Lyapunov-Schmidt Reduction for Unfolding Heteroclinic Networks of Equilibria and Periodic Orbits with Tangencies
This article concerns arbitrary finite heteroclinic networks in any phase space dimension whose vertices can be a random mixture of equilibria and periodic orbits. In addition, tangencies in the intersection of un/stable manifolds are allowed. The main result is a reduction to algebraic equations of the problem to find all solutions that are close to the heteroclinic network for all time, and their parameter values. A leading order expansion is given in terms of the time spent near vertices and, if applicable, the location on the non-trivial tangent directions. The only difference between a periodic orbit and an equilibrium is that the time parameter is discrete for a periodic orbit. The essential assumptions are hyperbolicity of the vertices and transversality of parameters. Using the result, conjugacy to shift dynamics for a generic homoclinic orbit to a periodic orbit is proven. Finally, equilibrium-to-periodic orbit heteroclinic cycles of various types are considered.
Heteroclinic networks in ordinary differential equations organise the nearby dynamics in phase space for closeby parameters. They thus act as organising centres and explain qualitative properties of solutions, and predict variations upon parameter changes. This makes heteroclinic networks a valuable object in studies of models for applications. When all vertices in the network are equilibria much about such bifurcations is known. Recently, heteroclinic networks whose vertices can also be periodic orbits have found increasing attention.
This article concerns the unfolding of finite heteroclinic networks consisting of hyperbolic equilibria and periodic orbits in an ordinary differential equation
with , and parameter for arbitrary and sufficiently large .
Any solution that remains close to the heteroclinic network of (1) assumed at for all time can be cast in terms of its itinerary in the heteroclinic network. See Figure 1 for a simple example. For any itinerary, bifurcation equations will be derived for the locus of parameters of all corresponding solutions to (1). The idea to formulate the unfolding in this way is borrowed from previous studies of heteroclinic chains of equilibria [26, 34].
Bifurcation studies from heteroclinic chains with equilibria mainly concerned homoclinic orbits and generated a huge amount of literature, see, e.g., [6, 10, 14, 16, 26, 34, 37, 40] just to name a few, and heteroclinic loops between two equilibria, see, e.g., [3, 5, 9, 12, 18, 26, 37, 41, 42]. Heteroclinic cycles with periodic orbits have found increasing attention recently, see, e.g., [2, 4, 19, 21, 32, 33, 38, 39].
The main new contributions of the present work are rigorous results allowing for periodic orbits in general heteroclinic networks, for tangent heteroclinic connections, and to formulate the bifurcation equations in a general form that can be used as the basic building block for a specific study. It is hoped that this makes the results useful for readers with applications to specific cases in mind.
A large field of applications are travelling waves in evolutionary partial differential equations in one space dimension whose profiles solve an equation of the form (1), but also for instance Laser models are often reduced to this form. There is a very large amount of such analytic and numerical studies in the literature, e.g., [8, 19, 22, 25, 31, 35, 36, 38, 39] to hint at some.
In the applied literature such bifurcation equations are frequently derived formally by a geometric decomposition in terms of local and global maps, e.g., [4, 12, 31]. The justification in particular of the local map is an issue and, if linear, requires non-resonance conditions on eigenvalues or else dimension dependent normal form computations. Other issues are the form of parameter dependence and the persistence of solutions upon inclusion of the higher order terms of the original vector field. These problems do not arise in the approach taken here, and the results can provide a rigourous foundation of formal reductions.
The reduction to bifurcation equations in this paper is motivated by the so-called ‘Lin-method’ described in , which is a Lyapunov-Schmidt reduction for boundary value problems of the itinerary. This method has been used and modified in a number of ways and contexts for equilibria, e.g., [17, 16, 40]. Tangent intersection of stable and unstable manifolds have been considered mostly for homoclinic orbits, i.e., homoclinic tangencies, a paradigm of chaotic dynamics; Lin’s method in this context has been used in . Periodic orbits introduce technical complications and for Lin’s method these have been overcome in  and, using Poincaré-maps, in . Transversality studies with respect to parameters in related cases were done in . An ergodic theory point of view is taken in [1, 11, 23, 24, 27, 29], and further papers by these authors, looking for instance at properties of non-wandering sets. More recently  treated periodic orbits in a very promising way using Fenichel-coordinates.
