Separatrix splitting at a Hamiltonian bifurcation
We discuss the splitting of a separatrix in a generic unfolding of a degenerate equilibrium in a Hamiltonian system with two degrees of freedom. We assume that the unperturbed fixed point has two purely imaginary eigenvalues and a double zero one. It is well known that an one-parametric unfolding of the corresponding Hamiltonian can be described by an integrable normal form. The normal form has a normally elliptic invariant manifold of dimension two. On this manifold, the truncated normal form has a separatrix loop. This loop shrinks to a point when the unfolding parameter vanishes. Unlike the normal form, in the original system the stable and unstable trajectories of the equilibrium do not coincide in general. The splitting of this loop is exponentially small compared to the small parameter. This phenomenon implies non-existence of single-round homoclinic orbits and divergence of series in the normal form theory. We derive an asymptotic expression for the separatrix splitting. We also discuss relations with behaviour of analytic continuation of the system in a complex neighbourhood of the equilibrium.
1 Set up of the problem
Normal form theory provides a powerful tool for studying local dynamics near equilibria. The normal form theory uses coordinate changes in order to represent equations in the simplest possible form. The normal form often possesses additional symmetries which are not present in the original system. For a Hamiltonian system a continuous family of symmetries implies existence of an additional integral of motion due to Noether theorem. In the case of two degrees of freedom, an additional integral of motion makes the dynamics integrable. In this way the normal form theory looses information on non-integrable chaotic dynamics possibly present in the original system.
The accuracy of the normal form theory depends on the smoothness of the original vector field and, in the analytic theory, the error becomes smaller than any order of a small parameter and in some cases an exponentially small upper bound can be established.
In this paper we illustrate this situation considering a classical generic bifurcation of an equilibrium in an one parameter analytic family of Hamiltonian systems with two degrees of freedom (see e.g. ). Let us describe our set up in more details. Let be a real-analytic family of Hamiltonian functions defined in a neighbourhood of the origin in endowed with the canonical symplectic form . The dynamics are defined via the canonical system of Hamiltonian differential equations
where . The corresponding flow preserves the Hamiltonian and symplectic form . It is convenient to write down the Hamiltonian equations in the vector form
where is the standard symplectic matrix and .
We assume that for the origin is an equilibrium of the Hamiltonian system with a pair of purely imaginary eigenvalues and a double zero one . More precisely, we assume that and the Hessian matrix is already transformed to the diagonal form , which can be achieved by a linear canonical transformation provided is not semi-simple. Then for any integer there is an analytic canonical change of variables which transforms the Hamiltonian to the following form
where , is polynomial in and , and the remainder term has a Taylor expansion which starts with terms of order , i.e., . In the analytic case, this remainder can be made even exponentially small [18, 17, 15]. Our assumptions imply that
Then the lower order terms of have the form
where we have explicitly written down all quadratic and some of the cubic terms. In a generic family . Without loosing in generality and for greater convenience we assume .
Since the remainder term in (2) is small, it is natural to make a comparison with the dynamics of the normal form described by the truncated Hamiltonian function
Obviously, the Poisson bracket and consequently is an integral of motion for the normal form. For every fixed equation (4) represents a natural Hamiltonian system in a neighbourhood of the origin on the -plane. Depending on the values of and , the potential takes one of three shapes shown on Figure 1.
Respectively, the equation has either none, one or two solutions located in a small neighbourhood of the origin. These solutions correspond either to a periodic orbit (if ) or to an equilibrium (if ) of the normal form. We note that for Figure 1 (a), (b) and (c) correspond respectively to , and . When the potential has the shape of Figure 1 (c) the corresponding Hamiltonian system has a separatrix loop similar to the one shown on Figure 2.
This separatrix loop looks similar to the separatrix of the equation defined by the Hamiltonian
The situation can be summarised in the following way. The plane is invariant for the normal form Hamiltonian . The restriction of the normal form Hamiltonian onto this plane defines a Hamiltonian system with one degree of freedom. As crosses the zero, a pair of equilibria is created on this plane, one saddle point and one elliptic one. The normal form system has a separatrix solution which converges to the saddle equilibrium both at and . Trajectories located inside this separatrix loop are periodic. All other trajectories escape from a small neighbourhood of the origin and their behaviour cannot be studied using only the local normal form theory presented here.
