Geometric determination of heteroclinic and unstable periodic orbit classical actions
Semiclassical sum rules, such as the Gutzwiller trace formula, depend on the properties of periodic, closed, or homoclinic (heteroclinic) orbits. The interferences embedded in such orbit sums are governed by classical action functions and Maslov indices. For chaotic systems, the relative actions of such orbits can be expressed in terms of phase space areas bounded by segments of stable and unstable manifolds, and Moser invariant curves. This also generates direct relations between periodic orbits and homoclinic (heteroclinic) orbit actions. Simpler, explicit approximate expressions following from the exact relations are given with error estimates. They arise from asymptotic scaling of certain bounded phase space areas. The actions of infinite subsets of periodic orbits are determined by their periods and the locations of the limiting homoclinic points on which they accumulate.
The properties of sets of rare classical orbits can be extremely important in the study of chaotic dynamical systems Poincaré (1899). For example, classical sum rules over unstable periodic orbits describe various entropies, Lyapunov exponents, escape rates, and the uniformity principle So (2007). The information which enters these classical summations are the stability properties and densities. Such sets of orbits are also linked to the properties of the analogous quantized systems through the derivation of semiclassical sum rules. A few cases are given by periodic Gutzwiller (1971); Balian and Bloch (1971); Berry and Tabor (1976) and closed orbit sum rules Du and Delos (1988a, b); Friedrich and Wintgen (1989) that determine quantal spectral properties, and homoclinic (heteroclinic) orbit summations Tomsovic and Heller (1991, 1993) generating wave packet propagation approximations. The interferences in such sum rules are governed by the orbits’ classical action functions and Maslov indices, and thus this information takes on greater importance in the context of the asymptotic properties of quantum mechanics. Various resummation techniques have been given to work with series which are often divergent in nature Tanner et al. (1991); Cvitanović and Eckhardt (1989); Berry and Keating (1990). Other studies exploring a fuller understanding of the interferences have also been carried out Argaman et al. (1993); Ozorio de Almeida (1989); Bogomolny (1992); Sieber and Richter (2001); Müller et al. (2004); Turek et al. (2005); Müller et al. (2005). Our interest in this paper is establishing a framework for understanding the relationships between periodic and homoclinic (heteroclinic) orbit actions and their action correlations.
A periodic orbit in a two degree-of-freedom system becomes either a single fixed point or an invariant set of points visited periodically in a two dimensional Poincare surface of section. It is sufficient to concentrate on symplectic mappings on a plane and study the unstable fixed or periodic points under their application. The fixed points play an advantageous role in this work due to convergence theorems in normal form coordinates. The normal form transformation was first proved by Moser to converge inside a disk-shaped neighborhood of the fixed point denoted by hereafter Moser (1956). Later, da Silva Ritter extended the convergence zone along the stable and unstable manifolds out to infinity da Silva Ritter et al. (1987).
Within the convergence zone are Moser invariant curves, which are images of invariant hyperbolas. As already noted by Birkhoff Birkhoff (1927); Moser (1956); Harsoula et al. (2015), the self or mutual intersections between such invariant curves can support periodic orbits with arbitrarily large periods. These periodic orbits accumulate alternatively on one or multiple homoclinic (heteroclinic) points in a homoclinic (heteroclinic) tangle. In the limit of the orbital period going to infinity, the invariant curves become infinitely close to the stable and unstable manifolds of the fixed points. The periodic orbits of such kind are said to be satellite to their respective homoclinic (heteroclinic) points da Silva Ritter et al. (1987); Ozorio de Almeida (1989). da Silva Ritter developed a method for the numerical computation of satellite orbits supported by such curves in the quadratic map da Silva Ritter et al. (1987). Therefore, every periodic orbit inside the convergence zone must be satellite to some homoclinic points, with its classical action closely related to that of the homoclinic orbit. Recent work shows that the size of the convergence zone can be quantified in terms of the outermost Moser curves Harsoula et al. (2015), i.e. ones with the largest normal form coordinates product, and the convergence zone can be numerically estimated using the outermost Moser curves as boundaries Contopoulos and Harsoula (2015).