Here  is used as a starting point, and equilibria or periodic orbits as vertices are treated in an essentially unified manner. Symmetries or conserved quantities are not used, but a generic setting is assumed. In contrast to , winding numbers of heteroclinic sets are not considered, and the underlying heteroclinic network is held fixed. Together with  this exposition is self-contained, but somewhat technical, and parts of  have to be repeated and improved in order to track higher order terms in this extended setting. The precise statement of the main result Theorem 4 can only be given rather late after a number of preparatory steps, notation and definitions. In particular, this includes §3 where we obtain suitable coordinates near the vertices. We next describe the main result, and refer to §5 for sample applications.
1.1 Description of the main result
For a chosen itinerary the method is a Lyapunov-Schmidt reduction which yields algebraic equations that relate system parameters to certain geometric characteristics , at each heteroclinic connection that the solution follows, perhaps repeatedly, and which connects vertex to . The time spent between Poincaré sections at , and at is for if is an equilibrium. If is a periodic orbit, this time is in general only approximately since we normalise for the minimal period of vertex . For tangent heteroclinic connections or more than one-dimensional heteroclinic sets, un/stable manifolds of and have a more than one-dimensional common tangent space at , and are the coordinates on that space, except the flow direction. The location of in the itinerary is illustrated in Figure 2.
The system parameters can generically be assumed to unfold each heteroclinic connection by a separate set of parameters which, however, must coincide at repeated connections. Here is the codimension of the th connection and without repeating the same connection in the sum. The geometric characteristics couple these parameter sets, but to leading order only to the nearest neighbours . If for all , then all heteroclinic connections are transverse, and the result proves the existence of solutions for any itinerary, and an expansion for the coordinates in the Poincaré-sections. Otherwise, an expansion of in terms of , is provided as described next.
Let and be the real and imaginary parts of the leading un/stable eigenvalue or Floquet exponent at vertex . For let . For all with and for sufficiently large and small , there exist , and linear maps , as well as quadratic maps (zero if ), so that a solution follows the chosen itinerary, if and only if
Here whenever the and in the itinerary correspond to the same actual heteroclinic connection. The coupling to the nearest neighbours is given by
where and are linear maps.
Negative Floquet multipliers, i.e., negative eigenvalues of the period map, have Floquet exponent with imaginary part . Since , , the argument in the cosine terms is which generates an oscillating sign as is incremented.
Note that if the itinerary has repetitions, then have to satisfy solvability conditions from repeating the corresponding equations. Each repetition yields new parameters , but all other quantities in (2) are the same if the underlying heteroclinic connections is the same.
The significance of (2) lies in the order of the remainder term , which, for certain , and arbitrary , is given by
In particular, is higher order with respect to at least one of the cosine terms if , .
The application of the above result to a Shil’nikov-homoclinic orbit to an equilibrium yields the same bifurcation equations as in , and as in that paper, many of the seminal results by Shil’nikov  follow from leading order analyses. Note that the resonant case can be treated by the above result as well. See  for resonance at homoclinic bifurcations.
Under the ‘flip’ condition or for the leading eigenvalue, the next leading terms need to be taken into account for an unfolding. Viewing , as parameters, the bifurcation of solutions can be understood from Theorem 4 if for or for . To overcome the barriers involving and to the next order eigenvalues requires a more refined setup beyond the scope of this article. See  and also [14, 20, 28] for such considerations in case of equilibria applied to homoclinic bifurcations.
The main result unifies the treatment of equilibria and periodic orbits as vertices of a network: for the reduced equations the only difference between equilibria and periodic orbits is that is a semi-infinite interval for an equilibrium, but the above defined discrete infinite sequence for a periodic orbit. The discrete sequence essentially counts the number of rotations that the solution makes about the periodic orbit. Note that replacing a periodic orbit by an equilibrium has consequences for the codimensions of heteroclinic connections.