The remainder term in the Hamiltonian (2) breaks the symmetry of the normal form and it is expected that in general the full equations do not possess neither an additional integral nor an invariant plane [10, 22]. Nevertheless, a part of the normal form dynamics survives. In particular, for the Hamiltonian system has a saddle-centre equilibrium with eigenvalues and , where is close to and is of order of . There are 4 solutions (separatrices) of the Hamiltonian system which are asymptotic to this equilibrium. These solutions converge to as or and are tangent asymptotically as to eigenvectors of , which correspond to the eigenvalues respectively.
Two of these separatrices are close to the separatrix loop of the normal form. The main objective of this paper is to study the difference between these separatrices. Our main theorem implies that the unstable solutions returns to a small neighbourhood of but, in general, misses the stable direction by a quantity which is exponentially small compared to . Consequently, the system (2) generically does not have a single-round homoclinic orbit for all sufficiently small . We also point out that our Main theorem implies existence of homoclinic trajectories for Lyapunov periodic orbits located on the central manifold exponentially close to the saddle-centre equilibrium (compare with similar statements for reversible systems in  and with the recent preprint ).
The dimension arguments [21, 5] show that the existence of a single-round homoclinic orbit to a saddle-centre equilibrium of a vector field in is expected to be a phenomenon of co-dimension between one and three depending on the presence (or absence) of Hamiltonian structure and reversible symmetries. In particular, the codimension one corresponds to a symmetric homoclinic orbit for a symmetric equilibrium in a reversible Hamiltonian system, and the codimension three corresponds to a non-symmetric homoclinic orbit to a symmetric equilibrium in a reversible non-Hamiltonian system. Treating and as independent parameters, Champneys  provided an example of a reversible vector field where lines of homoclinic points bifurcate from on the plane of .
The splitting of the one-round separatrix loop does not prevent existence of “multi-round” homoclinics, i.e., homoclinic orbits which make several rounds close to the separatrix of the normal form before converging to the equilibrium. Generically, if the system is both Hamiltonian and reversible, we expect the existence of reversible multi-round homoclinics for a sequence of values of the parameter which converges to . The study of these phenomena is beyond the goals of this paper.
Since in the limit the separatrix loop disappears and the ratio , the problem of the separatrix splitting near the bifurcation can be attributed to the class of singularly perturbed systems characterised by the presence of two different time-scales, similar to the problems considered in [10, 11, 22]. The difficulty of a singularly perturbed problem is related to the exponential smallness of the separatrix splitting in the parameter which requires development of specially adapted perturbation methods (see for example [28, 25, 6] and references in the review ).
The difficulties related to the exponential smallness do not appear in problems of the regular perturbation theory. At the same time dynamics of such systems share many qualitative properties with the singularly perturbed case.
If a reversible Hamiltonian system has a symmetric separatrix loop associated with a symmetric saddle-center equilibrium, then its one-parameter reversible Hamiltonian unfolding has multi-round homoclinic orbits for a set of parameter values which accumulate at the critical one [26, 14].
A generic two parameter unfolding of a Hamiltonian system which has a homoclinic orbit to a saddle-centre equilibrium was studies in , where countable sets of parameter values for which 2-round (and multi-round) loops are found.
The splitting of the separatrix loop has important consequences for the dynamics. The problem of constructing a complete description of the dynamics in a neighbourhood of a homoclinic loop to a saddle-centre was stated and partially solved in . Later this result was extended and improved in [21, 14, 26].
These papers do not directly cover the situation described in this paper (see  for a discussion of relations between these two classes in the Hamiltonian context). The main difference is related to the exponential smallness of the separatrix splitting in the bifurcation problem discussed in the present paper. The presence of exponentially small phenomena hidden beyond all orders of the normal form theory is also observed in other bifurcation problems (see for example [25, 7, 9, 2]).
Finally, we note that the problem of existence of small amplitude single- and multi-round homoclinic orbits arises in various applications. These applications include dynamics of the three-body-problem near libration point . Homoclinic solutions also appear in the study of traveling wave or steady-state reductions of partial differential equations on the real line which model various phenomena in mechanics, fluids and optics (for more details see [4, 5]). These solutions are of particular interest as they represent localized modes or solitary waves, these problems are often of the singular perturbed nature .
