Dynamics of A Re-Parametrization of A 2-Dimensional Mapping Derived from Double Discrete Sine-Gordon Mapping

We study the dynamics of a two dimensional map which is derived from another two dimensional map by re-parametrizing the parameter in the system. It is shown that some of the properties of the original map can be preserved by the choice of the re-parametrization. By means of performing stability analysis to the critical points, and also studying the level set of the integrals, we study the dynamics of the re-parametrized map. Furthermore, we present preliminary results on the existence of a set where iteration starts at a point in that set, in which it will go off to infinity after finite step.


Introduction
Arguably, one of the most important and general integrable maps is known in the literature as the Quispel-Roberts-Thompson map (QRT). It is a two-dimensional map depending on 18 parameters. The QRT map is closely related to so called soliton equations (Quispel et al., 1988;Quispel et al., 1989). More recent studies have focused on generalizations of QRT maps. One of them was proposed by Joshi and Kassotakis (2019). Main result of their paper is a new connection between two major theories that generalize QRT maps. They provide a new formulation of QRT involutions in terms of Hirota derivatives and discover conditions under which each involution can be factorized into two further involutions.
The sine-Gordon equation is a partial differential equation, which is known to have soliton solutions; hence, it is a soliton equation (Quispel et al., 1991). Discretizations of the sine-Gordon equation have been done in various ways (Quispel et al., 1988;Quispel et al., 1989;Quispel et al., 1991). The reduction of the sine-Gordon equation to a two dimensional ordinary difference equations using a standard staircase (see Van der Kamp and Quispel, 2010 for the method) is known as being a special case of the celebrated QRT map. Recently, Celledoni et al. (2019) have studied a new systematic approach for calculating the preserved measures and integrals of a rational map in which a two-dimensional sine-Gordon (standard) map was chosen as an example (see Celledoni et al., 2019 §4.3 Example 4).
Generating a new family of mapping from a known one is an interesting topic to study. A piece-wise linear map from a known integrable map by using the ultra discretization technique is generated by Tuwankotta et al. (2004). The number of independent integrals are preserved by the transformation which implies that the integrability is preserved. In our article (Zakaria and Tuwankotta, 2016) a straight-forward generalization by adding parameters in the Lax pair of the ordinary discrete sine-Gordon partial difference equation has done. By using the standard staircase method, the resulted equation is then reduced to system of ordinary difference equations (see Van der Kamp et al., 2007 for the method). Note that this generalized sine-Gordon system is also analyzed in (Duistermaat, 2010).
Another novel method to introduce a new class of discrete systems from an integrable discrete system, is by introducing the concept of dual (Quispel et al., 2005). This works beautifully for a single discrete equation, although the resulting equation may not be new nor integrable. This idea of dual is extended to system of discrete equations in (Tuwankotta et al., 2019). The latter is interesting in the sense that the method proposed there produces in general more than one system. Roberts et al. (2002) constructed a new family of mapping by interchanging the parameter and the integral. For example, consider a discrete dynamical system in n which is denoted by: x f x , where the prime denotes the upshift,  is a parameter in the system, and : nn   f . We assume that the exists a smooth function : nn G  such that: , which is called an integral for the system. Suppose that we can solve the Then by substituting this solution to () f  x (and call it () g x ), we derive a new discrete system: We will follow this technique and apply it to a generalized sine-Gordon equation. The aim of this paper is to show a number of properties of the new mapping (after re-parameterization) and to compare them with the original mapping. Furthermore, the qualitative behavior of the new mapping is studied by means of obtaining fixed points and their stability, and also the base points.
The outline of this paper is the following. In Section 2, a system of first order difference equations derived from generalized double discrete sine-Gordon (  sine-Gordon) equations is formulated by restricting to traveling wave solution. In Section 3 a new integrable mapping derived from  sine-Gordon mapping by interchanging the role of the integral and the parameter in the original system is presented. The new system is then analyzed by means of describing its symmetry and measure preservation. In Section 4 finding fixed points and periodic points (and their linear stability) and also computing the base points are discussed. These are done in the remaining subsections in Section 4. We have divided Section 4 into five subsections. This paper ends with some concluding remarks in Section 5.

