multistability, basin boundary structure, and …

March 31, 2004 9:38 00963

Papers

International Journal of Bifurcation and Chaos, Vol. 14, No. 3 (2004) 927–950c© World Scientific Publishing Company

MULTISTABILITY, BASIN BOUNDARY

STRUCTURE, AND CHAOTIC BEHAVIOR IN A

SUSPENSION BRIDGE MODEL

MARIO S. T. DE FREITAS∗ and RICARDO L. VIANADepartamento de Fısica, Universidade Federal do Parana,

C.P. 19081, 81531-990, Curitiba, Parana, Brazil

CELSO GREBOGIInstituto de Fısica, Universidade de Sao Paulo, C.P. 66318,

05315-970, Sao Paulo, SP, Brazil

Received July 3, 2002; Revised October 16, 2002

We consider the dynamics of the first vibrational mode of a suspension bridge, resulting fromthe coupling between its roadbed (elastic beam) and the hangers, supposed to be one-sidedsprings which respond only to stretching. The external forcing is due to time-periodic vorticesproduced by impinging wind on the bridge structure. We have studied some relevant dynamicalphenomena in such a system, like periodic and quasiperiodic responses, chaotic motion, andboundary crises. In the weak dissipative limit the dynamics is mainly multistable, presentinga variety of coexisting attractors, both periodic and chaotic, with a highly involved basin ofattraction structure.

Keywords : Multistability; chaos; basin boundaries; suspension bridge.

1. Introduction

A well-known and quite dramatic example of theresonance effects on structures under the action oftime-periodic forcing is the Tacoma Narrows bridgefailure, that occurred on 7 November, 1940 [Amannet al., 1941]. The source of an external driving forceis the aeolian harp effect caused by a periodic forcegenerated by a von Karman street of staggered vor-tices due to impinging wind on the bridge structure[Blevins, 1977]. However, the standard textbook ex-planation based on the linear resonance betweenthe frequency of the vortex street and the bridgenatural frequency has been recently questioned bymany authors [Billah & Scanlan, 1991]. Basically,linear resonance is a rather narrow phenomenon,unlikely to occur in an irregularly changing environ-

ment, like that encountered in the Tacoma Narrowsbridge failure [Lazer & McKenna, 1990].

The possible inadequacy of a linear explana-tion for this disaster has been already questionedby the board of experts, including von Karmanhimself, that have produced a report to the U.S.Federal Works Agency [Amann et al., 1941]. Nowa-days the role of nonlinear effects in the TacomaNarrows disaster (as well as of other similar sus-pension bridges) has been widely accepted. Two as-pects of the problem have been pointed out: firstly,the Tacoma Narrows Bridge was originally builtwith a slender, light and flexible roadbed. As a re-sult, large-amplitude transversal oscillations werepossible under widely different wind conditions.Secondly, as can be seen in the famous movie shot at

∗Permanent address: Departamento de Fısica, Centro Federal de Educacao Tecnologica do Parana, Curitiba, PR, Brazil.

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928 M. S. T. de Freitas et al.

the moment of disaster,1 torsional oscillations werealso observed just before the bridge collapsed [Lazer& McKenna, 1990]. Coupling between such differ-ent modes is a typical nonlinear feature [Nayfeh &Mook, 1979]. The torsional mode is particularly sen-sitive to the nonlinearly elastic nature of the hang-ers that connect the roadbed to the main suspensioncable.

A key point to explain the suspension bridgecollapse is that, in this and other related situations,a nonlinear response can lead to various complexdynamical phenomena, such as different modes ofvibration, traveling waves and chaotic motion. Non-linearity appears in a rather robust way in suspen-sion bridges, since the myriad of cables that con-nect the roadbed to the main suspended cable havea peculiar characteristic: a cable strongly resists tostretching but does not to compression. This leadsto a piecewise linear stiffness for the cables, and theoscillator dynamics is governed by a nonsmooth lin-ear vector field, with many features common to non-linear systems [Blazejczyk-Okulewska et al., 1999].

The piecewise linear stiffness found in the os-cillations of a suspension bridge is just one of thevarious examples of mechanical systems with dis-continuities [Wiercigroch & DeKraker, 2000]. Re-lated examples of technological interest are impactoscillations due to clearances or gaps between mov-ing parts like rotors and their bearings or shafts[Aidanpaa et al., 1994; Wiercigroch, 2000;Blazejczyk-Okulewska, 2000; Jerrelind & Stensson,2000], and stick-slip vibrations [Popp & Stelter,1990]. This has motivated many analytical and nu-merical studies on such oscillators, as the seminalpaper of Shaw and Holmes [1983] on the bilinearoscillator, and the ensuing work by Thompson et al.

[1983], Whiston [1987], Nordmark [1991], and Kimand Noah [1991], among many others.

In a model proposed by Lazer and McKenna inthe early 90’s [Lazer & McKenna, 1990], a suspen-sion bridge deck is assumed to be a one-dimensionalelastic beam connected to the main suspended ca-ble by a large number of hangers, treated as one-sided springs. According to the behavior displayedby the Tacoma Narrows bridge just before its fail-ure, the first transversal harmonic was the domi-

nant one and it resulted in a vibration amplitude of1.5 ft, excited by 35 mph winds. After three hours,the wind increased to 42 mph and the growing os-cillation amplitude caused a hanger to escape out ofits roadbed connection, resulting in an unbalancedloading and to a 0.2 Hz torsional vibration modelwhich ultimately caused the bridge to collapse.2

Instead of solving a boundary value prob-lem for the partial differential equation describingthe full spatio-temporal problem [Heertjes & Vande Molengraft, 2001], the strategy of [Lazer &McKenna, 1990] was to isolate and analyze thetime evolution of the first vibrational mode alone,by means of an initial value problem for an or-dinary differential equation. The Lazer–McKennamodel was further investigated by Doole and Hogan[1996], who have considered the periodic responseof the bridge to variations of the external drivingparameters, in which the action of the vortex trailon the structure was modeled by a periodic force.Another feature investigated was the role of thepreload caused by the proper weight of the roadbed.In a later work, this analysis was extended to thetorsional vibration modes of the bridge [Doole &Hogan, 2000].

