Gonzalez-Buelga, A

Krauskopf, B

Neild, SA

Sieber, J

Wagg, DJ

English

We present a continuation method that enables one to track or continue branches of periodic orbits directly in an experiment when a parameter is changed. A control-based setup in combination with Newton iterations ensures that the periodic orbit can be continued even when it is unstable. This is demonstrated with the continuation of initially stable rotations of a vertically forced pendulum experiment through a fold bifurcation to find the unstable part of the branch

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                          Sieber, J., Gonzalez-Buelga, A., Neild, S. A., Wagg, D. J., & Krauskopf, B.(2008). Experimental continuation of periodic orbits through a fold.Link to publication record in Explore Bristol ResearchPDF-documentUniversity of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-terms.htmlTake down policyExplore Bristol Research is a digital archive and the intention is that deposited content should not beremoved. However, if you believe that this version of the work breaches copyright law please contactopen-access@bristol.ac.uk and include the following information in your message:• Your contact details• Bibliographic details for the item, including a URL• An outline of the nature of the complaintOn receipt of your message the Open Access Team will immediately investigate your claim, make aninitial judgement of the validity of the claim and, where appropriate, withdraw the item in questionfrom public view.Experimental continuation of periodic orbits through a foldJ. SieberSchool of Engineering, University of Aberdeen, Kings College, Aberdeen, AB24 3UE, U.K.A. Gonzalez-Buelga, S.A. Neild, D.J. Wagg, and B. KrauskopfFaculty of Engineering, University of Bristol, Queen’s Building, University Walk, Bristol, BS8 1TR, U.K.We present a continuation method that enables one to track or continue branches of periodic orbitsdirectly in an experiment when a parameter is changed. A control-based setup in combination withNewton iterations ensures that the periodic orbit can be continued even when it is unstable. Thisis demonstrated with the continuation of initially stable rotations of a vertically forced pendulumexperiment through a fold bifurcation to find the unstable part of the branch.PACS numbers: 05.45.Gg,45.80.+r,02.30.OzKeywords: experimental bifurcation analysis, vertically forced pendulum, fold bifurcationCharacterizing a nonlinear dynamical system typicallyrequires the systematic investigation of stable and un-stable steady-states and periodic orbits in the relevantparameter region of the system. When a mathemati-cal model is available this task can be tackled efficientlyby performing a bifurcation analysis with the method ofnumerical continuation. It allows one to find and fol-low (or continue) solutions when varying a parameter —a technique that can also be used to map out stabilityboundaries (bifurcations) in multiple parameters. Sev-eral software packages are available for this task; see thereview papers [1, 2] as an entry point to the literature.Although highly developed as a numerical method forthe study of mathematical models, the use of continua-tion methods in physical experiments has proved muchmore difficult. One approach is a combination of systemidentification and feedback control as applied by [3, 4] toequilibria. In principle, it is also applicable to periodicorbits [5] but, as is reported in [6], these methods do notgenerally work well when applied to real physical experi-ments. An alternative is extended time-delayed feedback(ETDF) [7, 8], where the system is subject to a feedbackloop with a delay that is given by the period of the peri-odic orbit one wishes to stabilize. This approach avoidssystem identification and, thus, is easier to implementin real experiments [9]; see also the recent collection ofreviews [10].An important prototype problem for experimental con-tinuation is the continuation