Assuming a system is fully chaotic, the convergence zone should cover most, if not all, of the accessible phase space. In that case, nearly all of the periodic orbits lie on Moser invariant curves, and each one can be treated as a satellite orbit of some particular set of hyperbolic fixed points. Even if the system is not fully chaotic, the convergence zone can cover nearly all of the available phase space. Figure 4 of Harsoula et al. (2015) gives an excellent example of the convergence zone covering almost all of the complex region of the homoclinic tangle of the Hénon map Hénon (1969), avoiding only a small region inside the last KAM curve. Thus, a study of satellite orbits may often encompass nearly all periodic orbits of the system; i.e. satellite orbits are not typically a small subset of the periodic orbits.
In the quantum Baker’s map wave packet autocorrelation functions can equivalently be expressed as a sum over periodic fixed points or homoclinic orbit segments with an exact one-to-one correspondence between terms O’Connor et al. (1992). Similarly, there is the same, though not exact, correspondence for the stadium billiard as there may be problems with orbits which approach bifurcations points too closely, i.e. some of the orbits that come too close to the joint between the straight edge and curved hard walls Tomsovic and Heller (1993). Thus, it is of significant interest to understand how the homoclinic (heteroclinic) and periodic orbits are related.
This work develops a framework for expressing the actions of satellite orbits in terms of the relative actions of homoclinic (heteroclinic) orbits, phase space areas bounded by stable and unstable manifolds, and Moser invariant curves. These areas scale down with increasing periods, and the determination of the action of a leading satellite periodic orbit with small period is sufficient to approximate satellite orbits with larger periods; the numerical calculation of individual orbits becomes unnecessary to an excellent approximation. As a final remark, note that Maslov indices can be incorporated into this framework, but are not considered in this paper in order to focus on the classical actions. Previous studies of Maslov indices can be found in Creagh et al. (1990); Esterlis et al. (2014).
This paper is organized as follows. Section II sets the notation and basic definitions of homoclinic (heteroclinic) orbits and their actions. Section III is a generalization of the MacKay-Meiss-Percival action principle MacKay et al. (1984) for heteroclinic orbits, and expresses their actions as phase space integrals. Section IV concerns relative actions between two hyperbolic fixed points, and expresses them as phase space areas bounded by segments of the stable and unstable manifolds. Section V studies the satellite periodic orbits, and expresses their actions using phase space areas bounded by segments of the Moser invariant curves together with stable and unstable manifolds. An approximation for orbits with large periods is also given, together with a numerical calculation with the Hénon map. Some basic information on homoclinic (heteroclinic) tangles Wiggins (1992); Easton (1986); Rom-Kedar (1990), the MacKay-Meiss-Percival action principle MacKay et al. (1984); Meiss (1992), and normal form theory with satellite period orbits da Silva Ritter et al. (1987) can be found in Appendices A and B.
Ii Homoclinic (heteroclinic) orbits and relative actions
This section lays out the paper’s notation and a few basic concepts of homoclinic (heteroclinic) orbits in classical dynamical systems.
ii.1 Homoclinic (heteroclinic) orbits
Let be an analytic and area-preserving map on the -D phase space , and be a hyperbolic fixed point under with stability exponent . Denote the unstable and stable manifolds of by and respectively. Typically, its unstable and stable manifolds intersect infinitely many times and form a complicated pattern called a homoclinic tangle Poincaré (1899); Easton (1986); Rom-Kedar (1990) as partially shown in Fig. 1.
The intersection points belong to both and for all times. The homoclinic orbit, denoted by , is the bi-infinite collection of images:
where both and converge to . If the unstable and stable segments connecting with intersect only at , then is a primary homoclinic orbit. There must be at least two such orbits Wiggins (1992), such as and in Fig. 1. Of particular interest are the unstable segments and stable segments , which enclose the so-called “lobe regions” and , which are extensively studied in transport problems Bensimon and Kadanoff (1984); MacKay et al. (1984); Rom-Kedar (1990); Wiggins (1992). The region bounded by and is called the complex region, which is the main region of interest in transport theory. Notice that for open systems such as the Hénon map, any point out side the complex region will escape to inifinity, and thus the lobes and with will extend to infinity and never come back into the complex region. This ensures that there are no homoclinic points on segments and . The homoclinic points in such systems are distributed only on segments and .
A more general scenario is to have two hyperbolic fixed points with their own stable and unstable manifolds intersecting one another, forming a heteroclinic tangle Wiggins (1992). Consider and , with their unstable [stable] manifolds  and ; see Fig. 2.
