Delay-induced resonance suppresses damping-induced unpredictability
Mattia Coccolo, Julia Cantisán, Jesús M. Seoane, S. Rajasekar, Miguel A.F. Sanjuán
DDelay-induced resonance suppresses damping-inducedunpredictability
Mattia Coccolo, Julia Cantis´an, Jes´us M.Seoane, S. Rajasekar, and Miguel A.F. Sanju´an Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de F´ısicaUniversidad Rey Juan Carlos, Tulip´an s/n, 28933 M´ostoles, Madrid, Spain School of Physics, Bharathidasan University,Tiruchirapalli 620024, Tamilnadu, India (Dated: September 25, 2020)
Abstract
Combined effects of the damping and forcing in the underdamped time-delayed Duffing oscillatorare considered in this paper. We analyze the generation of a certain damping-induced unpredictabil-ity, due to the gradual suppression of interwell oscillations. We find the minimal amount of theforcing amplitude and the right forcing frequency to revert the effect of the dissipation, so that theinterwell oscillations can be restored, for different time delay values. This is achieved by using thedelay-induced resonance, in which the time delay replaces one of the two periodic forcings presentin the vibrational resonance. A discussion in terms of the time delay of the critical values of theforcing for which the delay-induced resonance can tame the dissipation effect is finally carried out. a r X i v : . [ n li n . AO ] S e p . INTRODUCTION The effects of the linear dissipation on both linear and nonlinear oscillators are wellknown[1–3]. In this sense, the amplitude of the oscillations decays and eventually goesto zero more or less rapidly depending on the magnitude of the damping term. However,when the dissipation competes with an external forcing, oscillations may survive or decay,depending on the intensities of both terms. In fact, this competition gives birth to differenteffects on the dynamics of an oscillator, as studied in Ref. [4], where it has been analyzedhow the dissipation can introduce uncertainty in the topology of the phase space of thesystem and how the forcing can counter it. Here, we aim to extend the current knowledgeon the effects of dissipation for time-delayed oscillators. This kind of systems include a termthat depends on a time interval [ − τ,
0] of the history of the system, where τ is the timedelay. This term may destroy stabilities [5] and induces oscillations on the system dependingon its parameters. In physical and biological systems, the time delay accounts for the finitepropagation time as information is not immediately propagated in nature. This means thatthe future evolution of the system depends not only on its present state, but on its previousstates. This is why we talk about history functions instead of initial conditions. Historyfunctions are sets of initial conditions in the continuous time interval [ − τ,
0] (see Refs. [6]).For simplicity, in this paper, we use as history function: u = 1. Time-delayed systems canbe found in many practical problems. Among others, they are present in neural networks[7], population dynamics [8, 9], electronics [10] or meteorology [11].We use the underdamped time-delayed Duffing oscillator due to the paradigmatic roleplayed in nonlinear dynamics. In the first part of this work, we explore the unpredictabilityinduced by the dissipation and relate it with different values of the time delay. Thus, weshow that the uncertainty can be reverted when the phenomenon of resonance is triggered.The phenomenon of resonance in nonlinear systems has been deeply studied, for instance,in Ref. [12]. In particular, vibrational resonance (VR) [13, 14] is a well-known resonancephenomenon in which the amplitude enhancement is triggered by two periodic externalforcings with different frequencies. The delay term can play the role of one of the periodicforcings typically used in vibrational resonance, and the resonance can be triggered by thecooperation of the time delay and just one external periodic forcing as shown in Ref. [15].The latter phenomenon has been called delay-induced resonance and has been studied among2ther fields in the context of meteorology, as for example the ENSO model in Ref. [16]. Otherbranches of science where time delay is relevant (neural networks, population dynamics orelectronics, as cited before) will be an important focus of study in the following years.Here, we start