Physica A (

)

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Time-delay effects on dynamics of a two-actor conflict model A. Rojas-Pacheco a , B. Obregón-Quintana b , L.S. Liebovitch c , L. Guzmán-Vargas a,∗ a

Unidad Profesional Interdisciplinaria en Ingeniería y Tecnologías Avanzadas, Instituto Politécnico Nacional, Av. IPN No. 2580, L. Ticomán, México D.F. 07340, Mexico b Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, México, D. F., Mexico c Department of Physics, Queens College, City University of New York. 65-30 Kissena Boulevard, Remsen Hall 125, Flushing, NY 11367, USA

article

info

Article history: Received 31 January 2012 Received in revised form 12 July 2012 Available online xxxx Keywords: Social conflicts Local stability Time delays Differential systems Conflicts Two actors Differential equations

abstract We present a study of time-delay effects on a two-actor conflict model based on nonlinear differential equations. The state of each actor depends on its own state in isolation, its previous state, its inertia to change, the positive or negative feedback and a time delay in the state of the other actor. We use both theoretical and numerical approaches to characterize the evolution of the system for several values of time delays. We find that, under particular conditions, a time delay leads to the appearance of oscillations in the states of the actors. Besides, phase portraits for the trajectories are presented to illustrate the evolution of the system for different time delays. Finally, we discuss our results in the context of social conflict models. © 2012 Elsevier B.V. All rights reserved.

1. Introduction The study of conflicts has been the object of researchers from social sciences and, recently, has attracted the attention of investigators from other areas of science. A conflict can be described as the opposition of individuals or groups related to different competing interest, opinions or identities. An important question about the evolution of conflict is to identify the main mechanisms by which conflict — between individuals, groups or nations — evolves toward similar or opposite states after transitory periods. In recent years, several models of conflict have been proposed, which are based on qualitatively defined reaction functions between actor linear models [1]. In Ref. [2], Liebovitch et al. proposed a nonlinear differential equation model of the conflict between two actors. This model is based on Gottman et al.’s [3] proposal and Deutsch’s [4] suggestions with particular attention to capture the cooperative or competitive behavior of actors through a limited number of parameters. Besides, their local stability analysis together with numerical simulations revealed important characteristics similar to those observed in real situations. On the other hand, differential equation models with time delays have been proposed to analyze systems ranging from regulatory genetic processes to control theory [5–12]. It is recognized that time delays play an important role in changing the stability of a fixed point. For example, in biological systems time delays are known to be important because they are involved in many regulation processes [13,14]. Within the context of control theory, time delays have been used to explore the possibility of inducing chaotic behavior for a time-continuing system with exponentially stable equilibrium points [15]. Besides, delayed-feedbacks controlling the spreading of epidemics in smallworld evolving networks have been reported to play an important role [16], and also, for the study of dynamical behavior of delayed neural networks [17].

∗

Corresponding author. E-mail addresses: [email protected], [email protected] (L. Guzmán-Vargas).

0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2012.09.021

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A. Rojas-Pacheco et al. / Physica A (

c1 = 0.5, c2 = 0.5, T = 0.001

4

x y

5

10

5

10

x0= 1, y0 = 2

-2 -4

0

5

10

x y

x0= 1, y0 = 2 0

5

10

x y

0 x0= 1, y0 = 2

-2 15

15

c1 = 0.5, c2 = -0.5, T = 2

2

0

15

c1 = -0.5, c2 = -0.5, T = 2

4

x y

2

10

0

-4

15

c1 = 0.5, c2 = -0.5, T = 0.001

4

5

-2

x and y

x and y x and y

x and y 0

0

2

x0= 1, y0 = 2

-2

x0= 1, y0 = 2

4

0

-4

-4

15

x y

2

0 -2

c1 = -0.5, c2 = -0.5, T = 0.001

4

x and y

x and y

x0= 1, y0 = 2 0

x y

2

0 -2

–

c1 = 0.5, c2 = 0.5, T = 2

4

2

-4

)

-4

0

time t

5

10

15

time t

Fig. 1. Results of numerical integration of the states x and y for two different time delays.