2 Symplectic approach to measuring the separatrix splitting
Let be an equilibrium of the Hamiltonian system with eigenvalues . Then for each (where is a positive constant) there is an analytic change of variables such that the equilibrium is shifted to the origin and the Hamiltonian function is transformed to its Birkhoff normal form which can be presented in the form
where is an analytic function of two variables
A statement equivalent to the convergence of the normal form was originally obtained in . Of course, since as , the size of domains of convergence shrinks to zero both for the normal form and for the normalising transformation. Nevertheless, it is possible to refine the estimates of  in order to establish that the sizes of the domains are sufficiently large to be used in the following arguments.
Obviously and, consequently, both and are constant along trajectories of the Hamiltonian system. Both functions are local integrals only and in general do not have a single-valued extension onto the phase space. The transformation which transforms the original Hamiltonian to the normal form is not unique. Nevertheless the values of and are unique as they do not depend on this freedom.
As the eigenvalues of the equilibrium are preserved, the Taylor expansion of the transformed Hamiltonian has the form
The structure of the phase space in a neighbourhood of the origin is illustrated by Figure 3 where . In particular, in the normal form coordinates points with correspond to Lyapunoff periodic orbits.
A trajectory which converges to as or without leaving the domain of the normal form has . In the normal form coordinates all these trajectories are easy to find explicitly.
Let be separatrix solutions of the Hamiltonian system (2) which converge to the equilibrium
being close to the separatrix loop described in the introduction. Since are solutions of an autonomous ODE, these assumptions define the functions up to a translation in time . We will eliminate this freedom later. At the moment it is sufficient to note that will be chosen to be in a small neighbourhood of the intersection of the normal form separatrix with the plane . Note that the curve may have more than one intersection with , in this case we chose a “primary” one.
The unstable separatrix leaves the domain of the normal form, makes a round trip near the ghost separatrix loop, and at a later moment of time comes back close to the stable direction of the Hamiltonian vector field at . Let and be the values of the elliptic and hyperbolic energies obtained after this round-trip. Conservation of the energy implies that , so the values of and are not independent. Traditionally the elliptic energy is used to measure the separatrix splitting. In particular, if , then the trajectory is homoclinic. If , the trajectory will eventually leave the neighbourhood of for the second time.
Theorem 1 (Main theorem)
We note that and . Then the asymptotic expansion (9) implies that is exponentially small compared to . Moreover, if this theorem implies the splitting of the separatrix and, hence, non-existence of a single-loop homoclinic orbit. We do not know an explicit formula to compute . Nevertheless, numerical methods of  can be adapted for evaluating the constants in the asymptotic series with arbitrary precision. The arguments presented in section 9 can be used to prove that is generically non-vanishing (as the map is a non-trivial analytic (non-linear) functional). Indeed, if the Hamiltonian analytically depends on an additional parameter , then it can be proved that is analytic. Then the Melnikov method can be used to show that for values of which correspond to an integrable Hamiltonian. Finally the analyticity implies that zeroes of are isolated and, consequently, the coefficient does not vanish for a generic Hamiltonian .
The proof of the main theorem is based on ideas proposed by V. Lazutkin in 1984 for studying separatrix splitting for the standard map and later used in  for studying separatrix splitting of a rapidly forced pendulum. This paper contains a sketch of the proof for the main theorem.
The Melnikov method is often used to study the splitting of separatrices. In general the Melnikov method does not produce a correct estimate for the problem discussed in this paper. Section 9 contains a discussion of the applicability of the Melnikov method.
3 Elliptic energy and the variational equation
As a first step of the proof we provide a description of the elliptic energy in terms of the splitting vector
which describes the difference between the stable and unstable separatrix solution, and a solution of a variational equation around . This description allows us to compute without explicit usage of a transformation to the normal form in a neighbourhood of the saddle-centre equilibrium.
