aW + 3abY + b~V + 3b&Y) . bl = ~1 (aW + 3a&Y - bW - 3b~U) Substituting Eq. (42) into (39) and solving Eq. (39), the third-order homogeneous polynomial (2S) is then determined by (42) where 1 SWo 1 4wo o o -3a\Y - b&Y + 3a~U + bW - 3abY + b\Y + 3aW + b~U alY - bbY + a~U - bW 3abY + 5b~U + aW - bIY alY + 5bbY + 3a~U + bW abY + b~U + aW + bIY (43) Higher Order Normal Form The higher order normal form in a rectangular coordinate system is derived accordingly and can be written as follows, A [(a:: wo) -(a~: wo)], W2i+ l (y) = (YT + y~)i [:: ~~i][::J Y = and L is a given degree of the normal form equa­ tion. The validity of such simplification is guar­ anteed by the implicit function theorem, as for each /-t near /-to, there will be an equilibrium p(/-t) near p(/-to) that varies smoothly with /-t. The nor­ mal form of Eq. (44) in polar coordinates can then be written as the form YI = r cos (), Y2 = r sin () L ;- = r(f3/-t + 2: air21) + hot. L {} = Wo + a/-t + 2: bir2i + hot i=l (45) A general formula for each ai, bi is not pres­ ently available but we can derive their expres­ sions up to any desired higher order using the algorithm mentioned above recursively, such that, { { a3 = 1~8 (35a~d + 5a~~ + 3a~~ + 5am + 5b~) + 3b¥~ + 5b~{ + 35b~J), b3 = ;;~ (5a~) + 3a~~ + 5a~~ + 35a~i - 35b~~ - 5bf{] - 3b~~ - 5b~~); a4 = _2156_ (63a~~ + 7aW + 3a~~ + 3a~~ + 7a\~ + 7b~~) + 3b~7] + 3b~-g + 7b~7 + 63bb'J), b4 = ;~ (7a~V + 3a~7] + 3a~-g + 7a&V + 63ab'J - 63b~~ - 7bW - 3b~~ - 3b~~ - 7bm); and for the eleventh order, a5 = 10~4 (231aJld + 21a?] + 7afJl + 5a~~ + 7aW, + 21ai1l + 21bfI) + 7b~J + 5b~{ + 7bf/ + 21b~J + 231bJ1)), b5 = 1~;4 (21afI) + 7a~J + 5a~{ + 7arti + 21a~J + 231aJY- 231bJlrl - 21b?] - 7bfJl - 5b~ - 7bCfJ - 21M1l); here the subscripts A, B refer to the indices 10 and 11, respectively. The complexity of higher order normal forms rapidly becomes apparent as pointed out by Leung and Zhang (1994) . Thus, the algorithm is required to be implemented with the symbolic manipulation in Mathematica (Wolfram, 1991) . In the case of m > 5, however, it may also cause overflow in the computers equipped with con­ ventional memory if we apply the nearly identity normal transformations directly in Eqs. (23) and (34). A general explicit formula representing the homogeneous polynomial terms is therefore very useful to reduce the size of the problem. If the nearly identical change of coordinate of order k (46) is applied then the new homogeneous polynomi­ als of order n can be obtained by n = k, F:(y) = Fn(Y) + J. hk(y) - Dhk(y) . J . y; (48) n > k, F:(y) = Fn(Y) + [DFn-k+1(y) . hk(y) - Dhk(y) . F n-k+l(Y)]n"'2k-l where F:(y) denotes the resulting homogeneous polynomials, km = min[kh k2], and kl = floor [~ =~]. k2 = floor [IJ The operator floor['] gives the greatest integer less than or eual to the varible in the bracket. The result confirms that only odd-order terms would appear in the normal form equation, correspond­ ing to the Poincare resonance (Guckenheimer and Holmes, 1983) . OSCILLATORS WITH ODD NONLINEARITY Here we consider the nonlinear differential equa­ tions with flu) to be an odd function of u. To illustrate this, two examples are presented. The first one is the well-known Duffing oscillator, which arises from various physical and engineer­ ing problems. The solution of this oscillator has been thoroughly studied so that the accuracy of the results can be compared with existing meth­ ods. In the second example, we work with an oscillator having a term of the fifth power. (50) (51) (52) Consider the Duffing oscillator