Normal Forms of C ∞ Vector Fields based on the Renormalization Group
aa r X i v : . [ m a t h . D S ] S e p Normal Forms of C ∞ Vector Fields based on theRenormalization Group
Department of Applied Mathematics and PhysicsKyoto University, Kyoto, 606-8501, JapanHayato CHIBA October 20 2008
Abstract
The normal form theory for polynomial vector fields is extended to those for C ∞ vectorfields vanishing at the origin. Explicit formulas for the C ∞ normal form and the near identitytransformation which brings a vector field into its normal form are obtained by means of therenormalization group method. The dynamics of a given vector field such as the existenceof invariant manifolds is investigated via its normal form. The C ∞ normal form theory isapplied to prove the existence of infinitely many periodic orbits of two dimensional systemswhich is not shown from polynomial normal forms. The Poincar´e-Durac normal form is a fundamental tool for analyzing local dynamics ofvector fields near fixed points [1, 9, 11]. It gives a local coordinate change around a fixedpoint which transforms a given vector field into a simplified one in some sense. The normalform theory have been well developed for polynomial vector fields; if we have a system ofordinary di ff erential equations dx / dt = ˙ x = f ( x ) on R n with a C ∞ vector field f vanishingat the origin (i.e. f (0) = x = Ax + g ( x ) + g ( x ) + · · · , x ∈ R n (1.1)where A is a constant matrix and g k ( x )’s are homogeneous polynomial vector fields of degree k . Then, normal forms, simplified vector fields, for polynomials g , g , · · · are calculated oneafter the other as summarized in Section 2. A coordinate transformation x y which bringsa given system into a normal form is of the form x = h ( y ) = y + h ( y ) + h ( y ) + · · · , (1.2)where h k ’s are homogeneous polynomials on R n of degree k that are also obtained step bystep. It is called the near identity transformation . Since h ( y ) is constructed as a formal E mail address : [email protected] ff eomorphism only on a small neighborhood of the origin. In orderto investigate the local dynamics of a given system, usually its normal form and the nearidentity transformation are truncated at a finite degree. We will refer to this method as the polynomial normal form theory .In this paper, we establish the C ∞ normal form theory for systems of the form ˙ x = Ax + ε f ( x ) by means of the renormalization group (RG) method, where A is a diagonalmatrix, f is a C ∞ vector field vanishing at the origin and ε is a small parameter. The RGmethod has its origin in quantum field theory and was applied to perturbation problems ofdi ff erential equations by Chen, Goldenfeld and Oono [3, 4]. For a certain class of vectorfields, the RG method was mathematically justified by Chiba [5, 6, 7]. Our method basedon the RG method allows one to calculate normal forms of vector fields without expandingin a power series. For example if f is periodic in x , its C ∞ normal form and a near identitytransformation are also periodic. As a result, the C ∞ normal form may be valid on a largeopen set or the whole phase space and it will be applicable to detect the existence of invariantmanifolds of a given system.In Sec.2, we give a brief review of the polynomial normal forms. In Sec.3.1, we providea direct sum decomposition of the space of C ∞ vector fields vanishing at the origin, whichextends the decomposition of polynomial vector fields used in the polynomial normal formtheory. Properties of the decomposition will be investigated in detail to develop the C ∞ normal form theory. In Sec.3.2, we give a definition of the C ∞ normal form and explicitformulas for calculating them are derived by means of the RG method. In Sec.3.3, weconsider the case that the linear part of a vector field is not hyperbolic. In this case, it isproved that if a C ∞ normal form has a normally hyperbolic invariant manifold N , then theoriginal system also has an invariant manifold which is di ff eomorphic to N . This theoremwill be used to prove the existence of infinitely many periodic orbits of a two-dimensionalsystem in Section 4. In this section, we give a brief review of the polynomial normal forms for comparison withthe C ∞ normal forms to be developed in the next section. See Chow, Li and Wang [9],Murdock [11] for the detail.Let us denote by P k ( R n ) the set of homogeneous polynomial vector fields on R n of degree k . Consider the system of ordinary di ff erential equations on R n dxdt = ˙ x = Ax + ε g ( x ) + ε g ( x ) + · · · , x ∈ R n , (2.1)where A is a constant n × n matrix, g k ∈ P k ( R n ) for k = , , · · · , and where ε ∈ R is a dummyparameter which is introduced to clarify steps of the iteration described below. Note that if2e have a system ˙ x = f ( x ) with the C ∞ vector field f satisfying f (0) =