Here, we focused on the effect of time delays on dynamics of a two-actor conflict model [2]. In particular, we evaluate the effect of time delays on the stability of fixed points by means of both theoretical and numerical approaches. We observe that under specific situations sustainable oscillations appear when a time delay is considered. This paper is organized as follows. In Section 2, a brief description of the two-actor conflict model with time delays is presented. The results of the effects of time delays on the stability of the system are described in Section 3. Finally, in Section 4 discussions and some concluding remarks are given. 2. Two-actor conflict model with time delays We consider the two dimensional model described in Ref. [2] given by: dx dt dy dt

= f (x, y) = m1 x + c1 tanh(yT )

(1)

= g (x, y) = m2 y + c2 tanh(xT )

(2)

where x and y represent the state of each actor at time t , m1 and m2 are constants related to decaying rates, c1 and c2 represent the strength of the feedback between the groups and T is a time delay. Given the values of constants m1 , m2 (representing the inertia moments), and c1 , c2 (the strength of the feedback), we are interested in evaluating the effects of time delays on three different cases: (i) positive feedback (cooperation) between actors; (ii) negative feedback (competition) between actors; and (iii) mixed feedback (cooperation-competition). For T = 0, the dynamics is characterized by the stability properties of the fixed points leading to two main cases: weak feedback (|c | < |m|) and strong feedback (|c | > |m|). For positive or negative weak feedback, there is only one stable fixed point located at the origin, and both groups tend to the neutral state as the time evolves. As the strength of the feedback increases (positive or negative) two new fixed points appear and the origin becomes an unstable saddle. For the special mixed case, positive–negative feedback, there is only one fixed point at the origin whose eigenvalue is a complex number with negative real part, indicating that both groups tend to the neutral state presenting oscillations with decaying amplitude.

A. Rojas-Pacheco et al. / Physica A (

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–

x y

x0 = 1, y0 = 1

0

c1 =3, c2 = 3, T = 2 x and y

x and y

c1 =3, c2 = 3, T = 0.001 4 2 0 -2 -4

10

5

4 2 0 -2 -4

15

0

x and y

10

5

x y

x0 = 1, y0 = -2 0

10

5

0

10

5

15

x0 = 1, y0 = -2

x y

10

5

15

c1 =3, c2 = 3, T = 2 x y

10

5

0

0

x and y

x and y

x0 = 1, y0 = -1

x y

x0 = 1, y0 = 2

4 2 0 -2 -4

15

c1 =3, c2 = 3, T = 0.001 4 2 0 -2 -4

15

c1 =3, c2 = 3, T = 2

c1 =3, c2 = 3, T = 0.001

4 2 0 -2 -4

4 2 0 -2 -4

15

x and y

x and y x and y

0

10

5

c1 =3, c2 = 3, T = 2 x y

x0 = 1, y0 = 2

x y

x0 = 1, y0 = 1

c1=3, c 2 = 3, T = 0.001 4 2 0 -2 -4

3

4 2 0 -2 -4

x0 = 1, y0 = -1

0

15

x y

10

5

15

time t

time t

x and y

x and y

x and y

x and y

Fig. 2. Numerical integrations of the states x and y for positive–positive feedback. For a time delay T = 2 and initial conditions (x0 , y0 ) = (1, −2), the states x and y exhibit oscillations around the origin.

4 2 0 -2 -4

4 2 0 -2 -4

4 2 0 -2 -4

4 2 0 -2 -4

c1 =3, c2 = 3, T = 5 x y

x0 = 1, y0 = 1 0

5

10

15

c1 =3, c2 = 3, T = 5 x y

x0 = 1, y0 = 2 0

5

10

15

c1 =3, c2 = 3, T = 5 x y

x0 = 1, y0 = -2 0

5

10

15

c1 =3, c2 = 3, T = 5 x y

x0 = 1, y0 = -1 0

5

10 time t

Fig. 3. Numerical integrations as in Fig. 2 but for a time delay T = 5.

