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

Abstract

<jats:p>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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mo>+</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>λ</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:mi mathvariant="normal">ln</mml:mi><mml:mi>a</mml:mi><mml:mi>x</mml:mi><mml:mo>;</mml:mo><mml:mi>x</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math>, depending on two parameters in one dimension, where assume that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>λ</mml:mi></mml:math> is a continuous positive real parameter and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>a</mml:mi></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mfenced></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mfenced></mml:math> are shown. The existence of chaos in the dynamics of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mfenced></mml:math> is explored by looking period-doubling in the bifurcation diagram, and chaos is to be quantified by determining positive Lyapunov exponents.</jats:p>

Chaotic behavior in real dynamics and singular values of family of generalized generating function of Apostol-Genocchi numbers

“…For λ = 1, a = 1, b = e a , t = z, the left hand side function in this relation becomes function of our family F. This paper is a generalization of work [9] of the family of functions λ 2z e z +1 which is a generating function of the Genocchi numbers. The paper [9] contains almost similar results and proofs from [18,21]. Besides we provide simpler proof of the result on the number of fixed points.…”

“…The exploration of dynamics of a real function has become an important topic, partially due to the fact that it deduce the iterations of the function in the complex plane which is mainly influenced by its real dynamics [6,12,16,23]. The real dynamics of the cubic polynomials, generalized logistic maps and oneparameter family of transcendental functions are given in [1,10,15,21] respectively. The real fixed points are described for one-parameter family of function x b x −1 in [17] and for a two-parameter family λ( x b x −1 ) n in [8].…”

Stabilization in chaotic maps using hybrid chaos control procedure

“…In 2019, the control of chaos using superior feedback method was studied by Ashish et al [38] and also in a series of article they examined the dynamic properties in various generalized logistic maps (see also Ashish et al [39] , [40] , [41] , [42] ). In a series of articles, M. Sajid et al [43] , [44] , [45] studied the chaotic properties of two-parameter functions with the logarithmic system, control of chaos in fractional order maps, and control in microscopic chaos using adaptive control technique. In 2023, F. Wang et al [46] examined the coexistence of heteroclinic cycles in 3D piecewise dynamical systems with three discontinuous switching manifolds using various mathematical analysis.…”

A Brief Study on Julia Sets in the Dynamics of Entire Transcendental Function Using Mann Iterative Scheme

“…These variations may include the periodic point structure, including other changes too. Sajid [32] obtained the chaotic behavior and bifurcation in the real dynamics of a newly proposed family of functions that depends on two parameters in one dimension. A recent review on some advances in dynamics of one-variable complex functions has been given in [33].…”