Dynamical systems on the Liouville plane and the related strictly contact systems
aa r X i v : . [ m a t h . D S ] J un Dynamical systems on the Liouville plane and therelated strictly contact systems
Stavros Anastassiou
Center of Research and Applications of Nonlinear Systems (CRANS)University of Patras, Department of MathematicsGR-26500 Rion, [email protected]
Abstract.
We study vector fields of the plane preserving the form of Liouville. Wepresent their local models up to the natural equivalence relation, and describe localbifurcations of low codimension. To achieve that, a classification of univariate functionsis given, according to a relation stricter than contact equivalence. We discuss, inaddition, their relation with strictly contact vector fields in dimension three. Analogousresults for diffeomorphisms are also given.
Keywords systems preserving the form of Liouville, strictly contact systems,classification, bifurcations
MSC[2000]
Primary 37C15, 37J10, Secondary 58K45, 53D10 ynamical systems on the Liouville plane and the related strictly contact systems Introduction
Dynamical systems preserving a geometrical structure have been studied quiteextensively. Especially those systems preserving a symplectic form have attracted alot of attention, due to their fundamental importance in all kinds of applications.Dynamical systems preserving a contact form are also of interest, both in mathematicalconsiderations (for example, in classifying partial differential equations) and in specificapplications (study of Euler equations).The 1–form of Liouville may be associated both with a symplectic form (by takingthe exterior derivative of it) and with a contact form (by adding to it a simple 1–formof a new variable). We wish here to study dynamical systems respecting the form ofLiouville. As we shall see, they are symplectic systems which may be extented to contactones.To set up the notation, let M be a smooth (which, in this work, means continuouslydifferentiable the sufficient number of times) manifold of dimension 2 n + 1. A contactform on M is a 1-form α such that α ∧ ( dα ) n = 0. A strict contactomorphism is adiffeomorphism of M which preserves the contact form (their group will be denoted as Dif f ( M, a )) while a vector field on M is called strictly contact if its flow consists ofstrict contactomorphims (we denote their algebra as X ( M, a )). In terms of the definingcontact form α , we have f ∗ α = α for a strict contactomorphism f and L X α = 0 for astrictly contact vector field X, where L X α denotes the Lie derivative of α in the directionof the field X. The classical example of a strictly contact vector field associated to α is the vector field of Reeb, R a , uniquely defined by the equations α ( R α ) = 1 and dα ( R α , · ) = 0.Associated to every contact vector field X is a smooth function H : M → R , calledthe contact Hamiltonian of X, which is given as H = α ( X ). Conversely, every smoothfunction H gives rise to a unique contact vector field X , such that a ( X ) = H and dα ( X, · ) = ( L R α H ) α ( · ) − dH ( · ). Usually we write X H to denote the dependence ofvector field X H on its (contact) Hamiltonian function H .Results conserning the local behavior for systems of this kind may be found in[6, 11, 9, 4], where the authors provide explicit conditions for their linearization, in theneighborhood of a hyperbolic singularity. The study of degenerate zeros, and of theirbifurcations, remains, however, far from complete.Here, in section 1, we recall the form of strictly contact vector fields of R , and theirrelation with symplectic vector fields of the plane. We show that the albegra X ( R , xdy )of plane fields preserving the form of Liouville xdy may be obtained by projecting on R stictly contact fields with constant third component. We begin the classificationof vector fields belonging in X ( R , xdy ) (we shall call them Liouville vector fields) byintroducing the natural equivalence relation, and by showing that the problem of theirclassification is equivalent to a classification of functions up to a specific equivalencerelation.In section 2, (germs at the orign of) univariate functions are classified up to ynamical systems on the Liouville plane and the related strictly contact systems X ( R , xdy ) are also given.Next, in section 5, we see that there is only one polynomial member of the group ofplane diffeomorphisms preserving the form of Liouville ( Dif f ( R , xdy ) stands for thisgroup). This is the linear Liouville diffeomorphism, and we show the linearization ofplane diffeomorphisms of this kind at hyperbolic fixed points.In section 6, we return to members of X ( R , a ) to observe that the models obtainedabove are members of a specific base of the vector space of homogeneous vector fields.Their linearization is again shown, albeit using classical methods of normal form theory.Last section contains some observations concerning future directions.For a classical introduction to symplectic and contact topology the reader shouldconsult [8], while [12] offers a more complete study of the contact case. Singularities ofmappings are treated in a number of textbooks; we recommend [7, 3] and [5] (see [15]for a recent application of singularity theory to problems of dynamics).