with the initial conditions x + w5x = -8X3 x(O) = a, i(O) = 0, where the parameter 8 > 0, Wo is the linear fre­ quency. In classical perturbation methods, it is usually assumed that 8 is small. In the present case, however, 8 need not be small. With the transformation x = XI. i = -WOX2, we rewrite Eq. (50) as -WoO][XI] + [8 0 ]. X2 -Wo xi Comparing Eq. (52) with the standard third-order normal form {i} = {x} + F3(X) , we have b30 = (8Iwo) and all the other coefficients are zero. If we substitute these coefficients into Eq. (42), we obtain al = 0, b l = (38/8wo). The third-order nonlinear transformation of coordinate {x} = [J]{y} + h3(y) can be deter­ mined by Eq. (43) and the coefficients of h3(y) are C30 = C03 = C21 = d30 = dl2 = 0, 8 58 8 CI2 = -W8o2' d21 = -W8o2' d03 = -42· Wo Here we use y to denote the new coordinates and x the old ones. While under the fourth order non­ linear transformation, all the coefficients of the homogeneous polynomial are identically zero, C40 = C31 = C22 = C\3 = C04 = d40 = d31 = d22 = d\3 = d04 = O. According to the method mentioned in the last section, such computation can be repeated up the the desired orders. Finally, we found that all the coefficients of the even-order coordinate homo­ geneous polynomials h2n(Y) are equal to zero and all the even-order terms are already in their sim­ plest forms. Thus, only the odd-order homoge­ neous polynomials need to be handled in our computation. The coefficients of normal form a2 and b2 can be obtained by taking the change of variable -2182 {y} = {z} + hs(z); a2 = 0, b2 = 256w6. Then, the fifth-order nonlinear coordinate change hs(z) is given by CSO = Cos = C23 = C41 = dso = d l4 = d32 = 0, If further transformations beyond the ninth order were performed, it is interesting to note that all the following higher order coefficients of normal forms ai, hi (i = 5,6, 7 . . .) as well as the coordi­ nate transformation hk(y) (k = 10, 11, 12, . . .) vanish. Therefore, the normal form of our cubic nonlinear differential system has an ultimate de­ gree of nine. The dynamic system can be repre­ sented by an exact polynomial differential ampli­ tude equation in a finite number of terms. The asymptotic solution of this normal-form equation is l UI = r coso r = 0 To get the steady-state periodic solution in the original coordinate, we could trace back all the transformations, for example, {x} = {y} + h3(y) = {z} + h5(z) + h3(z + h5(z» = {w} + h7(W) + h5(W + h7(W» + h3({W} = {u} + h9(U) + h7({U} + h9(U» + h5({U} + h7(W) + h5(U + h7(W») + h9(U) + h7({U} + h9(U») + h9(U) + h7({U} + h9(U»». (55) + h3({U} + h9(U) + h7({U} + h9(U» + h5({U} The steady-state periodic solution of the Duffing equation is x= ( ea3 23e2a5 167e3a7 3431e3a9) a - 32w6 + 1024w3 - 16384wg + 524288w3 coso ( ea3 3e2a5 30ge3a7 7033e4a9 ) + 32w6 - 128wg + 32768wg - 1048576w3 cos 30 ( e2a5 31e3a7 191e4a9 ) + 1024w3 + 32768wg + 524288w6 cos 5(J. _ ( (J + (65~~a:W6) cos 9(J It is clear to see that the coefficients of the first three orders of e in Eq. (57) are exactly the same as those given in Eq. (55). Nevertheless, Eq. (55) is different from Eq. (57) in the higher order terms. The result derived by normal form theory gives a finite, asymptotic power series, but using the L-P method the solution was represented by an infinite one. Oscillator with Fifth-Power Nonlinearity Consider an oscillator with a fifth-power nonlin­ earity as the second example, i + w6X = -ex5 with initial conditions (51). Using the normal form method described previously, one obtains the following results: The terms of asymptotic normal form series ex­ pansion of Eq. (59) is also finite with all higher order normal forms equal to zero. Ql = 0, and the fifth-order nonlinear coordinate change is given by Then, taking seventh-order transformation of co­ ordinates, we obtain a2 = 0, b2 = 0, with h7(W) = {o}. After that, we directly take the ninth-order change of variable which yields a3 = 0, b3 = (-215e2/3072wfi), and The steady-state periodic solution of the oscilla­ tor with fifth power non-linearity is and OSCILLATORS WITH EVEN NONLINEARITY The normal form method can also be easily ex­ tended to analyze an oscillator with even nonlin­ earity. A quadratic nonlinearity equation is con­ sidered in the following calculation. Quadratic Nonlinear Oscillator Consider a system having quadratic nonlinearity, with initial conditions (51). According to the nor­ mal form method, one needs to do a second-or­ der homogeneous polynomial transformation of coordinates to simplify the second-order terms in the original equations. The coefficients of this change can be evaluated by Eq. (26), the results are, -e -2e C20 = -W3o2' CII = 0, C02 = -W3o2' d20 = 0, 2e d ll = -W3o2' d02 = 0. The subsequent third-order variable change gives (60) al = 0, + ( 98e320a49w6) cos 90. The fourth-order homogeneous polynomial of transformation cannot be overlooked in this case and finally its coefficients are found to be C31 = CI3 = d40 = d22 = 0, e3 8e3 C40 = --96' C22 = --96' C04 = Wo Wo Then normal-form coefficients a2, b2 and the fifth-order homogeneous polynomial of coordi­ nate transformation can be evaluated as a2 = 0, The coefficients of the normal form a3, b3 are Substituting the transformation into the original equation, we have the steady-state periodic solu­ tion of the oscillator with quadratic nonlinearity __ ea2 _ 13e3a4 31ge5a6 ( _ e2a3 2w2 24w3 + 1728wAo + a 48w~ 143e4a5 35005e6a7) + 20736w~ - 20736wb2 cosO [YI] [eA -1][YI] Y2 = 1 eA Y2 -e [YIY~ + Y~] 0 , (63) in which, the first-order near identity transforma­ tion is given by where we choose XI = X, X2 = -x. Note that the higher order terms in e are neglected during the computation. We take transformations up to or­ der five. The resultant normal form of Eq. (63) in polar coordinates is expressed as follows, YI = r cosO, Y2 = r sinO f = er(A - ~ r2) The coefficients of the intermeidate coordinate transformations are 3e C30 = C03 = 0, C21 = CI2 = 8' -e -3e -e d30 = d03 = 4 ' d21 = -8-' dl2 = 8 ' respectively. The asymptotic solution ofEq. (65) could also be obtained if we have traced back all the nearly identity coordinate transformations x = ea(-A + ~ a2) cos 0 + ..£. a 3cos 30 32 32 + ( -a + -ge 3) . 3. a sm 0 + -e a sm 30 32 32 (65) (66) Y. --+-+~'-I'--- Y, Sink Center Y, A. <0 where the constant a denotes the amplitude de­ fined by the initial condition. Further discussion of the normal-form amplitude equation of (65) reveals the Hopf bifurcation of the Duffing-van der Pol equation on a parametric plane r - A. As for A < °and a source at (0, 0) surrounded by a shown in Figure 1, there is a spiral sink at (0, 0) limit cycle for A > o. The limit cycle evolves continuously from the center at (0,0) for A = o. The Hopf bifurcation is of importance in situa­ tions where a flow-induced oscillator is subjected to flutters or self-exciting movements. At such circumstances, the orbits of the steady-state peri­ odic solutions stay on the surface of the para­ boloid rotated by A = (l/8) r2. CONCLUSION We have presented an arithmetic algorithm to compute the higher order normal