In this work, we do not restrict ourselves toperiodic behavior of the bridge [Doole & Hogan,1996], but we consider the more complex behaviorexpected in a nonlinear system. We explore a wideregion of the forcing-damping parameter space, forwhich we find periodic, quasiperiodic and chaoticbehavior. Moreover, for weak dissipation we observea predominance of multistable behavior, involvingboth periodic and chaotic attractors, with a highlyconvoluted basin of attraction structure. In spite ofthis, the basin boundaries are not necessarily fractalcurves, as it typically happens in driven oscillators.Abrupt changes of chaotic behavior also occurs forthis system and they are related to boundary crises.

The rest of this paper is organized as fol-lows: in the second section, we outline the dy-namical model of the first vibrational mode of asuspension bridge as a piecewise-linear driven anddamped one-dimensional oscillator, following [Lazer& McKenna, 1990] and [Doole & Hogan, 1996]. Sec-tion 3 considers the harmonic, or periodic, response

1Images of the Tacoma Narrows bridge disaster are available on the website http://cee.carleton.ca/Exhibits/TacomaNarrows/,taken from the 20-minute silent movie.2See the website http://www.vibrationdata.com/Tacoma.htm for details.

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Multistability and Chaos in a Suspension Bridge 929

of the undamped oscillator. The fourth section isdevoted to the analysis of the multistable periodicbehavior observed for wide regions of the forcing-damping parameter space. The basin boundarystructure is considered in Sec. 5, whereas Sec. 6analyzes the onset of chaotic behavior and abruptchanges in the chaotic attractors due to a boundarycrises. The last section contains our conclusions.

2. Suspension Bridge Model

The deck, or roadbed, of a suspension bridge is as-sumed to be an elastic vibrating beam sustainedby hangers, which are steel cables attached to amain suspended cable (Fig. 1). The elastic beam ishinged at both ends to an anchorage block, and themain cable is supported by high towers. We shallconsider a simplified model of a suspension bridgewhich takes into account only the elasticity of itsdeck and of the hangers that support it. We neglectdeflections of the main suspended cable and otherstructural components.

We use x and z as the longitudinal andtransversal coordinates of the vibrating beam, re-spectively, whereas u is the beam deflection alongy (Fig. 1), assuming a downward deflection as pos-itive (u > 0). In this paper we consider only thetransversal beam vibrations so that we ignore theinfluence of the transverse coordinate z. Hence,the beam deflection is taken to be only a func-tion of the longitudinal coordinate and the time,i.e. u = u(x, t). The hangers are assumed as be-ing one-sided strings: they do not withstand com-pression efforts, but they oppose a linear restoringforce when stretched, provided the deformations aresmall enough to be treated in the elastic regime[Lazer & McKenna, 1990]. Hence, the stiffness re-sponse of a hanger is piecewise linear and asym-metric, this nonlinear response being the source ofthe complex dynamical behavior in the system. Theelastic restoring force offered by the hangers can beexpressed as −k′u+, where k′ is the spring constantand u+ = max{u, 0}.

In this case, the elastic response of the bridgehas two distinct features: for downward deflections,we take into account the combined response of thebeam and stretched hangers, whereas, for upwarddeflections, only the beam elasticity is considered.Moreover, we add a preloading term W (x) in thebridge model, due to the proper beam weight and

Fig. 1. Schematic figure of a suspension bridge. The hang-ers connect an elastic beam representing the bridge deck tothe main suspension cable.

its loading. The partial differential equation govern-ing the beam vibration is [Hartog & Pieter, 1987]

M∂2u

∂t2+ EI

∂4u

∂x4+ δ′

∂u

∂t

= −k′u+ + W (x) + F (x, t) , (1)

where M is the beam mass per unit length, E isthe Young modulus of the beam, and I the mo-ment of inertia of its transversal section. The dissi-pative effect on the beam vibration is modeled bya viscous damping term δ′ut. The external forceF (x, t) represents the periodic driving effect of aVon Karman vortex street. If the wind incidence isalong the transversal direction z, the external forceis collinear to the deflection with a well-defined pe-riod T = 2π/ω′

F (x, t) = F0(x) sin(ω′t) . (2)

The boundary conditions for Eq. (1) take into ac-count the hinging of the beam at its ends (x = 0and x = L):

u(0, t) = u(L, t) =∂2u

∂x2

∣

∣

∣

∣

(0,t)

=∂2u

∂x2

∣

∣

∣

∣

(L,t)

= 0 . (3)

Instead of solving (1) directly, we will considerthe time evolution of the bridge transversal vibrat-ing modes. In particular, only the first transversalharmonic is to be considered in this work (Fig. 2).Besides being the most commonly observed modefor low velocities in the Tacoma Narrows Bridge[Amann et al., 1941], the loss of stability of thismode was responsible for the torsional oscillationsthat eventually led to its failure [Billah, 1991]. Thisassumption implies that the preloading and thespatial part of the external force can be written,respectively, as

W (x)=W ′ sin(πx

L

)

, F0(x)=B′ sin(πx

L

)

. (4)

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Fig. 2. First transversal vibration mode of the beam repre-senting the suspension bridge roadbed.

Since the preloading W is usually taken to be a con-stant value, the decomposition for W (x), Eq. (4),should be intended as the first term in the harmonicexpansion of a constant function. The relative errorin taking only the lowest order is less than 10% inthe deflections [Lazer & McKenna, 1990].