by countering the effects of the dissipation through the introduction of aperiodic forcing as small as possible in our time-delayed oscillator. In particular, we showthat a certain value of the forcing frequency is the more suitable to induce the resonance.Then, we continue analyzing the possibility of using the forcing to enhance the single-welloscillations, related with particular values of the time delay τ , so that they are no longerconfined in one of the wells. In this case, we find that there is a critical value of the forcingamplitude that shifts the system sensitivity towards another value of the forcing frequency.For visualization purposes, we plot color gradients in the parameter ( p i , p j ) space, where p means a generic parameter, each color related to the peak to peak amplitude of theoscillations so that every couple of ( p i , p j ), i (cid:54) = j with i, j ∈ N is colored with the asymptoticamplitude generated. We call this kind of plot amplitude basins . With the use of thoseamplitude basins we show how the damping-induced unpredictability is connected withthe suppression of the interwell oscillations due to the appearance of fractal regions inthe amplitude basins. Moreover, they permit to easily visualize when the unpredictabilityis reverted or the single-well oscillations disappear. Also, the analysis of those kind ofplots makes it easy to determine the critical values, commented above. Every numericalintegration in this paper has been performed using the method of steps to reduce our DDEto a sequence of ODEs which are solved by a Runge-Kutta algorithm with adaptative stepsizecontrol (Bogacki-Shampine 3 / II. THE MODEL AND THE DAMPING EFFECT
For our analysis, we consider an underdamped Duffing oscillator with a restoring force αx + βx , a damping term µ ˙ x , a periodic forcing F cos Ω t , and a time delay term γx ( t − τ ).3hus, the equation reads as follows¨ x + µ ˙ x + αx + βx + γx ( t − τ ) = F cos Ω t. (1)For convenience, we fix the parameters as α = − β = 0 . γ = − . x ∗ ≈ ± . x ( t ) = x ( t − τ ) = x ∗ , and we obtainthe following results x ∗ = 0 , x ∗± = ± (cid:114) − α − γβ = ±√
13 = ± . . (2)Before exploring the combined effects of the forcing, the time delay and the dissipation,we proceed to analyze the dynamics of the oscillator without forcing and delay term. Thismeans that assuming no forcing, F = 0, and assuming no delay term, γ = 0, so that theoscillations are damped and confined to one single well due to the dissipation. As previouslymentioned, the time delay induces sustained oscillations in the system for certain valuesof the parameters ( γ, τ ). The dependence of the amplitude of these oscillations on τ hasbeen studied in the case of absence of dissipation, µ = 0, in Ref. [16]. The results aresummarized in figure 1a, where we show the peak to peak values of the amplitude A versusthe time delay τ . The figure shows several regions where the amplitude behaves differentlyfor different intervals of time delay values. As it can be seen, there is a first region, RegionI , τ ∈ (0 , . τ is increased, oscillations are created but confined to one well. This happens for τ ∈ [1 . , .
68) and we call this interval as
Region II . In
Region III , where τ ∈ [2 . , . Region IV located at the interval τ ∈ (3 . , . Region V indicated in figure 1a themotion becomes again confined to one well.For clarity, the dynamics of the system in all the above regions, when µ = 0, is displayedin figures 2a-j. On the left column, we display the orbits in the phase space, while on theright column we display the frequency spectra calculated through the Fast Fourier Transform(FFT), for each of the regions. Naturally, for the first region, for which the oscillations decay4 IG. 1: The maximum peak to peak amplitude A versus the delay term τ for equation 1 with F = 0. (a) The figure shows the variation of the amplitude versus τ for µ = 0, showing the 5regions for the different patterns of behaviour of the oscillations amplitude. Panels (b) and (c) plotthe same for µ = 0 .
02 and µ = 0 .
04, respectively. Panel (d) is a zoom of panel (b) to better visualizethe first τ values for which there are fluctuations in the trajectories amplitude in the Region IV.In panels (b)-(c) the vertical dotted red lines are the values of τ predicted by the stability analysisat which the fixed point x ∗ = ± .
