Some Maps

This chapter deals with some important maps and their elementary properties. In particular, we are interested in finding fixed points, their stability behaviors, and formation of periodic cycles, stabilities of the periodic cycles, and bifurcation phenomena of some special maps. Maps and their compositions represent many natural phenomena or engineering processes. For example, dynamical models have been used for the study of population of species over centuries. In general, we would like to know how the size, say at (n + 1)th generation of a population model is related to the preceding generations of that model. Often a growth rate or reproductive rate of a population appears in the model. This may be expressed by a relationship $$ x_{n + 1} = f(x_{n} ,r) $$ x n + 1 = f ( x n , r ) , where x n denotes the population at nth generation and r is the population growth parameter. Simple population model for species can be formulated through mathematical modeling where the reproductive rate is a function r(x) which decreases with increasing population x, from an initial value r(0) = r 0 to r(x) = 0 at some limiting value of population. A simple population model with a linear decrease of growth rate r(x) with increasing x (known as logistic growth rate) can be expressed mathematically by the function $$ f(x) = rx(1 - x),x \in [0,1] $$ f ( x ) = r x ( 1 - x ) , x ∈ [ 0 , 1 ] . Starting from some initial population x 0, the sequences of population at successive generations are given by $$ x_{n + 1} = rx_{n} (1 - x_{n} ),\,{\kern 1pt} n = 0,1,2, \ldots $$ x n + 1 = r x n ( 1 - x n ) , n = 0 , 1 , 2 , … . Similarly, the tent, Euler’s shift, and Hénon maps have importance in many contexts and are discussed in the following sections. The branching of a solution at a critical value of the parameter of a map is called bifurcation. It is a qualitative change of dynamics or orbits for changing values of parameters of a map. Bifurcation theory in discrete systems is vast and we shall introduce few particular bifurcations, viz., saddle-node, period-doubling, and transcritical bifurcations.