Analysis of a Fractional-Order Couple Model with Acceleration in Feelings

Abstract

A fractional-order nonlinear dynamical model of couple has been introduced. Upper bounds are obtained for a fractional-order nonlinear dynamical model. Also different from other models, a model with the order 2α is discussed. We are expecting an acceleration in feelings; that is why we increase the order of the derivative between 1<2𝛼≤2. Stability analysis of the fractional-order nonlinear dynamical model of involving two persons is studied using the fractional Routh-Hurwitz criteria. By using stability analysis on fractional-order system, we obtain sufficient condition on the parameters for the locally asymptotic stability of equilibrium points. Finally, numerical simulations are presented to verify the obtained results.

1. Introduction

The first noninteger order differentiation and integration notion was considered in 1695 by Leibniz and L’Hôpital. In a letter to L’Hôpital in 1695, Leibniz raised the following question: “Can the meaning of derivatives with integer order be generalized to derivatives with noninteger orders?” L’Hôpital was somewhat curious about that question and replied by another question to Leibniz: “What if the order will be 1/2?” After the letter was answered by Leibniz, fractional order in the concept of derivative was formed [1].

There are lots of topics on fractional modeling, but in recent decades the study of interpersonal relationships has begun to be popular. Interpersonal relationships appear in many contexts, such as in family, kinship, acquaintance, work, and clubs [2]. Mathematical modeling in interpersonal relationships is very important for capturing the dynamics of people, but there are few models in this area and models have been limited to integer order differential equations. Another interesting dynamic is marriage. Marriage has been studied scientifically for the past sixty years [3]. Researchers are trying to understand why some couples divorce, but others do not, and why, among those who remain married, some are happy and some are miserable with one another [4]. Since experiments in these areas are difficult to generate, mathematical models may play a role in explanation of the dynamics of a couple and behavioral features.

Recently, a fractional-order system for the dynamics of love affair between a couple has been considered [5]. In this paper, different from [5], a model with the order 2𝛼 is discussed. We are expecting an acceleration in feelings; that is why we increase the order of the derivative between 1<2𝛼≤2. Also, upper bounds are discussed for the system.

We begin by giving the definitions and properties of fractional-order integrals and derivatives [6].

2. Preliminaries and Definitions

The three most common definitions for fractional derivative can be given as the Grünwald-Letnikov definition, the Riemann-Liouville definition, and the Caputo definition.

Definition 1. The Riemann-Liouville type fractional integral of order 𝛼>0 of a function 𝑓∶(0,∞)→𝑅 is defined by 𝐼^𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑓(𝜏)𝑑𝜏, (1) where Γ(⋅) is the gamma function.

Definition 2. The Grünwald-Letnikov definition is given as _𝑎𝐷^𝛼_𝑡𝑓(𝑡)=lim_ℎ→0ℎ^{[()−𝛼𝑡−𝑎/ℎ]}_𝑗=0(−1)^𝑗𝛼𝑗𝑓𝑡−𝑗ℎ. (2)

Definition 3. The Riemann-Liouville type fractional derivative of order 𝛼>0 of a function 𝑓∶(0,∞)→𝑅 is defined by 𝐃^𝛼𝑓(𝑡)=𝑑^𝑛𝑑𝑡^𝑛1Γ(𝑛−𝛼)^𝑡₀(𝑡−𝜏)^{𝑛−𝛼−1}𝑓(𝜏)𝑑𝜏, (3) where 𝑛=[𝛼]+1 and [𝛼] is the integer part of 𝛼.

Definition 4. The Caputo type fractional derivative of order 𝛼>0 of a function 𝑓∶(0,∞)→𝑅 is defined by 𝐷^𝛼𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡₀(𝑡−𝜏)^{𝑛−𝛼−1}𝑓^𝑛(𝜏)𝑑𝜏, (4) where 𝑛=[𝛼]+1 and [𝛼] is the integer part of 𝛼.