606 undergo change of stability. with time to one of the two fixed points x ∗± , the FFT calculation of the frequency showsno peaks, i.e., no periodicity. For the rest of the regions, it is important to notice that theamplitude does not correspond to the peak to peak amplitude A as before. On the contrary,this amplitude M is calculated as x M − x ∗ , where x M is the maximum amplitude value and x ∗ is the stable fixed point. To better show the difference between the amplitudes A and M we have depicted figure 3. In the legend of the panels, we show the values of the maximumamplitude, M max , of the peak and the corresponding frequency, ω max , which is the frequencyof the delay-induced oscillations.Now, we want to go further and explore the effect of the time delay on the dynamics5 IG. 2: The representation of the phase space orbits and frequency spectra belonging to the dif-ferent regions shown in figure 1a. The computations are done with equation (1), for the parametervalues F = 0 and µ = 0. From left to right we represent the orbit in phase space and the frequencyspectrum of the oscillations. In panels (a) and (b), we take τ = 1, and the asymptotic solutionfalls into a fixed point and the frequency spectrum shows no oscillations. In panels (c)-(d), τ = 2,and the solutions are periodic and confined in one well. In panels (e)-(f), τ = 3, and the solutionsare aperiodic. In panels (g)-(h), τ = 4 .
5, and the solutions are sustained interwell periodic orbits.Finally, in panels (i)-(j), τ = 6 .
5, and the solutions are again confined in one well. In the legendthe maximum amplitude, M max , and the trajectories related frequencies, ω max , are displayed. IG. 3: The difference between the peak to peak amplitude A and the amplitude M shown in theFFT spectrum. The dotted line shows the fixed point x ∗ = 3 . when the damping µ (cid:54) = 0. In the figures 1b-c, we can observe the amplitude variation with τ for the damping parameter values µ = 0 .
02 and µ = 0 .
04. As in the case µ = 0 describedabove, a similar pattern of regions for the amplitude versus τ is observed, though for highervalues of µ the dynamics changes. As a matter of fact, we can see how the high-amplitudeoscillations, corresponding to Region IV, either disappear and the system remains at rest ata low energy as in the case µ = 0 .
04 or it happens for a few τ values as in the case µ = 0 . µ = 0 .
02 case.In order to provide analytical support for the numerical results presented above, weperform a linear stability analysis [17, 18] for the fixed points x ∗ = x ∗± = ± . x ∗ = +3 . x ∗ = − . λ + µλ + α + 3 β ( x ∗ ) + γ e λτ = 0 . (3)We take λ = ρ + i ω as the eigenvalue associated with the equilibria x ∗± . The critical stabilitycurve is the one for which ρ = 0 as it implies a change of sign in Re( λ ). For Re( λ ) ¡ 0 , thefixed point is stable while for Re( λ ) ¿0 is unstable. Substituting λ = i ω in equation 3 andseparating both the real and imaginary parts, we obtain the following equations ω − α − β ( x ∗ ) = γ cos ωτ (4) µω = γ sin ωτ . (5)7fter squaring and adding both equations, and substituting the parameter values α = − β = 0 . γ = − .
3, and x ∗ ≈ ± . ω in terms of the damping parameter µ given by ω , = (cid:115) − µ ∓ (cid:112) µ − µ + 9 + 2910 , (6)where ω and ω are the two positive values. We can easily derive τ = arccos( ω − α − β ( x ∗ ) γ ) ω (7)from equation 4, providing the τ values for which the stability changes in terms of theparameters and the frequency. Note that due to the periodicity of arccos function, wecan consider as solutions either τ or πω − τ . From equation 6, we can obtain the values ω = 1 .
613 and ω = 1 .
788 when µ = 0 .
02, and ω = 1 .
615 and ω = 1 .
786 when µ = 0 . τ in the range considered that is obtained from these frequenciesis 1 . µ = 0 .
02 and 1 . µ = 0 .
