Journal of Applied Nonlinear Dynamics

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1 Journal of Applied Nonlinear Dynamics 5(4) (6) Journal of Applied Nonlinear Dynamics Delay erms in the Slow Flow Si Mohamed Sah, Richard H. Rand Nanostructure Physics, KH Royal Institute of echnology, Stocholm, Sweden Dept. of Mathematics, Dept. of Mechanical & Aerospace Engineering, Cornell University. Ithaca, NY 485, USA. Submission Info Communicated by A.C.J. Luo Received January 6 Accepted 6 April 6 Available online January 7 Keywords Slow flow Delay Duffing Van der Pol Hopf bifurcation Abstract his wor concerns the dynamics of nonlinear systems that are subjected to delayed self-feedbac. Perturbation methods applied to such systems give rise to slow flows which characteristically contain delayed variables. We consider two approaches to analyzing Hopf bifurcations in such slow flows. In one approach, which we refer to as approach I, we follow many researchers in replacing the delayed variables in the slow flow with non-delayed variables, thereby reducing the DDE slow flow to an ODE. In a second approach, which we refer to as approach II, we eep the delayed variables in the slow flow. By comparing these two approaches we are able to assess the accuracy of maing the simplifying assumption which replaces the DDE slow flow by an ODE. We apply this comparison to two examples, Duffing and van der Pol equations with self-feedbac. 6 L&H Scientific Publishing, LLC. All rights reserved. Introduction It is nown that ordinary differential equations (ODEs) are used as models to better understand phenomenon occurring in biology, physics and engineering. Although these models present a good approximation of the observed phenomenon, in many cases they fail to capture the rich dynamics observed in natural or technological systems. Another approach which has gained interest in modeling systems is the inclusion of time delay terms in the differential equations resulting in delay-differential equations (DDEs). DDE s have found application in many systems, including rotating machine tool vibrations [4], gene copying dynamics [5], laser dynamics [] and many other examples. Despite their simple appearance, delay-differential equations (DDEs) have several features that mae their analysis a challenging tas. For example, when investigating a delay-differential equation (DDE) by use of a perturbation method, one is often confronted with a slow flow which contains delay terms. It is usually argued that since the parameter of perturbation, call it ε, is small, ε <<, the delay terms which appear in the slow flow may be replaced by the same term without delay, see e.g. [ 6,,]. he purpose of the present paper is to analyze the slow flow with the delay terms left Corresponding author. address: smsah@th.se ISSN , eissn /$-see front materials 6 L&H Scientific Publishing, LLC. All rights reserved. DOI :.589/JAND.6..7

2 47 Si Mohamed Sah, Richard H. Rand /Journal of Applied Nonlinear Dynamics 5(4) (6) in it, and to compare the resulting approximation with the usual one in which the delay terms have been replaced by terms without delay. he general class of DDEs that we are interested in is of the form ẍ + x = ε f (x,x d ), () where x d = x(t ), where =delay. As an example we choose the Duffing equation with delayed self-feedbac. ẍ + x = ε [ αẋ γx + x d ]. () he situation here is that when there is no feedbac ( = ), the Duffing equation does not exhibit a limit cycle. However it turns out that for > α a stable limit cycle is born in a Hopf bifurcation for a critical value of delay that depends on. Further increases in produce another Hopf, which sees the stable limit cycle disappear. See Fig. which shows a plot of the Hopfs in the parameter plane, obtained by using the DDE-BIFOOL continuation software [7 9]. In this wor we are interested in the details of predicting the appearance of the Hopf bifurcations using approximate perturbation methods. We offer two derivations of the associated slow flow, one using the two variable expansion perturbation method, and the other by averaging. 8 ɛ = Fig. Numerical Hopf bifurcation curves for ε =.5, α =.5 and γ = for Eq.() obtained by using DDE-BIFOOL. Derivation of slow flow he two variable method posits that the solution depends on two time variables, x(ξ,η), where ξ = t and η = εt. henwehave x d = x(t )=x(ξ,η ε). () Dropping terms of O(ε ), Eq.() becomes x ξξ + εx ξη + x = ε [ α x ξ γ x + x(ξ,η ε) ]. (4)