Some properties of the Caputo derivative and the Riemann-Liouville derivative formulas are given below: ^𝐶_𝑎𝐷^𝛼_𝑡^𝐶_𝑎𝐷^𝑚_𝑡𝑓(𝑡)=^𝐶_𝑎𝐷^𝑚_𝑡^𝐶_𝑎𝐷^𝛼_𝑡𝑓(𝑡)=^𝐶_𝑎𝐷_𝑡^𝛼+𝑚𝑓(𝑡),𝑓^(𝑠)(0)=0,𝑠=𝑛,𝑛+1,…,𝑚𝑚=0,1,2,…;𝑛−1<𝛼<𝑛,_𝑎𝐃^𝑚_𝑡𝑎𝐃^𝛼_𝑡𝑓(𝑡)=_𝑎𝐃^𝛼_𝑡𝑎𝐃^𝑚_𝑡𝑓(𝑡)=_𝑎𝐃_𝑡^𝛼+𝑚𝑓(𝑡),𝑓^(𝑠)(0)=0,𝑠=0,1,2,…,𝑚𝑚=0,1,2,…;𝑛−1<𝛼<𝑛. (5) We see that, contrary to the Riemann-Liouville approach, in the case of the Caputo derivative, there are no restrictions on the values 𝑓^(𝑠)(0) (𝑠=0,1…,𝑛−1).

3. Equilibrium Points and Their Locally Asymptotic Stability

In this section, we consider a fractional-order nonlinear two-dimensional system as follows: 𝐷^2𝛼𝑥₁(𝑡)=−𝛼₁𝑥₁+𝛽₁𝑥₂1−𝜀𝑥²₂+𝐴₁,𝐷^2𝛼𝑥₂(𝑡)=−𝛼₂𝑥₂+𝛽₂𝑥₁1−𝜀𝑥²₁+𝐴₂,𝑥₁(0)=0,𝑥₂(0)=0, (6) where 𝐷^2𝛼 is the fractional derivative of order 1<2𝛼≤2.  𝛼_𝑖>0, 𝛼_𝑖, 𝛽_𝑖, and 𝐴_𝑖 (𝑖=1,2) are real constants. These parameters are oblivion, reaction, and attraction constants. In the equations above, we assume that feelings decay exponentially fast in the absence of partners. The parameters specify the romantic style of individuals 1 and 2. In the beginning of relationships, because they have no feelings towards each other, initial conditions are considered zero.

We note that, with zero initial conditions, the following equation is valid: 𝐷^𝛼𝐷^𝛼𝑥(𝑡)=𝐷^2𝛼(𝑥(𝑡)). (7)

In that case, the system can be considered as follows: 𝐷^2𝛼𝑥₁(𝑡)=𝐷^𝛼𝐷^𝛼𝑥₁(𝑡,𝐷_2𝛼)𝑥₂(𝑡)=𝐷^𝛼𝐷^𝛼𝑥₂(𝑡),𝑥₁(0)=0,𝑥₂(0)=0. (8)

Let us make the following changes of variables: 𝑥₁=𝑦₁,𝐷^𝛼𝑥₁=𝑦₂,𝑥₂=𝑦₃,𝐷^𝛼𝑥₂=𝑦₄. (9)

Then, transformed system is given below: 𝐷^𝛼𝑦₁(𝑡)=𝑦₂,𝐷^𝛼𝑦₂(𝑡)=−𝛼₁𝑦₁+𝛽₁𝑦₃1−𝜀𝑦²₃+𝐴₁,𝐷^𝛼𝑦₃(𝑡)=𝑦₄,𝐷^𝛼𝑦₄(𝑡)=−𝛼₂𝑦₃+𝛽₂𝑦₁1−𝜀𝑦²₁+𝐴₂, (10) with initial conditions 𝑦₁(0)=0,𝑦₂(0)=0,𝑦₃(0)=0,𝑦₄(0)=0, (11) where 0.5<𝛼≤1, 𝛼_𝑖>0, 𝛼_𝑖, 𝛽_𝑖, and 𝐴_𝑖 (𝑖=1,2) are real constants.

