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

• Tarini Kumar Dutta
• Debasmriti Bhattacherjee
• Debasish Bhattacharjee
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 755)

## 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.

## Keywords

Period-Doubling bifurcation Periodic points Horizontal visibility graph Limit cycle

## 2010 AMS Classification

37 G 15 37 G 35 37 C 45 05 C 07

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© Springer Nature Singapore Pte Ltd. 2019

## Authors and Affiliations

• Tarini Kumar Dutta
• 1
• Debasmriti Bhattacherjee
• 1
• Debasish Bhattacharjee
• 1
1. 1.Gauhati UniversityGuwahatiIndia