1. Strictly contact vector fields and fields of Liouville
Let M be a closed smooth manifold of dimension 2n+1 equipped with a contact form α . The contact form is called regular if its Reeb vector field, R α , generates a free S action on M. In this case, M is the total space of a principal S bundle, the so calledBoothby-Wang bundle (see [2] for more details): S k −→ M π −→ B ,where k : S → M is the action of the Reeb field and π : M → B is the canonicalprojection on B = M/ S . B is a symplectic manifold with symplectic form ω = π ∗ da .The projection π induces an algebra isomorphism between functions on the base B andfunctions on M which are preserved under the flow of R α (such functions are calledbasic). It also induces a surjective homomorphism between strictly contact vector fieldsX of ( M, α ) and hamiltonian vector fields Y of (
B, ω ) (that is, fields Y with L Y ω = 0),the kernel of which homomorphism is generated by the vector field of Reeb.In our local, three dimensional, case, things are of course simpler. Using alocal Darboux chart, consider the euclidean space R equipped with the standard ynamical systems on the Liouville plane and the related strictly contact systems α = dz + xdy . Its Reeb vector fiel, R α = ∂∂z , induces the action ϕ t ( x, y, z ) = ( x, y, z + t ), and the quotient of R by this action, that is, the plane R withcoordinates ( x, y ), inherits the symplectic form ω = π ∗ dα = dx ∧ dy . Strictly contactvector fields of R project to hamiltonian fields on this plane (for a direct analogy withthe volume–preserving case the reader should consult [10]).Basic functions now depend, as one may easily verify, only on the first two variables,while the kernel of the above mentioned projection contains the multiples of ∂∂z . Studyingequation L X α = 0 we get the general expression of X = X ∂∂x + X ∂∂y + X ∂∂z ∈ X ( R , α ): X = ( − ∂∂y R X ( x, y ) dx ) ∂∂x + X ( x, y ) ∂∂y + ( − xX ( x, y ) + R X ( x, y ) dx ) ∂∂z . Its contact Hamiltonian is of course H ( x, y, z ) = R X ( x, y ) dx (recall that it does notdepend on the third variable), thus: X = − ∂H ( x,y,z ) ∂y ∂∂x + ∂H ( x,y,z ) ∂x ∂∂y + ( H ( x, y, z ) − x ∂H ( x,y,z ) ∂x ) ∂∂z . Observe that all vector fields of the ( x, y )–plane, preserving the symplectic structure dx ∧ dy , may be obtained in this way.In this work we restrict our attention to those members of X ( R , α ), which preservethe form of Liouville xdy (we shall denote their set as X L ( R , α )). The reason for thischoise will become clear in section 6. In this case, equation L X α = 0 becomes: X ( x, y ) = − x dX ( y ) dy , (1)while X ( x, y ) = c ∈ R . Thus, their general form is − x dh ( y ) dy ∂∂x + h ( y ) ∂∂y + c ∂∂z , for someunivariate function h ( y ) and a constant c . Observe that all vector fields of the planepresrving the form of Liouville may be obtained by projecting the members of X L ( R , α )on the z = 0 plane. We have, therefore, the following: To every h ∈ C k ( R , R ) , k ≥
2, corresponds a unique X ∈ X ( R , xdy ),namely − x dh ( y ) dy ∂∂x + h ( y ) ∂∂y . Members of X L ( R , α ) are trivially obtained by addingconstant multiples of ∂∂z to members of X ( R , xdy ).This lemma provides the general form of the vector fields we are interested in.Our goal is the classification of these vector fields according to the natural relationdefined in the obvious way: two fields X, Y ∈ X ( R , xdy ) are Liouville conjugate if thereexists a diffeomorphism of the plane preserving the form of Liouville, φ ∈ Dif f ( R , xdy ),such that φ ∗ X = Y , while two fields Z, W ∈ X ( R , a ) are strictly contact conjugateif a ψ ∈ Dif f ( R , α ) exists, such that ψ ∗ Z = W . Observe that classifying members of X ( R , xdy ) leads to a classification of fields belonging in X L ( R , a ); one needs only toextend φ to R as ψ ( x, y, z ) = ( φ ( x, y ) , z ).To proceed with the classification of Liouville vector fields of the plane, we shallexploit their dependence on real valued functions. Let f be a univariate function and ϕ a diffeomorphism of R . The Liouvillevector field corresponding to function f may be transformed, via a diffeomorphismrespecting the form xdy , to the Liouville vector field corresponding to the function φ ′ ( y ) f ( φ ( y )). ynamical systems on the Liouville plane and the related strictly contact systems Proof.