15

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A. Rojas-Pacheco et al. / Physica A (

a

c

c1 = -3, c2 = -3, T = 0.001 4

x and y

x and y x0 = 1, y0 = 1

-2

x y

2

0

–

c1 = -3, c2 = -3, T = 2

4

x y

2

)

0 -2 x0 = 1, y0 = 1

-4 0

b

2

6

c1 = -3, c2 = -3, T = 0.001

4

d

x and y

50

100

150

c1 = -3, c2 = -3, T = 2 x y

2

0 -2

0

4

x y

2 x and y

4

-4

0 -2

x0 = 1, y0 = 1.001

x0 = 1, y0 = 1.001 -4

-4 0

2

4 time t

6

0

50

100

150

time t

Fig. 4. Numerical integration of states x and y for T = 0.001 and T = 2. (a) For T = 0.001 the origin is a stable fixed point and both groups evolve toward the stable state. (b) As in (a) but for a small change in the initial condition. (c) For T = 2, both states oscillate around the neutral state. (d) When the initial condition is changed the system oscillates during a certain period of time and then x evolves to a positive state and y to a negative state.

3. Results 3.1. Linear stability analysis with time delays Now, we consider the case T > 0, that is, when a time delay is present in the system given by Eqs. (1) and (2). Following the linear stability analysis for systems with delays [18], we arrive to the characteristic equation which provides information about the time evolution of small perturbations about fixed points (see the Appendix). More specifically, the characteristic equation is of the form g (z ) + h(z ) e−2zT = 0

(3)

where z is an eigenvalue, and g (z ) and h(z ) polynomials of second and zero order, respectively. The solutions to this equation are not obvious because it has an infinite number of roots [18,19]. One way to overcome this situation is to consider the fact that a common effect of time delays to destabilize stable fixed points or to stabilize unstable fixed points by inducing sustained oscillations. If we assume that z is pure imaginary (z = iω), substitution into the characteristic equation leads to P (ω) + iQ (ω) = e−iωT

(4)

where P (ω) and Q (ω) are second and first order polynomials, respectively (see Eq. (A.7) in the Appendix). We observe that the right hand side of this equation represents the unitary circle whereas the left hand side describes a parabola. The intersection of these two curves could represent a change in the stability of the system. The analysis of intersections between the parabola and the unitary circle leads to the following classification: 1. If the parabola does not intersect the unit circle, and the system is stable for T = 0, then the system is stable independent of delay. 2. If the system is stable for T = 0 and the parabola intersects the unit circle, then the system can be affected by delays. Next, we explore possible changes in the stability of the system for the cases studied in Ref. [2].

A. Rojas-Pacheco et al. / Physica A (

a

x and y

x and y

-4

x0 = 1,y0 = 2

0

2

4

0

d

x y

x0 = 1,y0 = -2

0

2

20

30

x0 = 1,y0 = -2

2

0 -2

10

c1 = 3, c2 = -3, T = 2 4

x and y

x and y

x y

0

-4

6

c1 = 3, c2 = -3, T = 0.001

-4

x0 = 1,y0 = 2

-2

4 2

5

2

0 -2

–

c1 = 3, c2 = -3, T = 2

4

x y

2

b

c

c1 = 3, c2 = -3, T = 0.001

4

)

x y

0 -2

4 time t

6

-4

0

10

20

30

time t

Fig. 5. Numerical integrations for states x and y corresponding to strong feedback (positive–negative case). For T > 0, oscillations in the states are observed.

3.2. Weak feedback In this case we have |c | < |m|. We find that the intersection of the parabola with the real axis occurs at



2

it crosses the imaginary axis at 0, − 2m c2





m2 c2

 , 0 , whereas

(see the Appendix). It follows that the parabola never crosses the unit circle. Thus,

the fixed point at the origin is stable independent of the delay. We also perform numerical integration to compute the time evolution of x and y for particular values of constants and two specific time-delay values T = 0.001 and T = 2. The results are presented in Fig. 1. We observe that for T = 0.001, the system evolves towards the neutral state (the results are identical to those in Fig. 2 from Ref. [2]). For T = 2, the stability of the system is quite similar to the case T = 0.001, except for the appearance of a transient with small oscillations around the original evolution. 3.3. Strong feedback (positive–positive case) m2 c2