3 Si Mohamed Sah, Richard H. Rand /Journal of Applied Nonlinear Dynamics 5(4) (6) Expanding x in a power series in ε, x = x + εx + O(ε ), and collecting terms, we obtain Lx x ξξ + x =, (5) Lx α x ξ γ x + x (ξ,η ε) x ξη. (6) From Eq.(5) we have that x (ξ,η)=a(η)cosξ + B(η)sinξ. (7) In Eq.(6) we will need x (ξ,η ε): x (ξ,η ε)=a d cos(ξ )+B d sin(ξ ), (8) where A d = A(η ε) andb d = B(η ε). Substituting (7) and (8) into (6) and eliminating resonant terms gives the slow flow: da dη = α A + γ B + γ A B 8 8 A d sin B d cos, (9) db dη = α B γ A γ AB 8 8 B d sin + A d cos. () ransforming (9),() to polars with A = Rcosθ, B = Rsinθ, we obtain the alternate slow flow: dr dη = α R R d sin(θ d θ + ), () dθ γ R = + R d dη R cos(θ d θ + ). () where R d = R(η ε) andθ d = θ(η ε). Note that the slow flow (),() contains delay terms in R d and θ d in addition to the usual terms R and θ. Could this phenomenon be due to some peculiarity of the two variable expansion method? In order to show that this is not the case, we offer the following slow flow derivation by the method of averaging. WeseeasolutiontoEq.()intheform: x(t)=r(t)cos(t θ(t)), ẋ(t)= R(t)sin(t θ(t)). () As in the method of variation of parameters, this leads to the (exact) equations: dr = ε sin(t θ) f, (4) dt dθ = ε cos(t θ) f. (5) dt R where f = α Rsin(t θ) γ R cos (t θ)+r d cos(t θ d ), and where R d = R(t ) andθ d = θ(t ). Now we apply the method of averaging which dictates that we replace the right hand sides of Eqs.(4),(5) with averages taen over π in t, in which process R,θ,R d and θ d are held fixed. his gives dr = ε( α R dt R d sin(θ d θ + )), (6) dθ = ε( γ R + R d dt R cos(θ d θ + )). (7) Note that Eqs.(6),(7) agree with (),() when t is replaced by η = εt.

4 474 Si Mohamed Sah, Richard H. Rand /Journal of Applied Nonlinear Dynamics 5(4) (6) Analysis of slow flow A problem with the slow flow (),() is that they are DDEs rather than ODEs. Since ODEs are easier to deal with than DDEs, many authors (e.g. Wirus [], Morrison [], Atay [5]) simply replace the delay terms by terms with the same variables, but non-delayed. It is argued that such a step is justified if the product ε is small: A d = A(η ε) A(η)+O(ε), B d = B(η ε) B(η)+O(ε). (8) In what follows, we shall refer to this as approach I. For example, if we replace A d by A, andb d by B, Eqs.(9),() become: da dη = α A + γ B + γ A B 8 8 Asin Bcos, (9) db dη = α B γ A γ AB 8 8 Bsin + Acos. () hese ODEs have an equilibrium point at the origin. Linearizing about the origin, we obtain: [ ] d A = dη B [ α sin cos cos α sin ][ ] A. () B For a Hopf bifurcation, we require imaginary roots of the characteristic equation, or equivalently (Rand [], Strogatz []) we require the trace of the matrix in Eq.() to vanish when the determinant>. his gives Condition for a Hopf Bifurcation: sin = α. () Since this condition is based on the bold step of replacing the delay quantities in the slow flow by their undelayed counterparts, the question arises as to the correctness of such a procedure and the validity of Eq.(). See Fig. where Eq.() is plotted along with the numerically-obtained conditions for a Hopf. 8 ɛ = Fig. Numerical Hopf bifurcation curves (blue/solid) and analytical Hopf condition Eq.() (blac/dashdot) for ε =.5, α =.5 and γ = for Eq.().