The reduced equations for a specific case can be determined in three steps. First, choose the itinerary of the solution type of interest. Second, determine the codimensions, including tangencies, of all visited heteroclinic connections. Third, copy the equations from Theorem 4 for each element in the itinerary with positive codimension, and remove geometric characteristics that do not occur according to the type of itinerary and tangencies. In case of repetition in the itinerary, the locus of parameter values for the solutions should be found by analysing the arising algebraic solvability conditions (which can be highly non-trivial). Similarly, in case of tangencies, the locus of turning points or folds can be determined.
To illustrate this and the applicability of the abstract results, some sample applications for specific heteroclinic networks are presented in §5. In particular, for a generic homoclinic orbit to a periodic orbit conjugacy to (suspended) shift dynamics is proven. In addition, equilibrium-to-periodic orbit heteroclinic cycles of various types are considered, and 2-homoclinic orbits are studied for the first time in this context.
Note that the main result separately concerns each solution type as encoded in each itinerary. This is suitable, for instance, when looking for the aforementioned travelling waves. In some cases a whole class of solutions or even the entire invariant set can be characterised directly. However, our results do not provide stability information of the bifurcating solutions or hyperbolicity of invariant sets, or other ergodic properties. See  for stability results in homoclinic bifurcations using Lin’s method (where additionally PDE spectra are considered). As mentioned, the above result does not provide an expansion in general in the case of vanishing leading order terms (‘flip conditions’).
This paper is organised as follows. Section 2 contains details about the setting and some preparatory results. In §3 a suitable coordinate system is established for trajectories that pass near an equilibrium or periodic orbit or lie in un/stable manifolds. The main result is formulated and proven in §4. Finally, §5 contains sample bifurcation analyses and illustrate how to use the main result.
Acknowledgement. This research has been supported in part by NWO cluster NDNS+. The author thanks Björn Sandstede, Ale Jan Homburg and Alan Champneys for helpful discussions.
2 Setting and Preparation
The basic assumption is that at (1) possesses a finite heteroclinic network with vertices being equilibria or periodic orbits, , and edges , , being heteroclinic connections. Rather than unfolding as a whole, we consider the following paths within separately. Here we set (where if there is no minimum).
A possibly infinite set with , and , , is called itinerary if for all the edge is a heteroclinic connection from to (or homoclinic if ) and either , , or for some .
For ease of notation we say that a sequence of ‘objects’ (numbers, vectors or maps given in the context) with has reducible indexing (with respect to ) if whenever for .
Note that an itinerary can cycle arbitrarily and perhaps infinitely long within the heteroclinic network viewed as a directed graph, and the labelling can differ from that in . Any itinerary has a (possibly non-unique) reduced index set so that as well as for , with , and , respectively. Associated to this is , which may not itself be an itinerary (though it contains one).
Let be the index set of all equilibria in and that of all periodic orbits. We set and . Finally, denotes the minimal period of for , and we set for .
In the following, unless noted otherwise, we consider an arbitrary fixed itinerary . However, until §4 only neighbours and are relevant.
For let be the Floquet representation of the evolution of the linearization of (1) in . Here , and the matrices satisfy , for and , and for (in which case ).
The basic assumption about (1) and the heteroclinic network is
The vector field in (1) is of class for in and . The equilibria or periodic orbits , , are hyperbolic at , i.e., for any the matrices have no eigenvalues on the imaginary axis, except for a simple eigenvalue at the origin (modulo ) if .
Here a simple eigenvalue has algebraic and geometric multiplicity one. Hypothesis 1 implies that the spectrum of has the un/stable gaps
Since is finite, the gaps are uniformly bounded from below in . For convenience we choose arbitrary , with reducible indexing, such that . We also need the gap to the next leading eigenvalues/Floquet exponents. Let , be the eigenvalues of and define
Leading stable eigenvalues of a matrix are those with the largest strictly negative real part, and leading unstable those with the smallest strictly positive real part. For the main result, we will assume that these leading eigenvalues are simple as expressed in the following hypotheses.
