Parameter-dependent normal forms for bifurcation equations of dynamical systems Angelo uongo Dipartento di Ingegneria delle Strutture delle Acque e del Terreno Università dell Aquila Monteluco Roio 67040 Aquila Italia. E-mail: angelo.luongo@univaq.it Keywords: Normal forms bifurcation equations dynamical systems SUMMARY. Normal form for bifurcation equations of finite-densional dynamical systems are studied. First the general theory for no-parameter systems is restated in a form amenable to further developments. Then the presence of bifurcation parameters in the equations is accounted for. The drawbacs affecting two classical methods usually employed in literature are commented. A new method based on a perturbation expansion which is free from these difficulties is proposed and illustrated by a wored example. 1 INTRODUCTION The main goal of the local bifurcation analysis from equilibrium points consists in classifying the long-te behavior of a dynamical system in the parameter space. The tas requires obtaining bifurcation equations capturing the asymptotic essential dynamics. The bifurcation equations are usually built-up via the Center Manifold Theory [1] that permits to reduce the dension of the original system to that of the critical (generalized) eigenspace of the associated linear system. These equations describe the asymptotic motion of the nonlinear system on an invariant manifold which is tangent to the critical subspace. Once the bifurcation equations have been obtained and in order to mae their analysis easier they are conveniently transformed according to the Normal Form Theory [1]. The normal form of a nonlinear ordinary differential equation is the splest form it can assume under a suitable change of coordinates. Of course the best occurrence would be to transform the nonlinear equation in a linear one. This is indeed possible whenever the eigenvalues of the Jacobian matrix are non-resonant i.e. none of them can be expressed as a linear combination with positive integers of the eigenvalues. owever this favorable circumstance never occurs in bifurcation analysis since the Jacobian matrix admits zero and/or purely aginary eigenvalue for which resonance always exists; therefore bifurcation equations cannot be linearized. In spite of this the equations can be strongly splified by removing all the nonresonant terms and leaving only the unmovable resonant terms. The procedure of elination of the non-resonant terms is often described in literature through successive changes of coordinates able to (partially) remove in turn quadratic cubic and higherorder terms [1-3]. A nicer illustration of the procedure is presented in [4] in the framewor of a perturbation method where a unique change of coordinates is introduced and linear perturbation equations are solved in chain. The previous treatments however are devoted to systems in which no parameters appear in the equation. In contrast the main interest of bifurcation analysis just relies on equations in which parameters do appear able to span the neighborhood of the bifurcation point. To account for parameters it is suggested in literature to consider them as
dummy state-variables constant in te silarly to that done in the Center Manifold Theory [13]. The procedure although sple in principle leads to cumbersome calculations since increases the dension of the dynamical system. An alternative algorithm is suggested in [] and illustrated with reference to a sple opf bifurcation. There the parameters as considered as fixed quantities included in the coefficients and therefore they do not increase the dension of the system. As a drawbac the procedure leads to linear equations which are quasi-singular close to bifurcation so that resonances become quasi-resonances. These latter if not properly tacled would entail the occurrence of small divisors both in the coordinates transformation and in the normal form. The problem can be easily overcome in the case in which the Jacobian matrix is diagonalizable namely in the example illustrated in [] but in contrast is not trivial in the case in which the Jacobian matrix contains a Jordan bloc. In this paper parameter-dependent dynamical systems are considered close to bifurcation points at which their Jacobian matrix admitting non-hyperbolic