Let 𝛼∈(0.5,1] and consider the system 𝐷^𝛼𝑦₁(𝑡)=𝑓₁𝑦₁,𝑦₂,𝑦₃,𝑦₄,𝐷^𝛼𝑦₂(𝑡)=𝑓₂𝑦₁,𝑦₂,𝑦₃,𝑦₄,𝐷^𝛼𝑦₃(𝑡)=𝑓₃𝑦₁,𝑦₂,𝑦₃,𝑦₄,𝐷^𝛼𝑦₄(𝑡)=𝑓₄𝑦₁,𝑦₂,𝑦₃,𝑦₄, (12) with the initial values 𝑦₁(0)=0,𝑦₂(0)=0,𝑦₃(0)=0,𝑦₄(0)=0. (13) Here, 𝑓₁𝑦₁,𝑦₂,𝑦₃,𝑦₄=𝑦₂,𝑓₂𝑦₁,𝑦₂,𝑦₃,𝑦₄=−𝛼₁𝑦₁+𝛽₁𝑦₃1−𝜀𝑦²₃+𝐴₁,𝑓₃𝑦₁,𝑦₂,𝑦₃,𝑦₄=𝑦₄,𝑓₄𝑦₁,𝑦₂,𝑦₃,𝑦₄=−𝛼₂𝑦₃+𝛽₂𝑦₁1−𝜀𝑦²₁+𝐴₂. (14)

To evaluate the equilibrium points, let 𝐷^𝛼𝑦_𝑖(𝑡)=0⟹𝑓_𝑖𝑦^∗₁,𝑦^∗₂,𝑦^∗₃,𝑦^∗₄=0,𝑖=1,2,3,4, (15) from which we can get the equilibrium points 𝐾₀=(0,0,0,0) for 𝐴₁=𝐴₂=0 and 𝐾₁=(𝑦^∗₁,𝑦^∗₂,𝑦^∗₃,𝑦^∗₄).

The Jacobian matrix 𝐽(𝐾₁) for the system given in (14) is 𝐽𝐾₁=0100−𝛼₁0𝑎00001𝑏0−𝛼₂0, (16) where 𝑎=𝛽₁1−3𝜀𝑦^∗₃²,𝑏=𝛽₂1−3𝜀𝑦₁^∗2. (17) To discuss the local stability of the equilibrium 𝐾₁=(𝑦^∗₁,𝑦^∗₂,𝑦^∗₃,𝑦^∗₄) of the system given by (14), we consider the linearized system at 𝐾₁. The characteristic equation of the linearized system is of the form 𝑃(𝜆)=𝜆⁴+𝛼₂+𝛼₁𝜆²+𝛼₁𝛼₂−𝑎𝑏=0. (18) If 𝜆² is taken as 𝑘, we have the following reduced equation: 𝑃(𝜆)=𝑘²+𝑎₁𝑘+𝑎₂=0, (19) where 𝑎₁=𝛼₂+𝛼₁,𝑎₂=𝛼₁𝛼₂−𝑎𝑏. (20) According to the fractional Routh-Hurwitz criteria, we have the following theorem.

Theorem 5. If 𝑎₁>0 and 𝑎₂>0, then the equilibrium point 𝐾₁=(𝑦^∗₁,𝑦^∗₂,𝑦^∗₃,𝑦^∗₄) is locally asymptotically stable for all 𝛼∈(0,1).

Proof. 𝐾₁=(𝑦^∗₁,𝑦^∗₂,𝑦^∗₃,𝑦^∗₄) equilibrium of the system given by (12) is asymptotically stable if all of the eigenvalues, 𝑘_𝑖, 𝑖=1,2, of 𝐽(𝐾₁), satisfy the following condition (negative real part) [7, 8]: arg𝜆_𝑖>𝛼𝜋2. (21) For 𝑛=2, the Routh-Hurwitz criteria are just 𝑎₁>0 and 𝑎₂>0. The characteristic polynomial 𝑃(𝜆)=𝑘²+𝑎₁𝑘+𝑎₂=0 satisfies eigenvalues as below: 𝑘_1,2=−𝑎₁±𝑎²₁−4𝑎₂2. (22) Now, suppose that 𝑎₁ and 𝑎₂ are positive. It is easy to see that if the roots are real, they are both negative, and if they are complex conjugates, they have a negative real part.
Next, to prove the converse, suppose that the roots are either negative or have a negative real part. Then, it follows that 𝑎₁>0. If the roots are complex conjugates, 0<𝑎²₁<4𝑎₂, which implies that 𝑎₂ is also positive. If the roots are real, then since both of the roots are negative, it follows that 𝑎₂>0.