04. These values match with the numericallycalculated values of τ for which the fixed points lose stability as shown by the vertical dottedred lines in figures 1b-c. Besides, they mark the end of Region I. Notice that the stabilityalso changes for higher values of τ . FIG. 4: Amplitude basins: the figure shows a color gradient plot of the peak to peak amplitudeof the oscillations in the parameter space ( µ, τ ) for the case when the forcing amplitude is F = 0.Panel (b) is a zoom corresponding to Region IV, where 0 . < µ < .
04 and 3 . < τ < .
25. It isinteresting to notice how the fractalization of the yellow basin becomes more and more importantas the dissipation term grows.
8e analyze in detail the effect of the damping on the time-delayed system when F = 0in figure 4. The amplitude is plotted for different ( µ, τ ) values; from now on we refer to thiskind of plot as amplitude basins and the amplitude of the oscillations are calculated peakto peak for an easier display of the amplitude basins. For particular values of µ , verticalslices, we recover the same pictures as in figure 1. In this case, from the bottom to the top,the dark blue is the Region I without oscillations, in the light blue area live the single-welloscillations (Region II), the green area corresponds to interwell aperiodic oscillations (RegionIII) and the yellow one to high-amplitude and periodic interwell oscillations (Region IV).It is remarkable how the damping only appears to affect the latter region transforming itinto the dark blue region where the amplitude is zero. Finally, we have Region V, that alsoremains untouched by the damping.Figure 4b shows a zoom of figure 4a, for values of the time delay and dissipation thatgenerate a more complex structure in the parameter space. Regions I and IV do not presentsmooth basin boundaries like Regions I and II, for instance. On the contrary, it can be seenin this zoom that the yellow and blue basins are intermingled leading to a non-integer fractaldimension for the basin boundary. So that, a little variation in the parameters ( µ, τ ) canchange the motion of the system from high-amplitude oscillations to staying at rest, withoutpassing by intermediate states. In other words, the damping produces a fractalization of theamplitude basins, which implies a higher unpredictability for the system, showing only twopossible states, interwell oscillations or no oscillations at all. We refer to this phenomenonas damping-induced fractalization. III. RESTORING THE INTERWELL HIGH-AMPLITUDE OSCILLATION WITHA MINIMUM VALUE OF THE FORCING PARAMETER F In this section we address the following question: is it possible to restore the high-amplitude oscillations suppressed by the damping effect? The previous section showed howthe damping in our system (equation 1 with F = 0), reduces the high-amplitude oscillationsof Region IV (yellow in figure 4) making the trajectories to fall into the fixed point (darkblue in figure 4). This change may be undesired, specially because a small variation in thetime-delay may cause a dramatic change in the dynamics. In fact, in the amplitude basinsthe yellow and blue basins are intermingled due to the damping-induced fractalization of9he parameter space.The phenomenon of delay-induced resonance studied in [16] for the overdamped caseimplies that even for the parameter values for which the time delay does not induce sus-tained oscillations, a resonance may appear following a different mechanism. Our scenariois different, as the underdamped oscillator presents oscillations and the damping term, forcertain values, eliminates these oscillations. However, we show that even in this case, thedelay-induced resonance phenomenon appears and as a consequence a small forcing is a validmechanism to gain back the oscillations and reduce the fractality caused by the damping.In the following subsections, we explore the parameter values for which a small periodicforcing can restore the high-amplitude oscillations. To achieve our goal, we start analyzing,in the parameter space (Ω , τ ), the interaction between the damping parameter µ and theforcing amplitude F . Then, for fixed µ and τ values, we study the (Ω , F ) parameter spaceto evaluate their effect on the oscillations amplitude. Here, we are mainly interested in theRegion IV for µ ∈ [0 . , . A. Effect of the frequency of the forcing
As introduced before, we start analyzing the (Ω , τ ) parameter space to study the impactof a small periodic forcing amplitude, F = 0 . , for fixed damping parameter values. Infact, in figure 5 the forcing frequency Ω is varied for four values of µ . Remember that for µ = 0 .
