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# A fractional order nonlinear dynamical model of interpersonal relationships

- N Ozalp
^{1}and - I Koca
^{2}Email author

**2012**:189

https://doi.org/10.1186/1687-1847-2012-189

© Ozalp and Koca; licensee Springer 2012

**Received:**27 August 2012**Accepted:**19 October 2012**Published:**2 November 2012

## Abstract

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.

## Keywords

- fractional model
- fractional differential equations
- stability
- numerical solution

## 1 Introduction

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 [3].

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.

## 2 Model

First of all, we recall the definitions of fractional order integrals and derivatives [4].

**Definition 1**The Riemann-Liouville type fractional integral of order $\alpha >0$ for a function $f:(0,\mathrm{\infty})\to R$ is defined by

Here and elsewhere $\mathrm{\Gamma}(\cdot )$ denotes the gamma function.

**Definition 2**The Riemann-Liouville type fractional derivative of order $\alpha >0$ for a function $f:(0,\mathrm{\infty})\to R$ is defined by

where $n=\lceil \alpha \rceil +1$ and $\lceil \alpha \rceil $ is the integer part of *α*.

**Definition 3**The Caputo-type fractional derivative of order $\alpha >0$ for a function $f:(0,\mathrm{\infty})\to R$ is defined by

where $n=\lceil \alpha \rceil +1$ and $\lceil \alpha \rceil $ 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 [4].

where $0<\alpha \le 1$, ${\alpha}_{i}>0$, ${\alpha}_{i}$, ${\beta}_{i}$, 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. For instance, ${\alpha}_{i}$ describes the extent to which individual *i* is encouraged by his/her own feeling. In other words, ${\alpha}_{i}$ 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 ${\beta}_{i}$ 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 $-{\alpha}_{i}{X}_{i}$ says that the love measure of *i*, in the absence of the partner, decays exponentially and $\frac{1}{{\alpha}_{i}}$ is the time required for love to decay.

## 3 Equilibrium points and their asymptotic stability

with the initial values ${X}_{1}(0)={X}_{01}$ and ${X}_{2}(0)={X}_{02}$. Here ${f}_{1}({X}_{1},{X}_{2})=-{\alpha}_{1}{X}_{1}+{\beta}_{1}{X}_{2}(1-\epsilon {X}_{2}^{2})+{A}_{1}$ and ${f}_{2}({X}_{1},{X}_{2})=-{\alpha}_{2}{X}_{2}+{\beta}_{2}{X}_{1}(1-\epsilon {X}_{1}^{2})+{A}_{2}$.

from which we can get the equilibrium points ${K}_{0}=(0,0)$ for ${A}_{1}={A}_{2}=0$ and ${K}_{1}=({X}_{1}^{\ast},{X}_{2}^{\ast})$.

**Theorem 1**

*If one of the conditions below holds*,

*the mutual apathy equilibrium*${K}_{0}=(0,0)$

*of the system given in*(1)

*and*(2)

*is asymptotically stable*.

- (i)
$1<\frac{-4{\beta}_{1}{\beta}_{2}}{{({\alpha}_{1}-{\alpha}_{2})}^{2}}$,

- (ii)
$\frac{{\beta}_{1}{\beta}_{2}}{{\alpha}_{1}{\alpha}_{2}}<1$.

*The equilibrium point* ${K}_{0}=(0,0)$ *is otherwise unstable*.

*Proof*The mutual apathy equilibrium is asymptotically stable if all of the eigenvalues, ${\lambda}_{i}$, $i=1,2$ of $J({K}_{0})$, satisfy the condition [5, 6]

The equilibrium point ${K}_{0}=({X}_{1}^{\ast},{X}_{2}^{\ast})=(0,0)$ is asymptotically stable if both the eigenvalues of the Jacobian matrix for the system given in (1) are negative ($|arg({\lambda}_{1})|>\frac{\alpha \pi}{2}$, $|arg({\lambda}_{2})|>\frac{\alpha \pi}{2}$). It is clear that $B=({\alpha}_{1}+{\alpha}_{2})>0$. If ${B}^{2}-4C<0$, then all of the eigenvalues, ${\lambda}_{i}$, $i=1,2$, are negative and satisfy the condition given by (i). If ${B}^{2}>{B}^{2}-4C$, then both of the eigenvalues, ${\lambda}_{i}$, $i=1,2$, 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. □

**Theorem 2**

*We now discuss the asymptotic stability of the*${K}_{1}=({X}_{1}^{\ast},{X}_{2}^{\ast})$

*equilibrium of the system given by*(1).