In the same spirit, we separate the independentvariables in the beam deflection as

u(x, t) = y(t) sin(πx

L

)

, (5)

where y(t) = u(L/2, t) indicates the deflection ofthe bridge roadbed at its midpoint, assuming pos-itive (negative) values of y(t) for downward (up-ward) deflections, and with time evolution governed

by the following ordinary differential equation

Md2y

dt2+ δ′

dy

dt+ EI

(π

L

)4y + k′y+

= W ′ + B′ sin(ω′t) . (6)

We introduce nondimensional spatial and tem-poral variables as follows

x =πx

L, t =

(π

L

)2√

EI

Mt , (7)

as well as the normalized parameters

δ =

(

L

π

)2 δ′

2√

EIM, k =

(

L

π

)4 k′

EI, (8)

ω =

(

L

π

)2

ω′

√

M

EI, B =

(

L

π

)4 B′

EI,

W =

(

L

π

)4 W ′

EI,

(9)

such that Eq. (6) is rewritten as

y′′ + 2δy′ + my = W + B sin(ωt) , (10)

where the primes denote derivatives with respectto the scaled time, the hats on the variables wereremoved for ease of notation, and

m =

{

1 if y < 0,

(k + 1) if y > 0,(11)

(a) (b)

Fig. 3. One-dimensional oscillator with two springs, one of them with a clearance, equivalent to the suspension bridge flex-ional vibrations. (a) Upward deflections, for which only the beam stiffness is acting; (b) downward deflections, for which boththe beam and hanger stiffnesses occur. The string in the upper part of the figure plays no role in the elastic response of theoscillator.

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represents the slopes of the two pieces of the re-sponse curve.

Driven piecewise linear oscillators have been ex-tensively studied in the literature [Shaw & Holmes,

1983; Thompson et al., 1983; Whiston, 1987; Nord-mark, 1991], but in Eq. (10) the presence of thepreload W leads to some essential differences withthe already known cases. Figure 3 shows a mechan-ical oscillator equivalent to Eq. (10), in which thereare two springs with stiffness constants equal to 1

and k, respectively. Whereas the first spring is al-ways connected to the vibrating mass, the secondspring has a clearance with respect to it. This kindof oscillator has many applications in the impactvibration literature [Wiercigroch, 2000].

The nonsmoothness of the vector field (10) aty = 0 preserves the Lifshitz property, and thus

the existence and uniqueness theorem for differen-tial equations still holds [Guckenheimer & Holmes,1983]. However, while many standard results of

dynamical system theory remain applicable for thistype of systems, some important ones do not. Forexample, standard bifurcation theory does not ap-ply [Chin et al., 1994], as well as most techniques forcomputing Lyapunov exponents [Kapitaniak, 2000].

3. Periodic Orbits in the Undamped

Bridge Dynamics

Let us first consider that both the damping and theexternal force vanish (δ = 0, B = 0). In this case thesuspension bridge model (10), governing the deflec-tions of the first vibrational mode, can be writtenin the form y′′ = Feff (y) = −dVeff(y)/dy, where

Veff(y) = −Wy +1

2my2 (12)

is an effective potential which contains the effectsof the piecewise linear stiffness and the constantpreload.

(a) (b) (c)

(d) (e) (f)

Fig. 4. Restoring force (solid line) and the corresponding effective potential (dashed line) as a function of the vertical bridgedeflection y, without preload: (a) hanger stiffness; (b) beam stiffness; (c) combined stiffness. (d)–(f) Same cases, but withpreload.

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Figures 4(a)–4(c) show both the effective forceand potential for the hangers, beam and the com-bined system respectively, without preload (W =0). The hangers act as one-sided springs, accord-ing to Eq. (4), whereas the beam experiences alinear restoring force, such that the effective po-tential for the combined system is an asymmetriccurve. The existence of a preload (W 6= 0) causesthe appearance of a small harmonic well at thebottom of the effective potential curve, as can beseen in Figs. 4(d)–4(f). Small amplitude oscillations(called preloaded orbits by Doole and Hogan [1996])are strictly linear and can be analytically solved[Nayfeh & Mook, 1979]. We are, however, inter-ested in the large amplitude oscillations, for whichthe nonsmoothness of the effective potential playsa key role, and which requires a numerical solutionof Eq. (10), although approximate analytical meth-ods can be used to treat some particular cases [Kim& Noah, 1991; Cao et al., 2001]. Accordingly, wehave numerically solved (10) by using a 12th orderAdams method (from the LSODA package [Hind-march, 1983]).

In Fig. 5 we depict a typical phase portrait(downward displacement versus downward velocityof the bridge) for the free and undamped case. Thethree preloaded orbits shown are harmonic small-amplitude oscillations encircling the stable equilib-rium point located at y = y0 = W/(k + 1) andy = 0. Large amplitude bridge oscillations, on theother hand, are closed curves formed by arcs of el-lipses for y > 0 smoothly joined to arcs of circlesfor y < 0. Points for which y = 0 form a line ofnonsmoothness of the vector field (10), or a switch-

ing manifold, using the terminology of DiBernardoet al. [2001].

While the free and undamped case has anintegrable (in the Liouville sense) HamiltonianHu(p, y) = 1

2p2 + Veff(y) = const. (where p = y′ isthe momentum conjugate to the downward bridgedeflection y), the driven case is nonintegrable due tothe explicit time-dependence present in the forcingterm of the Hamiltonian

Hf (p, y, t) =1

2p2 +

1

2my2 − Wy − By sin(ωt) .