0, putting x ε x and expanding the system ε ˙ x = f ( ε x ) in ε yields the system (2.1).Let us try to simplify Eq.(2.1) by the coordinate transformation of the form x = y + ε h ( y ) , h ∈ P ( R n ) . (2.2)Substituting Eq.(2.2) into Eq.(2.1) provides id + ε ∂ h ∂ y ( y ) ! ˙ y = A ( y + ε h ( y )) + ε g ( y + ε h ( y )) + ε g ( y + ε h ( y )) + · · · . (2.3)Expanding the above in ε , we obtain˙ y = Ay + ε g ( y ) − ∂ h ∂ y ( y ) Ay + Ah ( y ) ! + ε e g ( y ) + · · · , (2.4)where e g ∈ P ( R n ). Let us define the map L A on the set of polynomial vector fields to be L A ( f )( x ) = ∂ f ∂ x ( x ) Ax − A f ( x ) . (2.5)Since L A keeps the degree of a monomial, it gives the linear operator from P k ( R n ) into P k ( R n ) for any integer k . Thus, the direct sum decomposition P k ( R n ) = Im L A | P k ( R n ) ⊕ C k (2.6)holds, where C k is a complementary subspace of Im L A | P k ( R n ) . One of the convenient choicesis C k = Ker ( L A | P k ( R n ) ) ∗ , where ( L A | P k ( R n ) ) ∗ is the adjoint operator with respect to a giveninner product on P k ( R n ). In particular, it is known that ( L A | P k ( R n ) ) ∗ = L A ∗ | P k ( R n ) holds for acertain inner product, where A ∗ denotes the adjoint matrix of A : P k ( R n ) = Im L A | P k ( R n ) ⊕ Ker L A ∗ | P k ( R n ) . (2.7)Here we note that the equality L A ( f )( x ) = f ( e At x ) = e At f ( x )for t ∈ R ; Ker L A ∗ | P k ( R n ) = { f ∈ P k ( R n ) | f ( e A ∗ x ) = e A ∗ f ( x ) } . Since Eq.(2.4) is written as˙ y = Ay + ε ( g ( y ) − L A ( h )( y )) + ε e g ( y ) + · · · , (2.8)there exists h ∈ P ( R n ) such that g − L A ( h ) ∈ Ker L A ∗ | P ( R n ) .Next thing to do is to simplify e g ∈ P ( R n ) by the transformation of the form y = z + ε h ( z ) , h ∈ P ( R n ) . (2.9)3t is easy to verify that this transformation does not change the term g − L A ( h ) of degreetwo and we obtain˙ y = Ay + ε ( g ( y ) − L A ( h )( y )) + ε ( e g ( y ) − L A ( h )( y )) + O ( ε ) . (2.10)In a similar manner to the above, we can take h so that e g − L A ( h ) ∈ Ker L A ∗ | P ( R n ) .We proceed by induction and obtain the well-known theorem. Theorem 2.1.
There exists a formal power series transformation x = z + ε h ( z ) + ε h ( z ) + · · · (2.11)with h k ∈ P k ( R n ) such that Eq.(2.1) is transformed into the system˙ z = Az + ε R ( z ) + ε R ( z ) + · · · , (2.12)satisfying R k ∈ Ker L A ∗ ∩ P k ( R n ) for k = , , · · · . The transformation (2.11) is called the near identity transformation and the truncated system˙ z = Az + ε R ( z ) + ε R ( z ) + · · · + ε m R m ( z ) (2.13)is called the normal form of degree m . Remark 2.2.
A few remarks are in order. The near identity transformation (2.11) is a di ff eo-morphism on a small neighborhood of the origin. Eqs.(2.11) and (2.12) are not convergentseries in general even if Eq.(2.1) is convergent. See Zung [14] for the necessary and suf-ficient condition for the convergence of normal forms. Note that a normal form (2.12) isnot unique. It is because there are many di ff erent choices of h in Eq.(2.10) which yieldthe same R : = g − L A ( h ), while such di ff erent choices of h may change R , R , · · · . Thesimplest form among di ff erent normal forms are called the hyper-normal form [11, 12].It is known that if A = diag ( λ , · · · , λ n ) is a diagonal matrix, Im L A and Ker L A ∗ ( = Ker L A ) are given byIm L A ∩ P k ( R n ) = span { x q x q · · · x q n n e i | n X j = λ j q j , λ i , n X j = q j = k } , (2.14)Ker L A ∗ ∩ P k ( R n ) = { f ∈ P k ( R n ) | f ( e At x ) = e At f ( x ) } = span { x q x q · · · x q n n e i | n X j = λ j q j = λ i , n X j = q j = k } , (2.15)respectively, where e , · · · , e n are the canonical basis of R n . Indeed, we can verify that L A ( x q x q · · · x q n n e i ) = ( n X j = λ j q j − λ i ) x q x q · · · x q n n e i . (2.16)The condition P nj = λ j q j = λ i is called the resonance condition . This implies that R k consistsof resonance terms of degree k . 4 C ∞ normal form theory In this section, we develop the theory of normal forms of the system dxdt = ˙ x = Ax + ε g ( x ) + ε g ( x ) + · · · , x ∈ R n , (3.1)for which g k is a C ∞ vector field, not a polynomial in general. We suppose that a matrix A isa diagonal matrix. If A is not semi-simple, by a suitable linear transformation and the Jordandecomposition, we can assume that A is of the form A = Λ + ε N , where Λ is diagonal and N is nilpotent. By replacing g ( x ) to g ( x ) + N x , we can assume without loss of generalitythat A is a diagonal matrix. C ∞ vector fields Let P ( R n ) be the set of polynomial vector fields on R n whose degrees are equal to or largerthan one. Define the linear map L A on P ( R n ) by Eq.(2.5). Then, Eq.(2.7) gives the directsum decomposition P ( R n ) = Im L A ⊕ Ker L A . (3.2)Note that Ker L A ∗ = Ker L A because A is diagonal by our assumption. By the completion,the direct sum decomposition (3.2) is extended to the set of C ∞ vector fields vanishing at theorigin. Theorem 3.1.
Let K ⊂ R n be an open set including the origin whose closure ¯ K is compact.Let X ∞ ( K ) be the set of C ∞ vector fields f on K satisfying f (0) =
0. Define the linear map L A : X ∞ ( K ) → X ∞ ( K ) by Eq.(2.5). Then, the direct sum decomposition X ∞ ( K ) = V I ⊕ V K (3.3)holds, where V I : = Im L A , (3.4) V K : = Ker L A = { f ∈ X ∞ ( K ) | f ( e A x ) = e A f ( x ) } . (3.5) Proof.