Constructing the fields corresponding to these two functions, according to therecipe given in lemma Lemma 1.1, we conclude that the diffeomorphism accomplishingthe desired transformation is ψ ( x, y ) = ( xφ ′ ( y ) , φ ( y )) which also preserves the Liouvilleform.This lemma ensures that the classification of Liouville vector fields, up todiffeomorphisms belonging in Dif f ( R , xdy ), reduces to a classification of univariatereal functions. In the next section, we turn our attention to this classification.
2. Restricted contact equivalence
Let f : ( R , → ( R ,
0) be the germ at the origin of a smooth function. Their ring willbe denoted as E . We introduce the following equivalence relation. Let f, g ∈ E . We shall call them restrictively contact equivalent ( RK -equivalent) if there exists a germ of a smooth diffeomorphism ϕ : ( R , → ( R ,
0) suchthat g = ϕ ′ ( f ◦ ϕ ). Let f, g ∈ E , with f ( x ) = x, g ( x ) = x + x . Define ϕ ( x ) = xx +1 . It iseasy to check that ϕ is a local diffeomorphism at ∈ R and g = ϕ ′ ( f ◦ ϕ ) . Let us recall here that two univariate function germs f, g ∈ E are called contactequivalent if f ( x ) = M ( x ) g ( ϕ ( x )), for some function germ M ( x ) and diffeomorphism ϕ . The equivalence relation we study here requires M ( x ) = ϕ ′ ( x ) . This explains why wecalled the above defined equivalence relation restricted contact.Suppose now that g s ∈ E is a curve of RK –equivalent germs, depending on thereal parameter s , with g = f . There exists thus a curve of local diffeomorphisms ϕ s :( R , → ( R , ϕ s (0) = 0 , ∀ s ∈ R and ϕ ( x ) = x , such that g s ( x ) = ϕ ′ s ( x ) f ( ϕ s ( x )).Differenting with respect to s and evaluating at s = 0 we get: ∂∂s g s ( x ) | s =0 = − X ′ ( x ) f ( x ) + f ′ ( x ) X ( x ),where X ( x ) is defined by the relation ∂∂s ϕ s ( x ) = X ( ϕ s ( x )). Note that X (0) = 0, thus X ( x ) ∈ m , the ideal of E generated by x ∈ E . Let f ∈ E . The ideal generated from the germs − X ′ ( x ) f ( x ) + f ′ ( x ) X ( x ) , X ∈ m , equals h f ( x ) i + f ′ ( x ) m . Proof.
It is obvious that, if X ( x ) ∈ m , then − X ′ ( x ) f ( x ) + f ′ ( x ) X ( x ) is a member of h f ( x ) i + f ′ ( x ) m . Let us prove the opposite inclusion.Let h ∈ h f ( x ) i + f ′ ( x ) m . Germs g ∈ E and k ∈ m exist, such that h ( x ) = g ( x ) f ( x ) + f ′ ( x ) k ( x ). We wish to find a germ X ∈ m such that: h ( x ) = − X ′ ( x ) f ( x ) + X ( x ) f ′ ( x ) ⇒ g ( x ) f ( x ) + f ′ ( x ) k ( x ) = − X ′ ( x ) f ( x ) + X ( x ) f ′ ( x ). ynamical systems on the Liouville plane and the related strictly contact systems X ( x ) = ( k ( x ) − f ( x ) R x g ( t )+ k ′ ( t ) f ( t ) dt if x = 00 if x = 0which is well defined and smooth in a neighborhood of the origin and, therefore, forevery h ∈ h f ( x ) i + f ′ ( x ) m a X ∈ m exists, such that h = − X ′ ( x ) f ( x ) + X ( x ) f ′ ( x ),hence the conclusion.Under the light of the lemma above, we proceed to the following: The tangent space of f ∈ E , with respect to RK –equivalence, is definedto be T RK f := h f ( x ) i + f ′ ( x ) m . The codimension of f is defined as codim RK ( f ) := dim ( m/T RK f ) . We calculate that, if f ( x ) = x , then T RK f = m , thus codim RK ( f ) = 0 ,while if g ( x ) = x , T RK g = m and codim RK ( g ) = 1 . As usual, the germ f ∈ E is called k –determined, with k ∈ N , if every other g ∈ E having the same k –jet with f is RK –equivalent to f. If such a finite k does not exist,we say that f is not finitely determined. The germ f ∈ E is k –determined, with respect to RK -equivalence, if m k +1 ⊆ mT RK f .Proof. We have to prove that if h ∈ m k +1 ⊆ mT RK f , the germs f and f + h are RK –equivalent.Towards this end, define f s = f + sh, s ∈ [0 , ϕ s ( x ), defined in a neighborhood of the origin, such that f s = ϕ ′ s f ( ϕ s ( x )). Differentiating with respect to s , we get: h ( x ) = − ϕ ′ s ( x ) X ′ ( ϕ s ( x )) f ( ϕ s ( x )) + ϕ ′ s ( x ) X ( ϕ s ( x )) f ′ ( ϕ s ( x )).Note that, for s = 0, we get the relation h ( x ) = − X ′ ( x ) f ( x ) + X ( x ) f ′ ( x ), which, by theprevious lemma, has a solution X ( x ) ∈ m since m k +1 ⊆ mT RK f . We need to show thata solution exists for all s ∈ [0 , R × [0 , R be the ring of function germs at 0 × [0 ,
