Osinga, Hinke M

English

Boundary crisis is a mechanism for destroying a chaotic attractor when one parameter is varied. In a two-parameter setting the locus of boundary crisis is associated with curves of homoclinic or heteroclinic bifurcations of periodic saddle points. It is known that this locus has nondifferentiable points. We show here that the locus of boundary crisis is far more complicated than previously reported. It actually contains infinitely many gaps, corresponding to regions (of positive measure) where attractors exist

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                          Osinga, H. M. (2006). Boundary crisis: mind the gaps!.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.Boundary Crisis: Mind the Gaps !Hinke M. Osinga∗Bristol Centre for Applied Nonlinear Mathematics,Department of Engineering Mathematics, University of Bristol,Queen’s Building, University Walk, Bristol BS8 1TR, United Kingdom(Dated: April 8, 2006)Boundary crisis is a mechanism for destroying a chaotic attractor when one parameter is varied.In a two-parameter setting the locus of boundary crisis is associated with curves of homoclinic orheteroclinic bifurcations of periodic saddle points. It is known that this locus has nondifferentiablepoints. We show here that the locus of boundary crisis is far more complicated than previouslyreported. It actually contains infinitely many gaps, corresponding to regions (of positive measure)where attractors exist.PACS numbers: 05.45.-a, 02.30.Oz, 02.60.-xThe experimental study of nonlinear dynamical sys-tems is necessarily limited to the observation of attract-ors and the transitions that they undergo as one or moreparameters are varied. herefore, it is important to studypossible transitions also from a more theoretical point ofview in order to predict what one expects to see in exper-iments. For this reason, transitions from steady-state orperiodic attractors to a regime where attracting chaoticdynamics occurs are an active field of research; see, forexample, [1].It is well known that the existence of a chaotic at-tractor can end abruptly in a boundary crisis when aparameter is varied [2]. A boundary crisis occurs whenthe attractor collides with a periodic orbit on its basinboundary. This periodic orbit is called the crisis orbit.Boundary crisis is a common phenomenon that can beobserved in experiments. For example, boundary criseswere reported in a leaky faucet experiment [3], for Alfve´nwaves in cosmic plasmas [4], for cold trapped ions in aPaul trap [5], in power systems [6], in lasers [7], in tri-trophic food chain models [8], and in a model of a measlesoutbreak in New York [9].Since boundary crisis is so common, one would expectthat slight perturbations in the experimental setup, i.e.small variations of other parameters in the system, donot influence the occurence of this phenomenon. Indeed,it should be possible to trace the boundary crisis in aplane of two parameters. Such a two-parameter studywas done in [10, 11], where the authors reported the ex-istence of a piecewise-smooth curve of boundary crises.They found that the curve is only piecewise smooth dueto changes in the period of the crisis orbit associated withthe boundary crisis. Along a smooth segment of the curvethe boundary crisis is organised by, effectively, the samecrisis orbit. At a so-called double-crisis vertex the curveis not differentiable, and the boundary crisis switches toa crisis orbit with a different period.From a mathematical point of view, the boundary∗Electronic address: H.M.Osinga@bristol.ac.ukcrisis is a global change in the dynamics that is causedby a tangency between a stable and an unstable man-ifold. The chaotic attractor is contained in the closureof this unstable manifold, which is either associated withthe crisis orbit or a periodic orbit on the attractor. Thestable manifold is always associated with the crisis orbitand forms the basin boundary of the attractor. There arethree global phenomena that can occur if two such man-ifolds become tangent. First, a boundary crisis occursif the stable manifold is indeed the basin boundary andthe closure of the unstable manifold is equal to the at-tractor at the moment of tangency. Second, if the latterholds, but the stable manifold is not the basin boundary,that is, the crisis orbit lies inside the basin of attraction,then an interior crisis occurs and the chaotic attractorsuddenly grows in size [2]. Third, if the attractor is con-tained in, but smaller than the closure of the unstablemanifold, then a basin boundary metamorphosis occurs,where the basin of attraction becomes fractal [12–15].The locus of tangency between a stable and an un-stable manifold is a smooth curve in a two-parameterplane. However, the global manifestation of this tan-gency changes type at points where another tangencycurve intersects it. Indeed, at a double-crisis vertextwo smooth arcs intersect: one arc changes type froma boundary crisis to an interior crisis, and the otherfrom a boundary crisis to a basin boundary metamorph-osis [10, 11].It seems that everything is known about the boundarycrisis, but this is actually not the case. By definition, achaotic attractor implies the existence of infinitely manyperiodic orbits with stable and