Graph Theoretic Scenario in Period Doubling and Limit Cycle Circumstances in Two-Dimensional Maps

Abstract

In this paper two-dimensional discrete dynamical systems have been considered. The period doubling bifurcation points of period $2^{n}$ corresponding periodic points of the dynamical system $x_{k + 1} = 1 - ax_{k}^{2} + y_{k}$, $y_{k + 1 } = \beta x_{k}$ where $a$ is a parameter and $\beta$ is constant are calculated for three different values of $\beta$, i.e., $\beta = 0.2$, $\beta = 0.02$ and $\beta = 0.01$. It has been seen that the relative position of the $x$ coordinate of the Henon map follows a Mathematical model, which can be used to discuss some graph theoretic properties, up to some values of $n$ and this value of $n$ increases as value of $\beta$ decreases. For the dynamical system $x_{n + 1} = ax_{n} \left( {1 - x_{n} } \right) - bx_{n} y_{n}$, $y_{n + 1} = - cy_{n} + dx_{n} y_{n}$ a limit cycle has been considered for a particular value of $a,b,c,d$ and graph theoretical scenario has been put forward where degree of every points have been calculated by using a suitable computer program.