1] and denote by m s the ideal of R consisting of those germs vanishing at 0 × [0 , m k +1 ⊆ m k +1 s ⊆ m s h f i R + f ′ ( x ) m s ⊆ m s h f s i R + m s h h i R + f ′ s m s + h ′ m s ⊆ m s h f s i R + f ′ s m s + m k +2 s ⊆ m s h f s i R + f ′ s m s + m s m k +1 s ⊆ m s ( h f s i R + f ′ s m s ), ynamical systems on the Liouville plane and the related strictly contact systems s ∈ [0 , h ∈ m k +1 ⊆ m s T RK f s . We can therefore find X s ( x ) ∈ m s ,defining the germ of diffeomorphism ϕ s which, for s = 1, establishes an equivalencebetween f and f + h .The classification of the elements of E now follows. We begin with germs that either donot vanish at the origin, or have a regular point there. Let f ∈ E . If f (0) = 0, it is RK –equivalent to 1, while if f (0) = 0 and f ′ (0) = a = 0, f is RK − equivalent to ax . Proof.
Let f ∈ E , with f (0) = 0. To show that it is RK –equivalent to 1, we must finda local diffeomorphism k ( x ) such that k ′ ( x ) = f ( x ), which is the same as k ′ ( x ) = f ( x ) ,which is a differential equation with smooth right hand side, at least in a neighborhoodof the origin, thus, such a smooth k ( x ) exists.On the other hand, let f (0) = 0 and f ′ (0) = a = 0. It is 1–determined, thus RK –equivalent to its linear part ax , while, as may be easily verified, the germ ax is RK –equivalent to bx only if a = b .Let us know proceed to germs with critical points. Let f ∈ E , with f (0) = f ′ (0) = ... = f k − (0) = 0 and f k (0) = 0 , k > f is RK –equivalent to x k , if k is an even number and to x k or − x k , if k is an oddnumber. Proof. If f is such a germ, then, in a neighborhood of the origin, we may write f ( x ) = x k g ( x ), with g (0) = 0. Thus T RK f = m k , and f is k –determined. It is thus RK –equivalent to ax k , while, as may easily be verified, the germ of a diffeomorphism ϕ ( x ) exists such that ϕ ′ ( x ) aϕ k ( x ) = x k , for every a ∈ R \ { } , if k is even, while if k isodd then ax k is RK –equivalent to − x k , for a < x k , for a > E . If a member of E does not vanish at the origin it is RK -equivalent tothe constant function . Members of E having codimension are RK -equivalent to ax ( a being the value of their derivative there). A member of E of odd codimension k is RK -equivalent to x k +1 , while if it is of even codimension k it is RK -equivalent to ± x k +1 ,depending on the sign of the value of its first non–vanishing derivative at the orgin. Table 1 contains the local models of members of E having codimension up tofive. We note that there are differences with the classical classification list for rightequivalence (in which list the A , A and A models may have both negative andpositive sign) and for contact equivalence (in which, for example, the A model doesnot depend on the constant a , see [7, 5]). The interested reader should consult [14] fora relation of contact and right equivalence, while the equivalence relation studied hereprovides more models than right and contact equivalence since it is stricter than both. ynamical systems on the Liouville plane and the related strictly contact systems symbol codimension function1 A a ayA y A ± ± y A y A ± ± y A y
3. Local models for members of X ( R , xdy )We return now to our study of vector fields of the plane, which preserve the form ofLiouville. To construct their local models, we make use of lemma Lemma 1.2 along withtheorem Theorem 2.9. (of regular points) Let X ∈ X ( R , xdy ) be such that X (0) = 0. Then,in a neighborhood of zero, it is conjugate, via a diffeomorphism preserving the form ofLiouville, to the constant vector field ∂∂y . Proof.