Since the set of polynomial vector fields is dense in X ∞ ( K ) equipped with the C ∞ topology (Hirsch [10]), for any u ∈ X ∞ ( K ), there exists a sequence u n in P ( R n ) such that u n → u as n → ∞ in X ∞ ( K ). Let u n = v n + w n with v n ∈ Im L A | P ( R n ) , w n ∈ Ker L A | P ( R n ) be the decomposition along the direct sum (3.2). Since u n is a Cauchy sequence in X ∞ ( K ), u n ( x ) − u m ( x ) is su ffi ciently close to zero with its derivatives uniformly on any compact sub-sets in K if n and m are su ffi ciently large. Hence, u n − u m is a polynomial whose coe ffi cientsare su ffi ciently close to zero. Since v n and w n consist of non-resonance and resonance terms,respectively, they do not include common monomial vector fields. This shows that v n − v m w n − w m are also Cauchy sequences in X ∞ ( K ), thus v n and w n converge to v and w ,respectively. Since L A is a continuous operator on X ∞ ( K ), L A w n = w ∈ Ker L A .For v n ∈ Im L A , take F n ∈ P ( R n ) satisfying v n = L A F n and F n ∈ Im L A that is uniquelydetermined through Eq.(2.16);( L A | Im L A ) − ( x q x q · · · x q n n e i ) = ( n X j = λ j q j − λ i ) − x q x q · · · x q n n e i . This proves that F n is also a Cauchy sequence converging to F ∈ X ∞ ( K ) and v = L A F ∈L A X ∞ ( K ). The desired decomposition u = v + w is obtained. (cid:3) We define the projections P I : X ∞ ( K ) → V I and P K : X ∞ ( K ) → V K . For g ∈ V I , thereexists a vector field F ∈ X ∞ ( K ) such that L A ( F ) = ∂ F ∂ x ( x ) Ax − AF ( x ) = g ( x ) . (3.6)Such F ( x ) is not unique because if F satisfies the above equality, then F + h with h ∈ V K also satisfies it. We write F = Q ( g ) if F satisfies Eq.(3.6) and P K ( F ) =
0. Then Q definesthe linear map from V I to V I . In particular, we have Q ◦ L A ( F ) = F , L A ◦ Q ( g ) = g , (3.7)for any F , g ∈ V I . We show a few propositions which are convenient when calculatingnormal forms. Proposition 3.2.
The following equalities hold for any g ∈ V I .(i) P K ◦ Q ( g ) = , (3.8)(ii) Q [ Dg · Q ( g ) + D Q ( g ) · g ] = P I [ D Q ( g ) · Q ( g )] , (3.9)(iii) e − As g ( e As x ) = ∂∂ s (cid:16) e − As Q ( g )( e As x ) (cid:17) , s ∈ R , (3.10)where D denotes the derivative with respect to x . Proof.
Part (i) of Prop.3.2 follows from the definition of Q . To prove (ii), we write F = Q ( g ).By using Eq.(3.6), it is easy to verify the equality ∂∂ x ∂ F ∂ x ( x ) F ( x ) ! Ax − A ∂ F ∂ x ( x ) F ( x ) ! = ∂ g ∂ x ( x ) F ( x ) + ∂ F ∂ x ( x ) g ( x ) . (3.11)It is rewritten as L A [ D Q ( g ) · Q ( g )] = Dg · Q ( g ) + D Q ( g ) · g . Q in the both sides and using (3.7) proves (ii). Part (iii) of Prop.3.2 is shown as ∂∂ s (cid:16) e − As Q ( g )( e As x ) (cid:17) = − Ae − As Q ( g )( e As x ) + e − As D Q ( g )( e As x ) · Ae As x = e − As L A ◦ Q ( g )( e As x ) = e − As g ( e As x ) . (cid:3) We define the Lie bracket product (commutator) [ · , · ] of vector fields by[ f , g ]( x ) = ∂ f ∂ x ( x ) g ( x ) − ∂ g ∂ x ( x ) f ( x ) . (3.12) Proposition 3.3. If g , h ∈ V K , then Dg · h ∈ V K and [ g , h ] ∈ V K . Proof.
It follows from a straightforward calculation. (cid:3)
Proposition 3.4.
For g ∈ V I and h ∈ V K , the following equalities hold:(i) ∂ g ∂ x h ∈ V I , Q ∂ g ∂ x h ! = ∂ Q ( g ) ∂ x h , (3.13)(ii) ∂ h ∂ x g ∈ V I , Q ∂ h ∂ x g ! = ∂ h ∂ x Q ( g ) , (3.14)(iii) [ g , h ] ∈ V I , Q ([ g , h ]) = [ Q ( g ) , h ] . (3.15) Proof.
