A Cherif

D Matignon

E Ahmed

E Demirci

I Koca

I Podlubny

MW Ahmad

N Ozalp

V Lakshmikantham

Advances in Difference Equations

English

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Ozalp and Koca Advances in Diﬀerence Equations 2012, 2012:189http://www.advancesindifferenceequations.com/content/2012/1/189RESEARCH Open AccessA fractional order nonlinear dynamical modelof interpersonal relationshipsN Ozalp1 and I Koca2**Correspondence:ibaltaci@gantep.edu.tr2Department of Mathematics,Faculty of Sciences, GaziantepUniversity, Gaziantep, TurkeyFull list of author information isavailable at the end of the articleAbstractIn this paper, a fractional order nonlinear dynamical model of interpersonalrelationships has been introduced. The stability of equilibrium points is studied.Numerical simulations are also presented to verify the obtained results.Keywords: fractional model; fractional diﬀerential equations; stability; numericalsolution1 IntroductionIn recent decades the study of interpersonal relationships has begun to be popular. In-terpersonal relationships appear in many contexts such as family, kinship, acquaintance,work, and clubs []. Mathematical modeling in interpersonal relationships is very impor-tant for capturing the dynamics of people. But there are fewmodels in this area, andmod-els have been restricted to integer order diﬀerential equations. Since experiments in thisarea are diﬃcult to design andmay be constrained by ethical considerations,mathematicalmodels can play a vital role in studying the dynamics of relationships and their behavioralfeatures []. In this paper, we consider a system of nonlinear fractional diﬀerential equa-tions. This fractional system of equations is obtained by replacing a derivative term by afractional derivative of order α > . The integer order model reported in [] is given asdXdt = –αX + βX( – εX)+A,dXdt = –αX + βX( – εX)+A.A fractional order system instead of its integer order counterpart has been consideredbecause fractional order diﬀerential equations are generalizations of integer order diﬀer-ential equations and fractional order models possess memory. Also, the fact that interper-sonal relationships are inﬂuenced by memory makes fractional modeling appropriate forthis kind of dynamical systems [].In this paper, ﬁrstly a fractional order nonlinear dynamical model of interpersonal re-lationships has been introduced. A detailed analysis for the asymptotic stability of equi-librium points has been given. Finally, numerical simulations are presented to verify theobtained results.2 ModelFirst of all, we recall the deﬁnitions of fractional order integrals and derivatives [].© 2012 Ozalp and Koca; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly cited.Ozalp and Koca Advances in Diﬀerence Equations 2012, 2012:189 Page 2 of 7http://www.advancesindifferenceequations.com/content/2012/1/189Deﬁnition  The Riemann-Liouville type fractional integral of order α >  for a functionf : (,∞)→ R is deﬁned byIαf (t) = (α)∫ t(t – τ )α–f (τ )dτ .Here and elsewhere (·) denotes the gamma function.Deﬁnition  The Riemann-Liouville type fractional derivative of order α >  for a func-tion f : (,∞)→ R is deﬁned byDαf (t) = dndtn(n – α)∫ t(t – τ )n–α–f (τ )dτ ,where n = α +  and α is the integer part of α.Deﬁnition  The Caputo-type fractional derivative of order α >  for a function f :(,∞)→ R is deﬁned byDαf (t) = (n – α)∫ t(t – τ )n–α–f n(τ )dτ ,where n = α +  and α is the integer part of α.Generally, in mathematical modeling we use Caputo’s deﬁnition. Themain advantage ofCaputo’s deﬁnition is that the initial conditions for fractional diﬀerential equations withCaputo derivatives take on the same form as for integer-order diﬀerential equations [].The model that we study in this paper is a fractional order nonlinear dynamical modelof interpersonal relationships. The model is a nonlinear dynamical system with two statevariables. The variables X and X are the measures of love of individuals  and  for theirrespective partners, where positive and negative measures represent feelings. The follow-ing model is considered:DαX(t) = –αX + βX( – εX)+A,DαX(t) = –αX + βX( – εX)+A()with initial conditionsX() = X, X() = X, ()where  < α ≤ , αi > , αi, βi, and Ai (i = , ) are real constants. These parameters areoblivion, reaction, and attraction constants. In the equations above, we assume that feel-ings decay exponentially fast in the absence of partners. The parameters specify the ro-mantic style