The intersecting stable and unstable manifolds of different fixed points generate heteroclinic orbits. In Fig. 2, and have the limiting points:
Unlike homoclinic tangles, there may be only one primary heteroclinic orbit; an example is shown ahead. Homoclinic and heteroclinic orbits play an important role in chaotic dynamics as they provide clues for the entire structure of the chaotic region. As shown in Birkhoff (1927); da Silva Ritter et al. (1987), infinite families of satellite periodic orbits accumulate on the homoclinic (heteroclinic) orbits, and the determination of the periodic orbit actions rely on those of the homoclinic (heteroclinic) orbits.
ii.2 Relative actions
The mapping can be viewed as a canonical transformation that maps a point to while preserving the symplectic area, therefore a generating (action) function can be associated with this process such that MacKay et al. (1984); Meiss (1992):
The total action of an orbit is the sum of the generating functions:
and is divergent in general. However, the MacKay-Meiss-Percival action principle MacKay et al. (1984); Meiss (1992) can be applied to obtain well defined action differences for particular pairs of orbits. An important and simple case is the relative action between a fixed point and any of its homoclinic orbits , which turns out to be equal to an area bounded by unstable and stable manifold segments as
where is the segment of the unstable manifold from to , and the segment of the stable manifold from to . The superscript from the last line indicates that the area is interior to a path that forms a closed loop, and the subscript indicates the path: . As usual, clockwise enclosure of an area is positive, counterclockwise negative. denotes the generating function along that maps to , and denotes the generating function of in one iteration. Likewise, a second important case is for homoclinic orbit pairs, which results in
where is the segment of the unstable manifold from to , and the segment of the stable manifold from to . See Appendix A for further details.
It is also desirable to have geometric relations for the differences of any pair of periodic orbits. Since they may not have the same period, comparing each over its primitive period relative to a fixed point suffices. For an -period orbit, i.e. and
However, ahead it is shown that the geometric form also requires homoclinic orbits and Moser invariant curves.
Iii Relative heteroclinic orbit actions
Consider two hyperbolic fixed fixed points and , and a heteroclinic intersection ; see Fig. 2. Since the infinite past and the infinite future are asymptotic to different fixed points, it is convenient to consider the heteroclinic orbit in two semi-infinite halves, where is the dividing point. The past orbit relative to is
The future orbit is similarly
The action of the past [future] orbit is given relative to . In particular, the relative action between and is defined as
and similarly for the future orbit history
where the subscript . Since this path is not closed, the final point is added to the area notation. The integral gives the algebraic area in Fig. 3.
This result can be generalized by considering a change in the dividing point to some other point . The form of Eq. (12) must be unchanged. Thus:
where now the past and future relative actions are defined with respect to and the unstable and stable manifold integral paths change accordingly. This simple extension is quite useful ahead.
At this point we would like to make a remark on the difference between the areas defined in Eq. (5) and Eq. (12): upon canonical transformations, the former is a closed area, thus invariant; while the latter is an open algebraic area, therefore not invariant. This is also consistent with the action functions on the left sides of the equations. Despite that the action functions are modified by the canonical transformations, the modifications cancel out between successive steps for the relative homoclinic actions, but not for the heteroclinics, the net change from which should match the change in the algebraic area.
Standard map example
Consider the action of a primary heteroclinic orbit of the standard map as an example. The mapping equations are Chirikov (1979)
where our example is for the parameter , a value for which the system dynamics are overwhelmingly dominated by chaotic motion. Perhaps the simplest case is that of the two hyperbolic fixed points and . The first point is hyperbolic with inversion, which adds a new element to relations coming further ahead. The primary intersection of ’s unstable manifold with ’s stable manifold, and the area defined in Eq. (12) are drawn in Fig. 3. Calculating numerically the left hand side of Eq. (12) using the action function for , and the right hand side using a construction of the manifolds gives , which is as accurate as one could expect using double precision computation. In this example, the two fixed points both lie on the axis, and the algebraic area defined by Eq. (12) is relatively simple. Examples of more complicated heteroclinic orbits connecting fixed points with nonzero values, can be found in Fig. 11 of Li and Tomsovic (2017).