Put F = Q ( g ). Note that g and h satisfy the equalities Eq.(3.6) and ∂ h ∂ x ( x ) Ax − Ah ( x ) = . By using them, we can prove the following equalities ∂∂ x ∂ F ∂ x ( x ) h ( x ) ! Ax − A ∂ F ∂ x ( x ) h ( x ) ! = ∂ g ∂ x ( x ) h ( x ) , (3.16) ∂∂ x ∂ h ∂ x ( x ) F ( x ) ! Ax − A ∂ h ∂ x ( x ) F ( x ) ! = ∂ h ∂ x ( x ) g ( x ) , (3.17)which imply that ∂ g /∂ x · h ∈ V I and ∂ h /∂ x · g ∈ V I . The same calculation also shows that ∂ F /∂ x · h ∈ V I and ∂ h /∂ x · F ∈ V I . Since Q = L − A on V I , (3.16) and (3.17) give (i) and (ii)of Prop.3.4, respectively. Part (iii) immediately follows from (i) and (ii). (cid:3) Remark 3.5.
Props.3.3 and 3.4 imply [ V K , V K ] ⊂ V K and [ V I , V K ] ⊂ V I . However, [ V I , V I ] ⊂ V I is not true in general. C ∞ normal forms Let us consider the system on R n of the form˙ x = Ax + ε g ( x ) + ε g ( x ) + · · · , x ∈ R n , (3.18)7here A is a constant n × n diagonal matrix, g ( x ) , g ( x ) , · · · ∈ X ∞ ( R n ) are C ∞ vector fieldsvanishing at the origin, and ε ∈ R is a parameter. To obtain a normal form of Eq.(3.18),we use the renormalization group method. According to [5], at first, we try to construct aregular perturbation solution for Eq.(3.18). Put x = ˆ x ( t ) = x + ε x + ε x + · · · (3.19)and substitute it into Eq.(3.18) : ∞ X k = ε k ˙ x k = A ∞ X k = ε k x k + ∞ X k = ε k g k ( ∞ X j = ε j x j ) . (3.20)Expanding the right hand side with respect to ε and equating the coe ffi cients of each ε k , weobtain the system of ODEs ˙ x = Ax , (3.21)˙ x = Ax + G ( x ) , (3.22) ... ˙ x i = Ax i + G i ( x , x , · · · , x i − ) , (3.23) ... where the functions G k are defined through the equality ∞ X k = ε k g k ( ∞ X j = ε j x j ) = ∞ X k = ε k G k ( x , x , · · · , x k − ) . (3.24)For example, G , G and G are given by G ( x ) = g ( x ) , (3.25) G ( x , x ) = ∂ g ∂ x ( x ) x + g ( x ) , (3.26) G ( x , x , x ) = ∂ g ∂ x ( x ) x + ∂ g ∂ x ( x ) x + ∂ g ∂ x ( x ) x + g ( x ) , (3.27)respectively. Since all systems are inhomogeneous linear equations, they are solved step bystep. The zeroth order equation ˙ x = Ax is solved as x ( t ) = e At y , where y ∈ R n is an initialvalue. Thus, the first order equation is written as˙ x = Ax + g ( e At y ) . (3.28)A general solution of this system whose initial value is x (0) = h (1) ( y ) is given by x ( t ) = e At h (1) ( y ) + e At Z t e − As g ( e As y ) ds . (3.29)8ow we consider choosing h (1) so that x ( t ) above takes the simplest form. Put P I ( g ) = g I and P K ( g ) = g K . Then, Prop.3.2 (iii) is used to yield x ( t ) = e At h (1) ( y ) + e At Z t e − As g I ( e As y ) ds + e At Z t e − As g K ( e As y ) ds = e At h (1) ( y ) + e At Z t ∂∂ s (cid:16) e − As Q ( g I )( e As y ) (cid:17) ds + e At Z t g K ( y ) ds = e At h (1) ( y ) + Q ( g I )( e At y ) − e At Q ( g I )( y ) + e At g K ( y ) t . (3.30)Putting h (1) = Q ( g I ), we obtain x ( t ) = Q ( g I )( e At y ) + g K ( e At y ) t . (3.31)Note that the term g K ( e At y ) t is so-called the secular term . Next thing to do is to calculate x . A solution of the equation of x is given by x ( t ) = e At h (2) ( y ) + e At Z t e − As ∂ g ∂ x ( e As y ) (cid:0) Q ( g I )( e As y ) + g K ( e As y ) s (cid:1) + g ( e As y ) ! ds , (3.32)where h (2) ( y ) = x (0) is an initial value. By choosing h (2) appropriately as above, we canshow that x is expressed as x ( t ) = QP I ( R )( e At y ) + P K ( R ) + ∂ Q ( g I ) ∂ y g K ! ( e At y ) t + ∂ g K ∂ y ( e At y ) g K ( e At y ) t , (3.33)where R is defined by R ( y ) = G ( y , Q ( g I )( y )) − ∂ Q ( g I ) ∂ y ( y ) g K ( y ) = ∂ g ∂ y ( y ) Q ( g I )( y ) + g ( y ) − ∂ Q ( g I ) ∂ y ( y ) g K ( y ) . (3.34)These equalities are proved in Appendix with the aid of Propositions 3.2 to 3.4. By proceed-ing in a similar manner, we can prove the next proposition. Proposition 3.6.