of a stable periodic orbitthrough a fold (saddle-node bifurcation). As one variesa system parameter the stable periodic orbit graduallyloses stability and then becomes unstable as it ‘turnsaround’ at the fold point, which is either a local min-imum or maximum of the parameter. One problem isthat ETDF and its modifications such as described in [8]do not converge uniformly near a fold of periodic orbits,meaning that they can generally not be used to trackthe unstable periodic orbit through the fold point; for atreatment of the autonomous case see [11].We present and demonstrate here a continuation(a) (b)ym = 0FIG. 1: Photographs taken during continuation tests showingwhen the pendulum is at the top of a stable (a) and an un-stable (b) rotation; the horizontal line (ym = 0) denotes thezero position.method that can be used directly in an experiment tocontinue periodic orbits irrespective of their stability.Our method does not require a mathematical model northe setting of specific initial conditions. Instead it relieson standard feedback control. The feedback reference sig-nal is updated by a Newton iteration that converges to astate where the control becomes zero. The general ideasbehind this method are described and tested extensivelyin simulations in [12].The goal of this paper is to demonstrate that ourmethod can indeed be used in an actual experimentto track periodic orbits reliably through folds to revealbranches of unstable orbits. To this end, we considera classical mechanical experiment: the vertically forcedpendulum.In our experiment, a pendulum is attached to a pivotthat moves vertically along a trajectory ym(t), which iscontrolled via a servo-mechanical actuator; this setup is2as presented in [13] and shown in the photos in figure 1.The actuator takes a reference trajectory yr(t) as its in-put signal and aims to match its output displacementym(t) to this reference signal yr(t). Ifyr(t) = p sin(ωt) (1)then the pendulum is harmonically forced in the verticaldirection with forcing frequency ω and forcing amplitudep. The internal dynamics of the actuator translating thereference yr into the actual motion ym is only knownapproximately. However, when ω is less than 10 Hz and ifthe forces exerted by the pendulum are small, the outputym closely follows yr with a small time lag (≈ 20 ms) anda small amplitude discrepancy (less than 0.5 mm). Thedynamics of the angular displacement φ of the pendulumare approximately a single-degree-of freedom system.We consider here the period-one rotations of the ver-tically forced pendulum, which are periodic orbits wherethe pendulum goes over the top once per forcing period.For any fixed forcing frequency ω and sufficiently largevalue of the forcing amplitude p one finds a dynamicallystable period-one rotation. A characteristic feature ofthe stable rotations is the in-phase relationship betweenthe pendulum and the forcing: the pivot is up when thependulum is in the upside-down position; see Fig. 1(a).For the same values of ω and p one also finds an unsta-ble rotation, which is in anti-phase with the forcing; seeFig. 1(b). Both rotations are born (for a given, fixedω) in a fold bifurcation at some specific value pf (ω) ofthe forcing amplitude, where a Floquet multiplier passesthrough 1. Note that the fold point pf (ω) also dependson the damping; if the damping is small and viscous thenpf (ω) ∼ ω−1 for large frequencies. (In our experimentwith a pendulum of approximate effective length 0.28 many frequency ω/(2pi) ≥ 2 Hz is large in this sense.)In the experiment we measure the outputθ(t) = φ(t)− ωt, (2)which is periodic for a periodic rotation (period one cor-responds to a period of T = 2pi/ω). The rotations arefeedback stabilizable by adding control to the actuatorinput yr in (1) based on the difference between the mea-sured relative angle θ(t) and a periodic reference signalθ˜(t). Note that feedback control via yr cannot achieveglobal stabilization because the amount of control is lim-ited