The area-relative-action relation has the advantage of giving results without the necessity of calculating the heteroclinic orbit. Only the intersection point and manifold segments are needed. Otherwise, a long orbit segment of centered at must be determined to get high accuracy. As numerical iterations forward and backward of fail to follow after a logarithmically short time in the precision divided by the Lyapunov exponent, numerical orbits diverge in this example after just a few iterations. Although techniques can be constructed to evade the divergence problem Li and Tomsovic (2017), Eq. (12) makes it unnecessary.
Iv Relative actions between hyperbolic fixed points
A very interesting relation derives from comparing Eqs. (12,13). Subtraction generates a relation between the relative action between two fixed points with an area bounded by unstable and stable manifolds. Defining
and similarly for the stable manifold segments, Eq. (16) simplifies to ()
The case is schematically illustrated in Fig. 4.
In this case, Eq. (IV) reads
Standard map example
Applying Eq. (IV) to the standard map with the same fixed points as before highlights an intriguing situation due to being hyperbolic with inversion. It turns out to be convenient to consider the twice-iterated map , under which heteroclinic orbits stay on the same branch of the unstable manifold of . Therefore, to calculate the action difference between the two fixed points, we consider only the -even cases. In addition, there is only one primary heteroclinic orbit, one lobe, and thus one area, not two; see Fig. 5.
The fundamental lobe structure for the heteroclinic tangle does not look like a turnstile as it would for a homoclinic tangle Li and Tomsovic (2017). Though not visible in the figure, the unstable manifold wraps counterclockwise around the fixed point in order for this to be possible.
For -even all the heteroclinic points map back onto the same branch of the unstable manifold of . Therefore, with , and and in Eq. (IV):
where is defined in Fig. 5, and comes from the generating function of the standard map. is the action difference between the two fixed points under , the equality is verified to a high accuracy ().
V Satellite periodic orbit actions
In chaotic dynamical systems there is another generic class of unstable periodic orbits that are of great interest. They are identified as successive points on Moser invariant curves. Certain sequences of these orbits accumulate on particular homoclinic or heteroclinic orbits Ozorio de Almeida (1989); da Silva Ritter et al. (1987); Moser (1956); Birkhoff (1927) and have been referred to as satellite periodic orbits Ozorio de Almeida (1989); see Appendix B for more details.
v.1 Satellite periodic orbits
Consider periodic orbits, which each associated with a homoclinic orbit segment supported by an invariant Moser curve; see Fig. 6 where a period- orbit is schematically illustrated. The Moser curve extends along and out to infinity and converges to them. Every homoclinic intersection between the manifolds will produce a self-intersection point on the Moser curve. As argued by Birkhoff Birkhoff (1927) and numerically computed by da Silva Ritter et al. da Silva Ritter et al. (1987), special choices can be found for each sufficiently large integer to make a period- periodic orbit. As increases, the corresponding converges to , and the homoclinic orbit is itself the limiting case of the period- periodic orbit for . The set of taken from all integer periodic orbits gives a sequence converging to . In practice, for any homoclinic orbit, , a truncation into finite segments () is possible, for which a Newton-Raphson search in its neighborhood can be used to construct the satellite orbit of period associated with . This provides a convenient way to construct the satellite orbits without the need to calculate the normal form series or the Moser curves.
The relative action difference between a given and its satellite is determined by a roughly parallelogram shaped region bounded by the manifolds and the Moser curve. To see how this area arises, consider the homoclinic tangle in Fig. 7, which is an expanded view of the tangle in Fig. 1. A Moser invariant curve is drawn which supports a period- orbit , satellite to the homoclinic orbit . is the switching point from the future to the past homoclinic segment. The orbit segment follows , then switches at , after which follows .
The relative-action-area-relation derivation makes direct use of Eq. (42) four times, once for each iteration of the map :
Starting from the initial point , and map , follow the path ; is the segment of the Moser invariant curve from to . The path maps to . Substituting the paths into Eq. (42) yields:
: Let the paths be and giving
: Let the paths be and giving
: Let the paths be and . This gives
The total relative action is thus
where the compound closed path is
By adding and subtracting certain path segments, it is possible to deform such that it separates into a path for the relative action of the homoclinic orbit and two parallelogram like correction terms. This gives the final desired relation between the relative action of the periodic and homoclinic orbits,
The two areas in the above equation resembles two near-parallelograms bounded by the manifolds and the Moser curves. The satellite orbit action is then:
where is given by area . Although the segment used here is a primary homoclinic orbit, with a careful definition of the points and near each orbit point , a generalized Eq. (28) applies to satellite orbits associated with any homoclinic orbit segment.