(13)

The effect of the preload can be absorbed into theabove Hamiltonian by making a shift of the originfrom y = 0 to the equilibrium at y = y0, by meansof a canonical transformation (p, y, t) → (pz, z, t)

Fig. 5. Trajectories in the phase plane (y1 = y, y2 = y)for the undamped and free bridge oscillations, with k = 10and W = 1, corresponding to six different initial conditions.Preloaded orbits have all their points with y > 0.

using the generating function

F2(pz, y, t) = (y − y0)pz =

(

y − W

1 + k

)

pz , (14)

such that the new Hamiltonian is, up to an unessen-tial constant, equal to

Hf (pz, z, t) =1

2p2

z +1

2z2 − 1

2kΘ(y)z2

− B

(

z +W

1 + k

)

sin(ωt) , (15)

where we have written for the nonsmooth elasticfunction m the expression

m(y) = 1 + kΘ(y) , (16)

in which Θ(y) is the unit step function.We can formally treat the time dependence by

passing to an extended phase space, by setting t ≡ τas an additional coordinate, and with E = −H f

as its conjugate momentum [Lichtenberg & Lieber-man, 1997]. An auxiliary parameter ξ can be chosento play the role of time in this case. The Hamilto-nian of the forced system in the extended phasespace is

He(pz, E; z, τ) =1

2p2

z +1

2z2 + E +

1

2kΘ(z(y))z2

− B

(

z +W

1 + k

)

sin(ωτ)

= E , (17)

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and dynamically similar to an autonomous two-degree-of-freedom system. It is non-integrable forthere is no other independent integral of motionbesides E .

In the above expression, the two first terms re-fer to a simple harmonic oscillator, for which we canintroduce the corresponding action and angle vari-ables (J, θ), defined as [Lichtenberg & Lieberman,1997]

z =√

2J sin θ , pz =√

2J cos θ , (18)

in such a way that (17) reads

H(J, E; θ, τ)=J+E+kΘ(y(J, θ))J sin2 θ

− B

(√2J sin θ+

W

1+k

)

sin(ωτ) .

(19)

We complete the algebra by making a second canon-ical transformation to the action and angle vari-ables related to the energy-time pair (Jϑ, Jϕ; ϑ, ϕ),

using the generating function F2(Jϑ, Jϕ; θ, τ) =ωτJϕ + θJϑ, such that the Hamiltonian is

H(Jϑ, Jϕ; ϑ, ϕ)

= H0(Jϑ, Jϕ) + H1(Jϑ, Jϕ; ϑ, ϕ)

= Jϑ + ωJϕ + kΘ(y(Jϑ, ϑ))Jϑ sin2 ϑ

− B sin ϕ

(

√

2Jϑ sin ϑ +W

1 + k

)

. (20)

The (extended) phase space for the forced os-cillator has four dimensions, but the energy surface(on which the system trajectories lie) is only three-dimensional, since E = E(Jϑ, Jϕ; ϑ, ϕ) = const.and there are only three independent variables, oneof them being the angle ϕ = ϕ0 +ωt, mod2π. Thuswe can study the system using a Poincare surfaceof section constructed at ϕ = 0, which is equivalentto a stroboscopic time-T map, where T = 2π/ω.

The two angle variables (ϑ and ϕ) characterizethe foliation of the phase space into nested closedsurfaces with the topology of tori. Closed curves ina Poincare surface of section are thus intersectionsof invariant tori. We define two frequencies relatedto each angle parameterizing the dynamics on thesetori, defined in terms of the unperturbed part of theHamiltonian (20), as follows

ωϑ =∂H0

∂Jϑ

= 1 , ωϕ =∂H0

∂Jϕ= ω , (21)

where ωϕ is the frequency of the external forcing,and ωϑ = 1 is the frequency of the closed orbits ofthe unperturbed harmonic oscillator (H0(J) = J).

The dynamical response of the forced systemdepends in a large extent on the ratio between thesefrequencies, or the winding number

α =ωϑ

ωϕ=

1

ω. (22)

If the winding number is a rational number ofthe form m/n, where m and n are co-prime in-tegers, then the frequencies are commensurate, inthat there are integer values of m and n for whichmωϕ−nωϑ = 0, or a m : n primary resonance, withthe following cases [Wiggins, 1997]: harmonic re-sponse (m = n = 1); subharmonic response of orderm (m > 1 and n = 1); ultra-harmonic response oforder m (m = 1 and n > 1); and ultra-subharmonicresponse of order m, n (m > 1 and n > 1).

A m : n resonance is a trajectory that closes onitself after n complete turns in the long way alongthe torus (ϕ), and m turns in the short way (ϑ). Interms of the Poincare surface of section (Jϑ, ϑ) wethus have m distinct points for a m : n resonance.These are marginally stable periodic orbits for thetime-T map, whose eigenvalues are equal to one inabsolute value. In the nested tori structure the pe-riodic orbits are arbitrarily close to one another, form and n can both be large integers. On the otherhand, if the winding number α is irrational, a trajec-tory never closes on itself, and densely fills a torus,represented by closed curves in the Poincare section(quasiperiodic response).

Consider the example depicted in Fig. 6, wherewe have relatively weak forcing. We have a 1 : 1primary resonance located close to the preloadedequilibrium point of the unforced system, as wellas resonances with m equal to 4, 3, 7, and 2,in increasing order of relative distance to the1 : 1 central resonance. The outermost closed curveshown in Fig. 6 represents a quasiperiodic or-bit, just like the closed curves encircling the res-onances. This is a rather general behavior observedin integrable Hamiltonian systems subjected to asmall nonintegrable perturbation [Lichtenberg &Lieberman, 1997].

Canonical perturbation theory fails when ap-plied at resonances due to the divergent terms theycause in the perturbation series. A standard pro-cedure (secular perturbation theory) removes thesedivergences by using two time scales, and averagingover the fast angle. The structure around a given

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Fig. 6. Trajectories in the phase plane (y1 = y, y2 = y) forthe undamped and forced bridge oscillations, with B = 0.5,ω = 4.3, k = 10, and W = 1, corresponding to different initialconditions.

primary resonance is topologically similar to anonlinear pendulum, in the sense that there are“librations”, or closed trajectories which in Fig. 6encircle the various m : n subharmonic resonances(chain of order-m periodic islands).