eigenvalues is either diagonalizable or non-diagonalizable. A perturbation algorithm is developed to derive parameterdependent normal forms without extending the state space but in contrast by eeping unaltered the dension of the original system. In Sect the bacground relevant to the Normal Form Theory concerning the parameter-independent case is supplied. In Sect 3 a new algorithm accounting for parameter-dependent system is detailed and an example is wored out in Sect 4. PARAMETER-INDEPENDENT NORMA FORMS et us first consider a dynamical system independent of parameters. Its equation of motion Taylor-expanded around the equilibrium point read: (1) x = Jx + f ( x) N N where f ( x) = O( x ) x R orc and J is the Jacobian matrix recast in Jordan form. Our goal is to transform Eq (1) into the normal form: () y = Jy + g( y) where g( y) = O( y ) is the splest possible. To this end we introduce a near-identity transformation of coordinates x y in which y differs from x only by higher order-quantities: x = y + h( y ) (3) with h( y) = O( y ) an unnown function. With Eqs () and (3) since x = [ I + h y ( y)] y where I the N N identity matrix and h ( y) the Jacobian matrix of h(y) Eq (1) modifies into: y h ( y) Jy Jh( y) = f ( y + h( y)) [ I + h ( y)] g( y ) (4) y This is a differential equation for the unnown h(y) for any properly chosen g(y). Since it cannot be solved exactly we will solve it by series expansions [4]. We assume: y
f ( y) f( y) f3( y) g( y) = g( y) + g3( y) + h( y) h ( y) h ( y) 3 (5) where f ( y) g ( y) h ( y) are homogeneous polynomials of degree in y. By substituting the series expansions (5) in (4) and equating separately to zero terms of the same degree we obtain the following chain of equations: h y ( y) Jy Jh( y) = f( y) g( y) h3 y ( y) Jy Jh3( y) = f3( y) + f y ( y) h( y) h y ( y) g( y) g3( y) (6) which are linear in h ( y ). They can be solved recursively to furnish h( y) g( y) at the order- h3( y) g3( y) at the order-3 and so on. The generic equation (6) can be rewritten as: h ( y) = fˆ ( y) g ( y) = 3 (7) in which is a linear differential operator whose action is h ( y) : = h y ( y) Jy Jh ( y) ; moreover f ˆ ( y) is a nown-term since f ˆ ( y) f ( y) and fˆ ( y ) ( > ) is completely determined by the lower-order solutions..1 Solving the generic perturbation equation The three homogeneous polynomials of degree appearing in Eq (7) are: M M M fˆ ( y ) = α p ( y ) g ( y ) = β p ( y ) h ( y ) = γ p ( y ) (8) m m m m m m m= 1 m= 1 m= 1 where αm βm γ m are constants and { p 1( y) p ( y) p ( y)} M is a basis of M linearly independent vector-valued monomial of degree in the N coordinates y j. By substituting Eqs (8) in Eq (7) and equating separately to zero the coefficients of the same monomials we obtain a linear system of equations of type: γ = α β (9) in which is a M M matrix; moreover α : = { αm} β : = { βm} and γ : = { γ m} are M -vectors. In Eq (9) coefficients α are nown whereas coefficients β and γ are unnown.
owever since our goal is to render g ( ) choosing y the splest possible we have some freedoms in β. Of course the best choice would be to tae β = 0 (entailing g ( y ) 0 ); however this operation is subordinated to the ran of matrix is non-singular ; (b) matrix is singular. If det 0 we can tae β = 0 and solve (9) for Consequently from Eqs (8) g ( y) 0 and h ( ) det = 0 = in order for Eq (9) can be solved its now-term of matrix =. Two cases must be analyzed: (a) 1 γ to obtain γ = α. y is univocally determined. In contrast if α β must belong to the range (solvability or compatibility condition). This requirement is fulfilled if the nown term is orthogonal to the ernel of the adjoint operator Since this space is spanned by vectors h where is the geometrical multiplicity of the zero-eigenvalue of (transposed and conjugate of ). v satisfying v = h 0 ( h = 1 ) compatibility requires: V β = V α (10) where V : = [ v 1 v v ] is a matrix collecting the vectors v h. Equations M M (10) entail that only (suitable) entries of vector β can be taen equal to zero the remaining having to satisfy compatibility. Therefore g y cannot be removed monomials in ( ) by the near-identity transformation. The previous analysis must of course be carried out at each order up-to the maxum order K considered in the analysis. At