A fractional order nonlinear dynamical model of interpersonal relationships
© Ozalp and Koca; licensee Springer 2012
Received: 27 August 2012
Accepted: 19 October 2012
Published: 2 November 2012
In this paper, a fractional order nonlinear dynamical model of interpersonal relationships has been introduced. The stability of equilibrium points is studied. Numerical simulations are also presented to verify the obtained results.
A fractional order system instead of its integer order counterpart has been considered because fractional order differential equations are generalizations of integer order differential equations and fractional order models possess memory. Also, the fact that interpersonal relationships are influenced by memory makes fractional modeling appropriate for this kind of dynamical systems .
In this paper, firstly a fractional order nonlinear dynamical model of interpersonal relationships has been introduced. A detailed analysis for the asymptotic stability of equilibrium points has been given. Finally, numerical simulations are presented to verify the obtained results.
First of all, we recall the definitions of fractional order integrals and derivatives .
Here and elsewhere denotes the gamma function.
where and is the integer part of α.
where and is the integer part of α.
Generally, in mathematical modeling we use Caputo’s definition. The main advantage of Caputo’s definition is that the initial conditions for fractional differential equations with Caputo derivatives take on the same form as for integer-order differential equations .
where , , , , and () 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. For instance, describes the extent to which individual i is encouraged by his/her own feeling. In other words, indicates the degree to which an individual has internalized a sense of his/her self-worth. In addition, it can be used as the level of anxiety and dependency on other person’s approval in romantic relationships. The parameter 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 avoid closeness in a romantic relationship. Therefore, the term says that the love measure of i, in the absence of the partner, decays exponentially and is the time required for love to decay.
3 Equilibrium points and their asymptotic stability
with the initial values and . Here and .
from which we can get the equilibrium points for and .
The equilibrium point is otherwise unstable.
The equilibrium point is asymptotically stable if both the eigenvalues of the Jacobian matrix for the system given in (1) are negative (, ). It is clear that . If , then all of the eigenvalues, , , are negative and satisfy the condition given by (i). If , then both of the eigenvalues, , , are negative and satisfy the conditions given by (ii). If one of the conditions above does not hold, model gives rise to unbounded feeling, which is obviously unrealistic. □
The proof of Theorem 2 is similar to that of Theorem 1.
Theorem 3 Let be as given in (4a). If , then the positive equilibrium point of the system given in (1) and (2) is unstable.
Proof If , from Descartes’ rule of signs, it is clear that the characteristic equation has at least one positive real root. So, the equilibrium point of the system given in (1) and (2) is unstable. □
4 Numerical method
Moreover, every solution of (7) is a solution of IVP (6)  (Lakshmikantham and Vatsala 2007).
The following theorems are given for solving differential equations of fractional order in .
on , where , .
5 Numerical solutions and simulations
6 Concluding remarks
We have formulated and analyzed a fractional order nonlinear dynamical model of interpersonal relationships. We have obtained a stability condition for equilibrium points. We have also given a numerical example and verified our results.
We would like to thank the referees for their valuable comments.
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