of individuals  and . For instance, αi describes the extent to which individuali is encouraged by his/her own feeling. In other words, αi indicates the degree to which anindividual has internalized a sense of his/her self-worth. In addition, it can be used as thelevel of anxiety and dependency on other person’s approval in romantic relationships. TheOzalp and Koca Advances in Diﬀerence Equations 2012, 2012:189 Page 3 of 7http://www.advancesindifferenceequations.com/content/2012/1/189parameter βi represents the extent to which individual i is encouraged by his/her partner,and/or expects his/her partner to be supportive. It measures the tendency to seek or avoidcloseness in a romantic relationship. Therefore, the term –αiXi says that the love measureof i, in the absence of the partner, decays exponentially and αiis the time required for loveto decay.3 Equilibrium points and their asymptotic stabilityLet α ∈ (, ] and consider the systemDαX(t) = f(X,X),DαX(t) = f(X,X),with the initial valuesX() = X andX() = X. Here f(X,X) = –αX +βX(–εX )+A and f(X,X) = –αX + βX( – εX ) +A.To evaluate the equilibrium points, letDαXi(t) =  =⇒ fi(X∗ ,X∗)= , i = , ,fromwhich we can get the equilibrium pointsK = (, ) forA = A =  and K = (X∗ ,X∗ ).The Jacobian matrix J(X∗ ,X∗ ) for the system given in () isJ(X∗ ,X∗)= ∂(f, f)∂(X∗ ,X∗ )=[∂f∂X∗∂f∂X∗∂f∂X∗∂f∂X∗]=[–α β( – εX∗ )β( – εX∗ ) –α].Mutual apathy equilibrium for the system given in () is K = (X∗ ,X∗ ) = (, ) and theJacobian matrix at this point is as follows:J(, ) =[–α ββ –α].Theorem  If one of the conditions below holds, the mutual apathy equilibrium K = (, )of the system given in () and () is asymptotically stable.(i)  < –ββ(α–α) ,(ii) ββαα< .The equilibrium point K = (, ) is otherwise unstable.Proof The mutual apathy equilibrium is asymptotically stable if all of the eigenvalues, λi,i = ,  of J(K), satisfy the condition [, ]| argλi| > απ . ()These eigenvalues can be determined by solving the characteristic equationdet(J(K) – λI)= ,Ozalp and Koca Advances in Diﬀerence Equations 2012, 2012:189 Page 4 of 7http://www.advancesindifferenceequations.com/content/2012/1/189which leads to the equationλ + Bλ +C = ,whereB = (α + α),C = αα – ββ.The roots of the characteristic equation areλ, =–B ±√B – C .The equilibriumpointK = (X∗ ,X∗ ) = (, ) is asymptotically stable if both the eigenvaluesof the Jacobian matrix for the system given in () are negative (| arg(λ)| > απ , | arg(λ)| >απ ). It is clear that B = (α + α) > . If B – C < , then all of the eigenvalues, λi, i =, , are negative and satisfy the condition given by (i). If B > B – C, then both of theeigenvalues, λi, i = , , are negative and satisfy the conditions given by (ii). If one of theconditions above does not hold, model gives rise to unbounded feeling, which is obviouslyunrealistic. Theorem  We now discuss the asymptotic stability of the K = (X∗ ,X∗ ) equilibrium ofthe system given by (). The Jacobian matrix J(K) evaluated at the (X∗ ,X∗ ) equilibrium isgiven asJ(X∗ ,X∗)=[–α β( – εX∗ )β( – εX∗ ) –α].The characteristic equation of the linearized system is as follows:P(λ) = λ +Dλ +K = , ()whereK = αα – ββ( – εX∗)( – εX∗),D = (α + α).(a)The roots of the characteristic equation areλ, =–D ±√D – K . ()The equilibrium point K = (X∗ ,X∗ ) of the system given in () and () is asymptoticallystable if one of the following conditions holds for eigenvalues which are given as ().(i)  < –ββ(α–α) ( – εX∗ )( – εX∗ ),(ii) ββαα( – εX∗ )( – εX∗ ) < .Ozalp and Koca Advances in Diﬀerence Equations 2012, 2012:189 Page 5 of 7http://www.advancesindifferenceequations.com/content/2012/1/189The proof of Theorem  is similar to that of Theorem .Theorem  Let K = αα – ββ( – εX∗ )( – εX∗ ) be as given in (a). If K < , thenthe positive equilibrium point K = (X∗ ,X∗ ) of the system given in () and () is unstable.Proof IfK < , fromDescartes’ rule of signs, it is clear that the characteristic equation P(λ)has at least one positive real root. So, the equilibrium point K = (X∗ ,X∗ ) of the systemgiven in () and () is unstable. 