Theorem 6. Let 𝑎₂=(𝛼₁𝛼₂−𝑎𝑏) be as given in (20). If 𝑎₂<0, then the positive equilibrium point 𝐾₁=(𝑦^∗₁,𝑦^∗₂,𝑦^∗₃,𝑦^∗₄) of the system given in (12) is unstable.

Proof. If 𝑎₂<0, from Descartes' rule of signs, it is clear that the characteristic equation 𝑃(𝜆) has at least one positive real root. So, the equilibrium point 𝐾₁=(𝑦^∗₁,𝑦^∗₂,𝑦^∗₃,𝑦^∗₄) of the system given in (12) is unstable.

4. Analysis of a Model with Upper Bounds

In this section, we consider fractional-order system with the order 𝛼 between 0<𝛼<1: 𝐷^𝛼𝑥₁(𝑡)=−𝛼₁𝑥₁+𝛽₁𝑥₂1−𝜀𝑥²₂+𝐴₁,𝐷^𝛼𝑥₂(𝑡)=−𝛼₂𝑥₂+𝛽₂𝑥₁1−𝜀𝑥²₁+𝐴₂,𝑥₁(0)=0,𝑥₂(0)=0. (23) A detailed analysis of this model is given in [5]. With the help of the following lemmas, upper bounds are discussed for the system.

Before giving our results, we give some useful lemmas [9, 10].

Lemma 7. Let 𝛼, 𝛽, 𝛾, and 𝑝 be positive constants. Then, ^𝑡₀(𝑡−𝑠)^{𝑝(𝛽−1)}𝑠^{𝑝(𝛾−1)}𝑑𝑠=𝑡^𝜃𝐵𝑝𝛾−1+1,𝑝𝛽−1+1,𝑡∈𝑅₊, (24) where 𝐵[𝜉,𝜂]=∫¹₀𝑠^𝜉−1(1−𝑠)^𝜂−1𝑑𝑠(ℜ𝜉>0,ℜ𝜂>0) ve 𝜃=𝑝(𝛽+𝛾−2)+1.

Lemma 8. Let 𝑢, 𝑣, and 𝑓_𝑖∈𝐶(𝐼,𝑅₊), 𝑖=1,2, with 𝑓_𝑖 be nondecreasing; let 𝜑_𝑖𝑗∈𝐶(𝐼×𝐼,𝑅₊) be nondecreasing in a variable 𝑡 for every 𝑠 fixed (𝑖=1,2). If 𝑢(𝑡)≤𝑓₁(𝑡)+^𝑡₀𝜑₁₁(𝑡,𝑠)𝑢(𝑠)+𝜑₁₂()𝑣(𝑠𝑣(𝑡)𝑡,𝑠)𝑑𝑠,≤𝑓₂(𝑡)+^𝑡₀𝜑₂₁(𝑡,𝑠)𝑢(𝑠)+𝜑₂₂(𝑡,𝑠)𝑣(𝑠)𝑑𝑠,𝑡∈𝐼, (25) then, for 𝑡∈𝐼, one has 𝑢(𝑡)≤𝑓₁(𝑡)+𝑓₂(𝑡)^𝑡₀𝜑₁₂(𝑡,𝑠)Φ₂(𝑠)𝑑𝑠×exp^𝑡₀𝜑₁₁()+𝑡,𝑠𝑑𝑠_𝑡0𝜑₁₂(𝑡,𝑠)Φ₂(𝑠)×^𝑠₀𝜑₂₁(𝑠,𝜏)Φ₁(𝜏),𝑣(𝑡)≤𝑓𝑑𝜏𝑑𝑠₂(𝑡)+𝑓₁(𝑡)^𝑡₀𝜑₂₁(𝑡,𝑠)Φ₁(𝑠)𝑑𝑠×exp^𝑡₀𝜑₂₂()+𝑡,𝑠𝑑𝑠_𝑡0𝜑₂₁(𝑡,𝑠)Φ₁(𝑠)×^𝑠₀𝜑₁₂(𝑠,𝜏)Φ₂(𝜏)𝑑𝜏𝑑𝑠, (26) where Φ_𝑖(𝑡)∶=exp∫^𝑡₀𝜑_𝑖𝑖(𝑡,𝑠)𝑑𝑠, 𝑖=1,2.