5 Si Mohamed Sah, Richard H. Rand /Journal of Applied Nonlinear Dynamics 5(4) (6) Let us now return to Eqs.(9),() and treat them as DDEs rather than as ODEs. In what follows we shall refer to this as approach II. Again linearizing about the origin, we obtain where A d = A(η ε) andb d = B(η ε). We set da dη = α A A d sin B d cos, () db dη = α B B d sin + A d cos, (4) A = aexp(λη), B = bexp(λη), A d = aexp(λη ελ), B d = bexp(λη ελ), (5) where a and b are constants. his gives [ λ α exp( λε )sin exp( λε )cos exp( λε )cos λ α exp( λε )sin For a nontrivial solution (a, b) we require the determinant to vanish: ][ ] a = b [ ]. (6) ( λ α exp( λε)sin ) + 4 exp( λε )cos =. (7) We set λ = iω for a Hopf bifurcation and use Euler s formula exp( iωε )=cosωε isinωε. Separating real and imaginary parts we obtain 4 cosεω + 6ω sin sin εω + 8α sin cosεω 6ω + 4α =, (8) 4 sin εω 8α sin sinεω + 6ω sin cosεω + 6αω =. (9) he next tas is to analytically solve the two characteristic Eqs.(8)-(9) for the pair (ω, ). o this aim we use a perturbation schema by setting ω cr = cr = N n= N n= ε n ω n = ω + εω + ε ω + () ε n n = + ε + ε +. () Inserting Eqs. ()-() in Eqs.(8)-(9), aylor expanding the trig functions with respect to the small parameter ε <<, and equating terms of equal order of ε we obtain: ω cr = ω = α () ( cr = ± εω + ε ω ± ε ω + ε4 ω 4 ± ε5 ω ), () where is a solution to the equation sin = α/, thatis (Eqs.(4) (5) are the blac/dashdot curves in Fig..) = π + arcsin( α ) (4) = π arcsin( α ). (5)

6 476 Si Mohamed Sah, Richard H. Rand /Journal of Applied Nonlinear Dynamics 5(4) (6) ɛ =.5 = ɛ Fig. Critical delay vs. the feedbac magnitude for ε =.5 (left). Critical delay vs ε for = (right). Red/solid curves: Eqs.(8)-(9). Blue dots: numerical roots of Eqs.(8)-(9). hese results are for Eq.() with parameter α =.5. Eq.() appears to be the front end of a geometric series. Assuming the series () actually is a geometric series, we can sum it: ( cr = + εω + ε ω + ε ω +...) = (6) εω εω < ( cr = εω + ε ω ε ω +...) = (7) + εω Replacing in Eqs.(6),(7) by the derived values listed in Eq. (4)-(5), we obtain the following expressions for the critical values ω cr and cr for which Hopf bifurcations tae place: π + arcsin ( α/) cr = (8) εω cr εω cr < π arcsin ( α/) cr = (9) + εω cr where ω cr = ω = α /. Fig. shows a comparison of Eqs.(8),(9) with numerical solutions of Eqs.(8)-(9) for various parameters. he numerical solutions were obtained using continuation method. he excellent agreement indicates that Eqs.(8),(9) are evidently exact solutions of Eqs.(8)- (9). We now wish to compare the two approaches, namely I : the approach where we replace A d by A, andb d by B in the slow flow, which gave the condition (), and II : the alternate approach where the terms A d and B d are ept without change in the slow flow, resulting in Eqs.(8),(9). Fig.4 shows a comparison between the analytical Hopf conditions obtained via the two approaches and the numerical Hopf curves. he approach II plotted by red/dashed curves gives a better result than the approach I (blac/dashdot curves). herefore in the case of Duffing equation, treating the slow flow as a DDE gives better results than approximating the DDE slow flow by an ODE. In order to chec whether this is also the case for a different type of nonlinearity, we consider in the next section the van der Pol equation with delayed self-feedbac.