< 1, indicating that the intersection of the parabola with the real axis is located √ within the unit  circle. The crossing of the parabola with the unit circle is determined by the values ω∗ = −m2 + c 2 and  ω∗ T ∗ = ω1∗ arctan c 22m . With this information, the fixed point (0, 0) is a candidate for changes in the stability. For T = 0, −2m2 the origin is a saddle point and the system has two stable points (one positive and one negative). For T > 0, the appearance In this case |c | > |m|. We observe that

of sustainable oscillations is expected under specific initial conditions. We performed numerical integrations for the same situations described in Fig. 4 of Ref. [2]. The results are presented in Fig. 2. For T = 0.001 (left column of Fig. 2), both groups always evolve toward stable states where either both are positive or negative. When the initial condition is located along the stable eigendirection, the system (both groups) evolves toward the neutral state. For T = 2 and the initial conditions (x0 , y0 ) = (1, 1) and (x0 , y0 ) = (1, 2), the system evolves toward the stable state with positive values with a slight transient oscillation. Interestingly, for the initial conditions (x0 , y0 ) = (1, −2) and (x0 , y0 ) = (1, −1), the states x and y present oscillations about the origin. In Fig. 3 we present the same numerical results for the evolution of the states and the same initial conditions but for a time delay T = 5. We observe that the time delay seems to be related to the time spent by the group (x or y) in a positive or a negative state.

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A. Rojas-Pacheco et al. / Physica A (

a 10 5 0

d

c1 = c2 = 3, T = 0

b

0

-5

0

-10 -10

5

e

c1 = c2 = 3, T = 2

0

10 5

g(z) H(z)

0

c

-5

0

f

c1 = c2 = 3, T = 10

0

10 5

g(z) H(z)

0

g(z) H(z)

-5

0

5

c1 = 3, c2 = -3, T = 2

g(z) H(z)

-5

0

5

c1 = 3, c2 = -3, T = 10

g(z) H(z)

-5

-5 -10 -10

-10 -10

5

10 5

c1 = 3, c2 = 3, T = 0

-5

-5 -10 -10

–

-5

10 5

10 5

g(z) H(z)

-5 -10 -10

)

-5

0 z

5

-10 -10

-5

0

5

z

Fig. 6. Plot of the functions g (z ) and H (z ) = h(z ) e−2zT to illustrate the effect of strong time delays: (a)–(c) positive–positive feedback and (d)–(f) positive–negative feedback.

3.4. Strong feedback (negative–negative case) In this case |c | > |m|. The stability properties are similar to the positive–positive case, that is, the characteristic equation leads to an intersection between the parabola and the unit circle and changes in the stability are expected. Fig. 4 shows the results of numerical integration for specific initial conditions and for two time delays. For T = 0.001 (Fig. 4(a)), both groups evolve toward the neutral state when the initial condition is located along the stable eigendirection. However, when this initial condition is slightly changed (Fig. 4(b)), the system evolves toward two different states (one positive and one negative), that is, the system exhibits sensitivity to changes in the initial conditions. For T = 2, and for a initial condition along the stable eigendirection (Fig. 4(c)), both groups oscillate around the origin, whereas for a small variation in the initial condition (Fig. 4(d)), the groups evolve towards different stable states after an oscillatory period. 3.5. Strong feedback (positive–negative case) In this case the origin exhibits changes in its stability when a time delay is considered. We also find that there is an intersection between the parabola and the unit circle. Results of numerical integrations are presented in Fig. 5 for two time delays. For T = 0.001, we recover the case where the states of both actors oscillate with decaying amplitude as they evolve toward their neutral steady state (Fig. 5(a) and (b)). When T = 2 (Fig. 5(c) and (d)), both groups oscillate with an approximately constant amplitude related to the fact that the eigenvalue is complex. We notice that these oscillations reflect the fact that, as the time delay increases, the actors tend to alternate their states in such a way that, whereas one actor remains around one stable state, the second one performs the transition to the opposite state. These behaviors are contained within a continuing limit cycle when they are observed in a phase-space plot (see Section 4 for details). We also remark that the appearance of these oscillatory behaviors may represent the main characteristics of repeating conflict episodes. 4. Discussion and phase–space analysis In order to go further in the analysis of the effect of time delays on the stability properties of the system, we construct representative plots of the behavior of the functions g (z ) and H (z ) = h(z ) e−2zT , defined in Eqs. (A.4) and (A.5), respectively