02, when F = 0, the yellow amplitude basin is still present and the fractalization ofthe basins only begins at its boundary. On the other hand, for µ = 0 .
05 most of the yellowbasin has disappeared. Therefore, we can see that by adding the forcing for the first case,the dynamics does not barely change (figure 5a), except for a widening at around Ω = 1. Onthe other hand, by taking a look to figure 4 we can see that for µ = 0 .
04 the yellow basins iscompletely gone. In fact, figure 5 shows us, as we increase the values of µ , the erosion of thehigher amplitude basin. But, it also shows that the effect of introducing a forcing is to createa yellow ‘island’ around Ω = 1. This means that near a specific frequency the forcing is ableto counter the damping effects and to recover the high-amplitude interwell oscillations. Infigure 5c, where µ = 0 .
04, we can see that still the small high-amplitude island resists and it10
IG. 5: The erosion of the yellow amplitude basin in the parameter space ( τ, Ω) are represented inpresence of a forcing of amplitude F = 0 .
02. The color bar relates the color with the oscillationsamplitude. The panels (a-d) show the basins for µ = 0 .
02, 0 .
03, 0 .
04 and 0 .
05, respectively. is centered around that frequency value. So, we call it the resonance frequency Ω r = 1. It isinteresting to note that this frequency value is different from the ω max observed in figure 2h.Now the yellow island disappears for µ = 0 .
05, figure 5d, suggesting a shift for the minimalvalue for which the interwell oscillations disappear when the forcing is present.Additionally, it is remarkable that the yellow amplitude basin ( τ ∈ [3 . , . . Effect of the amplitude of the forcing In the previous subsection, we have studied how the enhancement of the dissipation, witha fixed small amplitude forcing, suppresses the high-amplitude oscillation except around anarea centered in Ω r = 1. In particular, for the value of µ = 0 .
04, that makes the interwelloscillations to disappear for F = 0, so that only a little island of high-amplitude oscillationsremains. This is the most interesting case to study, that is, whether the interwell oscillationscan be restored with the right forcing amplitude. Therefore, we fix µ = 0 .
04 and plot,in figure 6, the amplitude basins in the parameter space (Ω , τ ) but changing the forcingamplitude. In these last figures, it is possible to see that the value of F , again, only affects FIG. 6: The reconstruction of the amplitude basins in the parameter space ( τ, Ω) are shown for µ = 0 .
04 and in presence of a forcing with F = 0 .
04 (a), F = 0 .
08 (b), F = 0 . F = 0 . the yellow amplitude basin. For the values of F considered ( F = 0 . , . , . , . ≈ r , reducing the fractalization. That is, near the resonancefrequency the damping-induced unpredictability can be suppressed. On the other hand,for different frequency values even though it is possible to appreciate a reconstruction ofthe yellow basin, the structure is complex and intermingled. This gives us the relevance ofthe frequency selection to trigger the delay-induced resonance and to restore the interwelloscillations. It is also worth mention that in the Region IV, there are values of τ for which,independently of the forcing parameters, the oscillations reach the two wells. We refer tothe yellow stripe around τ ≈ C. Effect of the forcing amplitude and frequency for fixed µ and τ values At this point we know the µ and τ values where we focus our attention and we aim toexplore the parameter space (Ω , F ). Therefore, we fix the value τ = 4 .
5, which is in themiddle of the yellow islands that has been created thanks to the damping forcing competition.In figures 7a-b we start focusing our attention around the frequency values that generatedthe yellow ‘island’ described before, so Ω changes in the range of values [0 . , .
4] while F in the range [0 , . µ = 0 .
03 and µ = 0 .
04, respectively. For µ = 0 .
03, we do notneed a high forcing, that is F (cid:38) .