These islands are formed by quasiperiodic or-bits which wind around the torus in the same wayas the periodic orbits do. For example, the m = 4island in Fig. 6 completes four turns in the long wayalong the torus, before closing on itself. At each in-termediate crossing a trajectory visits a differentisland belonging to the chain, which is roughly de-limited by a locally chaotic layer. The island widthincreases with the square root of the perturbationstrength [Lichtenberg & Lieberman, 1997], whichis proportional to the amplitude of the externalforcing B. Primary islands are, in turn, surroundedby smaller secondary islands corresponding to har-monics between one of the fundamental frequenciesand the frequency of the librations inside a primaryisland, and so on.

Figure 7 presents a situation found for highervalues of B and k. Differently from the large num-ber of resonances shown in Fig. 6, now only the 2 : 1one is still visible, what leads us to conclude thatother islands were either engulfed or are simply toonarrow to be resolved. Besides the quasi-integrablecurves encircling the entire system as well as the 2 : 1

Fig. 7. Phase space trajectories for higher forcing ampli-tude (B = 3.0) and hanger stiffness (k = 50), the remainingparameters being the same as in the preceding figure.

resonance, the noteworthy feature is the existenceof a wide area-filling chaotic region.

The fact that most nonchaotic trajectories arequasiperiodic is not by chance, being rather pre-dicted by the KAM theory: under a sufficientlysmall perturbation most irrational tori are pre-served, even though with some alteration of theirshapes. Chaotic trajectories appear due to hetero-clinic intersections of stable and unstable manifoldsof the hyperbolic points connecting adjacent islandsof a same chain. These trajectories do not lie on in-variant tori and, for weak perturbation, form a nar-row area-filling layer around a given island. As theisland itself is enlarged for stronger perturbation, itschaotic layer similarly increases in width. Chaoticlayers belonging to adjacent layers may even fuseand engulf intermediate chains, as the perturbationbuilds up, as can be observed in Fig. 7.

If the perturbation is small enough such thatthere are surviving quasiperiodic trajectories (KAM

tori) between adjacent resonances, the chaotic lay-ers do not allow for large excursions of a chaotictrajectory. For higher perturbation, however, theseKAM tori are progressively destroyed and theirchaotic layers overlap, generating large scale excur-sions. There is an extensively studied barrier tran-sition between these two chaotic regimes [Chirikov,1979]. Even in this case, the wide chaotic regioncontinues to be bounded by KAM surfaces that

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encircle the overall structure, as shown in Fig. 7.In conclusion, even for higher forcing amplitudeswe cannot expect chaotic motion everywhere in theenergy surface, but rather a stochastic diffusion ofarea-filling trajectories over a limited portion of thephase-space.

4. Multistability in the Damped

Bridge Dynamics

In this section, we consider the effects of adding asmall viscous damping in the otherwise conservativesystem represented by the suspension bridge model.The discussion is based on the properties the sys-tem displays in the absence of dissipation, and arerelated to the periodic, quasiperiodic or chaotic na-ture of the orbits. Being now a dissipative system,

there is no longer an energy surface to work with,but as the system still has a time-periodic variablerelated to the external forcing, we can also make aPoincare section by a stroboscopic, or time-T sam-pling: (y(t), y(t)) 7→ fT (y, y) = (y(t+T ), y(t+T )).

An example of stroboscopic plot of a harmonicresponse, with nonzero damping and forcing, isshown in Fig. 8, where the different colors repre-sent the basins of attraction of stable orbits withperiods equal to 1 (black), 2 (yellow), 3 (red) and4 (cyan). In Figs. 9(a)–9(c) we present phase por-traits, for both continuous and discrete time, fororbits with periods equal to two, three and four, re-spectively; whereas in Fig. 9(d) the four orbits areshown together.

The existence of four observed periodic or-bits corresponding to subharmonics of the periodic

Fig. 8. Phase portrait (y − y) for the stroboscopic (time-T ) map for W = 1, δ = 0.01, k = 10, B = 0.5, and ω = 2π/T = 4.3.We indicate the existence of basins of attraction of multiple coexisting periodic orbits by using colors: period-1 (black), period-2(yellow), period-3 (red), period-4 (cyan), period-5 (dark green). The central region has a higher resolution than the rest of thefigure.

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(a) (b)

(c) (d)

Fig. 9. Trajectories in the phase plane (y, y) corresponding to coexisting periodic orbits whose basins are depicted in Fig. 8:(a) period-2 (squares); (b) period-3 (diamonds); and (c) period-4 (stars). The marked points are from the correspondingstroboscopic map, and are shown together in (d).

driving term, illustrates multistability in the oscil-lator response to the external forcing. The generalfeatures shown in Fig. 8 are expected for conserva-tive systems with small dissipation, and can be ex-plained in rather general grounds. The island cen-ters, which were marginally stable periodic orbits(sources) of the conservative system, become peri-odic attractors (sinks or stable foci) for the strobo-

scopic map, whose eigenvalues are smaller than onein absolute value. The periodic orbits in Fig. 8 willbe labeled F1 to F4. However, while in the Hamil-tonian case there is an infinite number of periodicorbits, we find only a finite number of attractorsin the weakly dissipative system, their number de-caying as the damping increases from zero [Feudelet al., 1996]. For the dissipative standard map it was

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Fig. 10. Bifurcation diagram for the stroboscopic map, showing the vertical position y(t = nT ) at discrete times versus thedamping coefficient δ. The remaining parameters are the same as in Fig. 8. Coexisting period-N orbits are represented by thesame colors as in Fig. 8, and are indicated as pN in the figure.

found that the number of attractors associated withprimary islands scales as the inverse of the dampinglevel [Schmidt & Wang, 1985]. In fact, Fig. 8 ex-hibits only four periodic orbits. Orbits with higherperiods may exist, but their basins would be toonarrow to be numerically resolved.