Formulation of the Problem
Consider a three parameters family of partial difference equation on two dimensional lattice:   This equation is derived from the compatibility condition of the generalized Lax pair of the classical discrete sine-Gordon equation (see Zakaria and Tuwankotta, 2016). The travelling wave solutions of (1) can be obtained by considering the following form:  , .
Using this, we can write (3) as: (4) where the prime denotes the upshift. The mapping (4) has an integral, i.e.
Substituting (6) into (4), we derive a new mapping: Note that the integral (8) is nothing but the function ( , ) xy  in (6). The new map, ˆ( , ) f x y , has the following properties:  The orbits of ˆ( , ) f x y is 4-periodic. This is simple to show that the linear map in (7) of the plane, which for the linear map in equation (7) is immediate (the Jacobian determinant is constant and equal to 1), as well as the so-called "reverse symmetry", since . Alternatively, we can use different procedure to show that an integrable map is measure preserving (Roberts et al., 2002). For our map, ˆ( , ) f x y , the procedure can be followed as shown below.
The mapping ˆ( , ) f x y is measure preserving (area preserving) because there is the density xy Df x y xy    when the density  is given by The dynamics of the mappings in (7) Consequently, we transform the integral (5) to For the case where: 1 0 a  , let us add a constant in (10) which takes a special form: This implies that we can look at the zero level curve of F and solve it for  : Substituting this expression of ( , ) xy  into f in (9) gives: is an integral for the system: The mapping (13) has some properties:  f is measure preserving and orientation-reversing (or anti-measure preserving), which means (Roberts et al., 2002) Meanwhile, from the right-hand side, we have where the so-called density  is given by

Critical Point and Base Point of the Integral
There are two important elements in analyzing the dynamics of system (13)

The Critical Point
The critical points of the integral function (12) It is easy to verify that: (1,1) and ( 1, 1)  are two fixed points while (1, 1)  and ( 1,1)  are two 2-periodic points.

The Base Point
Apart from the critical points, the so-called base points also play a crucial role in the dynamics of (13). Note that the invariant (12) Figure 1 shows the base points (red-circle) together with fixed points and 2-periodic points (blue-circle). Three diagrams in Figure 1 are some level sets of integral (12) together with the fixed points, the 2-periodic points, and the base points for 0 = 1.5.

Preimages of the Base Points
A base point is mapped to infinity by the mapping f . Furthermore, there are two base points which do not refer to a base point. But after one iteration they are mapped to base points. This means that if we start at those points, the iteration of f will be sent to infinity after two iterations.
Let us name the set of points which are mapped by f to a base point by 1 The solution is 1 10 00 The graphs of 2 P to 6 P for 0 =4 a are presented in Figure 2. The graph of 2 P , 3 P , 4 P , 5 P , and 6 P are plotted using green, red, blue, orange, and black, respectively. It is interesting to note that if

Generic Situation
To study the dynamics of the system (13), we have plotted some of the level sets of the integral (see Figure 3). These level sets contain solution of the system (13). Furthermore, by studying how the level sets deformed as we vary the value parameter, the bifurcations in the system can be studied. This is however beyond the scope of this paper.
By setting 0 = 4 and 1 = 0.7, we have all of the curves around the fixed points, (1,1) and (-1,-1), are of hyperbolic type. On the other hand we have all of the curves around the 2-periodic points, (-1,1) and (1,-1), are of elliptic type. But for 0 = 4 and 1 = −0.875, the situation is reverse, i.e. all of the curves around the fixed points are of elliptic type while all of the curves around the 2-periodic points are of hyperbolic type.

Conclusion
After re-parametrizing the parameter in (9) and in (10)  measure preserving, and reversible. Furthermore, it has two fixed points and two 2-periodic points which are of an elliptic type and a hyperbolic type. It is interesting to underline the fact that we have the set of k P consisting of points in 2 which is mapped to infinity after k -iterates of the map.
Then we can consider the set of 1 = k P  . For an arbitrary level set of the integral (12), the question should be whether the intersection between the set of with the level set is finite or infinite (could it be dense on the level set). This is a subject to future investigation.