Take the example of Fig. 8, where a period- satellite orbit is associated with a non-primary homoclinic orbit segment . Since the Moser curve approaches the stable and unstable manifolds as it extends along them to infinity, it is forced to make a self-intersection at as the stable and unstable manifolds intersect at . The particular Moser curve is the one for which the mapping of gives back . Therefore, can be thought as being induced by . Following the same logic, define to be the intersection between and the Moser curve that is induced by : as the Moser curve extends along , it intersects with in the same way that intersects, so the homoclinic point induces a on the Moser curve. Similarly, is defined as the intersection between and the Moser curve that is induced by . All and ’s can be located in the same way using as the inducing point. It follows that all previous derivation steps continue to hold with the resulting more general expression of satellite orbit action:
where can be any homoclinic point. This formula expresses the satellite action in terms of the fixed point action, the homoclinic relative action, and two four-segmented simple closed curves bounded by stable/unstable manifolds and the Moser curves. The calculation of the two areas require the construction of the Moser curve, as well as the orbit points and , which can be difficult to compute. However, a simple approximation scheme is possible.
v.2 Geometric area approximation
Equation (29) can be approximated with a wedge product form that only requires the location of the homoclinic points and . In this way, it is possible to calculate the full action of a period- () satellite orbit without its reconstruction or its Moser invariant curve. Assuming the action and the area of some homoclinic orbit point are known, then the first two terms on the right-hand-side of Eq. (29) do not depend on knowing , and only the two areas are needed. Notice from Fig. 8 that is mapped to under iterations, so that the areas of the two are identical. Thus,
The final approximate closed path has only one side which depends on a Moser invariant curve. Furthermore, as shown in Fig. 7, is exceedingly close to , where is a point on a different homoclinic orbit. Consider that
and the mean expansion rate of the map is estimated by the positive stability exponent of the fixed point under one iteration of the map, . After iteractions, the unstable segment is stretched into with an expansion factor of roughly . This implies that the ratio of areas
For all but the smallest values of , the small final area term of Eq. (31) can be dropped.
At this point, one can calculate just by following the manifolds, which is very straightforward. However, there is a further approximation one can make. The manifolds are highly constrained in their behaviors in the local neighborhood of . They must run along nearly parallel, nearly straight lines. This is approximately a parallelogram with area
where , and similarly for ; i.e. the coordinates are just those of and relative to . With this approximation, to a high degree of accuracy the full satellite orbit action is determined knowing only , , and in general:
where is the satellite orbit associated with . is the switching point at which the orbit switches from to .
A possible confusion arises from the fact that the same satellite orbit can also be viewed as associated with any shift in the truncation of the homoclinic orbit: , where is any integer. Furthermore, for , the Newton iteration using as trail orbit will also converge, and one can verify that it leads to the same satellite orbit as using . Therefore, the choice of the switching point along the satellite orbit seems not unique. This ambiguity can be resolved by defining the switching point to be the one that minimizes the error from approximation Eq. (34) in the original coordiate system, which is the error from replacing by the wedge product . In practice, the switching point is easy to identify. Since the error is the difference between the curvy “trapezoid” and its linear interpolation, the minimization is achieved by choosing the orbit point that is “closest” to the fixed point. Therefore, in the example of Fig. 7, the switching point can be identified graphically to be , which is the closest point along relative to . By ranging over all possible choices of , and , Eq. (34) suffices to calculate the classical actions of all periodic orbits inside the convergence zone.
v.3 Hénon map example
Consider the action of a satellite orbit in the area-preserving Hénon map Hénon (1976):
with parameter value . We have numerically computed a period- orbit satellite to one of the primary homoclinic orbit segments , where and . This gives:
whereas the wedge product gives:
The difference is in the decimal place. On the other hand, without the knowledge of , the full action can be calculated using Eq. (34):
to be compared with the actual action:
The relative error equals , demonstrating the high accuracy of the wedge product approximation.