03, to restore the high-amplitude oscillations for almostany value of Ω, in the interval kept in consideration. However, for µ = 0 .
04, when theforcing is not present, the high-amplitude oscillations disappear completely for the range of τ considered and they are only restored for a certain range of values of Ω (cid:38)
1. Also, theminimum amplitude of the forcing to restore the high–amplitude oscillations, for almost allthe frequency values in the interval, is F ≈ . F , we show in fig-ures 7c-d an expansion of figure 7b. This gives us a better understanding of the amplitudebasins, by showing them for higher values of the forcing F and for a more complete rangeof frequencies Ω. In fact in figure 7c the value of the forcing F goes from 0 to 1 and Ω goesfrom 0 to π . In figure 7d the value of the forcing F goes from 0 to 10 and Ω goes from 0 to2 π . In all the figures it is possible to appreciate that the effect of the forcing is either noneor it suddenly makes the amplitude jump into a sustained oscillation between the two wells13 IG. 7: Amplitude basins in the parameter space (Ω , F ) for τ = 4 . µ = 0 .
03 and (b) µ = 0 .
04. The interval of frequencies has been chosen to enclose the yellow ‘island’ observed infigure 5. Panels (c) and (d) are expansions of the panel (b) for further values of the maximumamplitude F and for a wider band of frequencies. of the potential. If we compare the figures and the gradient bar on the right side of each ofthem it is possible to check the previous affirmation.Finally, in figure 7d it can be seen that the amplitude basins for higher values of F becomemore complex and that the role played by the frequency becomes really complicated. Also,for F (cid:38) µ = 0 .
03, but the figureis not in the panel for lack of further information. It appears that the parameter space canbe divided in two areas, for small forcing values on the bottom of the figure, the systemis more sensitive to the Ω = 1 frequency, while for
F > F IVc ≈
2. 14 . Forcing amplitude to recover the interwell oscillations in Region IV
Finally, it is relevant to study the effects of the forcing amplitude in the same way we didfor the dissipation. Therefore, we decided to plot, in figure 8a, the (
F, τ ) parameter space,for µ = 0 .
04 and Ω = Ω r , and compare it with the ( µ, τ ) plot of figure 4a. The value ofthe damping term has been fixed in order to study the possibility to restore the interwelloscillations for a case where the oscillations are completely damped.It can be seen in figure 8a how the high-amplitude oscillations (yellow region) are restoredfor increasing values of F . Interestingly, the figure shows a mirror symmetry with respect FIG. 8: (a) The amplitude basins in the (
F, τ ) parameter space for µ = 0 .
04, Ω = 1, τ ∈ [0 . , F ∈ [0 , . τ values of panel (a) for F = 0 .
14, tobetter visualize the reconstructed amplitudes and the similarities with figure 1a. to figure 4a. The system’s dynamics can be checked in figure 8b, where we represent aslice of figure 8a, for F = 0 .
14. It is possible to see that the figure is similar to figure 1a,that displays the evolution of the amplitude versus τ for the case µ = 0. In fact, in bothfigures 8a-b, it is possible to appreciate that all the regions that remained unaffected by thedamping, are also not affected by the forcing. On the other hand, the system’s behaviorin Region IV is the same as with µ = 0, once the forcing amplitude reaches values close to F = 0 . F . Also, in Region IV, the no-dampingsituation is fully recovered due to the interaction of the time delay and the forcing, i.e., the15elay-induced resonance phenomenon.In figure 9, we show, from left to right, the phase space orbits and the FFT of two trajec-tories with different F values, for a better understanding of actual result. In the first case FIG. 9: Representation, from left to right, of the phase space orbits and the FFT of two trajectories.On panels (a) and (b) the parameters are set to fall before the beginning of the yellow basin offigure 8a, while in the others to fall inside it. In particular for both trajectories we set τ = 4 . µ = 0 .
04, but for the upper one F = 0 .