A. Rojas-Pacheco et al. / Physica A (

a

)

–

7

c1 =3, c2 = 3

Eigenvalues

2 0 Re(z) Im(z)

-2 -4 -6 -8 0

b

0.05

0.1 Time delay T

0.2

0.15

c1 =3, c2 = -3 4 3 Re(z) Im(z)

Eigenvalues

2 1 0 -1 -2 -3 -4 0

0.5

1 Time delay T

1.5

2

Fig. 7. Numerical solutions of the characteristic equation for several values of T . (a) Positive–positive feedback. (b) Positive–negative feedback.

(see the Appendix). Eigenvalues z of the linearized system (Eqs. (A.1) and (A.2)) correspond to intersections of the two functions g (z ) and H (z ), that is, the intersection of both functions represents a real value root of the characteristic equation, related with the stability properties of the fixed point. Fig. 6 shows the behavior of both functions for three values of the time delay and different values of the feedback strength. For T = 0 and positive–positive feedback (Fig. 6(a)), H (z ) is a constant and the intersection confirms that the eigenvalues z1 , z2 are real (one positive and one negative). For T > 0 (Fig. 6(b) and (c)), the function H (z ) is a decreasing function that approaches 0 exponentially with a positive intersection value with g (z ). In contrast, for the positive–negative feedback and T ≥ 0 (Fig. 6(e) and (f)), there is no intersection between the functions, indicating that both eigenvalues are complex conjugate. To explore the behavior of the real and imaginary parts of these eigenvalues for different values of the delay, we used a numerical approach. We separate real and imaginary parts of Eq. (3) and then solve for Re(z ) and Im(z ). The Newton method was used iteratively for different time-delay values. We use a Monte Carlo approach for the initial guesses of zinit . The results are showed in Fig. 7. For positive–positive feedback (Fig. 7(a)), both eigenvalues are real (one positive and one negative) when T = 0; as the time delay increases, the positive eigenvalue remains almost constant whereas the negative one becomes more negative, in agreement with numerical integrations. For positive–negative feedback (Fig. 7(b)), the two eigenvalues are complex conjugates for T = 0. As the time delay increases, the real part becomes positive for a specific value of the delay but remains small whereas the imaginary parts slightly decrease. All these results are in agreement with numerical integrations described before. We also have constructed phase-space portraits to illustrate the evolution of the system for the situations discussed in the paper. In Fig. 8, the cases corresponding to weak feedback are presented. For positive–positive (Fig. 8(a) and (b)) and negative–negative (Fig. 8(c) and (d)) feedback, the effect of the time delay is not important. For positive–negative feedback (Fig. 8(e) and (f)), when one increases the time delay the trajectories are slightly modified (Fig. 8(f)). The corresponding phase-space plots for positive–positive and negative–negative strong feedback are presented in Fig. 9. As we discussed in the previous section, both groups always evolve toward stable states where either both are positive or negative, as we can see in Fig. 9(a) and (c) for a time delay T = 0, and when the delay increases (Fig. 9(b) and (c)) the system decays faster along the stable eigendirection. Finally, Fig. 10 shows the phase-space plots of x vs. y for strong positive–negative feedback with different time delays. A stable spiral is observed when the time delay takes the value T = 0.001 (Fig. 10(a)). In contrast, as

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A. Rojas-Pacheco et al. / Physica A (

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c 1 = 0.5, c 2 = 0.5, T = 0

c

2

)

