Some Properties Of Quadratic Maps On Plane

In recent decades, more and more scholars have been interested in studying qualitative properties, such as stability, oscillations, bifurcations and chaotic behavior etc. of discrete dynamical systems, particularly, the difference equations [cf. 17-26,28], and many significant results have been obtained. But due to its complexity, its theoretical frame has not been completed yet. Moreover, some deep qualitative studies are just setting out, or even blank space.In qualitative theory of planar vector fields and the study of the 16th Hilbert's problem, the theory of the planar quadratic vector fields, such as the existence conditions and distribution of limit cycle, takes a very important position. In this paper, we study some general properties of quadratic maps on plane, the simplest nonlinear discrete dynamical systems. In section 2, we investigate the existence and the maximum number of invariant line. Then, we compute the maximum number of tangent point in a line when which is not invariant in section 3, and prove that the number of foci and center is at most 2. In section 4, after introducing focal quantity, which was used in vector field theory formerly, into planar maps, we use normal form theory to compute the first two focal quantities of focus and study the Naimark-Sacker bifurcation.