Since X ∈ X ( R , xdy ), it is of the form X ( x, y ) = − xf ′ ( y ) ∂∂x + f ( y ) ∂∂y , for asmooth, real valued, function f(y). Since X (0 ,
0) = f (0) ∂∂y = 0, we get f (0) = 0, whichmeans that f is RK –equivalent to the constant function 1, thus, by lemma Lemma 1.2,a diffeomorphism preserving xdy exists, transforming X to ∂∂y .Let us now turn our attention to hyperbolic singularities. (hyperbolic singularities) Let X ∈ X ( R , xdy ) having a hyperbolicsingularity at the origin. Then, in a neighborhood of zero, it is conjugate, via adiffeomorphism preserving the form of Liouville, to the vector field − ax ∂∂x + ay ∂∂y . Proof.
The vector field is of the form − xf ′ ( y ) ∂∂x + f ( y ) ∂∂y , and it is easy to check thatthe eigenvalues of zero are − f ′ (0) and f ′ (0). Thus zero is a hyperbolic singularityif, and only if, f ′ (0) = 0, and f is therefore RK –equivalent to ay , a = f ′ (0). Theexistence of a diffeomorphism transforming X to − ax ∂∂x + ay ∂∂y is guarantied, by lemmaLemma 1.2.We see that, at a hyperbolic singularity, all members of X ( R , xdy ) are topologicallyequivalent: they are of the saddle type. Up to diffeomorphisms respecting the form ofLiouville, however, their equivalence classes are classified by a real number.The lemmata above ensure that the first non–vanishing jet of members of X ( R , xdy ) completely determine their local behavior, at least in the simplest cases.Actually, this holds in general. Let
X, Y ∈ X ( R , xdy ) . If j k X (0) = j k Y (0) = 0 , k = 0 , .., i − , and j i X (0) = j i Y (0) = 0 , for some i ∈ N \ { } , there exists a diffeomorphism preserving xdy which conjugates X and Y .ynamical systems on the Liouville plane and the related strictly contact systems X ( R , xdy ), up to codimension 5.Table 2 symbol codimension local model ∂∂y A a − ax ∂∂x + ay ∂∂y A − xy ∂∂x + y ∂∂y A − xy ∂∂x + y ∂∂y A − xy ∂∂x + y ∂∂y A − xy ∂∂x + y ∂∂y A − xy ∂∂x + y ∂∂y For the cases A ± , A ± we have ommited writing the vector fields for the negativeand the positive sign since one may be obtained from the other after a multiplicationwith − x –axis). Othen than that, topologically their behavior is quitesimple to analyze, since fuction xf ( y ) serves as a first integral.It remains to analyze the behavior of pertubations of these vector fields.
4. Bifurcations of low codimension
At regular points, members of X ( R , xdy ) are all conjugate to each other, via adiffeomorphism preserving the form of Liouville. At hyperbolic singularities all suchvector fields may be transformed to their linear part; these linear parts are not conjugateto each other, since the eigenvalues there are a conjugacy invariant. However, up totopological equivalence, they are all saddle points, thus hyperbolic singularities arestructurally stable.This is no more the case when we analyze vector fields belonging to the classes A k , k ≥
1. To describe their local bifurcations we should first compute their transversalunfoldings.
Let X be the germ at the origin of a Liouville vector field. Denoteby S its singularity class (that is, the set of all germs at the origin of vector fields ofLiouville which are Liouville equivalent to X ). A transversal unfolding of X consists ofa set of germs at the origin of Liouville vector fields, which set intersects S transversallyat X .Thus, to construct transversal unfoldings of Liouville vector fields, we must first computethe tangent spaces of singularity classes. Let X f ∈ X ( R , xdy ) (where f is the function defining X f ) and S itssingularity class. We have:ynamical systems on the Liouville plane and the related strictly contact systems T X f S = { X g ∈ X ( R , xdy ) /g ∈ h f i} .Proof. Let X f = − xf ′ ( y ) ∂∂x + f ( y ) ∂∂y be the germ at the origin of a Liouville vectorfield and ψ s ( x, y ) = ( xϕ ′ s ( y ) , ϕ s ( y )) the germ at the origin of a family of diffeomorphismspreserving the Liouville form, where ϕ ( y ) = y , ϕ s (0) = 0 and ϕ ′ s (0) = 0. Define: X s = ψ s ∗ X f = ( − xf ′ ( y ) ϕ ′ s ( y ) − xf ( y )( ϕ ′ s ( y )) ϕ ′′ s ( y )) ∂∂x + ϕ ′ s ( y ) f ( y ) ∂∂y .It is a curve of Liouville vector fields belonging to S , and we have X = X f . To calculatethe tangent space T X f S we need to evaluate at s = 0 the derivative with respect to theparameter s of X s . It is: ∂∂s X s | s =0 = ( − xf ′ ( y )Φ ′ ( y ) − x Φ ′′ ( y ) f ( y )) ∂∂x + Φ ′ ( y ) f ( y ) ∂∂x .We have denoted as Φ( y ) the vector field defined by ∂∂s ϕ s ( y ) = Φ( ϕ s ( y )). Note that ∂∂s X s | s =0 is a Liouville vector field, corresponding to the function Φ ′ ( y ) f ( y ), whichbelongs to h f i E , since Φ ∈ m . Thus, the tangent space of S at X f consists of thoseLiouville fields corresponding to functions belonging in the ideal h f i E .The theorem above allows us to study bifurcations of Liouville vector fields. Toillustrate this, we present here such bifurcations of low codimension.We begin with the singularity class A . The members of this class form asubset of codimension 1 in the set of those members of X ( R , xdy ) vanishing at