In the normal form coordinates the Hamiltonian is described by equation (5), thus the corresponding equations of motion take the form
where and . Since on the local stable trajectory , equation (7) implies that , and we can find this trajectory explicitly:
where is a constant. Then the variational equation around this solution takes the form
where and . A fundamental system of solutions for the variational equation is found explicitly:
Note that we have chosen . The first two solutions are mutually complex conjugate, so real-analytic solutions can be easily constructed when needed. A direct computation shows that and . For all other pairs the symplectic form vanishes. Later we will study those solutions for non-real values of . So it is interesting to note that the function exponentially grows in the complex upper half-plane , while exponentially decays there.
For each fixed value of we can consider the collection of vectors , , as a basis in . Then the function
where and are components of in the normal form coordinates (for the values of corresponding to the first return of the unstable trajectory to the small neighbourhood of the saddle-centre ). Taking into account that is real-analytic and using the definition of of (6), we obtain that the equality
holds for real values of . Since is a local integral, also stays constant for real while the unstable solution remains inside the domain of the normal form.
The equation (15) provides a relation between and . While is defined using the normal form coordinates, the function is defined by (14) and can be evaluated in other canonical systems of coordinates. This computation relies on accurate analysis of the way the splitting vector and the solutions of the variational equation are transformed under coordinate changes. It is important to note that although canonical coordinate changes do preserve the symplectic form, does not take the same value when evaluated in a different coordinate system but can differ by a value of the order of .
Slightly overloading the notation, let and be respectively the splitting vector and the solutions of the variational equation written in the original coordinates. The splitting vector is defined by equation (10). The solutions can be fixed by asymptotic conditions described in the next section to ensure that they represent the same solutions of the variational equation as in (14) but expressed in the other coordinates. Then we define a function in a way similar to (14)
It is easy to check that , where the constant bounds the -norm of the transformation between the systems of coordinates. For the real , the function is uniformly bounded and we conclude that . We conclude that . Then equation (15) implies that
A refinement of the arguments from  implies that . This factor does not break the approximation as is of the same order as and is exponentially small compared to . We will use the equation (17) to obtain an estimate for .
4 Variational equation
On the next step of the proof we study solutions of the variational equation near the unstable separatrix solution :
This is a linear homogeneous non-autonomous equation. Since the variational equation comes from a Hamiltonian system, it is easy to check that for any two solutions and of equation (18) the value of the symplectic form is independent of . This property together with asymptotic behaviour of solutions at are used to select a fundamental system of solutions.
Let be an eigenvector of the linearised Hamiltonian vector field at ,
such that . Note that the complex conjugate vector is also an eigenvector, but it corresponds to the complex conjugate eigenvalue . Let with being vectors with real components. Then our normalisation condition is equivalent to . The vector is defined uniquely up to multiplication by a complex constant of unit absolute value, in other words, for any real the vector also satisfies our normalisation assumption. We assume that this freedom is eliminated in the same way as in the linear part of the normal form theory near the saddle-centre. In particular, is a smooth function of (including the limit ).
Now we are ready to define fundamental solutions of the variational equation. One solution is selected by the assumption
for . The other one is defined using the real symmetry:
These two solutions are not real on the real axis and . Sometimes it is useful to consider their linear combinations
which are real-analytic.
The third solution is given by
The last solution is chosen to satisfy the following normalisation conditions:
We note that since the original system is Hamiltonian, for any two functions and which satisfy the variational equation, is independent of . Consequently the vectors , , , form a standard symplectic basis for every :
Then we can write the splitting vector in this basis:
The normalisation condition (19) implies that
For the future use we also define
which involve the non-real solutions of the variational equation. The real symmetry implies that .
We note that in general the coefficients depend on time. Indeed, the equation (1) implies that
where is a remainder of a Taylor series. Then differentiating the definition of with respect to and taking into account that is a symplectic matrix we get
where is the index of the canonically conjugate variable (e.g. and ). So are -close to being constant. Moreover, are much smaller than and . Indeed, taking into account the definition of we get
where is the differential of at the point . Since , we conclude that .
Initial condition for can be chosen in such a way that (by translating time in the stable solution in order to achieve the zero projection of on the direction of the Hamiltonian vector field at represented by ). Thus .
Taking into account the real symmetry we see that the problem of the separatrix splitting is reduced to the study of a single complex constant (via the equation (17)).