7 Si Mohamed Sah, Richard H. Rand /Journal of Applied Nonlinear Dynamics 5(4) (6) ɛ = Fig. 4 Numerical Hopf bifurcation curves (blue/solid) for Eq.() for ε =.5, α =.5 and γ =. Alsoshown are the results of approach I, the analytical Hopf condition Eq.() (blac/dashdot), and the results of approach II, Eqs.(8),(9) (red/dashed) 4 Another example: Van der Pol equation As another example we choose the van der Pol equation with delayed self-feedbac. his system has been studied previously by Atay and by Suchorsy et al. ẍ + x = ε [ ẋ( x )+x d ]. (4) In the case of van der Pol, when there is no feedbac ( = ), this system is well nown to exhibit a stable limit cycle for ε >. It turns out (Atay [5], Suchorsy [6] ) that as delay increases, for fixed >, the limit cycle gets smaller and eventually disappears in a Hopf bifurcation. Further increases in produce another Hopf, which sees the stable limit cycle get reborn. Fig.5 shows a plot of the Hopfs in the parameter plane. As for the case of Duffing equation we are interested in the details of predicting the appearance of the Hopf bifurcations using approximate perturbation methods. We follow the same procedure as for the case of Duffing equation, that is by deriving the slow flow using the two variable expansion method, and the averaging method. However, for simplicity we omit the averaging method analysis since we obtain the same slow flow by both methods. he obtained slow flow in the cartesian coordinates has the following expression: da dη = A A 8 AB 8 A d sin B d cos, (4) db dη = B B 8 A B 8 B d sin + A d cos, (4) where A d = A(η ε) andb d = B(η ε). Replacing A d by A, andb d by B, Eqs.(4),(4) become: da dη = A A 8 AB 8 Asin Bcos, (4) db dη = B B 8 A B 8 Bsin + Acos. (44)

8 478 Si Mohamed Sah, Richard H. Rand /Journal of Applied Nonlinear Dynamics 5(4) (6) ɛ = Fig. 5 Numerical Hopf bifurcation curves for ε =. for Eq.(4) obtained by using DDE-BIFOOL. ɛ = Fig. 6 Numerical Hopf bifurcation curves (blue/solid) and approach I analytical Hopf condition Eq.(45) (blac/dashdot) for ε =. for Eq.(4).

9 Si Mohamed Sah, Richard H. Rand /Journal of Applied Nonlinear Dynamics 5(4) (6) Linearizing (4) and (44) about the origin and looing for the condition where Hopf bifurcation taes place, we find: Condition for a Hopf Bifurcation: sin =. (45) his condition is plotted in Fig.6 along with the numerically-obtained conditions for a Hopf. If we now treat Eqs.(4),(4) as DDEs rather than as ODEs and linearize about the origin, we obtain where A d = A(η ε) andb d = B(η ε). We set da dη = A A d sin B d cos, (46) db dη = B B d sin + A d cos, (47) A = aexp(λη), B = bexp(λη), A d = aexp(λη ελ), B d = bexp(λη ελ, (48) where a and b are constants. his gives [ λ + exp( λε )sin exp( λε )cos exp( λε )cos λ + exp( λε)sin For a nontrivial solution (a, b) we require the determinant to vanish: ][ ] a = b [ ]. (49) ( λ + exp( λε )sin) + 4 exp( λε )cos =. (5) We set λ = iω for a Hopf bifurcation and use Euler s formula exp( iωε )=cos ωε isinωε. Separating real and imaginary parts we obtain cos ωε sin ω sin ωε sin + 4 cosωε + 4 ω =, (5) ω cosωε sin + sinωε sin sin ωε ω =. (5) 4 As in the case of Duffing equation we proceed by using a perturbation schema to analytically solve the two characteristic Eqs. (5)-(5) for the pair (ω, ). We set the critical frequency and delay to be: ω cr = cr = N n= N n= where is a solution to the equation sin = /, thatis ε n ω n = ω + εω + ε ω +, (5) ε n n = + ε + ε +, (54) (Eqs.(55) (56) are the blac/dashdot curves in Fig.6.) = arcsin( ), (55) = π arcsin( ). (56)

10 48 Si Mohamed Sah, Richard H. Rand /Journal of Applied Nonlinear Dynamics 5(4) (6) ɛ = = ɛ Fig. 7 Comparison of numerical versus analytic results obtained by approach II for Eq.(4). Critical delay vs. the feedbac magnitude for ε =.5 (left). Critical delay vs ε for = (right). Red/solid curves: Eq.(58)-(59). Blue dots: numerical roots of Eqs.(5)-(5). ɛ =. ɛ = Fig. 8 Numerical Hopf bifurcation curves (blue/solid) for Eq.(4), for ε=. (left) and ε=.5 (right). Also shown are the results of approach I, the analytical Hopf condition Eq.(45) (blac/dashdot), and the results of approach II, Eqs.(58)-(59) (red/dashed). Inserting Eqs. (5)-(54) in Eqs.(5)-(5), aylor expanding the trig functions with respect to the small parameter ε <<, and equating terms of equal order of ε we obtain: ω cr = ω =, (57) arcsin (/) cr =, (58) εω cr εω cr < π arcsin (/) cr =. (59) + εω cr Figure 7 shows a comparison of Eqs.(58),(59) with numerical solutions of Eqs.(5)-(5) for various parameters. he excellent agreement indicates that Eqs.(58),(59) are evidently exact solutions of Eqs.(5) (5). Figure 8 shows a comparison between the numerically-obtained Hopf conditions and the Hopf conditions obtained by following the two approaches I and II. Whenε =., Eq. (55) (blac/dashdot