each of these orders the number M of independent vectors p ( ) m y increases this entailing large-densional matrix and vectors α β γ. It is liely to occur for example that matrix is non-singular but + 1 is singular. In this case we will be able to remove all nonlinearities of degree but just some of degree +1. As final result of the procedure the vector g( y) furnishes the normal form () of the original system (1) truncated at the order K. Moreover the associated vector h( y ) permits to come bac to the original variable x via the near-identity transformation (3).. Resonance conditions An portant question concerns the possibility to predict in advance on the ground of the sole nowledge of the spectral properties of the Jacobian matrix J if matrix is singular or not and in addition which terms in f ( ) x can be removed and which instead have to remain in the normal form. It is possible to show (see e.g. [1]) that if the Jacobian matrix J is diagonalizable then is also diagonal: = diag[ Λ ] (11)
where: and where { m m m } N 1 m + m + + m = N (1) Λ : = ( m λ λ ) i = 1 N j j i j= 1 are all the combinations of positive integers such that 1 N ) 1. Equation (11) states that is nonsingular (and therefore all nonlinearities of degree are removable) if all Λ 0. In contrast is singular (and therefore some nonlinearities of degree are not removable) if one or more nearly zero). In this latter case the monomial associated with Λ Λ are zero (or (i.e. the m-th independent monomial in the i-th equation) survives in the normal form. The condition Λ = 0 is also referred to as resonance condition (of order ) since it entails that one of the eigenvalue e.g. λ i is as a linear combination of all the eigenvalues. It is portant to stress that when the eigenvalues at an equilibrium point are on the aginary axis resonance always occur due to the fact that the aginary eigenvalues are conjugate in pairs. This confirms the well nown results that the flow around a non-hyperbolic equilibrium point cannot be linearized. Finally we mention the fact that when the Jacobian matrix is not diagonalizable the matrix appearing in the equation (9) is no more diagonal. Nevertheless it can be proved that its eigenvalues are still given by (1) so that singularity of resonance conditions 0 their a priori selection is not trivial as in the diagonal case. is due to the occurrence of the Λ =. Therefore only resonant terms appear in the normal form but.3 Normalization conditions Equations (9) in the case of interest in which As we have already observed due to the is singular are now analyzed in greater detail. compatibility condition (10) M components of vector β can be chosen freely best if equal to zero. Therefore normal form is not unique but it depends on the specific components that are zeroed. In order to state a criterion that determines β uniquely it needs to introduce a normalization condition that establishes the space of belonging of the vector namely: β = Bb (13) 1 For example if N= and =3 there are M3=8 independent monomials in Eq (8); therefore from Eqs (11) (1): = diag[3 λ λ λ + λ λ λ + λ λ 3 λ λ ; 3 λ λ λ + λ λ λ + λ λ 3 λ λ ] 3 1 1 1 1 1 1 1 1 1 1 = diag[ λ λ + λ λ 3 λ λ ; 3 λ λ λ λ + λ λ ] 1 1 1 1 1 1
where M B is a matrix listing columnwise a basis for b : = { bh} ( h = 1 ) are the components of the compatibility conditions (Eq (10)) become: and β in this basis. With this normalization V B b = V α (14) which is a system of requiring B Once equations in the unnowns b. A common choice consists in β has zero-projection onto the range ( ) i.e. β ( ) V ; however this choice not always give the splest normal form. β has been determined Eqs (9) give solutions for γ namely: this entailing γ = γˆ + U c c (15) where U : = [ u 1 u u ] is a matrix collecting the right eigenvectors u h = h satisfying 0 M u. The arbitrary constants normalization condition of the type: c can be chosen by enforcing a second C γ = 0 (16) which requires γ is orthogonal to the space spanned by the columns of the C. Again one can tae C V from which c ˆ = V γ since V U = I ; if M matrix is nondiagonalizable the same result holds if the proper left eigenvectors in V are substituted by the higher-order generalized eigenvectors. It should be noticed that different choices for C entail different coordinate transformations h ( y ) all leading to the same normal form g ( y ) ; however these different choices have repercussions on higher-order terms. 