unstable manifolds. Wereport here that, in a two-parameter setting, there areactuatlly infinitely many curves of tangencies betweentwo manifolds, and there are infinitely many intersectionsbetween tangency curves. In effect, it is impossible tomark a curve segment that is associated with a boundarycrisis. The locus of boundary crisis is interrupted byinfinitely many gaps, so that it forms a fractal set.In this Letter, we consider the He´non map [16], theparadigm example of a two-dimensional dissipative map.2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2−1.5−1−0.500.511.52xyp0W s(p1)Wu(p0)p1W s(p1)FIG. 1: Manifolds of the He´non map (1) for (α, β) =(1.4269212, 0.3). The unstable manifold Wu(p0) of the fixedpoint p0 is tangent to the stable manifold Ws(p1) of the fixedpoint p1.We use the definition, given in [17],(x, y) 7→ (α+ β y − x2, y) (1)that is obtained by a parameter-dependent linear scalingof the variables from the standard definition [16]. Theparameters α and β remain unchanged in this coordinatetransformation so that all crisis and tangency curves inthe (α, β)-plane are identical for both definitions. TheHe´non map has the property that the determinant of theassociated Jacobian matrix is constant and equal to −β.Hence, the map is dissipative for |β| < 1. An attractorexists as soon as α > − 14 (1 − β)2, and it is believedthat the upper bound for α is a (β-dependent) boundarycrisis.Simo´ [18] has extensively studied the He´non map forfixed β = 0.3 and varying α. As α increases, a period-doubling route to chaos occurs and a strange attractorexists over a range of α. However, this interval is in-terspersed with so-called periodic windows where theattractor is not chaotic. A boundary crisis occurs atα ≈ 1.4269212 via a tangency between the manifolds oftwo fixed points [18]. Figure 1 shows the moment of tan-gency and illustrates how the strange attractor, which isformed by the closure of the unstable manifold Wu(p0)of the fixed point p0 touches its own basin boundary,formed by the stable manifold W s(p1) of the other fixedpoint p1. Hence, p1 is the crisis orbit.We focus our attention on the locus of tangencybetween Wu(p0) and W s(p1) in the two-parameter set-ting. As α and β are varied continuously, the fixedpoints p0 and p1 vary continuously, and so do the man-ifolds Wu(p0) and W s(p1). The moment where thesetwo manifolds are tangent forms a curve in the (α, β)-plane, which we denote by T1. Reference [10] reports1 1.2 1.4 1.6 1.8 200.10.20.30.40.5(a)αx8 9 7 65 6 5 3(b)αβV0V1 	T1899776 5653FIG. 2: Bifurcations along the curve T1 of tangency betweenWu(p0) and Ws(p1) from the double-crisis vertices V1 to V0.The chaotic attractor that exists just to the left of T1 con-tains periodic windows (a), the start of which are initiated bysaddle-node bifurcations that cross T1 transversely (b). Thelabels in panel (a) indicate the periods of the periodic win-dows and the corresponding bifurcating periodic orbits areindicated in panel (b).that T1 corresponds to a boundary crisis on the seg-ment between the double-crisis vertices V0 = (2, 0) andV1 ≈ (0.973005, 0.55861). This segment of T1, shown inFig. 2(b), is almost a straight line; see also [10, Fig. 3].(The trained eye can observe ‘ripples’ along the curve inFig. 3 of [10], which are clarified in this Letter.)Figure 2(a) shows a projection onto the (α, x)-plane ofthe attractor just to the left of T1, that is, just before theboundary crisis, along the curve segment (αT −0.01, βT ),for (αT , βT ) on T1 in between V0 and V1. The pictureclearly shows periodic windows, where, for a range of α,3the attractor does not densely fill one large interval ofx-values. Inside these periodic windows the attractor is,in fact, a k-periodic orbit pk that undergoes a sequenceof period-doubling bifurcations until a chaotic attractoremerges that consists of k pieces. The period k, i.e. thenumber of pieces, is labelled in Fig. 2(a).The start of a k-periodic window is a saddle-node bi-furcation. We computed curves of saddle-node bifurca-tions of k-periodic orbits in the (α, β)-plane using con-tinuation with Content [19]. Figure 2(b) shows howthese curves (in grey for the periods labelled) intersectT1 transversely. The range of α for the two figures isthe same so that one can compare the start of the peri-odic windows in Fig. 2(a) with the intersections of thesaddle-node bifurcation curves with T1 in Fig. 2(b). Sincethe attractor is a periodic orbit just to the right of eachsaddle-node bifurcation curve, each intersection point onT1 is an end point of the boundary crisis. The ‘ripples’that one can observe in [10, Fig. 3] appear to lie exactlyin the regions where the periodic windows exist.The end of each k-periodic window, by which we meanthe moment where the attractor becomes one large at-tractor again, is an interior crisis. The interior crisis iscaused by a tangency of the manifolds of the k-periodicsaddle orbit that appeared in the saddle-node bifurca-tion at the start of a k-periodic window. Each of thesek-periodic tangencies lies on a curve Tk of k-periodic tan-gencies in the (α, β)-plane. The tangency curves Tk liealmost parallel to the k-periodic saddle-node bifurcationcurves, and also intersect T1 transversely. All such in-tersections points are double-crisis vertices, that is, tothe left of T1 each curve Tk is an interior crisis, butto the right of T1 it is a boundary crisis with a crisisorbit of period k. The segment of T1 in between a k-periodic saddle-node bifurcation curve and the associatedtangency curve Tk does not correspond to a boundarycrisis. Namely, the attractor is relatively small here andfar away from its basin boundary. Indeed, close to thesaddle-node bifurcation curve, the attractor is not evenchaotic. Hence, instead of a boundary crisis, these seg-ments of T1 instead cause a basin boundary metamorph-osis.In order to illustrate what happens along T1, wechecked the dynamics in a neighbourhood of the double-crisis vertex at the end of the first 5-periodic window inFig. 