Let 𝛼∈(0,1] and consider the system 𝐷^𝛼𝑥₁(𝑡)=𝑓₁𝑡,𝑥₁,𝑥₂,𝐷^𝛼𝑥₂(𝑡)=𝑓₂𝑡,𝑥₁,𝑥₂, (27) with the initial conditions 𝑥₁(0)=0 and 𝑥₂(0)=0. Here, 𝑓₁(𝑡,𝑥₁,𝑥₂)=−𝛼₁𝑥₁+𝛽₁𝑥₂(1−𝜀𝑥²₂)+𝐴₁ and 𝑓₂(𝑡,𝑥₁,𝑥₂)=−𝛼₂𝑥₂+𝛽₂𝑥₁(1−𝜀𝑥²₁)+𝐴₂. Now, upper bounds for a fractional-order nonlinear system are discussed with the following theorem.

Theorem 9. Let 𝑓₁ and 𝑓₂∈𝐶(𝐼×𝑅²,𝑅) and satisfy the following inequality: 𝑓₁𝑡,𝑥₁,𝑥₂≤𝛼₁(𝑡)𝑥₁+𝛽₁(𝑡)𝑥₂,𝑓₂𝑡,𝑥₁,𝑥₂≤𝛽₂(𝑡)𝑥₁+𝛼₂(𝑡)𝑥₂, (28) where 𝛼_𝑖 and 𝛽_𝑖∈𝐶(𝐼,𝑅₊) (𝑖,𝑗=1,2) and 𝑥₁,𝑥₂∈𝑅. Then, one has the following upper bounds for system of fractional order: 𝑥₁(𝑡)≤𝑡^𝛼−11×exp𝑞𝑘^∗(𝑡)×^𝑡₀𝛼^𝑞₁(𝑠)𝑑𝑠+^𝑡₀𝛽^𝑞₁(𝑠)Ψ₂(𝑠)×𝑘^∗(𝑠)^𝑠₀𝛽^𝑞₂(𝜏)Ψ₁(𝜏),𝑥𝑑𝜏𝑑𝑠₂(𝑡)≤𝑡^𝛼−11×exp𝑞𝑘^∗(𝑡)×^𝑡₀𝛼^𝑞₂(𝑠)+𝑑𝑠_𝑡0𝛽^𝑞₂(𝑠)Ψ₁(𝑠)×𝑘^∗(𝑠)^𝑠₀𝛽^𝑞₁(𝜏)Ψ₂(𝜏)𝑑𝜏𝑑𝑠, (29) for 𝑡>0, where 𝑝=1+4𝛼1+3𝛼,𝑞=1+4𝛼𝛼,𝑘^∗(𝑡)=𝑡^𝑞𝛼−1𝐵^𝑞/𝑝𝑝(𝛼−1)+1,𝑝(𝛼−1)+1Γ^𝑞(𝛼),Ψ₁(𝑡)=exp𝑘^∗(𝑡)^𝑡₀𝛼^𝑞₁(𝑠),Ψ𝑑𝑠₂(𝑡)=exp𝑘^∗(𝑡)^𝑡₀𝛼^𝑞₂(𝑠)𝑑𝑠. (30)