3 PARAMETER-DEPENDENT NORMA FORMS The previous treatment is devoted to systems at bifurcation in which parameters (if any) assume fixed values. In contrast the main interest of bifurcation analysis is to study systems close to a bifurcation in which parameters are liely to vary. There exist two approaches in literature to include parameters in Normal Forms. They are shortly commented ahead; then an alternative method is illustrated. 3.1 Usual approaches The commonest procedure followed in literature consists in considering the parameters as dummy state-variables constant in te (silarly to that done in the Center Manifold Theory [13]). Accordingly the state-space is extended to parameters µ and trivial equations are
appended to the bifurcation equations that therefore read: x = Jx + f ( xµ ) { µ = 0 (17) where J is the Jacobian matrix at the bifurcation point ( xµ ) = ( 0 0 ) and bilinear (and higher- J µx have been shifted into f ( xµ ). Accordingly the normal form and the order) terms µ near-identity transformation must be taen respectively as: (18) y = Jy + g( yµ ) x = y + h( yµ ) The procedure although sple in principle leads to cumbersome calculations due to the larger number of independent monomials in the ( yµ ) -space. An alternative procedure is suggested in []. According to this method the parameters are incorporated in the coefficients and therefore they do not increase the dension of the system. Consequently Eqs (1)-(3) change into: and Eq (9) into: x = Jµx ( ) + f ( xµ ; ) y = Jµy ( ) + g( yµ ; ) x = y + h( yµ ; ) (19) ( µ ) γ ( µ ) =α ( µ ) -β ( µ ) (0) The dependence on µ of the eigenvalues of J of course entails a silar dependence of the eigenvalues of ( µ ). Since due to the resonance ( 0) is singular for some and since µ is small ( µ ) is nearly-singular i.e. a quasi-resonance occurs. If this occurrence were not properly tacled small denominators would appear in the normal form leading to series not uniformly valid. The drawbac can be overcame in the diagonal case by considering the nearly- µ as they were exactly zero by exploiting the arbitrariness of zero eigenvalues ( ) the quantities Λ µ of ( ) β s. owever the problem is not so trivial if the Jacobian matrix is not diagonalizable and a solution of the problem available in the literature is not nown to the author. 3. A perturbation algorithm A perturbation algorithm is proposed to derive parameter-dependent normal forms. In order to eep unaltered the dension of the original system the state space is not extended. Equation (0) is again taen into account but instead to solve it as is the parameters appearing in the equations are rescaled as µ εµ where ε > 0 is a small booeeping parameter. Consequently after expanding in series all quantities it follows:
where: α α 0 α 1 α = 0 + ε 1 + ε + β = β 0 + ε β 1 + ε β + (1) γ γ 0 γ 1 γ d 1 d 0 µ µ 0 : = ( ) 1 : = : = dµ dµ µ = 0 µ = 0 dα 1 d α α α 0 α µ α µ 0 : = ( ) 1 : = : = dµ dµ µ = 0 µ = 0 () and silar. By substituting Eqs (1) and () in Eq (0) and equating separately to zero terms with the same power of ε the following chain of perturbation equations is obtained: 0 ε : 0 γ 0 =α 0 -β 0 1 ε : γ =α -β γ n ε : 0 1 1 1 1 0 γ =αˆ -β 0 n n n (3) ˆ n where α is a vector nown from the previous steps. The generating equation (3 1 ) coincides with that of the no-parameter case; therefore quasi-resonances are ruled out at any orders since 0 is (exactly) singular. By solving in sequence Eqs (3) and enforcing compatibility and normalization at each step all the terms of the series (1 ) for β and desired order. An example of the procedure is wored out in the next section. γ are evaluated up-to the 4 AN EXAMPE: TE DOUBE-ZERO BIFURCATION As an example let us consider the following two-densional dynamical system governing the motion reduced to the Center Manifold of a larger system around a double-zero (Taens- Bogdanov) bifurcation: x 1 0 1 x1 α1x1 + αx1x + α3x = x ν µ + x α4x1 + α5x1x + α6x (4) µ are bifurcation parameters vanishing at the bifurcation point. To mae the where = ( µ ν ) example splest as possible the coefficients α m are assumed to be independent of µ.