theorigin. To transversally unfold them, we only need to add to their local model,linear terms preserving the form of Liouville. We arrive thus at the vector field Q a ( x, y ) = ( − ax − xy ) ∂∂x + ( ay + y ) ∂∂y , where a a real parameter. We have thefollowing: The set of X ∈ X ( R , xdy ) with j X (0) = j X (0) = 0 and j X (0) = 0 has codimension 1 in the set of Liouville vector fields vanishing at theorigin. Its members are all conjugate to the A model given above. The curve of vectorfields Q a ( x, y ) intersects at a = 0 this set transversally.Proof. The codimension and the conjugacy to the A model follows easily fromthe analysis given in the previous sections. Note that Q ( x, y ) is the A model,corresponding to the function y . The intersection is transversal, since: ∂∂a Q a ( x, y ) | a =0 = − x ∂∂x + y ∂∂y .This is a Liouville vector field corresponding to the function y which is the only function(up to a constant) which vanishes at the origin and belongs to E / h y i .Thus, Q a ( x, y ) is a transversal unfolding of the A singularity. Vector fields dependingon a single parameter undergoe, for isolated values of this parameter, the bifurcationdepicted in Figure 1; this bifurcation is therefore the codimension 1 bifurcation occuringin vector fields of interest. ynamical systems on the Liouville plane and the related strictly contact systems x xy xy Figure 1.
Bifurcation of codimension one: a < , a = 0 , a >
0. The dotted line inthe center picture stands for the line of singularities. xy xy xy
Figure 2.
Bifurcations of codimension two: a n , b n < a = b = 0 (center), a > , b = 0 (right). The dotted line in the center picture stands for the line ofsingularities. We proceed to bifurcations of codimension two. Consider the A model, and add to itterms of lower degree. We arrive at T a,b = ( − ax − bxy − xy ) ∂∂x + ( ay + by + y ) ∂∂y ,where a, b real parameters. We have the following: The set of X ∈ X ( R , xdy ) with j X (0) = j X (0) = j X (0) = 0 and j X (0) = 0 has codimension 2 in the set of Liouville vector fields vanishing at theorigin. Its members are all conjugate to the A model given above. The surface of vectorfields T a,b ( x, y ) intersects at a = b = 0 this set transversally. Its proof goes along the lines of the previous proposition, and it is therefore omitted.In figure 2 we present the bifurcations system T a,b system undergoes, for characteristicparameter values.Before discussing the diffeomorphism case, let us note that we could studybifurcations of arbitrary, finite, codimension following the exact same approach. ynamical systems on the Liouville plane and the related strictly contact systems
5. Plane diffeomorphisms preserving the form of Liouville
Let us now turn our attention to diffeomorphisms of the plane respecting the form ofLiouville. As we saw, they are of the general form f ( x, y ) = ( xh ′ ( y ) , h ( y )). Diffeomorphism h ( y ) of R uniquely defines such a diffeomorphism.The unique linear diffeomorphism preserving the form of Liouville (and the origin)is thus ( x, y ) ( ax, a y ). Aside from this, there are no other polynomial members of Dif f ( R , xdy ); as a consequence, finite jets (of any order) of Liouville diffeomorphismsstudied here do not belong to the same group.The classification of strict contactomorphisms, according to the natural equivalencerelation, is of course our purpose; f, g ∈ Dif f ( R , xdy ) are Liouville conjugate if thereexists a third Liouville diffeomorphism φ such that f ◦ φ = φ ◦ g . To continue, and sincewe focus on fixed points, we impose the conditions f (0) = g (0) = 0.Generically, such diffeomorphisms may be linearized in a neighborhood of the origin. There exists a codimension zero subset of those members of
Dif f ( R , xdy ) vanishing at the origin, every member of which may be transformed,via a change of coordinates preserving the Liouville form, to its linear part.Proof. Let us consider the set of Liouville diffeomorphisms having linear part( ax, a y ) , a = ±
1. Its codimension is zero (in the set of Liouville diffeomorphismsvanishing at the origin) and its members are of the form f ( x, y ) = ( xh ′ ( y ) , h ( y )) where h ( y ) = a y + h.o.t. a local diffeomorphism (we use h.o.t. as an abbreviation for ”higherorder terms”).We have supposed that a = ±
1; therefore a local diffeomorphism ψ of R existssuch that ψ ◦ h ◦ ψ − = a y (this is the content of the Sternberg linearizationtheorem, see [1]). Using this diffeomorphism define φ ( x, y ) = ( xψ ′ ( y ) , ψ ( y )) and observethat it is a diffeomorphism, preserving the Liouville form, with inverse φ − ( x, y ) =( x ( ψ − ( y )) ′ , ψ − ( y )).As is easy to confirm, ψ ◦ f ◦ ψ − = ( ax, a y ).We have thus found the generic model for the mappings under study,that is ( x, y ) ( ax, a y ). As already remarked, it is actually the unique polynomial model for membersof Dif f ( R , xdy ); thus Liouville diffeomorphisms either may be linearized or arenot finitely determined (at least finitely determined under the relation of Liouvilleconjugacy).