by the physical restriction of the reference signal yrto amplitudes less than 3 cm. However, local feedbackstabilization is sufficient for our purposes. Namely, wesuperimpose the feedback on the harmonic forcing (1) bysetting the requested pivot trajectory yr to the solutionofy¨r(t) = −ω2p sin(ωt) + S(φ(t)) PD[θ − θ˜](t) (3)where S(φ) = 1/ sinφ if | sinφ| > 0.2 and 0 otherwise.The factor S ensures that control is only applied at non-zero rotation angles (φ 6= 0, pi). The second term in (3) isa standard proportional-plus-derivative (PD) controllerdefined by PD[x] = kpx + kdx˙ (kp = kd = 0.4 in thisexperiment). Since the angular velocity φ˙ is not directlymeasured in the experiment, the term x˙ is approximatedby a linear filter xv = N ·(x−xf ) where xf is the solutionof x˙f = N · (x− xf ) and N is a large quantity (N = 100in this experiment). The differential equation (3) andthe filter are linear and are solved in real-time in parallelwith the experiment on a dSpace DS1104 RD real-timecontroller board. To ensure that the solution of (3) meetsthe physical restrictions on the actuator amplitude (ym ≤3 cm) we reset y˙r whenever φ = 0.The introduction of feedback control into the experi-ment via (3) adds a parameter to the overall system: the(periodic) reference signal θ˜(t). We introduce the scalarparameter θ˜0 and determine θ˜(t) using the recursion re-lation (also evaluated in real time)θ˜h(t) = (1−R)θ˜h(t− T ) +R [θ(t− T )− avg[θ](t− T )]θ˜(t) = θ˜0 + θ˜h(t) (4)where T = 2pi/ω is the period of the forcing, R ∈ (0, 1]is a relaxation factor and avg[θ](t) = 1/T∫ tt−T θ(τ) dτ isthe average of the output θ over the last forcing period (itis a constant scalar for T -periodic functions). We definethe limitΘ(p, θ˜0) := limt→∞ avg[θ](t), (5)which exists (and the convergence of the time profile isuniform) for all pairs (p, θ˜0) that are in the vicinity of the(unknown) family of rotations near fold points. Choos-ing R closer to zero enlarges the region where the limit(5) exists but slows down the convergence. Equation (5)defines a smooth map Θ : R2 7→ R that maps the sys-tem parameter pair (p, θ˜0) to the asymptotic average ofthe output of the experiment. The map Θ is not knownanalytically but can be evaluated for any (p, θ˜0) by run-ning the experiment with control (3) and (4) until thetransients have died out. In practice the limit Θ(p, θ˜0) isreached after 2–3 seconds during our experimental runs.The reference signal θ˜(t) corresponds to a natural pe-riodic rotation of the original (uncontrolled) verticallyforced pendulum if and only if the difference θ − θ˜ iszero, making the feedback control non-invasive. This isthe case when the fixed point equationΘ(p, θ˜0)− θ˜0 = 0 (6)is satisfied. For parameter pairs (p, θ˜0) satisfying (6) theparameter θ˜0 is equal to the average of the phase differ-ence between the rotation and the forcing.Our scheme is a modification of the classical ETDFscheme [7, 14]. The core of this modification is thesolution of the fixed point problem (6) by means ofa Newton iteration. Classical ETDF corresponds for3small R and a fixed p to a relaxed fixed point iterationθ˜0,new = (1 − R)θ˜0,old + RΘ(p, θ˜0,old) for equation (6),which is known to diverge for the unstable rotations [10].At the fold point (pf , θ˜0,f ) the partial derivative ∂2Θequals 1, and this makes the fixed-point problem (6) sin-gular.To overcome this singularity we embed (6) into apseudo-arclength continuation [1]. The pairs of (p, θ˜0)satisfying (6) form a curve. We introduce y = (p, θ˜0)T ,and extend (6) by the pseudo-arclength conditionyTt (y − yold) = h (7)where h is the (small) stepsize along the curve, yold isthe previous point along the curve and yt is the unit se-cant through the previous two points along the curve (asa practical approximation of the tangent to the curve).Equations (6) and (7) define a system of equations of theform F (y) = 0. They can be solved by a relaxed quasi-Newton recursion and we choose the recursion with Broy-den’s