According to Eqs (18) we assume a normal form and a near-identity transformation as follows: y 1 0 1 y1 β1( µ ) y1 + β( µ ) y1 y + β3( µ ) y = y ν µ + + y β4( ) y1 β5( ) y1 y β6( ) y µ + µ + µ x1 y1 γ 1( µ ) y1 + γ ( µ ) y1 y + γ 3( µ ) y = + + x y γ 4( ) y1 + γ 5( ) y1 y + γ 6( ) y µ µ µ (5) By substituting Eqs (5) in Eq (6 ) and zeroing separately the coefficients of the three independent monomials in the two equations we obtain six algebraic equations of the type (0) (with = ): 0 ν 0 1 0 0 γ ( µ ) α β ( µ ) 1 1 1 µ ν 0 1 0 γ ( ) α β( ) µ µ 0 1 µ 0 0 1 γ 3( µ ) α3 β3( µ ) = ν 0 0 µ ν 0 γ 4( µ ) α4 β4( µ ) 0 ν 0 0 ν γ ( µ ) α β ( µ ) 5 5 5 0 0 ν 0 1 µ γ 6( ) µ α6 β6( µ ) (6) The matrix ( ) µ appearing in Eq (6) admits the eigenvalues Λ = ( µ ± 3 µ + 4 ν ) / Λ Λ = ( µ ± µ + 4 ν ) / which all tend to zero when 1 34 56 µ 0 ; thus ( ) µ cannot be inverted if small denominators must be avoided. Therefore according to the proposed method we resort to the perturbation equations (3). The generation equation (3 1 ) is obtained by putting µ = 0 in Eq (6). Since Ran[ ] = 4 ( ) = span{ u u } where 0 0 has a two-densional ernel 0 1 u 1 = (010001) T and = (001000) T u ; moreover ( T 0) = span{ v1 v} where v 1 = (00010) T and v = (000100) T nown term must satisfy two compatibility conditions (Eqs (10)): 1 10 5 50 4 40. In order Eq (3 1 ) can be solved the ( α β ) + ( α β ) = 0 α β = 0 (7) Since β0 β30 β60 are not involved in compatibility they are taen zero; moreover β = 0 β = α β = α + α is chosen to satisfy Eqs (7); therefore 10 40 4 50 5 1 β span[ e e ] 4 5 is taen as normalization condition. Solution to Eq (3 1) furnishes T γ = (( α + α ) / α + c c α α c ) c c ; we chose c1 = c = 0 thus 0 6 3 1 1 6 1 1
γ span[ e e e e ]. adopting the normalization 1 4 5 Passing to the ε -order perturbation equation (3 ) we first evaluate the right hand member. α ( 0 ( ) / 0) T = 0 γ = α ν α µ α µ + α α ν α ν and according Since 1 1 0 3 3 1 6 3 to normalization adopted β 1 = (000 β41 β510) T solvability furnishes β = α µ + ( α α ) ν / β = α ν. Finally the normalized solution to Eq (3 ) 41 1 6 51 3 γ = ( α µ / 00 α ν 00) T. reads 1 3 3 By substituting the results obtained in Eqs (5) to within an error of have the following normal form and coordinate transformation: where: 3 O( y µ y ) we y 1 0 1 y1 0 x1 y1 γ 1y1 + γ y1 y = y ν µ + = + y β x 4 y1 + β5 y1 y y γ 4 y1 + γ 5 y1 y (8) β = α α µ + ( α α ) ν / β = [ α + α α ν ] 4 4 1 6 5 1 5 3 γ = ( α + α α µ ) / γ = α γ = α + α ν γ = α 1 6 3 3 4 1 3 5 6 (9) It should be noticed that parameters only alter the coefficients not the structure of the two expressions as a consequence of having used the same normalizations at all orders. CONCUSIONS By following a perturbation approach we proposed a new method for evaluating parameterdependent normal forms of bifurcation equations both for diagonalizable and non-diagonalizable Jabobian matrices. The method avoids the extension of the state space to parameters as well as the occurrence of small denominators in the solution which are drawbacs encountered in classical methods. The efficiency of the procedure was illustrated by an example relevant to a double-zero bifurcation. Acnowledgments: This wor was supported by the Italian Ministry of University (MIUR) through a PRIN co-financed program (www.disg.uniroma1.it/fendis). [1] Gucenheer J. olmes P. Nonlinear Oscillations Dynamical Systems and Bifurcations of Vector Fields Springer New Yor (1983). [] WigginsW. Introduction to Applied Nonlinear Dynamical Systems and Chaos Springer NewYor (1996). [3] Troger. Steindl A. Nonlinear Stability and Bifurcation Theory SpringerWien New Yor (1991). [4] Nayfeh A.. Method of Normal Forms J. Wiley & Sons New Yor (1993).