With dissipation, the tori inside a periodic is-land are entirely destroyed (since there is no longeran energy surface), and the region they occupiedin the Poincare section becomes roughly the basinof attraction of the stable focus corresponding tothat period. The chaotic orbits of the conservativesystem become chaotic transients in the weakly dis-sipative case, eventually asymptoting to some ofthe coexisting attractors [Feudel & Grebogi, 1997].The set of points that do not asymptote to any ofthe sinks has Lebesgue measure zero and includespoints on the boundaries of the basins of differentsinks [Feudel et al., 1998]. It has been numericallyobserved that attractors of high periods have com-paratively small basins of attraction, hence they are

somewhat difficult to detect [Feudel et al., 1996].Chaotic attractors also follow this rule and theyare found only very rarely in weakly dissipative sys-tems, for their basins are extremely small. In spiteof this, chaotic dynamics may be present for trajec-tories restricted to fractal basin boundaries, and re-sults in long chaotic transients for trajectories nearthe boundaries [Feudel & Grebogi, 1997].

In the bridge oscillation problem, we investi-gated the dependence of coexisting periodic orbitsas a function of the damping coefficient δ, for a fixedvalue of the forcing amplitude. In Fig. 10, we showperiodic orbits by plotting the y-coordinate (of thestroboscopic map) versus dissipation. For values ofδ less than 0.005, we observe orbits with periodsup to five, indicated in Fig. 10 by different colors.As δ increases, the orbits of higher period suddenlydisappear at well-defined values of the damping co-efficient, until there remains only the period-1 at-tractor for strongly damped case, for which there isa simple entrainment.

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Fig. 11. Bifurcation diagram for the stroboscopic map, showing the vertical position y(t = nT ) at discrete times versus theforcing frequency ω. The remaining parameters are the same as in Fig. 8. Coexisting period-N orbits are represented by thesame colors as in Fig. 8, and are indicated as pN in the figure.

The abrupt disappearance of the stable attrac-tor in Fig. 10 is due to a saddle-node, or tangentbifurcation (of eigenvalue +1) at the death point,so that another unstable orbit with the same pe-riod approaches the stable orbit and both disappearafter the bifurcation [Feudel & Grebogi, 1997]. Forexample, a stable period-3 orbit dies at δ ≈ 0.025because it coalesces with an unstable orbit of thesame period at this point. If the orbit death is dueto a saddle-node bifurcation, in the neighborhood ofthe death point δ = δSN the bifurcation curve hasthe shape of a quadratic branching (only the stablebranch being plotted in Fig. 10) with the normalform [Wiggins, 1997] z 7→ z + (δ − δSN) + z2, wherez = y − y∗q , and y∗q is a point of the period-q orbitof the stroboscopic map.

Another example of the multistable behaviordisplayed is in Fig. 11, where we plot a similar bi-furcation diagram for the forcing frequency ω, thedamping coefficient being held constant. There isa concentration of coexisting attractors for values

of the driving frequency between 4.0 and 4.3, ap-proximately. The number of coexistent attractorsdecreases for both higher and smaller frequencies. Aperiod-3 orbit in Fig. 11 is born at ωSN ≈ 3.7 anddies at ωCR ≈ 4.9. The stable period-3 orbit un-dergoes a period-doubling cascade and evolves to achaotic orbit in the vicinity of the death point ωCR.This orbit is born through a saddle-node bifurcationat ωSN, such that an unstable orbit of the same pe-riod is simultaneously created and collides with achaotic orbit at the death point ωCR (cf. Fig. 12).This is an example of interior crisis, after which thechaotic attractor disappears, leaving in its place achaotic saddle, which is a nonattracting invariantset with a dense orbit [Grebogi et al., 1982]. An ini-tial condition close to this chaotic saddle will experi-ence a chaotic transient and eventually asymptotesto another attractor.

This period-doubling cascade is not visible inFig. 11 because the period-doubling accumulationoccurs so fast that it is extremely difficult to be

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Fig. 12. Schematic figure showing the birth and death of aperiodic orbit in a weakly dissipative system.

detected when the dissipation is weak. Even forsimpler systems, like the dissipative standard map,this cascade is barely visible due to the plottingresolution [Feudel & Grebogi, 1997]. Let pBIFF bethe value of the control parameter which indicatesthe onset of a period-doubling cascade ending atthe accumulation point pACC. In the parameterspace, we have a “lifetime” |pCR−pBIFF| comprisingthe period-doubling cascade and the correspondingchaotic regime (the latter being restricted to thesubinterval |pCR − pACC|). In our numerical sim-ulations, which use weak dissipation, this lifetimeis smaller than the computed resolution of the di-agram. These observations can be put in a moregeneral context. Since we are able to see that mostinitial conditions asymptote to periodic attractorswith low periods, it follows that either higher pe-riod periodic orbits do not exist, or else they havevery small basins of attraction. Moreover, typicallythe higher the period of the attractor, the shorteris its existence interval in the parameter space.

5. Basin Boundary Structure

The overall phase-plane structure observed in Fig. 8is very similar to that displayed by a forced Duff-ing oscillator [Zavodney et al., 1990]. The structureof the basins of attraction for the subharmonic or-bits comes from two components: one is the bubbleshape that comes from the pendular form of the

Fig. 13. Magnification of the small box depicted from Fig. 8,containing a part of the (black) basin of a period-1 stable fo-cus F1, and the basin bands (cyan) of a period-4 stable focusF4. The red region is the basin of a period-3 attractor F3,and S3 is a period-3 saddle belonging to the correspondingbasin boundary, with portions of its invariant manifolds beingshown.

