Chaotic Dynamics of the Fractional-Love Model with an External Environment

Abstract

Based on the fractional order of nonlinear system for love model with a periodic function as an external environment, we analyze the characteristics of the chaotic dynamic. We analyze the relationship between the chaotic dynamic of the fractional order love model with an external environment and the value of fractional order (α, β) when the parameters are fixed. Meanwhile, we also study the relationship between the chaotic dynamic of the fractional order love model with an external environment and the parameters (a, b, c, d) when the fractional order of the system is fixed. When the parameters of fractional order love model are fixed, the fractional order (α, β) of fractional order love model system exhibit segmented chaotic states with the different fractional orders of the system. When the fractional order (α = β) of the system is fixed, the system shows the periodic state and the chaotic state as the parameter is changing as a result.

Numerical Solutions for Multi-Term Fractional Order Differential Equations with Fractional Taylor Operational Matrix of Fractional Integration

“…In fact, fractional calculus provides the mathematical modeling of some important phenomena like social and natural in a more powerful way than the classical calculus. During the last few decades, many applications were reported in many branches of science and engineering such as chaotic systems [6,7], fluid mechanics [8], viscoelasticity [9], optimal control problems [10,11], chemical kinetics [12,13], electrochemistry [14], biology [15], physics [16], bioengineering [17], finance [18], social sciences [19], economics [20,21], optics [22], chemical reactions [23], rheology [24], and so on. Due to the importance of FDEs, the solutions of them are attracting widespread interest.…”

The Fractional Form of the Tinkerbell Map Is Chaotic

“…Recently, nonlinear oscillations on Riemannian manifolds that can exhibit a chaotic behavior were introduced in [7,8]. Other related works include an investigation of the chaotic dynamics in a fractional love model with an external environment, as in [9], and an extension using a fuzzy function [10].…”

Analysis of Chaotic Behavior in a Novel Extended Love Model Considering Positive and Negative External Environment

“…Bae et al [ 24 , 25 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 ] proposed that the love model of Romeo and Juliet had periodic motion and chaotic behavior through time series and phase portraits. It deals with the same and different time delays [ 24 , 25 ].…”

“…It deals with the same and different time delays [ 24 , 25 ]. It also applies several types of external environment such as sine wave, sine wave with Fourier series, discontinuous function, Gaussian, triangular and fuzzy membership function [ 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 ]. However, Bae et al did not consider detailed relationships as external environments such as qualitative or quantitative analysis of characteristics.…”

“…Most previously published papers did not demonstrate an exact chaotic phenomenon in the Romeo and Juliet model with time series, phase portrait, power spectrum, Poincare map, Lyapunov exponent, and bifurcation diagram. They demonstrated chaotic phenomena or behaviors by only using time series, phase portrait, and bifurcation diagram [ 24 , 25 , 34 , 35 , 36 , 37 , 38 , 39 , 40 ].…”