Define functions R k , k = , , · · · on R n to be R ( y ) = g ( y ) , (3.35)and R k ( y ) = G k ( y , QP I ( R )( y ) , QP I ( R )( y ) , · · · , QP I ( R k − )( y )) − k − X j = ∂ QP I ( R j ) ∂ y ( y ) P K ( R k − j )( y ) , (3.36)9or k = , , · · · . Then, Eq.(3.23) has a solution x i = x i ( t , y ) = QP I ( R i )( e At y ) + p ( i )1 ( e At y ) t + p ( i )2 ( e At y ) t + · · · + p ( i ) i ( e At y ) t i , (3.37)where p ( i ) j ’s are defined by p ( i )1 ( y ) = P K ( R i )( y ) + i − X k = ∂ QP I ( R k ) ∂ y ( y ) P K ( R i − k )( y ) , (3.38) p ( i ) j ( y ) = j i − X k = ∂ p ( k ) j − ∂ y ( y ) P K ( R i − k )( y ) , ( j = , , · · · , i − , (3.39) p ( i ) i ( y ) = i ∂ p ( i − i − ∂ y ( y ) P K ( R )( y ) , (3.40) p ( i ) j ( y ) = , ( j > i ) . (3.41)This proposition can be proved in the same way as Prop.A.1 in Chiba [5], in which Prop.3.6is proved by induction for the case that all eigenvalues of A lie on the imaginary axis.Now we have a formal solution of Eq.(3.18) of the form x = ˆ x ( t , y ) = e At y + ∞ X k = ε k x k ( t , y ) = e At y + ∞ X k = ε k (cid:16) QP I ( R k )( e At y ) + p ( k )1 ( e At y ) t (cid:17) + O ( t ) . (3.42)This solution diverges as t → ∞ because it includes polynomials in t . The RG method isused to construct better approximate solutions from the above formal solution as follows[3, 4, 5, 6, 7].We replace polynomials t k in Eq.(3.42) by ( t − τ ) k , where τ ∈ R is a new parameter.Next, we regard y = y ( τ ) as a function of τ to be determined so that we recover the originalformal solution :ˆ x ( t , y ) = e At y ( τ ) + ∞ X k = ε k (cid:16) QP I ( R k )( e At y ( τ )) + p ( k )1 ( e At y ( τ ))( t − τ ) (cid:17) + O (( t − τ ) ) . (3.43)Since ˆ x ( t , y ) is independent of the “dummy” parameter τ , we impose the condition dd τ (cid:12)(cid:12)(cid:12)(cid:12) τ = t ˆ x ( t , y ) = = e At dydt + ∞ X k = ε k ∂ QP I ( R k ) ∂ y ( e At y ) e At dydt − p ( k )1 ( e At y ) ! . (3.45)10ubstituting Eq.(3.38) yields0 = e At dydt + ∞ X k = ε k ∂ QP I ( R k ) ∂ y ( e At y ) e At dydt ! − ∞ X k = ε k P K ( R k )( e At y ) − ∞ X k = ε k k − X j = ∂ QP I ( R j ) ∂ y ( e At y ) P K ( R k − j )( e At y ) = e At dydt − ∞ X j = ε j P K ( R j )( y ) + ∞ X k = ε k ∂ QP I ( R k ) ∂ y ( e At y ) e At dydt − ∞ X j = ε j P K ( R j )( y ) . (3.46)Now we obtain the ODE of y as dydt = ∞ X j = ε j P K ( R j )( y ) , (3.47)which is called the RG equation . Since Eq.(3.43) is independent of τ , we put τ = t to obtainˆ x ( t , y ( t )) = e At y ( t ) + ∞ X j = ε j QP I ( R j )( e At y ( t )) , (3.48)where y ( t ) is a solution of Eq.(3.47). This ˆ x ( t , y ( t )) gives an approximate solution of thesystem (3.18) if the series is truncated at some finite order of ε . Since P K ( R j ) satisfies P K ( R j )( e At y ) = e At P K ( R j )( y ), putting e At y = z transforms Eqs.(3.47) and (3.48) into dzdt = Az + ∞ X j = ε j P K ( R j )( z ) , (3.49)ˆ x ( t , e − At z ( t )) = z ( t ) + ∞ X j = ε j QP I ( R j )( z ( t )) , (3.50)respectively. Since P K ( R j ) ∈ V K , we conclude that Eqs.(3.49) and (3.50) give a normal formof the system (3.18) and a near identity transformation x z . Indeed, the next theorem isreduced to Theorem 2.1 when g k ∈ P k ( R n ). Theorem 3.7.
Define the m-th order near identity transformation to be x = z + ε QP I ( R )( z ) + ε QP I ( R )( z ) + · · · + ε m QP I ( R m )( z ) . (3.51)Then, it transforms the system (3.18) into the system˙ z = Az + ε P K ( R )( z ) + ε P K ( R )( z ) + · · · + ε m P K ( R m )( z ) + ε m + S ( z , ε ) , (3.52)where S ( z , ε ) is a C ∞ function with respect to z and ε . We call the truncated system˙ z = Az + ε P K ( R )( z ) + ε P K ( R )( z ) + · · · + ε m P K ( R m )( z ) (3.53)11he m-th order normal form of Eq.(3.18). This system is invariant under the action of theone-parameter group z e As z , s ∈ R . Proof.