11 Si Mohamed Sah, Richard H. Rand /Journal of Applied Nonlinear Dynamics 5(4) (6) curve) gives a better match with the lower numerical branch (blue/solid curve) than does Eq. (58) (red/dashed curve). However for the upper numerical branch, Eq. (59) gives a better approximation than Eq. (56), see Fig.8. As ε is increased (ε =.5), Eq. (55) still gives a better approximation for the lower numerical branch than Eq. (58). On the other hand Eq. (59) succeeds in tracing the upper numerical branch, see Figure 8. 5 Discussion In the two studied examples we saw that the two approaches gave different results. In the Duffing equation, the approach II gave better results. his is expected since we did not approximate A d by A, andb d by B, and instead analyzed the slow flow as a DDE. However in the van der Pol example, we obtained unexpected results. From Fig.8, the upper Hopf branch obtained by the approach II gave a better approximation of the upper numerical Hopf curve than the one obtained from approach I, Eq. (56). his could be explained by the fact that as is increased, the term ε increases as well, which maes the approximation A d = A, andb d = B no longer valid. By contrast, in the the approach II the increasing of does not affect the condition (59). But unexpectedly, the condition (58) obtained by approach II fails to give a better result for the lower Hopf curve. his could be explained by the singularity that taes place in the lower numerical Hopf branch where the limit cycle disappears and the origin x = changes its nature as an equilibrium. For example when ε =.5 this singular behavior occurs for, see Fig.8. Both the method of averaging and the two variable expansion perturbation method are built on the assumption that the solution at O(ε ) is a periodic solution around the origin x =. However for increasing ε and the origin no longer exhibits this behavior, and our assumption of the periodicity of our unperturbed solution does not hold anymore. Note that Eq. (55) does not contain an ε term, thus it does not vary with increasing ε. Figure 9 shows a numerical simulation of the van der Pol equation (4) for =., wherethe origin has changed its nature. his figure corresponds to the lower Hopf curve in Fig.8 when ε =.5. his unexpected failure of approach II leads us to wonder if this happens because the system is a self-sustained one. In order to show that is not the case, we consider a limit cycle system studied by Erneux and Grasman [6]. In their wor, they looed for the Hopf curves in a limit cycle system with delayed self-feedbac: ẍ + x = ε [ ẋ( x )+x d x ]. (6) We apply the same procedure, approach II, as we did for Duffing and van der Pol examples to equation Eq. (6), and we obtain the following critical frequency and time delay: ω cr = 4, (6) ω cr = 4 +, (6) cr = π arcsin(/), (6) + εω cr cr = arcsin (/) + εω cr. (64) Figure shows a comparison between approach II, Eqs. (6),(6),(6),(64), and approach I, which again gives Eqs. (55),(56), and the numerical Hopf curves obtained by use of DDE-BIFOOL. Fig. shows that approach II gives better results than approach I. However approach I still gives a good fit for the lower Hopf curve as in the case of Eq. (4).

12 48 Si Mohamed Sah, Richard H. Rand /Journal of Applied Nonlinear Dynamics 5(4) (6) =. 4 x Fig. 9 Numerical integration for x as a function of time t in Eq.(4) for ε =.5, =. and delay =.4. Note that the motion grows large and there is no limit cycle. he origin has changed its nature. See Figure 8 and text. t Fig. Numerical Hopf bifurcation curves (blue/solid) for Eq.(6) for ε=. (left) and ε=.5 (right). Also shown are the results of approach I, the analytical Hopf condition Eq.(45) (blac/dashdot), and the results of approach II, Eqs.(6)-(64) (red/dashed).