Proof. Since 𝑓₁ and 𝑓₂ are assumed to be continuous functions, every solution of the initial value problem (IVP) given by (23) is also a solution of the following integral system for 0<𝛼<1: 𝑥₁(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑓₁𝜏,𝑥₁(𝜏),𝑥₂(𝜏)𝑥𝑑𝜏,₂(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑓₂𝜏,𝑥₁(𝜏),𝑥₂(𝜏)𝑑𝜏. (31) Moreover, every solution of integral system is a solution of the IVP [11]. Now, we derive from (28) and (33) the following: 𝛽(𝑡)≤𝑡^1−𝛼Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝜏^𝛼−1𝛼₁𝛽(𝜏)+𝛽₁𝛾(𝜏𝛾(𝑡)≤𝑡)𝑑𝜏,_1−𝛼Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝜏^𝛼−1𝛽₂𝛽(𝜏)+𝛼₂𝛾(𝜏)𝑑𝜏, (32) where 𝛽(𝑡)=𝑥₁(𝑡)𝑡^1−𝛼,𝛾(𝑡)=𝑥₂(𝑡)𝑡^1−𝛼. (33)
Using Hölder's inequality for (1/𝑝)+(1/𝑞)=1 with 𝑝=(1+4𝛼)/(1+3𝛼) in (32), we get the inequality below: 𝛽(𝑡)≤𝑡^1−𝛼Γ(𝛼)𝑘(𝑡)^𝑡₀[𝛼^𝑞₁𝛽^𝑞(𝜏)+𝛽^𝑞₁𝛾^𝑞(𝜏)]𝑑𝜏^1/𝑞,𝛾(𝑡)≤𝑡^1−𝛼Γ(𝛼)𝑘(𝑡)^𝑡₀[𝛽^𝑞₂𝛽^𝑞(𝜏)+𝛼^𝑞₂𝛾^𝑞(𝜏)]𝑑𝜏^1/𝑞, (34) where 𝑘(𝑡)=(∫^𝑡₀(𝑡−𝜏)^{𝑝(𝛼−1)}𝜏^{𝑝(𝛼−1)}𝑑𝜏)^1/𝑝. Then, we have 𝛽^𝑞(𝑡)≤𝑡^{𝑞(1−𝛼)}Γ^𝑞(𝛼)𝑘^𝑞(𝑡)^𝑡₀𝛼^𝑞₁𝛽^𝑞(𝜏)+𝛽^𝑞₁𝛾^𝑞(𝜏𝛾)𝑑𝜏,_𝑞(𝑡)≤𝑡^{𝑞(1−𝛼)}Γ^𝑞(𝛼)𝑘^𝑞(𝑡)^𝑡₀𝛽^𝑞₂𝛽^𝑞(𝜏)+𝛼^𝑞₂𝛾^𝑞(𝜏)𝑑𝜏. (35) By Lemma 7, the following inequality is obtained for 𝑘(𝑡): 𝑘(𝑡)=𝑡^{(2𝛼−2)+(1/𝑝)}𝐵^1/𝑝𝑝(𝛼−1)+1,𝑝(𝛼−1)+1. (36) By using the last relation in (35), we obtain 𝛽^𝑞(𝑡)≤𝑘^∗(𝑡)^𝑡₀𝛼^𝑞₁𝛽^𝑞(𝜏)+𝛽^𝑞₁𝛾^𝑞(𝜏𝛾)𝑑𝜏,_𝑞(𝑡)≤𝑘^∗(𝑡)^𝑡₀𝛽^𝑞₂𝛽^𝑞(𝜏)+𝛼^𝑞₂𝛾^𝑞(𝜏)𝑑𝜏, (37) where 𝑘^∗(𝑡)=𝑡^𝑞𝛼−1𝐵^𝑞/𝑝𝑝(𝛼−1)+1,𝑝(𝛼−1)+1Γ^𝑞(𝛼). (38) Since 𝑝(𝛼−1)+1=1+4𝛼()1+3𝛼𝛼−1+1=4𝛼²1+3𝛼>0,𝑞𝛼−1=1+4𝛼𝛼𝛼−1=4𝛼>0, (39) we have the following inequality: 0<𝐵𝑝(𝛼−1)+1,𝑝(𝛼−1)+1<+∞ (40) and the function 𝑘^∗(𝑡) is nondecreasing on 𝐼. Now, with an application of Lemma 8 to (37) combining with (33), upper bounds for a fractional-order nonlinear system are obtained.