6. Homogeneous members of X ( R , a ) and linearization Having completed the study of vector fields of Liouville we may now state results forstrictly contact vector fields of R . Indeed, one needs only to add constant multiples of ∂∂z to the local models presented above, to obtain vector fields which preserve both thecontact form a and the form of Liouville. ynamical systems on the Liouville plane and the related strictly contact systems X L ( R , a ) stems from the fact thatthey are the only strictly contact vector fields which may have homogeneous components.Indeed, recall from section 1 the general form of a strictly contact vector field: X = − ∂H ( x,y,z ) ∂y ∂∂x + ∂H ( x,y,z ) ∂x ∂∂y + ( H ( x, y, z ) − x ∂H ( x,y,z ) ∂x ) ∂∂z .Assuming that H ( x, y, z ) (remember it does not depend on z ) is a homogenouspolynomial of degree d , vector field X above is homogeneous of degree d − d = 1, or zero, for d ≥
2. Members of X L ( R , a )are therefore the only homogeneous members of X ( R , a ). We shall elaborate in thisobservation in this section, to show, using classical normal form theory, the linearizationof strictly contact vector fields respecting the form of Liouville.Consider members of X ( R , a ) vanishing at the origin. If X is such a field, let X = X + X + ... + X k be its k –jet at zero, for some natural number k , where each X i , i = 1 , ..k, is a homogeneous field of degree k . It is easy to see, equating terms ofthe same degree in equation L X ( a ) = 0, that each X i is itself a member of X ( R , a ).We denote as X d ( R , a ) the subset of X ( R , a ), the components of which arehomogeneous functions of degree d. We easily prove the following: The vector space X d ( R , a ) is one dimensional. For each d ∈ N \ { } , itsbase consists of the field X d = dxy d − ∂∂x − y d ∂∂y .The local models of Table 2 constitute, therefore, the basis generating the fields ofinterest.Linear fields (belonging to X ( R , a ) are of the form X = ax ∂∂x − ay ∂∂y , with a arbitrary constant. In our case, therefore, the existence of hyperbolic singularities isexcluded (actually, X is also the unique linear member of X ( R , α ); strictly contactvector fields do not possess hyperbolic singularities). Despite this fact, fields havingnon–zero linear part can be linearized, in a neighborhood of the origin. We shall proveit now using an approach different from the one indicated above.There are ( d + 3 d + 2) monomials depending on three variables, having degree d , as simple counting arguments may assure. Thereupon, the vector space X d ( R ) ofhomogeneous vector fields of degree d is of dimension ( d + 3 d + 2), and one may easilyverify that the fields appearing in Table 3, being ( d + 3 d + 2) independent vector fieldsof degree d , constitute a basis of it. Vector fields of interest here belong to this base (to obtain them, justset m = m = 0 to the first field of the second class). This base was presented, inthe general n–dimensional case, in [13], section 4 of which contains the arguments weshall use to prove the next proposition. The author wishes to thank Prof. J D Meissfor clarifying them to him. ynamical systems on the Liouville plane and the related strictly contact systems fields condition number y m z m ∂∂x x m z m ∂∂y x m y m ∂∂z m + m = d d + 3(1 + m ) x m +1 y m z m ∂∂x − (1 + m ) x m y m +1 z m ∂∂y (1 + m ) x m y m +1 z m ∂∂y − (1 + m ) x m y m z m +1 ∂∂z m + m + m + 1 = d d + dx m +1 y m z m ∂∂x + x m y m +1 z m ∂∂y + x m y m z m +1 ∂∂z m + m + m + 1 = d ( d + d ) If X ∈ X d ( R ), the vector field [ X , X ], where X is the unique, linear and non–zero, strictly contact vector field presented above, is also homogeneous of degree d (thebrackets [ · , · ] denote the usual commutator of vector fields). We may define thereforethe operator ad X : X d ( R ) → X d ( R ), X [ X , X ]. Vector fields belonging to thebase of X d ( R ) are eigenvectors of this operator; thus the subspaces generated by themare invariant under ad X , ensuring the diagonal form of its matrix. There exists a codimension zero subset of X L ( R , α ) every memberof which may be transformed to its linear part. The linearizing diffeomorphism is closeto the identity and preserves the