2(a), approximately at (α5, β5) = (1.49017, 0.26553).Figure 3 shows from top to bottom the attractors andtheir basins for parameter values just below, to the left,and above (α5, β5), respectively. Panel (a) shows the situ-ation for (α5, β5 − 0.005), which is as expected for para-meter values close to a boundary crisis: the attractor andits basin boundary are similar to the manifolds Wu(p0)and W s(p1) in Fig. 1. For a slightly larger value of α thecurve T1 is reached, the manifolds Wu(p0) and W s(p1)become tangent and a boundary crisis takes place.Figure 3(b) is for (α5 − 0.01, β5), which is inside thefirst 5-periodic window in Fig. 2(a). While the basinof attraction is virtually the same as in Fig. 3(a), the−2 −1 0 1 2−2−1012−2 −1 0 1 2−2−1012−2 −1 0 1 2−2−1012xyyy(a)(b)(c)FIG. 3: Basins of attraction with the attractors of the He´nonmap (1) for (α, β) near (α5, β5) = (1.49017, 0.26553). Thephase portraits are for (α5, β5 − 0.005) (a), (α5 − 0.01, β5)(b), and (α5, β5 + 0.005) (c).4attractor is a periodic orbit of period 5. Indeed, mov-ing from (α5, β5 − 0.005) to (α5 − 0.005, β5) is very sim-ilar to entering the 5-periodic window from the right inFig. 2(a): a (reversed) interior crisis occurs followed bya (reversed) period-doubling sequence to a 5-periodic at-tracting orbit.The transition from (α5 − 0.01, β5) to the situation inFig. 3(c) involves a crossing of T1. However, no boundarycrisis takes place. Figure 3(c) shows that the attractorpersists — it is a 10-periodic orbit here,— but the basinof attraction has changed dramatically. As predicted, abasin boundary metamorphosis takes place.The description of the 5-periodic window is typical forall other periodic windows along T1. Even though weonly labelled eight of them in Fig. 2(a), one must ex-pect infinitely many periodic windows that cannot beobserved at the scale of the figure. For example, there aretwo saddle-node bifurcation curves of period 9 and alsotwo of period 7 in Fig. 2(b), but there is only one peri-odic window each in Fig. 2(a). A zoom of the bifurcationdiagram does reveal the other two periodic windows inthe parameter region indicated by the intersection points.For each of these possibly infinitely many periodic win-dows the associated curve segment on T1 in between thedouble-crisis vertices V0 and V1 causes a basin boundarymetamorphosis, rather than a boundary crisis. Hence,each of these segments is a gap on T1, where this curveis not associated with a boundary crisis. The recursivestructure of the periodic windows suggest that the gapsare created in a Cantor like process, so that the locus ofboundary crisis along T1 is a fractal set.In summary, the overall structure of the (α, β)-planeis drastically different from what is reported in [10, 11],where it is suggested that an attractor exists up to apiecewise-smooth curve that ranges, roughly, over α ∈[0.56, 4.2]; see also [10, Fig. 3]. We have shown thatthere are likely infinitely many gaps along the curve seg-ment in between the double-crisis vertices V0 = (2, 0)and V1 ≈ (0.973005, 0.55861), where the boundary crisisis reportedly caused by a tangency T1 between the man-ifolds of fixed points [10]. In fact, the He´non map hasinfinitely many of such tangency curves that criss-crossthe (α, β)-plane. Gaps in the locus of boundary crisisoccur where tangency curves intersect. Hence, it is notcorrect to think that the locus of boundary crisis formsa single piecewise-smooth curve with a finite number ofnondifferentiable points (the double-crisis vertices). Therecursive process of removing gaps along tangency curveslikely leads to a fractal set as the locus of boundary crisis.In an experiment, including a numerical one, it is diffi-cult to avoid jumping over smaller gaps when attemptingto trace a locus of boundary crisis. As a result a mis-leading bifurcation diagram may be obtained. We pointout that the gaps do not necessarily correspond to at-tractors with a small basin, although there does seem tobe a relationship between the size of the basin and thewidth of the gap. In [20] a large gap was reported in athree-dimensional dynamical system that appears to bebounded in exactly the same way as discussed in thisLetter.[1] J. M. T. Thompson, H. B. Stewart, and Y. Ueda, Phys.Rev. E 49, 1019 (1994).[2] C. Grebogi, E. Ott, and J. A. 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Boundary crisis: mind the gaps!

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Boundary crisis: mind the gaps!

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