forms. By ap­ plying the explicit formula proposed in this work, we can achieve, in a standard manner, the de­ sired higher order normal forms for nonlinear dif­ ferential polynomial equations. We found that the steady-state solution of the undamped Duff­ ing equation can be represented by a finite cosine Y. Sink y • r Limit Cycle Source series with varying phase in finite polynomial terms. To show the versatility of the algorithm, we illustrated an example in which the order of nonlinearity is not restricted to odd numbers. The application to limit cycle bifurcation is also demonstrated by the Duffing-van der Pol oscilla­ tor. The second author wishes to thank Dr. Q.C. Zhang for his helpful discussion on the topic of normal form. The research was supported by the Research Grant Coun­ cil of Hong Kong. Algorithm/or Higher Order Hop/ Normal Forms Journal of Engineering Volume 2014 Volume 2014 Sensors Hindawi Publishing Corporation ht p:/ www.hindawi.com Rotating Volume 2014 International Journal of Hindawi Publishing Corporation ht p:/ www.hindawi.com Active and Advances in Civil Engineering Hindawi Publishing Corporation ht p:/ www.hindawi.com Journal of Robotics Hindawi Publishing Corporation ht p:/ www.hindawi.com Advances ctronics Submit your manuscr ipts VLSI Design Hindawi Publishing Corporation ht p:/ www.hindawi.com Hindawi Publishing Corporation Hindawi Publishing Corporation Navigation and Observation Hindawi Publishing Corporation ht p:/ www.hindawi.com Modelling Sim ulation & Engineering International Journal of Engineering Electrical and Computer International Journal of Aerospace Engineering Arnold , V. I. , 1983 , Geometrical Methods in the Theory of Ordinary Differential Equations , SpringerVerlag, New York. Birkhoff , G. D. , 1927 , Dynamical Systems , Vol. 9 , AMS Collection Publications . Blevins , R. D. , 1977 , Flow-Induced Vibration , Van Nostrand Reinhold, New York. Chow , S. N. , and Hale , J. K. , 1982 , Method of Bifurcation Theory, Springer-Verlag, New York. Guckenheimer , J. , and Holmes , P. , 1983 , Nonlinear Oscillations , Dynamical Systems, and Bifurcations of Vector Fields , Springer-Verlag, New York. looss , G., and Joseph , D. D., 1980 , Elementary Bifurcation and Stability Theory , Springer-Verlag, New York. Leung , A. Y. T. , and Zhang , Q. C. , 1994 , "Normal Form Analysis of Hopf Bifurcation Exemplified by Duffing's Equation," Shock and Vibration , Vol. 1 , pp. 233 - 239 . Poincare , H. , 1889 , Les Methods Nouvelles de la Mecanique Celeste , Gauthier-Villars, Paris. Sethna , P. R. , and Sell , G. R. , 1978 , "Review of the Hopf Bifurcation and Its Applications," Journal of Applied Mechanics Vol. 45 , pp. 234 - 235 . Takens , F. , 1973 , "Normal Form for Certain Sigularities of Vector Fields," Annals of the Institute of Fourier , Vol. 23 , pp. 163 - 195 . Vakakis , A. F. , and Rand , R. H. , 1992 , "Normal Mode and Global Dynamics of a Two-Degree-ofFreedom Nonlinear System-I. Low Energies," International Journal Non-Linear Mechanics , Vol. 27 , pp. 861 - 874 . van der Beek , C. G. A. , 1989 , "Normal Form and Periodic Solutions in the Theory of Nonlinear Oscillation-Existence and Asymptotic Theory," International Journal Non-Linear Mechanics , Vol. 24 , pp. 263 - 279 . Wolfram , S. , 1991 , Mathematica: A System for Doing Mathematics by Computer , Addison-Wesley, Redwood City, CA. and Volume 2014


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A.Y.T. Leung, T. Ge. An Algorithm for Higher Order Hopf Normal Forms, Shock and Vibration, DOI: 10.3233/SAV-1995-2405