Abstract
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.
Highlights
Arguably, one of the most important and general integrable maps is known in the literature as the Quispel-Roberts-Thompson map (QRT)
The QRT map is closely related to so called soliton equations (Quispel et al, 1988; Quispel et al, 1989)
4.4 Generic Situation To study the dynamics of the system (13), we have plotted some of the level sets of the integral
Summary
One of the most important and general integrable maps is known in the literature as the Quispel-Roberts-Thompson map (QRT). 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. By using the standard staircase method, the resulted equation is 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 analyzed in (Duistermaat, 2010). By substituting this solution to f (x) (and call it g(x) ), we derive a new discrete system: x = g(x) , with integral: (x) We will follow this technique and apply it to a generalized sine-Gordon equation.
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