contact form.Proof. The subset we refer to is the set of vector fields of interest with non zerolinearization, and its codimension is easily obtained.Classical normal form theory ensures that, by changing coordinates, we may discardall terms of X = X + X + ... ∈ X L ( R , α ) which are not contained in the complementof the range of this operator (an operator which leaves invariant the spaces X d ( R , α ),as well as the subspaces generated by the basic vector fields, the subspace of fields whichinterest us included).The matrix of X is self–adjoint, so a complement to the range of ad X is the kernelof this operator. This kernel however, as may easily be verified, is trivial, providingus with a diffeomorphism which transforms to its (non–zero) linear part every field of X L ( R , α ). This diffeomorphism preserves the 1–form defining the contact structure;this stems from the diagonal form of the matrix of ad X .Strictly contact vector fields project to symplectic fields of the plane; homogeneousstrictly contact vector fields project to fields of the plane preserving the form of Liouville.We have studied here the local behavior of the later; the local study of the first remainsa challenging task. ynamical systems on the Liouville plane and the related strictly contact systems
7. Conclusions
Contact systems have a long history, and attract a lot of attention, since they forma valuable tool in topological constructions, in Hamiltonian dynamics and in manyphysical applications (see [12] for a textbook account of these fields, and furtherreferences).Almost all contact systems possess hyperbolic singularities, as transversalityarguments show. In this case, conditions for linearization have been obtained ([11, 9, 4]).Results are much more rare, however, if the singularities are degenerate.We chose here to consider the simpler case of homogeneous strictly contact systems.This led us to the study of plane systems, preserving the form of Liouville, a subjectwhich has an interest of its own. To study these fields we had to classify univariatefunctions according to the restricted contact equivalence relation. All these admitgeneralizations and deserve more study.Indeed, extending the definition of restricted contact equivalence to arbitrarydimensions we get of course the differential conjugacy relation for vector fields. Onecould probably reobtain results of normal form theory, using this approach, which wouldpotentially help the problem of classifying vector fiels preserving the form of Liouvillein any dimension.And, as already mentioned, the general problem of analyzing the behavior of contactdynamical systems stands, both interesting and difficult. The author hopes to furthercomment on these subjects in the future.
Acknowledgments
This work is dedicated to my two professors, Tassos Bountis and Spyros Pnevmatikos,on the occasion of their 65th birthday. It is only a pleasure for the author to acknowledgethe influence they had on him and to thank them for their constant support. [1] Sternberg S, ”Local C n transformations of the real line” Duke Math. J., 24, 97-102, 1957.[2] Boothby W M, Wang H C, ”On contact manifolds”, Ann. of Math., 2, 68, 721-734, 1958.[3] Br¨ocker T, ”Differential Germs and Catastrophes”, Cambridge University Press, 1975.[4] Lyˇ c agin V V, ”On sufficient orbits of a group of contact diffeomorphisms”, Math. USSR Sbornik,33, 2, 223-242, 1977.[5] Damon J, “The unfolding and determinacy theorems for subgroup of A and K ”, Mem.ofAm.Math.Soc., 50, 306, 1984.[6] Chaperon Mark, ”G´eom´etrie Diff´erentielle et Singularit´es des Syst`emes Dynamiques”, Ast´erisque,138-139, 1986.[7] Arnol’d V I, Gusein–Zade S M, Varchenko A N, ”Singularities of Differentiable Maps“, Birkhauser,1986.[8] Arnol’d V I, ”Mathematical Methods of Classical Mechanics“, Springer, 1989.[9] Banyaga A, de la Llave R, Wayne C E, ”Cohomology equations near hyperbolic points andgeometric versions of Sternberg linearization theorem”, Journal of Geometric Analysis, 6(4),613-649, 1996.[10] Haller G, Mezic I, ”Reduction of three–dimensional, volume–preserving flows with symmetry”,Nonlinearity, 11, 319-339, 1998. ynamical systems on the Liouville plane and the related strictly contact systems16