rank-one update; see [12] for the technical details.To start a continuation we choose a large forcing am-plitude p (2 cm). For large p the basin of attraction of thestable rotation of the uncontrolled system is large enoughto find the rotation by swinging up the pendulum manu-ally. We measure the periodic output θ and set the initialparameter θ˜0 to the average of this output, thus definingthe initial y = (p, θ˜0)T . In the actual implementationwe scale p by a factor of 20 such that both componentsof the vector y are of order one; the approximate initialsecant to the curve is set to yt = (−1, 0)T . The initialguess for the quasi-Newton Jacobian is J =[−1 00 1].Figure 2(a) shows four branches of rotations in the(p, θ˜0)-plane as continued by our method. Each branchis for a different, fixed forcing frequency ω and vary-ing forcing amplitude p, continued from a stable rotationnear the point A through the fold to an unstable rotationnear the pointB. The upper part of a branch correspondsto stable and the lower part to unstable rotations. Thelarger circles on each of the branches in panel (a) are theapproximate values of the fold points pf (ω). Figure 2(b)shows the location of the fold points in the (ω, p)-planein comparison with the theoretical prediction (thin solidcurve) based on a viscous damping approximation.Each of the four branches in Fig. 2(a) is made upof points at which the quasi-Newton recursion has con-verged; in practice we accept a point when the differenceθ− θ˜ in equation (3) is below 5×10−3 (during one forcingperiod). An experimental continuation run is performedas one continuous experiment without stopping or man-ual intervention; it takes about 30 minutes for a curveresolution as in Fig. 2(a). The experimental continua-tion stops at the lower end point of the branches, wherewe found that the recursion (4) becomes unstable at a pe-riod doubling. This is a similar effect as for the classicalETDF recursion, which has been found to lose stabilityin a torus bifurcation [15].0 5 10 15 20−0.4−0.200.20.42 3 4 53456p[mm]p[mm](a)(b)3Hz2Hz4Hz5Hzθ˜ 0[pi]ABω[Hz]FIG. 2: Experimental one-parameter bifurcation diagrams(a) for 2 Hz, 3 Hz, 4 Hz, 5 Hz, respectively, showing experi-mentally measured rotations (small circles: hollow for sad-dle rotations, full for stable rotations) and estimated foldpoints (large full circles). Panel (b) shows the fold pointsin (ω, p)-plane (circles) in comparison with a viscous modelestimate (thin solid line). Parameters values in (3), (4), (7)were kp = kd = 0.4, R = 0.8, h = 0.02, and convergencetolerance 5× 10−3.4.5 5 5.5−0.2−0.10p[m×10−3]θ˜ 0[pi]FIG. 3: Variation of the phase compared to experimental ac-curacy near the fold for ω = 3 Hz. The error bars indicate themaximum of |p− pm|, where pm is the amplitude of the pivotdisplacement ym. Hollow circles: parameter p as obtained byquasi-Newton iteration; full circles: pm as measured.Figure 3 shows an enlargement of the branch near thefold for a forcing frequency of ω = 3 Hz. Horizontal errorbars have been attached to each point (the vertical errorin θ˜0 is invisibly small). The size of the horizontal errorbars highlights the extreme difference in the scale of theaxes: the range of p is 1 mm, which is of the order of a fewmultiples of the experimental accuracy, whereas θ˜0 spansa range of approximately 60 degrees. This implies thatin a small parameter region of p near the fold, between4−202468−0.2−0.10−10010570 580 590 6004.44.64.8−0.15−0.1−0.05θ[pi]t[s](a)(b)(c)(d)(e)θ−θ˜[pi×10−3]θ˜ 0[pi]y m[mm]p[mm]FIG. 4: Time profiles during the experimental continuationrun for ω/(2pi) = 3 Hz ((a, b, d) measured, (c, e) set by quasi-Newton iteration).4.5 mm and 5.5 mm, the response of the uncontrolled pen-dulum (the average phase avg[θ] of the rotation relativeto the forcing) changes by 60 degrees. Thus, the fold sce-nario presented in Fig. 3 is an example of a very sensitivedependence of the response (the phase of the rotation)of a nonlinear dynamical