4 Numerical methodConsider the initial value problem (IVP) with Caputo-type FDE given byDαx(t) = f(t,x(t)),x() = x,()where f ∈ C([,T]×R,R),  < α < . Since f is assumed to be a continuous function, everysolution of IVP given by () is also a solution of the following Volterra fractional integralequation:x(t) = x +(α)∫ t(t – τ )α–f(τ ,x(τ ))dτ , t ∈ [,T]. ()Moreover, every solution of () is a solution of IVP () [] (Lakshmikantham and Vat-sala ).The following theorems are given for solving diﬀerential equations of fractional orderin [].Theorem  Let ‖ · ‖denote any convenient norm on Rn. Assume that f ∈ C[R,Rn], whereR = [(t,X) :  ≤ t ≤ a and ‖X – X‖ ≤ b], f = (f, f, . . . , fn)T and X = (x,x, . . . ,xn)T , andlet ‖f (t,X)‖ ≤ M on R. Then there exists at least one solution for the system of FDE givenbyDαX(t) = f(t,X(t))()with the initial conditionX() = X ()on ≤ t ≤ β , where β =min(a, [ bM(α + )]α ),  < α < .Theorem  Consider IVP given by ()-() of order α ( < α < ). Letg(ν,X∗(ν))= f(t –(tα – ν(α + ))/α ,X(t – (tα – ν(α + ))/α))and assume that the conditions of Theorem  hold. Then a solution X(t) of () can be givenbyX(t) = X∗(tα/(α + )),Ozalp and Koca Advances in Diﬀerence Equations 2012, 2012:189 Page 6 of 7http://www.advancesindifferenceequations.com/content/2012/1/189where X∗(v) is a solution of the system of integer order diﬀerential equationsd(X∗(ν))dν = g(ν,X∗(ν))with the initial conditionX∗() = X.5 Numerical solutions and simulationsLettingα = ., α = ., β = ., β = –.,ε = ., A = ., A = ., α = .we consider the system ()-()DαX = –.X + .X( – .X)+ .,DαX = –.X( – .X)– .X + ..()Let the initial conditions beX() = , X() = . ()A positive equilibrium point for the problem ()-() is calculated asX∗ = ., X∗ = ..For the numerical solution of ()-(), Theorem  has been used. The corresponding in-teger order system given in Theorem  isdX∗dv = –.X∗ + .X∗( – .X∗)+ .,dX∗dv = –.X∗( – .X∗)– .X∗ + ..If the solution of this integer order system is in (X∗ (v),X∗ (v)), then the solution of IVP()-() is (X∗ (t./(.)),X∗ (t./(.))). We get the numerical solution of the in-teger order system by applying the fourth-order Runge-Kutta method. The approximatesolutions X(t) and X(t) are displayed in Figure  for α = .. Figure  shows the asymp-totic approximation of (X(t),X(t)) to the equilibrium point (., .).6 Concluding remarksWe have formulated and analyzed a fractional order nonlinear dynamical model of inter-personal relationships. We have obtained a stability condition for equilibrium points. Wehave also given a numerical example and veriﬁed our results.Ozalp and Koca Advances in Diﬀerence Equations 2012, 2012:189 Page 7 of 7http://www.advancesindifferenceequations.com/content/2012/1/189Figure 1 The graphs of X1(t) (above) and X2(t) (below).Figure 2 Approximation to the equilibrium point.Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsThe authors have achieved equal contributions. All authors read and approved the ﬁnal version of the manuscript.Author details1Department of Mathematics, Faculty of Sciences, Ankara University, Besevler, Ankara 06100, Turkey. 2Department ofMathematics, Faculty of Sciences, Gaziantep University, Gaziantep, Turkey.AcknowledgementsWe would like to thank the referees for their valuable comments.Received: 27 August 2012 Accepted: 19 October 2012 Published: 2 November 2012References1. Cherif, A: Stochastic Nonlinear Dynamical Models of Interpersonal and Romantic Relationships: Strange Attractions.Arizona State University, Temple (2009)2. Cherif, A, Barley, K: Stochastic nonlinear dynamics of interpersonal and romantic relationships. Appl. Math. Comput.217, 6273-6281 (2011)3. Ahmad, MW, El-Khazali, R: Fractional order dynamical models of love. Chaos Solitons Fractals 33, 1367-1375 (2007)4. Podlubny, I: Fractional Diﬀerential Equations. Academic Press, San Diego (1999)5. Ahmed, E, El-Sayed, AMA, El-Saka, HAA: Equilibrium points, stability and numerical solutions of fractional-orderpredator-prey and rabies models. J. Math. Anal. Appl. 325, 542-553 (2007)6. Matignon, D: Stability results for fractional diﬀerential equations with applications to control processing. In:Computational Eng. in Sys. Appl., vol. 2. Lille, France, 1996, p. 963 (1996)7. Demirci, E, Özalp, N: A method for solving diﬀerential equations of fractional order. J. Comput. Appl. Math. 236,2754-2762 (2012)8. Lakshmikantham, V, Vatsala, AS: Theory of fractional diﬀerential inequalities and applications. Commun. Appl. Anal.11(3-4), 395-402 (2007)doi:10.1186/1687-1847-2012-189Cite this article as: Ozalp and Koca: A fractional order nonlinear dynamical model of interpersonal relationships.Advances in Diﬀerence Equations 2012 2012:189.

A fractional order nonlinear dynamical model of interpersonal relationships

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A fractional order nonlinear dynamical model of interpersonal relationships

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