By putting z = e At y in Eqs.(3.51) and (3.52), we prove that the transformation x = e At y + ε QP I ( R )( e At y ) + · · · + ε m QP I ( R m )( e At y ) (3.54)transforms (3.18) into the system˙ y = ε P K ( R )( y ) + · · · + ε m P K ( R m )( y ) + ε m + e S ( t , y , ε ) . (3.55)The proof is done by a straightforward calculation. By substituting Eq.(3.54) into Eq.(3.18),the left hand side is calculated as dxdt = e At + m X k = ε k ∂ QP I ( R k ) ∂ y ( e At y ) e At ˙ y + Ae At y + m X k = ε k ∂ QP I ( R k ) ∂ y ( e At y ) Ae At y . (3.56)Since QP I ( R k ) satisfies the equality ∂ QP I ( R k ) ∂ y ( y ) Ay − A QP I ( R k )( y ) = L A QP I ( R k )( y ) = P I ( R k )( y ) , (3.57)Eq.(3.56) is rewritten as dxdt = e At + m X k = ε k ∂ QP I ( R k ) ∂ y ( e At y ) e At ˙ y + Ae At y + m X k = ε k (cid:16) P I ( R k )( e At y ) + A QP I ( R k )( e At y ) (cid:17) . (3.58)Furthermore, P I ( R k ) = R k − P K ( R k ), (3.36) and (3.58) are put together to yield dxdt = e At + m X k = ε k ∂ QP I ( R k ) ∂ y ( e At y ) e At ˙ y + Ae At y + m X k = ε k A QP I ( R k )( e At y ) + m X k = ε k (cid:16) G k ( e At y , QP I ( R )( e At y ) , · · · , QP I ( R k − )( e At y )) − k − X j = ∂ QP I ( R j ) ∂ y ( e At y ) P K ( R k − j )( e At y ) − P K ( R k )( e At y ) (cid:17) . (3.59)On the other hand, the right hand side of Eq.(3.18) is transformed as A ( e At y + m X k = ε k QP I ( R k )( e At y )) + ∞ X k = ε k g k ( e At y + m X j = ε j QP I ( R j )( e At y )) = Ae At y + m X k = ε k A QP I ( R k )( e At y ) + m X k = ε k G k ( e At y , QP I ( R )( e At y ) , · · · , QP I ( R k − )( e At y )) + O ( ε m + ) . (3.60)12hus Eq.(3.18) is transformed into the system˙ y = e At + m X k = ε k ∂ QP I ( R k ) ∂ y ( e At y ) e At − × m X k = ε k P K ( R k )( e At y ) + k − X j = ∂ QP I ( R j ) ∂ y ( e At y ) P K ( R k − j )( e At y ) + O ( ε m + ) = e − At id + ∞ X j = ( − j m X k = ε k ∂ QP I ( R k ) ∂ y ( e At y ) j × e At m X i = ε i P K ( R i )( y ) + m X k = ε k ∂ QP I ( R k ) ∂ y ( e At y ) e At m − k X i = ε i P K ( R i )( y ) + O ( ε m + ) = m X k = ε k P K ( R k )( y ) + e − At ∞ X j = ( − j m X k = ε k ∂ QP I ( R k ) ∂ y ( e At y ) j e At m X i = m − k + ε i P K ( R i )( y ) + O ( ε m + ) = m X k = ε k P K ( R k )( y ) + O ( ε m + ) . This proves that Eq.(3.18) is transformed into the system Eq.(3.55). (cid:3)
Remark 3.8.
Eq.(3.52) is valid on a region including the origin on which the near identitytransformation (3.51) is a di ff eomorphism. In the polynomial normal form theory describedin Section 2, since ε k QP I ( R k )( z ) is a polynomial in z of degree k , the near identity trans-formation may not be a di ff eomorphism when z ∼ O (1 /ε ) in general. For the C ∞ normalform, the near identity transformation may be a di ff eomorphism on larger set. For exampleif QP I ( R k )( z ) , k = , , · · · , m are periodic as Example 4.1 below, Eq.(3.51) is a di ff eomor-phism for any z ∈ R n if ε is su ffi ciently small. If the matrix A in Eq.(3.18) is hyperbolic, which means that no eigenvalues of A lie onthe imaginary axis, then the flow of Eq.(3.18) near the origin is topologically conjugate tothe linear system ˙ x = Ax and the local stability of the origin is easily understood. If A has eigenvalues on the imaginary axis, Eq.(3.18) has a center manifold at the origin andnontrivial phenomena, such as bifurcations, may occur on the center manifold. We considersuch a situation in this subsection. By using the center manifold reduction [2, 6], we assumethat all eigenvalues of A lie on the imaginary axis. We also suppose that A is diagonal asbefore. In this case, the operators P K and QP I are calculated as follows:13ecall that the equality Z t e − A ( s − t ) g ( e A ( s − t ) x ) ds = Z t e − A ( s − t ) P I ( g )( e A ( s − t ) x ) ds + Z t e − A ( s − t ) P K ( g )( e A ( s − t ) x ) ds = QP I ( g )( x ) − e At QP I ( g )( e − At x ) + P K ( g )( x ) t (3.61)holds. We have to calculate QP I ( g ) and P K ( g ) to obtain the normal form (3.53). Since e − As g ( e As x ) is an almost periodic function with respect to s , it is expanded in a Fourierseries as e − As g ( e As x ) = P λ i ∈ Λ c ( λ i , x ) e √− λ i s , where Λ is the set of the Fourier exponents and c ( λ i , x ) ∈ R n is a Fourier coe ffi cient. In particular, the Fourier coe ffi cient c (0 , x ) associatedwith the zero Fourier exponent is the average of e − As g ( e As x ): c (0 , x ) = lim t →∞ t Z t e − As g ( e As x ) ds . (3.62)Thus we obtain Z t e − A ( s − t ) g ( e A ( s − t ) x ) ds = Z t X λ i ∈ Λ c ( λ i , x ) e √− λ i ( s − t ) ds = X λ i , √− λ i c ( λ i , x )(1 − e − √− λ i t ) + c (0 , x ) t . (3.63)Comparing it with Eq.(3.61), we obtain P K ( g )( x ) = c (0 , x ) = lim t →∞ t Z t e − As g ( e As x ) ds , (3.64) QP I ( g )( x ) = X λ i , √− λ i c ( λ i , x ) = lim t → Z t (cid:16) e − As g ( e As x ) − P K ( g )( x ) (cid:17) ds , (3.65)where R t denotes the indefinite integral whose integral constant is chosen to be zero. Theseformulas for P K and QP I allow one to calculate the normal forms systematically.Now we suppose that the normal form for Eq.(3.18) satisfies P K ( R ) = · · · = P K ( R m − ) = m ≥
1. By putting z = e At y , Eq.(3.52) takes the form˙ y = ε m P K ( R m ) + O ( ε m + ) . (3.66)If ε is su ffi ciently small, some properties of Eq.(3.66) are obtained from the truncated sys-tem ˙ y = ε m P K ( R m ). In this manner, we can prove the next theorem. Theorem 3.9 [5, 7].