system on its system parameter(the forcing amplitude p). This implies that the rotationsshown in Fig. 3 would be extremely difficult to find bycareful parameter tuning with the available experimen-tal equipment even on the stable part of the branch nearthe fold. By contrast, our continuation method followsthe branch of rotations through the rapid change with-out difficulty: the dependence of the feedback controlledpendulum on the parameter pair (p, θ˜0) is not sensitiveand the resulting nonlinear system (6)–(7) is uniformlywell-conditioned near the fold.To provide more insight into how points on branchesare accepted, Fig. 4 shows a 30 s snapshot of the timeprofile of the experimental continuation run for 3 Hz.Panel (a) shows the difference θ − θ˜0 and panels (b)–(e)the quantities θ, θ˜0, ym and p as updated by the quasi-Newton iteration at discrete times. Filled circles alongthe time profile in Fig. 4(a) indicate when the differenceθ− θ˜0 is accepted as sufficiently small. Then the respec-tive point (p, θ˜0) is accepted and we start the next stepalong the branch (by updating yold and yt in the pseudo-arclength condition (7)). As a result, the difference θ− θ˜0jumps briefly to a much larger value. The Newton iter-ation then drives the system to convergence; the opencircles indicate that θ − θ˜0 is periodic. At these pointsavg[θ] is measured and new parameters p and θ˜ are setto initiate the next Newton iterate.In conclusion, we have presented a control-based con-tinuation method and demonstrated that it is capableof tracking periodic orbits through fold bifurcations ina vertically forced pendulum experiment. Our approachdoes not require knowledge of an underlying mathemat-ical model. Instead, residuals of a Newton iteration aredirectly measured in order to drive the control action tozero to find the next point on a branch. Importantly,this Newton iteration does not have to run in real-time.Hence, our method can be applied to any experimentthat is feedback stabilizable in the classical sense [16].Our ongoing work focuses on control-based continuationof bifurcations in more than one parameter and for hybridtests, where a laboratory experiment of a critical compo-nent is coupled bidirectionally to a numerical model ofthe remainder of the tested system.[1] E. Doedel, in Numerical Continuation Methods for Dy-namical Systems: Path following and boundary valueproblems, edited by B. Krauskopf, H. Osinga, andJ. Gala´n-Vioque (Springer-Verlag, Dordrecht, 2007), pp.51–75.[2] W. Govaerts and Y. Kuznetsov, in Numerical Contin-uation Methods for Dynamical Systems: Path followingand boundary value problems, edited by B. Krauskopf,H. Osinga, and J. Gala´n-Vioque (Springer-Verlag, Dor-drecht, 2007), pp. 51–75.[3] E. Abed, H. Wang, and R. Chen, Physica D 70, 154(1994).[4] C. Siettos, D. Maroudas, and I. Kevrekidis, Int. J. ofBifurcation and Chaos 14, 207 (2004).[5] E. Ott, C. Grebogi, and J. Yorke, Phys. Rev. Lett. 64,1196 (1990).[6] W. van de Water and J. de Weger, Phys. Rev. E 62, 6398(2000).[7] K. Pyragas, Phys. Lett. A 170, 421 (1992).[8] K. Pyragas, Phys. Rev. Lett. 86, 2265 (2001).[9] S. Schikora, P. Ho¨vel, H.-J. Wu¨nsche, E. Scho¨ll, andF. Henneberger, Phys. Rev. Lett. 97, 213902 (2006).[10] E. Scho¨ll and H. Schuster, eds., Handbook of Chaos Con-trol (Wiley, New York, 2007), 2nd ed.[11] B. Fiedler, V. Flunkert, M. Georgi, P. Ho¨vel, andE. Scho¨ll, Phys. Rev. Lett. 98, 114101 (2007).[12] J. Sieber and B. Krauskopf, Nonlinear Dynamics 51, 365(2008).[13] A. Gonzalez-Buelga, D. Wagg, and S. Neild, StructuralControl and Health Monitoring 14, 991 (2007).[14] D. Gauthier, D. Sukow, H. Concannon, and J. Socolar,Phys. Rev. E 50, 2343 (1994).[15] W. Just, T. Bernard, M. Ostheimer, E. Reibold, andH. Benner, Phys. Rev. Lett. 78, 203 (1997).[16] K. Nam and A. Arapostathis, IEEE Transactions on Au-tomatic Control 37, 378 (1992).

Experimental continuation of periodic orbits through a fold

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