periodic islands found in the undamped case. Theelliptic point at the center of the resonance becomesa stable focus, when we add damping. The secondcomponent is the twist of bands belonging to dif-ferent basins of attraction, which is a typical char-acteristic of forced oscillators. However, these basinboundaries are not necessarily fractal curves in thisand other related cases. Frequently the basins ofattraction become striated and spiral about thecentral resonance, forming bands which accumulatealong the invariant stable manifold of an unstablesaddle orbit belonging to a basin boundary. This ac-cumulation occurs geometrically, with a rate givenby the corresponding eigenvalue in the unstable di-rection of the saddle orbit, and creates a misleadingimpression of self-similarity or fractality. This canbe seen for example in Fig. 13, where the small boxshown in Fig. 8 is magnified and the cyan bandsof the period-4 attractor (F4) basin alternate withbands of the black region belonging to a period-1 (F1) basin. The bands accumulate towards theboundary of the red region, the latter being a basinof a period-3 stable focus (F3). There is also aperiod-3 saddle point S3 belonging to this bound-

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Fig. 14. Schematic representation of the accumulation of basin bands towards the stable manifold of a saddle fixed pointbelonging to the basin boundary.

ary. We also depict in Fig. 13 the invariant man-ifolds which stem from S3. In spite of resemblinga fractal band structure, it is nothing but a loga-rithmic accumulation of basin filaments [McDonaldet al., 1985].

In order to clarify the mechanism underlyingthis band accumulation process, in Fig. 14 we showan unstable period-3 saddle orbit S3 belongingto the boundary between F3 and F4 basins, andits corresponding invariant manifolds. The basinboundary is the closure of the stable manifold of thesaddle S3. Since the time-T map fT (y, y) obtainedby us is invertible, we can compute the forward andbackward images of points and sets in the phaseplane. Under the backward images of the map, theiterations of some set A of points intersecting theunstable manifold of S3, or f−n(A), approach S3as time goes to infinity.

Let the set A be a part of one of the basinbands in Fig. 8, represented by the boundary line.The image of this boundary line under the back-ward dynamics of the map fT is supposed to crossthe unstable manifold of S3. The backward images

of this boundary approach the fixed point S3 astime increases in such a way that: (i) the intersec-tion points between the unstable manifold and theboundary of the set A converge exponentially fast,according to the corresponding unstable eigenvalueof the linearized map at S3; (ii) the lengths of thelobes increase exponentially to compensate for thedecreasing of the lobes’ widths, and the lobes them-selves tend to follow the stable manifold of S3. Theunion of all images of the boundary is a curve whichoscillates as it approaches the unstable fixed point.

The net effect is that segments of the boundaryline will become extremely thin filaments accumu-lating on the stable manifold of S3 [Pentek et al.,1995], and does not result in a fractal basin bound-ary. This can be also checked by noting in Fig. 13that the stable and unstable manifolds of the saddleorbit S3 do not intercept themselves at points otherthan S3, ruling out the possibility of homoclinicpoints and consequently of a fractal basin boundary[McDonald et al., 1985]. This is a key observation,taken into account that a naive numerical estimateof the basin boundary dimension in this case, using

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the uncertainty exponent technique, would lead toa misleading fractionary result.

Let an initial condition for the time-T map(y(0), y(0)), determined to an error ε, be repre-sented in the Poincare section by a ball of radiusε centered at (y(0), y(0)). Suppose this initial con-dition belongs to the basin of one of the attractors,say the period-4 stable focus F4 in Fig. 8 (for whichthe basin has been painted cyan). Now choose atrandom another initial condition inside the ε-ball.If this new initial condition asymptotes to anotherattractor, say the period-1 focus F1, the first initialcondition is said to be ε-uncertain [McDonald et al.,1985]. An error ball encircling an uncertain initialcondition crosses at least the boundary between twobasins. If we repeat this process for a large numberof randomly chosen points in the phase plane, ac-cording to the Lebesgue measure, it is possible tocompute the ratio f(ε) between the number of ε-uncertain initial conditions to the total number ofpoints chosen. This uncertain fraction is expectedto scale with ε in a power-law fashion: f(ε) ∼ εα,where α is the uncertainty exponent [McDonaldet al., 1983], and gives the rate at which we have toincrease the accuracy in determining the initial con-dition, in order to reduce the uncertain fraction f(ε)by a given amount. The uncertainty exponent α isrelated to the basin boundary box-counting dimen-sion d by α = D−d, where D = 2 is the phase-spacedimension [McDonald et al., 1985]. Let us apply thisnumerical procedure to evaluate the dimension ofthe basin boundary undergoing band accumulationin Fig. 13. In Fig. 15, we plot the correspondinguncertain fraction versus the size of the error ballsε, which is well-fitted by a power-law scaling, withslope α = 0.8807 ± 0.0085. Since α < 1 it turns outthat a specified decrease in the uncertainty of ini-tial conditions yields only a relatively small reduc-tion of the uncertainty towards which attractor theresulting trajectory will asymptote (final state sen-

sitivity). The resulting dimension, however, leads toa misleading fractal value (d ≈ 1.12), whereas fora smooth basin boundary we would expect d = 1.How can we account for this apparently paradoxicalresult?

The logarithmic accumulation of bands de-picted in Fig. 8 is shown in Fig. 13, where a rectan-gle of sides v and w containing the saddle point S3has been singled out. The uncertain fraction f(ε)can be estimated as the ratio between the totaluncertain area A(ε) and the area of the selectedrectangle B = vw. Part of the uncertain area is

Fig. 15. Uncertain fraction of 4096 initial conditions ran-domly chosen in the phase space region depicted in Fig. 13,versus the radius of a small error ball centered at these points.The error bars refer to six different sets of initial conditions.The solid curve is a linear regression fit with slope α ≈ 0.88.In spite of this result, the basin boundary is not fractal.

