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

  • Tarini Kumar Dutta
  • Debasmriti Bhattacherjee
  • Debasish BhattacharjeeEmail author
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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Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

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

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