01 and for the lower one F = 0 .
1. Fromthe legend, note that in (b) ω max (marked in the figure as a red dotted line) matches the forcingfrequency Ω = 1, while this does not occur in (d). (figures 9a and b), for the chosen F value, the parameters correspond to a region before thebeginning of the yellow basin, while in the second one (figures 9c and d) the parameters cor-respond to the yellow basin. It can be seen that in figure 9b the oscillation frequency matchesthe forcing frequency, while in figure 9d, when the delay-induced resonance is triggered, theoscillations frequency is different. Actually, the frequency of the sustained interwell oscilla-tions is ω = 1 .
35 which is in agreement with the frequency of the delay-induced oscillationsof the Region IV ( F = µ = 0), as we can check in figure 2h and its legend.We depict, in figure 10, the Region IV oscillation frequencies, ω max calculated with the16FT, for different orbits changing the values of the parameters µ and F . In particular, FIG. 10: The variation of ω max (the frequency of the delay-induced oscillations) with the timedelay τ for (a) F = µ = 0, (b) F = 0 .
01, (c) F = 0 . F = 0 .
14 with µ = 0 .
04 and Ω = 1.In (b) and (d) the solid circles are the numerically computed ω max while the continuous cuve is thepolynomial fit (equation 8). in figure 10a we show the frequencies of the delay-induced oscillations setting F = µ = 0.Then, in figures. 10b-d we show the oscillation frequencies for µ = 0 .
04 and F = 0 . F = 0 . F = 0 .
14, respectively. We can appreciate that as the forcing amplitude grows,the oscillation frequencies are restored to the no-damping case. In fact, in figure 10b, thefrequencies are nothing similar to the non-damping case. On the other hand, in figure 10c,the frequencies curve is almost restored, except for some boundary τ values, that are relatedwith the zones of the region on the parameter space in which there is still some remains offractalization. Finally, figure 10d shows that the oscillation frequencies return close to thevalues of no-damping case. To corroborate it, we have plotted the curve fits on figures 10a-d.The relation is polynomial and reads ω max ( τ ) = p τ + p τ + p . (8)17he coefficients of the two polynomials are p = 0 . p = − . p = 3 .
6, withdeviations of small order.The point is that the case without dissipation is restored, not only regarding the oscil-lations amplitude, but also regarding the oscillations frequencies. This is the effect of theconjugate effect, Ref. [16], that suggests that if the oscillator is driven by a small forcing, wecan enhance those oscillations, by adding a delay term. In this context, the delay plays therole of the forcing in triggering the resonance, so that the final oscillation frequency matchesthe delay frequency ω instead of the forcing frequency Ω, like in our case. IV. DELAY-INDUCED RESONANCE FOR THE SINGLE-WELL OSCILLA-TIONS
In the previous section, we focused on the effect of the forcing as an element to restore thehigh-amplitude oscillations that the damping had eliminated. In other words, we focusedon the range of τ for which the yellow amplitude basin disappeared. Now, we consider theeffect of the forcing in the remainder of the range of τ . So, we shift our attention to thesingle-well oscillations of Regions II and V and explore the forcing parameter values, F andΩ that trigger the delay-induced resonance and generate interwell oscillations in that region.For values of τ outside the Region IV, in figure 8a, the forcing does not have any effect.Thus, we need to increase the magnitude of the forcing in order to change the dynamics forthe rest of the τ values. Particularly, we are interested in the possibility of increasing theenergy of the oscillations confined into one well, τ ∈ (2 ,
3) and τ > .