–

c 1 = -0.5, c 2 = -0.5, T = 0

e

2

c 1 = 0.5, c 2 = -0.5, T = 0 2

1

1

1

0

0

0

-1

-1

-1

-2 -2

c 1 = 0.5, c2 = 0.5, T = 0.5

b

-2 -2

2

0

d

0

-2 -2

2

c 1 = -0.5, c 2 = -0.5, T = 0.5

f

2

2

1

1

1

0

0

0

-1

-1

-1

0

2

-2 -2

0

2

2

c 1 = 0.5, c 2 = -0.5, T = 0.5

2

-2 -2

0

-2 -2

0

2

Fig. 8. Phase-space plot of the evolution of x vs. y for weak feedback. Positive–positive feedback, with a time delay (a) T = 0.001 and (b) T = 0.5. Negative–negative feedback with a time delay (c) T = 0.001 and (d) T = 0.5. Positive–negative feedback with a time delay (e) T = 0.001 and (f) T = 0.5.

the time delay increases, a closed trajectory is identified (Fig. 10(b)–(d)). We observe that these closed trajectories tend to be square-shaped as the time delay increases. 5. Conclusions The model analyzed here represents the interaction between two actors, which could be individuals, groups of people, or even whole nations. The time delays studied here could represent the transmission delays of information between actors that are far apart or the delay in their response to that new information. Our results suggest interesting features that may be present in the behavior of such conflicts. First, we have shown that when there is only weak interaction between the actors, namely when |c | < |m|, that these time delays do not introduce almost any new dynamical features. Second, for strong interactions, namely when |c | > |m|, these time delays destabilize fixed point attractors into oscillations. Sometimes these oscillations are transient and sometimes they result in continuing limit cycles. These limit cycles may well be present in the continual repeating episodes found in real world conflicts. Third, the shortest delays are least likely to generate oscillations. This suggests that the free flow of instantaneous information may reduce the volatility of conflicts. On the other hand, the longer the delays, the longer the period of the oscillations and the more stretched out in time is the volatility of the conflict. In summary, we have found that, in the context of the model, the time delays play an important role in the dynamics of conflicts between actors. Acknowledgments This work was partially supported by COFAA-IPN, EDI-SIP-IPN and CONACyT (Project 49128-F-26020, México). Appendix Given the two dimensional ODE (Eqs. (1) and (2)), where we have introduced a time delay T , we apply the typical procedure of linear stability analysis with delays [18]. If (xs , ys ) is a fixed point of the system, we consider small perturbations about the fixed point to obtain the linearized differential equation system given by,

△˙x = m1 △x + c1 sech2 (ys )△yT

(A.1)

△˙y = m2 △y + c2 sech (xs )△xT

(A.2)

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A. Rojas-Pacheco et al. / Physica A (

a

c 1 =3, c 2 = 3, T = 0

b

2

)

c 1 =3, c 2 = 3, T = 0.1

1

0

0

-1

-1

-2

0

-1

c

1

-2

2

c 1 = -3, c 2 = -3, T = 0

d

2

-2

0

0

-1

-1

0

-1

1

2

0

1

2

1

2

c 1 = -3, c 2 = -3, T = 0.1

1

-2

-1

2

1

-2

9

2

1

-2

–

-2

-2

-1

0

Fig. 9. Phase-space plot of time evolution of x and y for strong feedback. Positive–positive feedback (a) for a time delay T = 0.001 and (b) T = 0.1. Negative–negative feedback (c) for a time delay T = 0 and (b) T = 0.1.

where △x = x − xs , △y = y − ys . Assuming exponential solutions to the system (Eqs. (A.1) and (A.2)), the corresponding eigenvalue equation leads to the characteristic equation given by, g (z ) + h(z ) e−2zT = 0

(A.3)

with z an eigenvalue, and g (z ) and h(z ) are polynomials of second and zero order, respectively. The explicit expressions for g (z ) and h(z ) are, g (z ) = z 2 − (m1 + m2 )z + m1 m2

(A.4)

h(z ) = −c1 c2 sech (ys ) sech (xs ).