5. Numerical Simulation

In nonlinear dynamic systems, predictability can be possible with stability. Also relationship development would be predictable given the right parameters. In this model, parameters provide the condition for the locally asymptotic stability of equilibrium points by using stability analysis on fractional-order transformed system.

In this paper, we focus on couple dynamics depending on the parameters as below. Many scenarios are possible. But in this model, secure or cautious lover (individual 1 retreats from his own feelings but is encouraged by that of individual 2 (𝛼₁<0 and 𝛽₁>0)) and hermit (individual 1 retreats from his own feelings and that of individual 2 (𝛼₂<0 and 𝛽₂<0)) are considered.

Let 𝛼₁=0.005,𝛼₂=0.006,𝛽₁𝛽=0.0004,₂=−0.0001,𝜀=0.001,𝐴₁𝐴=0.02,₂=0.03,2𝛼=1.6. (41) Now, we consider the system 𝐷^2𝛼𝑥₁=−0.005𝑥₁+0.0004𝑥₂1−0.001𝑥²₂𝐷+0.02,_2𝛼𝑥₂=−0.0001𝑥₁1−0.001𝑥²₁−0.006𝑥₂+0.03. (42) Let the initial conditions be 𝑥₁(0)=0,𝑥₂(0)=0. (43)

After the system is transformed, the following system is obtained with the order of 𝛼=0.8: 𝑓₁𝑦₁,𝑦₂,𝑦₃,𝑦₄=𝑦₂,𝑓₂𝑦₁,𝑦₂,𝑦₃,𝑦₄=−0.005𝑦₁+0.0004𝑦₃1−0.001𝑦²₃𝑓+0.02,₃𝑦₁,𝑦₂,𝑦₃,𝑦₄=𝑦₄,𝑓₄𝑦₁,𝑦₂,𝑦₃,𝑦₄=−0.006𝑦₃−0.0001𝑦₁1−0.001𝑦²₁+0.03. (44) Let the initial conditions be 𝑦₁(0)=0,𝑦₂(0)=0,𝑦₃(0)=0,𝑦₄(0)=0. (45) Positive equilibrium point for the problem (44) and (45) is calculated as 𝑦^∗₁=4.38469,𝑦^∗₂=0,𝑦^∗₃=4.92833,𝑦^∗₄=0. (46) The approximate solutions 𝑦₁(𝑡) and 𝑦₃(𝑡) (resp., govern the feelings (𝑥₁) of 𝐴 to 𝐵 and the feelings (𝑥₂) of 𝐵 to 𝐴) are displayed in Figure 1 for 2𝛼=1.6 with acceleration in feelings. Figure 2 shows the asymptotic approximation of (𝑦₁(𝑡),𝑦₂(𝑡),𝑦₃(𝑡),𝑦₄(𝑡)) to the equilibrium point (4.38469,0,4.92833,0) for 𝛼=0.8. For the numerical solution of the system, we use the predictor corrector method [12].

Figure 1 

The graphs of 𝑦3(𝑡) (above) and 𝑦1(𝑡) (below) with the order 2𝛼.

Figure 2 

Approximation to the equilibrium point with the order 𝛼.

We have demonstrated via numerical simulations that the fractional-order nonlinear couple model (42) and (43) can exhibit asymptotic behavior in the presence of nonlinearity for an appropriate set of model parameters. We have observed that the model approaches the equilibrium points asymptotically.

6. Conclusions

In this paper, stability analysis of the fractional-order nonlinear dynamical model of couple is studied by using the fractional Routh-Hurwitz criteria. By using stability analysis on fractional-order system, sufficient condition on the parameters for the locally asymptotic stability of equilibrium points is obtained. A fractional-order nonlinear dynamical model of couple with the order 2𝛼 has been formulated and analyzed. In the discussed model, acceleration is observed in the solution. Also upper bounds for a system with the order 𝛼 have been obtained.

Finally, we have demonstrated via numerical simulations that a fractional-order nonlinear model of couple can exhibit asymptotic behavior in the presence of an appropriate set of model parameters.