5 Formal expansions
The proof of the main theorem requires construction of accurate approximations for the stable and unstable separatrix solutions of the Hamiltonian system (1) as well as the fundamental solutions of the variational equation (18). Taking into account that the Hamiltonian (2) can be formally transformed to the integrable normal form (4), we construct an approximation by finding a formal solution to the systems defined by the normal form Hamiltonian. Of course, the series of the normal form theory diverge in general, but they can be shown to provide asymptotic expansions for the true solutions restricted to properly chosen domains on the complex plane of the time variable .
5.1 Formal separatrix
In this section we find a formal separatrix for the normal form Hamiltonian
where is a formal series in three variables and with the lower order terms given by (3). The corresponding Hamiltonian system has the form
Obviously, the plane is invariant and we construct a formal separatrix located on this plane. Formal expansions can be substantially simplified with the help of an auxiliary small parameter . So instead of performing expansion directly in powers of , we look for a solution of the system (21) considering and as formal power series in . The coefficients of the series for and are assumed to be functions of the slow time . The following lemma establishes existence and uniqueness of a formal solution in a specially designed class of formal series.
It is important to note that the leading terms in the series and are of the form and respectively. This choice makes the expansions of Lemma 2 compatible with approximations for the separatrix obtained using the standard scaling, a traditional tool used in the bifurcation theory.
There are unique real coefficients , and such that the formal series
where is not constant and the coefficients , with have the form
Proof. The restriction of the system (21) onto the invariant plane is equivalent to a single equation of the second order
where denotes differentiation with respect to the variable and is used to denote the formal series . Our assumptions on the lower order terms of the Hamiltonian imply that
with and . Multiplying the differential equation by and integrating once we get
where is a formal series in (the first two terms in the sum are power series in by the assumptions of the lemma, so must be in the same class). This equation has a unique formal solution of the form
where is polynomial of order in . Indeed, differentiating with respect to we get
and taking the square
where we used the identity . Substituting the formal series and we get
After substituting these expressions into the equation we get
This equality is treated in the class of formal series in powers of . The leading order is of order of . Collecting all terms of this order we get
Looking for in the form we get
This equation is equivalent to the following system for the coefficients:
This system has a unique solution with , which leads to a non-constant :
Then we continue by induction. Suppose that for some all coefficients are defined uniquely up to , and . Then collect the terms of order to obtain an equation of the form
where is a polynomial of order in with coefficients depending on already known ones. We can find the coefficients of starting from the largest power of . We find from the linear term in and from the constant term. In the essence we solve a linear algebraic system with a triangle matrix with non-vanishing elements on the diagonal. So the coefficients are unique.
If is an analytic family defined by (2) then there is an analytic coordinate change which moves the remainder term beyond any fixed order . Neglecting this remainder we obtain a polynomial Hamiltonian of the form (4) and it is not too difficult to verify that for this Hamiltonian where is an exponent of the saddle-centre equilibrium of the truncated normal form. Since is not changed by smooth coordinate changes, the Taylor expansions of and in powers of coincide in the first terms (indeed, our formal computations show that the first -terms of this series are uniquely determined by the first orders of the Hamiltonian, these terms are the same as the remainder affects terms of higher order only).
5.2 Formal variational equation
In addition to the formal separatrix solution we will need to study formal solutions for the corresponding variational equation. These formal solutions will be used to approximate analytic solutions of the variational equation with an error being of a sufficiently high order in .
The variational equation near the formal solution provided by Lemma 2 has the form
This system is split into two independent pairs of linear equations. This property allows us to solve this system explicitly. Indeed, a direct substitution shows that the function
satisfies the system (24). In order to give precise meaning to this expression in the class of formal series we define the first formal solution of the formal variational equation by
where the formal series is defined by termwise integration
and , are real coefficients.
These coefficients can be computed by substituting the
formal series into
and integrating the result termwise.111In order to check that the result of integration has the stated form consider
. Then and integrating by parts we get
The other solution is defined using the real symmetry: .
The third solution is obtained by differentiating the formal separatrix with respect to :
Finally, the fourth solution is found from the normalisation assumptions
We sketch the derivation for its form. A solution to the equation
also satisfies the same variational equation as