Consider the leading stable eigenvalues or Floquet exponents at . Assume that this is either a simple real eigenvalue or a simple complex conjugate pair , with .
Consider the leading unstable eigenvalues or Floquet exponents at . Assume that this is either a simple real eigenvalue or a simple complex conjugate pair , with .
To emphasise where these hypotheses enter we will not assume them globally, which has the effect that a priori exponential rates for estimates are not , but for an arbitrary due to possible secular growth. In the following denotes a priori an arbitrarily small positive number, which may vanish under Hypothesis 2, 3.
Hence, for suitable as well as asymptotic phases of with respect to we obtain the estimates (see, e.g., )
where depends only on and . For the requirement (4) of equal asymptotic phase for and determines up to multiples of and uniquely in . For we have and set .
To distinguish in- and outflow at we denote for .
Let denote the evolution of and that of . Hyperbolicity of gives the following exponential dichotomies for and trichotomies for (see, e.g., ).
Notation. Indices separated by one or more ‘slashes’ as in indicate alternative choices for the statement with all these indices chosen equal at a time.
There exist projections , continuous in , and , continuous in , such that the following holds. Set and , respectively.
For : ,
For : ,
The projections are complementary: , , , ; analogous for ,
The spaces and are unique, the spaces and arbitrary complements such that the previous holds,
The projections commute with the linear evolution:
They distinguish un/stable and center direction: there is depending on and , such that for all
We denote the un/stable and center spaces, respectively, by
For a decomposition we denote by the unique projection with kernel and image .
In order to link the trichotomies of the in- and outflow near , we define
We also define the aforementioned sets of travel time parameters
It is shown in , Lemma 2, that there is such that for the are complementary projections, for , and the norms are uniform in .
In order to control the leading order terms in the bifurcations equations we make the following change of coordinates locally near all , . In the new ‘straight’ coordinates the un/stable manifolds locally coincide with the un/stable eigenspaces of the linearization in , respectively. For periodic orbits the strong un/stable fibers locally coincide with the un/stable eigenspaces. Since these are graphs over the eigenspaces and tangent at the equilibrium or periodic orbit this is straightforward. See e.g., (3.27) in . However, as in  this change of coordinates is an obstacle to apply the method within the class of semilinear parabolic partial differential equations. However, in  this problem has been circumvented in a way that should apply here as well.
To emphasise the effect of this coordinate change and to make the notation of estimates throughout the text more readable, we define for and any , and if explicitly mentioned, the terms
and set for .
Notation. In the following we use the usual order notation if there is a constant such that for all large or small enough and norms as given in the context. In terms of this is always as . In a chain of inequalities for such order computations we allow the constant to absorb other constant factors and take maximum values of several constants without giving explicit notice.
The next lemma is the basis for estimating error terms in the following sections.
There exist and depending only on and such that for the following holds for all .
For readability, we set , see (4). Let be the stable/center and unstable eigen- or trichotomy projections on the whole real line, , of
which trivially exist by the Floquet form. Note that these di/trichotomies differ from those of the linearization in .
First note that
so that is determined by . (An appropriate norm for estimating this difference goes via suitable bases of these linear spaces.) Since for it holds that for all (the projections are constant for ) we have
and we will estimate the two differences on the right hand side separately.
General perturbation estimates of dichotomies (e.g. Lemma 1.2(i) in ) imply and .
Hence, on the one hand .
On the other hand we can write
and, due to asymptotic phase we have in the center direction. Therefore, and . (And analogously ).
In combination, since the weak version of the first estimate follows. The strong version of this estimate in straight coordinates is a consequence of the fact that then for all for sufficiently large . Hence, for we have
which implies the stronger estimate.
The proof of the second estimate is completely analogous.
Since the stable manifold is a (at least) quadratic graph over the stable eigenspace for and center-stable trichotomy space for at we have that
On the other hand, as in the proof of the previous item, we can replace by with error of order . Since lies in the kernel of and there are non-negative constants and such that