18 (light blue regionsonline in figure 4a), namely Region II and V, so that they become interwell oscillations. Tothat end, the amplitude basins in the (
F, τ ) parameter space for higher values of the F aredepicted in figure 11a-b for Ω = 1 and Ω = 2, respectively. The first frequency choice hasbeen taken to carry on the previous section analysis; the second one is related to figure 7d,where for a higher value of the forcing amplitude, the higher-amplitude oscillations pop upfor precisely Ω = 2. In fact, in that figure, for τ = 4 . F >
2, the oscillationsamplitude rises to
A >
20. Thus, in figure 11, we can see how the light blue amplitude basinreaches the interwell oscillations.We conclude that it is possible to enhance the oscillations confined to one well of RegionsII or V, so that they become interwell oscillations when the amplitude of the forcing increases.18
IG. 11: The plot of the amplitude basins in the (
F, τ ) parameter space. Here µ = 0 .
04 and (a)Ω = 1 and (b) Ω = 2.
Also, figures 11a-b show that the system’s response for Region II is different depending onthe frequency value. In fact, in both figures we can divide the Region II in 2 different areas:the single-well oscillations area, on the left, and the interwell oscillations area, on the right.The two regimes have a clear boundary in the two figures but different critical F values. Itis possible to find those critical parameter values where the dark blue zone finish along withthe single-well oscillations, at τ ≈ . F ≈ . τ ≈ . F = 3 . A ≈
30 or more. It can beappreciated, just as we commented before for figure 7d, that for a certain critical F valuethe system sensitivity to the forcing frequency shifts from Ω = 1 to Ω = 2. For the RegionIV it was F IVc ≈
2, for the Region II it is F IIc ≈
5. So, we can say that those critical F c discriminate two different regimes in those two regions: small- and high-amplitude forcings.So that, the system is more sensitive to the frequency Ω (cid:48) r = 1 before the critical value F c andthen becomes more sensitive to Ω (cid:48)(cid:48) r = 2 for values of F beyond the critical one. Finally, alsoRegion V arises to interwell oscillations in both figures but does not show the same behaviorsas Region II. Therefore, the amplitude of the oscillations does not change significantly fromone plot to the other, although, for Ω = 1, the interwell oscillations start for a smaller valueof the forcing amplitude. 19 . CONCLUSIONS We have analyzed the effect of the damping on the dynamics of the underdamped time-delayed Duffing oscillator. Firstly, we have shown that for small damping parameter values,high-amplitude oscillations (related with Region IV of time delay values), are damped whilethe rest of the dynamics is not affected. This effect of the damping produces a fractalizationin the parameter space increasing the unpredictability of the system. We have demonstratedthat this unpredictability can be reverted by a very small forcing amplitude with a specificvalue of the resonance frequency, Ω (cid:48) r = 1, through the delay-induced resonance phenomenon.Note that this is a similar resonance effect as vibrational resonance where one of the externalperiodic forcings is substituted by the delay term. Moreover, not only the oscillations am-plitude is restored, also the oscillations frequencies of the case without forcing and dampingare restored, thanks to the effect of the conjugate phenomenon.Then, we have found a critical value of the forcing amplitude, F IVc = 2 that switches thesystem sensitivity to the forcing frequency towards Ω = 2. So that, for
F > F
IIc the higheroscillations amplitude are reached for Ω = 2, while for smaller values are reached for Ω = 1.Finally, we proved that the same resonance phenomenon may be used to produce interwelloscillations for τ values inside the Regions II and V, for which the oscillations are boundedto one well, even without dissipation. The forcing amplitude, in these cases, needs to beof a bigger magnitude to induce the intrawell oscillations. Moreover, the forcing amplitudeplays, again, a key role in the system sensitivity to the forcing frequency for Region II. Infact, we have found a threshold value of the forcing amplitude, namely F IIc ≈
5. For smallervalues of the forcing amplitude, the system resonates for Ω = 1. While for values of F higherthan F IIc , the resonance frequency is Ω = 2. Finally, we expect that this work can be usefulfor a better understanding of the delay-induced resonance phenomenon in presence of bothdissipation and forcing. On the other hand, Region V reaches the interwell oscillations fora smaller value of F when the forcing frequency is Ω = 1. VI. ACKNOWLEDGMENT
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