Suppose that all eigenvalues of the diagonal matrix A lie on the imagi-nary axis and that the normal form for Eq.(3.18) satisfies P K ( R ) = · · · = P K ( R m − ) = P K ( R m ) , m ≥
1. If the truncated system dy / dt = ε m P K ( R m )( y ) has anormally hyperbolic invariant manifold N , then for su ffi ciently small | ε | , the system (3.18)14as an invariant manifold N ε , which is di ff eomorphic to N . In particular the stability of N ε coincides with that of N .This theorem is proved in Chiba [7] in terms of the RG method and a perturbation theoryof invariant manifolds [13]. For many examples, m = dydt = ε P K ( R )( y ) = ε P K ( g )( y ) = ε · lim t →∞ t Z t e − As g ( e As y ) ds , (3.67)which recovers the classical averaging method. See [7, 8] for many applications for thedegenerate cases m ≥ In this section, we give a few examples to demonstrate our theorems.
Example 4.1.
Consider the system on R ( ˙ x = x + ε sin x , ˙ x = − x , (4.1)where ε > x = z + z , x = i ( z − z ) to diagonalize Eq.(4.1)as ddt z z ! = i − i ! z z ! + ε sin( z + z )sin( z + z ) ! , (4.2)where i = √−
1. We calculate the normal forms of this system in two di ff erent ways, thepolynomial normal form and the C ∞ normal form. (I) To calculate the polynomial normal form, we expand sin( z + z ) as ddt z z ! = iz − iz ! + ε z + z z + z ! − ε ( z + z ) ( z + z ) ! + ε ( z + z ) ( z + z ) ! − ε ( z + z ) ( z + z ) ! + · · · . (4.3)The fourth order normal form of this system is given by ddt y y ! = iy − iy ! + ε y y ! − ε y ( y y + i ) y ( y y − i ) ! + ε y y ( y y + i ) y y ( y y − i ) ! − ε y ( y y + iy y + y y + i ) y ( y y − iy y + y y − i ) ! . (4.4)Putting y = re i θ , y = re − i θ yields ˙ r = ε r − ε r + ε r − ε
144 ( r + r ) , ˙ θ = − ε + ε · r − ε
144 (39 r + . (4.5)15ixed points of the equation of r (i.e. the zeros of the right hand side) imply periodic orbitsof the original system (4.1). The near identity transformation is given by z z ! = y y ! + ε i y +
124 (2 y − y y − y ) + O ( y , y ) − y +
124 ( y + y y − y ) + O ( y , y ) , (4.6)and it is easy to see that this gives a di ff eomorphism only near the origin. (II) Let us calculate the C ∞ normal form of Eq.(4.2). The first term P K ( R ) of the normalform is given by using Eq.(3.64) as P K ( R )( y , y ) = lim t →∞ t Z t e − is e is ! sin( e is y + e − is y )sin( e is y + e − is y ) ! ds . (4.7)Thus the first order normal form is given by ddt y y ! = iy − iy ! + ε π R π e − it sin( e it y + e − it y ) dt R π e it sin( e it y + e − it y ) dt . (4.8)Putting y = re i θ , y = re − i θ yields ˙ r = ε π Z π cos t · sin(2 r cos t ) dt = ε J (2 r ) , ˙ θ = + ε π r Z π sin t · sin(2 r cos t ) dt = , (4.9)where J n ( r ) is the Bessel function of the first kind defined as the solution of the equation r x ′′ + rx ′ + ( r − n ) x =
0. By Eq.(3.65), it is easy to verify that the first order near identitytransformation is periodic in y and y although we can not calculate the indefinite integralin Eq.(3.65) explicitly. Thus there exists a positive number ε such that if 0 < ε < ε , thenear identity transformation is a di ff eomorphism on R . Since J (2 r ) has infinitely manyzeros, Thm.3.9 proves that the original system (4.1) has infinitely many periodic orbits. Example 4.2.
