Chaotic Behaviour and Bifurcation in Real Dynamics of Two-Parameter Family of Functions including Logarithmic Map

Mohammad Sajid

doi:10.1155/2020/7917184

Abstract

The focus of this research work is to obtain the chaotic behaviour and bifurcation in the real dynamics of a newly proposed family of functions fλ,ax=x+1−λxlnax;x>0, depending on two parameters in one dimension, where assume that λ is a continuous positive real parameter and a is a discrete positive real parameter. This proposed family of functions is different from the existing families of functions in previous works which exhibits chaotic behaviour. Further, the dynamical properties of this family are analyzed theoretically and numerically as well as graphically. The real fixed points of functions fλ,ax are theoretically simulated, and the real periodic points are numerically computed. The stability of these fixed points and periodic points is discussed. By varying parameter values, the plots of bifurcation diagrams for the real dynamics of fλ,ax are shown. The existence of chaos in the dynamics of fλ,ax is explored by looking period-doubling in the bifurcation diagram, and chaos is to be quantified by determining positive Lyapunov exponents.

Highlights

During the past four decades, there are significantly developments in computational power, advances in technology, and mathematical theory that have facilitated formulation of nonlinear approaches for intricate systems [1, 2] since many physical, socioeconomic, and natural systems are intrinsically nonlinear, so these systems show large range of characteristics
Chaos has been noted in several experimental works and it redesigns many researches in different fields of engineering and science [3,4,5]; the study of chaotic behaviour in dynamical systems has been the interest of many scientists, engineers, and mathematicians
The dynamics of a DC-AC resonant self-oscillating LC series inverter is analyzed in [13] from the point of view of piecewise smooth dynamical systems, and the bifurcation analysis is performed in a onedimensional parameter space

Summary

Introduction

During the past four decades, there are significantly developments in computational power, advances in technology, and mathematical theory that have facilitated formulation of nonlinear approaches for intricate systems [1, 2] since many physical, socioeconomic, and natural systems are intrinsically nonlinear, so these systems show large range of characteristics. It gives that f′λ,aðρ1Þf′λ,aðρ2Þf ′λ,aðρ3Þf′λ,aðρ4Þ = j0:04519 × ð−2:26683Þ × 0:90905 × ð−2:07780Þj ð17Þ ≈ 0:19347 < 1: the periodic 4-cycle of f3:35,0:9ðxÞ is attracting (iii) If λ = 3:4, the periodic points of 8-cycle of f λ,aðxÞ are determined as ρ1 = 0:61675, ρ2 = 1:26247, ρ3 = 0:84200, ρ4 = 1:35862, ρ5 = 0:63074, ρ6 = 1:27879, ρ7 = 0:80822, and ρ8 = 1:36456 It follows that f ′λ,aðρ1Þf ′λ,aðρ2Þf ′λ,aðρ3Þf ′λ,aðρ4Þf ′λ,aðρ5Þf ′λ,aðρ6Þf ′λ,a Á ðρ7Þf ′λ,aðρ8Þf ′λ,aðρ9Þf ′λ,aðρ10Þ × f ′λ,aðρ11Þf ′λ,a Á ðρ12Þf ′λ,aðρ13Þf ′λ,aðρ14Þf ′λ,aðρ15Þf ′λ,aðρ16Þj. It gives that f ′λ,aðτ1Þf ′λ,aðτ2Þf ′λ,aðτ3Þf ′λ,aðτ4Þf ′λ,aðτ5Þf ′λ,aðτ6Þf ′λ,a ððρτ172ÞÞff′λ′λ,a,aððττ81Þ3fÞ′λf,a′λð,aτð9τÞ1f4′λÞ,fað′λτ,a1ð0τÞ1f5′λÞ,fað′λτ,a1ð1τÞ16×Þf

Numerical Simulation of Real Cycles of Period 3 with Their Nature

Chaos by Bifurcation Diagrams and Lyapunov Exponents

Conclusion