The information about classical actions associated with homoclinic, heteroclinic and periodic orbits that come into various semiclassical sum rules play an important role in the study of quantum chaotic dynamical systems. Although the orbit actions can be calculated from the generating functions, the relations and correlations amongst their values cannot be discovered without an analysis of the type given in this paper. Furthermore, in the asymptotic limit of semiclassical mechanics, the actions must be known to high precision to understand the interferences that arise in quantum dynamics, and that otherwise requires the accurate determination of long orbit segments. Since any initial deviation due to the machine precision will diverge exponentially, it is a priori difficult to compute periodic orbits with long periods. The analysis given here gives an explicit mechanism from which correlations could emerge and avoids the numerical difficulties by making the detailed long orbit calculations unnecessary.
One interesting example is given by the heteroclinic tangle of the standard map, which arises from the two unstable fixed points of the map on the line. One of the fixed points is hyperbolic with reflection, which generates a single lobe fundamental structure in the tangle under a double iteration of the map. This lobe’s area must equal twice the action difference of the fixed points, a nontrivial relation to imagine without generating Eqs. (12,13).
For fully chaotic systems, the convergence zone can cover most of the accessible phase space Harsoula et al. (2015), and in that case nearly all of the periodic orbits fall into the category of satellite orbits, to which our analysis applies. Action differences between any pair of the satellite periodic orbits or between them and particular homoclinic (heteroclinic) orbits follow naturally. The simple, rather accurate geometric approximation involving a wedge product generates expressions that do not require the construction of the orbits or Moser invariant curves, only short sections of the stable and unstable manifolds (very simple and stable to calculate) and the endpoints of the homoclinic segments concerned. The error of this approximation scheme decreases exponentially as the length of the orbit increases and the instability exponent of the system increases.
All Moser invariant curves intersect in the untransformed phase space and as shown in Birkhoff (1927); da Silva Ritter et al. (1987); Ozorio de Almeida (1989), some satellite orbits lie on more than one Moser curve. In those cases, the actions of the satellite orbits are also related to homoclinic (heteroclinic) orbit actions, and possibly multiple fixed point actions Ozorio de Almeida (1989). It is of significant interest to understand the connections of the resulting multiple possible action relations, and this subject is left for future publication.
Appendix A MacKay-Meiss-Percival action principle
The MacKay-Meiss-Percival action principle discussed in this section was first developed in MacKay et al. (1984) for transport theory. A comprehensive review can be found in Meiss (1992). Generalization of the original principle beyond the “twist” and area-preserving conditions is discussed in Easton (1991). A higher-dimensional generalization using generating -forms and phase space volume forms is discussed in Lomelí and Meiss (2009).
Consider an arbitrary point and its orbit in phase space . The twist condition indicates the existence of a generating (action) function which brings into under the mapping , such that:
The total action is the sum:
The central step to obtain the MacKay-Meiss-Percival action principle is demonstrated by Fig. 39 along with Eq. (5.6) in Meiss (1992). Shown here in Fig. 9 are two arbitrary points , and their images , . Let be an arbitrary curve connecting and , which is mapped to a curve connecting and . Let and denote the algebraic area under and respectively. Then the difference between these areas is
i.e., the difference between the two algebraic areas gives the difference between the action functions for one iteration of the map. Starting from this, MacKay MacKay et al. (1984) derived a formula relating the action difference between a pair of homoclinic orbits to the phase space area of a region bounded by stable and unstable manifolds, as demonstrated by Fig. 10.
In this Figure, and is a pair of homoclinic points:
There exist unstable and stable manifolds connecting the two points shown by the solid and dashed curves. Those manifolds could be the manifolds of other fixed points, or manifolds associated with and themselves. Let and be the corresponding segments, then the action difference between and is given by:
where denotes the area shown in Fig. 10.
Appendix B Normal form coordinates, Moser invariant curves and satellite periodic orbits
There are infinite families of unstable periodic obits accumulating on every homoclinic orbit da Silva Ritter et al. (1987); Ozorio de Almeida (1989). These orbits are supported by Moser invariant curves, with the orbit points being successive self- or mutual-intersections between the invariant curves. The existence of such curves and orbits is a consequence of the Birkhoff-Moser theorem Birkhoff (1927); Moser (1956); da Silva Ritter et al. (1987). If the Poincaré map is invertible and analytic, there exists an analytic transformation (normal form transformation) from the normal form coordinates to the neighborhood of stable and unstable manifolds of the hyperbolic fixed point, for which the map takes the simple form:
where is a polynomial function of the product Harsoula et al. (2015):