Consider the system on R of the form ( ˙ x = x + ε g ( x ) , ˙ x = − x , (4.10)where the function g ( x ) is defined by g ( x ) = ( x , x ∈ [2 n , n + , − x , x ∈ [2 n + , n + , (4.11)for n = , , , · · · and g ( x ) = − g ( − x ) (see Fig.1 (a)).16 g ( x )(a) (b) xg ( x )~ Fig. 1: The graphs of the functions g ( x ) and e g ( x ).We add to Eq.(4.10) a small perturbation whose support is included in su ffi ciently smallintervals ( n − δ, n + δ ) , n ∈ Z so that the resultant system ( ˙ x = x + ε e g ( x ) , ˙ x = − x , (4.12)is of C ∞ class (see Fig.1 (b)). Like as Example 4.1, the first order C ∞ normal form of thissystem written in the polar coordinates is given by ˙ r = ε π Z π cos t · e g (2 r cos t ) dt : = ε π R ( r ) , ˙ θ = + ε π Z π sin t · e g (2 r cos t ) = . (4.13)On the outside of the support of the perturbation, the function R ( r ) is given by R ( r ) = ( π r , r ∈ (2 n + δ, n + − δ ) , − π r , r ∈ (2 n + + δ, n + − δ ) . (4.14)By the intermediate value theorem, R ( r ) has zeros near r = n ∈ Z . In particular, fixed pointsnear r = n + e g ( x ) at the origin,we obtain the normal form ˙ r = ε r , which is valid on a small neighborhood of the origin.17 Appendix
In this appendix, we derive Eq.(3.33) from Eq.(3.32). By integrating by parts, Eq.(3.32) iscalculated as x = e At h (2) ( y ) + e At Z t e − As ∂ g ∂ x ( e As y ) Q ( g I )( e As y ) + g ( e As y ) ! ds + e At Z t e − As ∂ g ∂ x ( e As y ) g K ( e As y ) ds · t − e At Z t ds Z s e − As ′ ∂ g ∂ x ( e As ′ y ) g K ( e As ′ y ) ds ′ = e At h (2) ( y ) + e At Z t e − As ∂ g ∂ x ( e As y ) Q ( g I )( e As y ) + g ( e As y ) ! ds + e At Z t e − As ∂ g K ∂ x ( e As y ) g K ( e As y ) ds · t + e At Z t e − As ∂ g I ∂ x ( e As y ) g K ( e As y ) ds · t − e At Z t ds Z s e − As ′ ∂ g K ∂ x ( e As ′ y ) g K ( e As ′ y ) ds ′ − e At Z t ds Z s e − As ′ ∂ g I ∂ x ( e As ′ y ) g K ( e As ′ y ) ds ′ . Since Dg K · g K ∈ V K and Dg I · g K ∈ V I by Props.3.3 and 3.4, we obtain x = e At h (2) ( y ) + e At Z t e − As ∂ g ∂ x ( e As y ) Q ( g I )( e As y ) + g ( e As y ) ! ds + e At ∂ g K ∂ x ( y ) g K ( y ) t + Q ∂ g I ∂ x g K ! ( e At y ) t − e At Q ∂ g I ∂ x g K ! ( y ) t − e At Z t ∂ g K ∂ x ( y ) g K ( y ) sds − e At Z t e − As Q ∂ g I ∂ x g K ! ( e As y ) − Q ∂ g I ∂ x g K ! ( y ) ! ds = e At h (2) ( y ) + e At Z t e − As ∂ g ∂ x Q ( g I ) + g − ∂ Q ( g I ) ∂ x g K ! ( e As y ) ds + e At ∂ g K ∂ x ( y ) g K ( y ) t + ∂ Q ( g I ) ∂ x ( e At y ) g K ( e At y ) t . Since R is defined by Eq.(3.34), the above is rewritten as x = e At h (2) ( y ) + e At Z t e − As P I ( R )( e As y ) ds + e At Z t e − As P K ( R )( e As y ) ds + e At ∂ g K ∂ x ( y ) g K ( y ) t + ∂ Q ( g I ) ∂ x ( e At y ) g K ( e At y ) t = e At h (2) ( y ) + QP I ( R )( e At y ) − e At QP I ( R )( y ) + e At P K ( R )( y ) t + e At ∂ g K ∂ x ( y ) g K ( y ) t + ∂ Q ( g I ) ∂ x ( e At y ) g K ( e At y ) t . Putting h (2) = QP I ( R ), we obtain Eq.(3.33).18 eferences [1] V. I. Arnold, Geometrical methods in the theory of ordinary di ff erential equations,Springer-Verlag, New York, 1988[2] J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, 1981[3] L. Y. Chen, N. Goldenfeld, Y. Oono, Renormalization group theory for global asymp-totic analysis, Phys. Rev. Lett. 73 (1994), no. 10, 1311-15[4] L. Y. Chen, N .Goldenfeld, Y. Oono, Renormalization group and singular perturba-tions: Multiple scales, boundary layers, and reductive perturbation theory, Phys. Rev.E 54, (1996), 376-394[5] H. Chiba, C approximation of vector fields based on the renormalization groupmethod, SIAM J. Appl. Dyn. Syst. Vol.7, 3 (2008), pp. 895-932[6] H. Chiba, Approximation of Center Manifolds on the Renormalization Group Method,J. Math. Phys. Vol.49, 102703 (2008)[7] H. Chiba, Extension and Unification of Singular Perturbation Methods for ODEsBased on the Renormalization Gourp Method, SIAM j. on Appl. Dyn.Syst., Vol.8,1066-1115 (2009)[8] H.Chiba, D.Pazo, Stability of an [N / ff erential topology, Springer-Verlag, New York-Heidelberg, 1976[11] J. Murdock, Normal forms and unfoldings for local dynamical systems, Springer-Verlag, New York, (2003)[12] J. Murdock, Hypernormal form theory: foundations and algorithms, J. Di ffff