Exact multiple inverses of the quadratic planar map

We developed a method for exactly determining the multiple inverses of quadratic planar maps. The quadratic planar map was transformed through an affine transformation into its conjugate domain, where all inverses can be exactly determined, and transform the inverses back. We showed that for all quadratic planar maps with non-vanishing critical curves, except for two exceptional cases of the map with a degenerate critical curve, there exists at least one available transformation. For both exceptional cases and for those maps with vanishing critical curves, the inverses are determinable without any transformation. This result can be applied to the calculation of stable manifolds of the quadratic planar map. It may help us further understand the dynamics, the basins of attraction, and the bifurcations associated with the quadratic planar map under iteration.


Introduction
The quadratic planar map is an important class of the two-dimensional non-invertible map called planar endomorphism. The most remarkable feature of planar endomorphism is the non-invertibility. Determination for preimages (inverses) of a point under the map plays an important role in studying the dynamical behaviour of nonlinear systems described by the iteration of the map [1][2]. For example, the basin boundary of attraction is often organized by the stable manifold associated with a saddle fixed point under the inverse of the map [3][4]. It is impossible to write down the inverses of such maps explicitly and the determination of the preimage often resorts to the Newton method numerically. However, whether the Newton method converges to the root (preimage) depends on the initial guess. The dynamics of the Newton method often settle down to periodic orbits or chaos when the root is near the critical point of the map. Furthermore, the Newton method gives no guarantee that how many preimages exist even though the root can be found very quickly. This work arises from the study on the calculation of the stable manifold of quadratic planar maps [5]. Kostelich et al. [6] have given a plot of the stable manifold of a non-invertible map, but still leave the problem of how to trace the stable manifold to traverse the critical curve of the map unresolved.
In this paper, we successfully developed an analytical method for determining the exact multiple inverses of quadratic planar maps. Generally speaking, the inverses of all quadratic planar maps with non-vanishing critical curves can be determined exactly by an affine transformation except for two exceptional cases of maps with critical curves degenerating to straight lines, for which no available transformation could be found. However, for both exceptional cases and for those maps with vanishing critical curve, inverses are determinable without any transformation. This result can be applied to the calculation of stable manifolds of the quadratic planar map. It may help us further understand the dynamics, the basins of attraction, and the bifurcations associated with the quadratic planar map under iteration. where T   β Q Ar β and det ( ) Since the matrix A is real-symmetric (we exclude the trivial case A=0 in which the inverse can be determined directly), it can always be diagonalized such that The LC is generally a hyperbola, but degenerates into two crossing lines as 0 However, when Similarly, (3) is reduced to (leave r to be arbitrary rather than Thus the LC of T can be classified into the third category.

Determination of the inverses of the quadratic planar map
It is convenient to rewrite (1) as , and all have real entries. Our goal is to find the exact multiple values of . That is, we wish to find all the points x such that ' x which takes the form explicitly 2  2  11  12  13  11  12  1   2  2  21  22  23  21  22  2 ' ' x a x a xy a y b x b y c y a x a xy a y b x b y c Generally, it is impossible to write down the inverses explicitly. Our aim is to solve this general problem exactly by an analytical method.
By introducing an affine transformation x ψ x φ. We say that the map ( ) T x is conjugate to the map ( ) T x and the two equations ' x are equivalent. If we can find the value of 1 ( ') T  x , we can also find the value of 1 x as shown in Figure 1. Note that the Taylor series expansion of any quadratic planar map is also a quadratic planar map and all the coefficients associated with the quadratic terms remain unchanged. We have ( Therefore, the conjugate map of T has the form

A special case of the quadratic planar equation
There is a special case of the quadratic planar equation ' ( ) T  x x that we can find its inverses

Quadratic planar map with a non-degenerate critical curve
In this subsection, we present the main result of this paper about how to find the affine transformation   x ψx φ such that the equivalent equation in the conjugate domain satisfies (11) and prove that there exists at least one available transformation for all quadratic planar maps with non-degenerate LCs.
Considering (8)  x a x a xy a y c y a x a xy a y y c In order to find the required affine transformation, we must find the critical point φ by imposing an additional constraint that, after the transformation, the constant term of the first scalar equation of ' Our main result is as follows.  , and the parameter  satisfies the following equation Proof. Since . Since e2 is the eigenvector of ( )  For all quadratic planar maps with non-degenerate critical curves, there exists at least one available affine transformation such that the multiple inverses of the map can be exactly determined via (13) and (14).

Quadratic planar map with a degenerate critical curve
Thus far we have ensured the existence of the transformation such that the multiple inverses of the quadratic planar map with a non-degenerate LC can be determined analytically. There left the maps with straight lines of LCs (hyperbolic and parabolic types), with an isolated single point of LC (elliptic type), and with vanishing LCs which we cannot guarantee the existence of the transformation. Essentially, for the maps with LCs degenerating to straight lines, we still can find an available transformation via (13) and (14), but, under certain ' x , two exceptional cases leading to no available transformation may occur. First, it is possible that 1 0 R  and (14) has only one real root 1