*The Jacobian matrix*$J({K}_{1})$

*evaluated at the*$({X}_{1}^{\ast},{X}_{2}^{\ast})$

*equilibrium is given as*

*The characteristic equation of the linearized system is as follows*:

*where*

*The roots of the characteristic equation are*

*The equilibrium point*${K}_{1}=({X}_{1}^{\ast},{X}_{2}^{\ast})$

*of the system given in*(1)

*and*(2)

*is asymptotically stable if one of the following conditions holds for eigenvalues which are given as*(5).

- (i)
$1<\frac{-4{\beta}_{1}{\beta}_{2}}{{({\alpha}_{1}-{\alpha}_{2})}^{2}}(1-3\epsilon {X}_{2}^{\ast 2})(1-3\epsilon {X}_{1}^{\ast 2})$,

- (ii)
$\frac{{\beta}_{1}{\beta}_{2}}{{\alpha}_{1}{\alpha}_{2}}(1-3\epsilon {X}_{2}^{\ast 2})(1-3\epsilon {X}_{1}^{\ast 2})<1$.

The proof of Theorem 2 is similar to that of Theorem 1.

**Theorem 3** *Let* $K={\alpha}_{1}{\alpha}_{2}-{\beta}_{1}{\beta}_{2}(1-3\epsilon {X}_{2}^{\ast 2})(1-3\epsilon {X}_{1}^{\ast 2})$ *be as given in* (4a). *If* $K<0$, *then the positive equilibrium point* ${K}_{1}=({X}_{1}^{\ast},{X}_{2}^{\ast})$ *of the system given in* (1) *and* (2) *is unstable*.

*Proof* If $K<0$, from Descartes’ rule of signs, it is clear that the characteristic equation $P(\lambda )$ has at least one positive real root. So, the equilibrium point ${K}_{1}=({X}_{1}^{\ast},{X}_{2}^{\ast})$ of the system given in (1) and (2) is unstable. □

## 4 Numerical method

*f*is assumed to be a continuous function, every solution of IVP given by (6) is also a solution of the following Volterra fractional integral equation:

Moreover, every solution of (7) is a solution of IVP (6) [8] (Lakshmikantham and Vatsala 2007).

The following theorems are given for solving differential equations of fractional order in [7].

**Theorem 4**

*Let*$\parallel \cdot \parallel $

*denote any convenient norm on*${R}^{n}$.

*Assume that*$f\in C[{R}_{1},{R}^{n}]$,

*where*${R}_{1}=[(t,X):0\le t\le a\phantom{\rule{0.5em}{0ex}}\mathit{\text{and}}\phantom{\rule{0.5em}{0ex}}\parallel X-{X}_{0}\parallel \le b]$, $f={({f}_{1},{f}_{2},\dots ,{f}_{n})}^{T}$

*and*$X={({x}_{1},{x}_{2},\dots ,{x}_{n})}^{T}$,

*and let*$\parallel f(t,X)\parallel \le M$

*on*${R}_{1}$.

*Then there exists at least one solution for the system of FDE given by*

*with the initial condition*

*on* $0\le t\le \beta $, *where* $\beta =min(a,{[\frac{b}{M}\mathrm{\Gamma}(\alpha +1)]}^{\frac{1}{\alpha}})$, $0<\alpha <1$.

**Theorem 5**

*Consider IVP given by*(8)-(9)

*of order*

*α*($0<\alpha <1$).

*Let*

*and assume that the conditions of Theorem*4

*hold*.

*Then a solution*$X(t)$

*of*(6)

*can be given by*

*where*${X}_{\ast}(v)$

*is a solution of the system of integer order differential equations*

*with the initial condition*

## 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.

## Declarations

### Acknowledgements

We would like to thank the referees for their valuable comments.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.