(A.5)

2

2

For the specific case (xs , ys ) = (0, 0), we get the characteristic equation z 2 − 2mz + m2 − c1 c2 e−2zT = 0

(A.6)

where we have assumed m = m1 = m2 . It is known that the typical effect of a time delay is to change the stability of a stable fixed point by inducing sustained oscillations. To test this effect, if we suppose that z is pure imaginary z = iω and substitute it into the characteristic Eq. (A.6) we get m2 − ω2

−i

c1 c2

2m c1 c2

ω = e−2iωT .

The parabola described by the real part of the left-hand side of (A.7) crosses the real axes at



m2

axes at 0, −2 c

1 c2



(A.7)



m2 c1 c2



, 0 and the imaginary

. From this information, assuming that c = c1 = c2 , we can classify the following cases:

• If |c | < |m|, the parabola intersects the real axes at a point located outside the unit circle. The parabola never cross the circle.

• If |c | > |m|, the intersection of the parabola and the real axes is located inside the unit circle. ofpoints where the  The set  1 2mω∗ ∗ ∗ 2 parabola and the unit circle intersect correspond to ω = c1 c2 − m and T = ω∗ arctan − m2 −(ω∗ )2 where sustained oscillations arise.

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c 1 =3, c 2 = –3, T = 0

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c

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0

1

-2

2

4

2

2

0

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-2

-2

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2

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4

0

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2

4

c 1 =3, c 2 = –3, T = 2 4

-2

-2

d

c 1 =3, c 2 = –3, T = 0.6

-4 -4

–

c 1 =3, c 2 = –3, T = 0.19

b

2

-2 -2

)

-4 -4

-2

0

Fig. 10. Phase-space plot of time evolution of x and y for strong feedback. Positive–negative feedback for a time delay: (a) T = 0.001, (b) T = 0.19, (c) T = 0.6 and (d) T = 2.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

P.T. Coleman, R.R. Vallacher, A. Nowak, L. Bui-Wrzosinska, The American Behavioral Scientist 50 (2007) 1454–1475. L.S. Liebovitch, V. Naudot, R. Vallacher, A. Nowak, L. Bui-Wrzosinska, P. Coleman, Physica A 387 (2008) 6360–6378. J. Gottman, C. Swanson, K. Swanson, Personality & Social Psychology Review 6 (4) (2002) 326–340. M. Deutsch, The Handbook of Conflict Resolution: Theory and Practice, John Wiley & Sons, San Francisco, 2006. T. Chen, H.L. He, G.M. Church, Pacific Symposium on Biocomputing 4 (1999) 29–40. Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, Inc., San Diego, 1993. D. Xu, Z. Yang, Journal of Mathematical Analysis and Applications 305 (1) (2005) 107–120. R.R. Kobak, H.E. Cole, Child Development 64 (1) (1993) 231–245. J.P. Richard, Automatica 39 (2003) 1667–1694. K. Gu, Journal of Dynamic Systems, Measurement, and Control 125 (2003) 158–165. S. Xu, J. Lam, International Journal of Systems Science 39 (2008) 1095–1113. R. Páez-Hernández, F. Angulo-Brown, M. Santillán, Journal of Non-Equilibrium Thermodynamics 31 (2006) 173–188. T. Müller, M. Lauk, M. Reinhard, A. Hetzel, C.H. Lücking, J. Timmer, Annals of Biomedical Engineering 31 (11) (2003) 1423–1439. W.H. Wu, F.S. Wang, M.S. Chang, BMC Bioinformatics 11 (2010) S12. X.F. Wang, G. Chen, K.F Man, IEEE Transactions of Circuits and Systems 48 (5) (2001) 641–645. X. Li, X. Wang, IEEE Transactions on Automatic Control 51 (3) (2006) 534–540. T. Chen, W. Lu, G. Chen, Neural Computation 17 (2005) 949–968. N. MacDonald, Biological Delay Systems: Linear Stability Theory, Cambridge University Press, Cambridge, 1989. N. Radde, EURASIP Journal on Bioinformatics & Systems Biology 2009 (327503) (2009) 1–14.