13 6 Conclusion Si Mohamed Sah, Richard H. Rand /Journal of Applied Nonlinear Dynamics 5(4) (6) When a DDE with delayed self-feedac is treated using a perturbation method (such as the two variable expansion method, multiple scales, or averaging), the resulting slow flow typically involves delayed variables. In this wor we compared the behavior of the resulting DDE slow flow with a related ODE slow flow obtained by replacing the delayed variables in the slow flow with non-delayed variables. We studied a sample system based on the Duffing equation with delayed self-feedbac and found that replacing the delayed variables in the slow flow by non-delayed variables introduced substantial error. Our conclusion is that the researcher must be cautious about replacing a DDE slow flow by a corresponding ODE. Note that we are not saying that the approximation involved in replacing a DDE slow flow by an ODE slow flow necessarily involves significant error, but rather that such an approximation may involve significant error. In fact when we analyzed limit cycle systems with delayed self-feedbac we found that the ODE approximation of the slow flow was adequate for the lower Hopf curve. However for one limit cycle system, Eq. (4), approach II failed to approximate the lower Hopf curve, while for a different limit cycle system, eq. (6), approach II gave a fit as good as approach I. Our conclusion is therefore that the researcher is advised to perform both the simplified ODE slow flow analysis, and the more lengthy analysis on the DDE slow flow, to mae sure that the simplified ODE approximation is acceptable. Acnowledgements he author S.M. Sah gratefully acnowledges the financial support of the Ragnar Holm Fellowship at the Royal Institute of echnology (KH). References [] Ji, J.C. and Leung, A.Y.. (), Resonances of a nonlinear SDOF system with two time-delays on linear feedbac control, Journal of Sound and Vibration, 5, [] Maccari A. (), he resonances of a parametrically excited Van der Pol oscillator to a time delay state feedbac, Nonlinear Dynamics, 6, 5-9. [] Hu, H., Dowell, E. H. and Virgin, L. N. (998), Resonances of a harmonically forced duffing oscillator with time delay state feedbac, Nonlinear Dynamics, 5, -7. [4] Wahi, P. and Chatterjee, A. (4), Averaging oscillations with small fractional damping and delayed terms. Nonlinear Dynamics. 8, -. [5] Atay, F.M. (998), Van der Pol s oscillator under delayed feedbac. Journal of Sound and Vibration, 8(), -9. [6] Suchorsy, M.K., Sah, S.M. and Rand, R.H. (), Using delay to quench undesirable vibrations. Nonlinear Dynamics, 6, 7-6. [7] Engelborghs, K., Luzyanina,. and Roose, D. (), Numerical bifurcation analysis of delay differential equations using: DDE- BIFOOL. ACM ransactions on Mathematical Software, 8(), -. [8] Engelborghs, K., Luzyanina,. and Samaey, G. (), DDE- BIFOOL v..: a Matlab pacage for bifurcation analysis of delay differential equations. echnical Report W-, Dept. Comp. Sci., K.U.Leuven, Leuven, Belgium [9] Hecman, C.R. (), An introduction to DDE-BIFOOL is available as Appendix B of the doctoral thesis of Christoffer Hecman: asymptotic and numerical analysis of delay- coupled microbubble oscillators (Doctoral hesis). Cornell University [] Wirus, S. and Rand, R.H. (), he dynamics of two coupled van der Pol oscillators with delay coupling. Nonlinear Dynamics,, 5-. [] Morrison,.M. and Rand, R.H. (7), : Resonance in the delayed nonlinear Mathieu equation. Nonlinear Dynamics, 5, 4-5. [] Rand, R.H. (), Lecture notes in nonlinear vibrations published on-line by the Internet-First University

14 484 Si Mohamed Sah, Richard H. Rand /Journal of Applied Nonlinear Dynamics 5(4) (6) Press ecommons.library.cornell.edu/handle/8/8989 [] Strogatz, S. H. (994), Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Addison-Wesley, Reading, Massachusetts) [4] Kalmar-Nagy,. and Stepan, G. and Moon, F.C. (), Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations, Nonlinear Dynamics, 6, -4. [5] Verdugo, A. and Rand, R. (8), Hopf Bifurcation in a DDE Model of Gene Expression, Communications in Nonlinear Science and Numerical Simulation,, 5-4. [6] Erneux. and Grasman J. (8), Limit-cycle oscillators subject to a delayed feedbac, Physical Review E, 78(), 69.