due to the basin of the F1 focus, and is formed by(black) strips of half-width ε accumulating in thestable manifold of S3, with a rate given by the un-stable eigenvalue at S3, denoted as λu. As thesefinite-width strips approach the stable manifold ofS3, they fuse themselves into a single strip A0(ε)with half-width ε, giving A0(ε) ∼ 2εv [McDonaldet al., 1985]. The rest of the uncertain area dependson the number r of the parts of the (cyan) basin ofthe F4 focus that are inside the rectangle B but out-side A0. Let Aj(ε) ∼ 2(2ε)v, j = 1, 2, . . . , r denoteeach part, where r is the number of backward itera-tions of the map fT necessary to put all the parts ofthe F4 basin from the rectangle B to the strip A0.Hence ε ≈ λ−r

u w, from which results r ∼ ln(1/ε).Combining these two contributions the total

uncertain area is A(ε) = A0(ε) + rAj(ε) ∼ ε +ε ln(1/ε). The uncertain fraction f(ε) ∼ A(ε)/B hasthus the same dependence. For large ε the first termdominates (f(ε) ∼ ε), whereas the second termdominates for small ε, resembling a power-law scal-ing (f(ε) ∼ εα). Hence, a numerical estimation ofthe uncertain fraction, when made in regions wherethere is a band accumulation, creates misleading re-sults if a too fine grid is used. If a slightly largermesh is used we recover the linear scaling charac-teristic of smooth basin boundaries.

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Other dynamical systems, like the weakly dis-sipative Henon map, have basins of attraction ofcoexisting attractors interwoven in a complex way.It was found that the uncertainty exponent for thismap is approximately 0.14, much smaller than thatcomputed for the bridge model [Feudel & Grebogi,1997]. In the Henon map case we would have a moreintense final state sensitivity, and in fact the basinboundaries are actually fractal curves. Since α inthis case is so small, it turns out that the box di-mension of the basin boundary is close to that ofthe phase space.

Nevertheless, it should be stressed that, whilethe basin boundary may not be a fractal, we stillhave final state sensitivity if error balls of a smallradius are used, and it is an issue of paramountimportance from the point of view of the actual be-havior of the system we are investigating. For ex-ample, imagine that in Fig. 8 we have parametersfor which the initial condition is located in a phaseplane region where there exist many basin bandsbelonging to different attractors. As the parameterspecifications are strongly dependent on the exter-nal noise, it may happen that a trajectory lying insome basin hops to another basin, and so on [Krautet al., 1999].

Each time any such hopping occurs, we have arather abrupt change of behavior which, if accom-panied by a large amplitude jump, may cause dam-ages to the bridge structure. This can be regarded asan example of complex behavior, even without con-sidering the spatial degrees of motion (which nat-urally play a significant role in the actual bridgedynamics). A complex system has many accessibledynamical regimes, and typically experiences hop-ping between them [Macau & Grebogi, 1999]. Fromthe point of view of a safe operation of the bridge,it should be better to avoid such possibilities andlocate the system far enough from regions wherebasins accumulate, even though they are not frac-tal curves.

6. Chaotic Behavior and Boundary

Crisis

According to Sec. 4, chaotic attractors in weaklydissipative systems are comparatively rare for theyhave small basins of attraction and smaller lifetimesin the parameter space. This explains why chaoticattractors are not easily observed in bifurcation di-agrams such as shown in Figs. 10 and 11, but this isno longer true if either the dissipation or the driving

amplitude is increased. Figure 16 shows an example,for the bridge oscillations, in which both parame-ters have been increased with respect to the casesanalyzed in the previous sections. In such cases, weare now able to observe chaotic trajectories [such asin Fig. 16(a)] belonging to a chaotic attractor withfractal dimension [Fig. 16(b)]. For this set of pa-rameters, there is also a coexisting period-2 attrac-tor [Figs. 16(c) and 16(d)] (the period-1 attractor isnot shown), and results from the relative basin dom-inance that the stable foci of periods 2 and 1 have,when compared with attractors of higher periods.This dominance is illustrated, in the conservativecase, by the large island width displayed by a 2 : 1resonance (see Fig. 6).

As the vector field in (10) is nonsmooth at y = 0we cannot resort to the usual methods for comput-ing the Lyapunov exponents in order to identifychaotic behavior [Kapitaniak, 2000]. A direct wayto prove the existence of a chaotic attractor likethat depicted in Fig. 16 is to show the appearanceof a topological horseshoe in the attractor dynamics[Shaw & Holmes, 1983]. A horseshoe is a topolog-ical rectangle whose image under the forward andbackward iterations of the time-T map fT forms afolded strip [Grebogi et al., 1983], the asymptoticset containing infinitely many unstable periodic or-bits. This chaotic invariant set is non-attracting andcontains a dense orbit [Guckenheimer & Holmes,1983], also called a chaotic (or strange) hyperbolicsaddle. Any trajectory originating from an initialcondition off this chaotic saddle, but very close toit, will experience a very complicated and irregularbehavior. For these orbits there is extreme sensitiv-ity to initial conditions, in the sense that two or-bits, originally very close to each other, will divergeexponentially as time increases. The unstable peri-odic orbits embedded in this nonattracting chaoticinvariant set are, for the time-T map, saddle pointsresulting from homoclinic and heteroclinic intersec-tions of their unstable and stable manifolds. Hence,it suffices to show that there is one such intersection,for a homoclinic point to be mapped on another.This generates an uncountably infinite number ofunstable periodic orbits, which support the natu-ral measure in the chaotic attractor [Grebogi et al.,1988].

Figure 17 shows the stable and unstable man-ifolds arising from an unstable periodic orbit Pembedded in the supposedly chaotic attractor de-picted in Fig. 16(b). This figure shows a part of the

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(a) (b)

(c) (d)

Fig. 16. Trajectories in the phase plane (y, y) for δ = 0.05, ω = 4.0, B = 3.0, W = 1, and k = 50. (a) Small segment of achaotic trajectory, the full attractor being depicted in (b). The marked points are from the stroboscopic map. (c) A coexistingperiod-2 orbit; (d) both attractors are shown together.

stable (W s(P )) and unstable (W u(P )) manifoldsthat stem from the saddle point P . The closureof the unstable manifold is the attractor in whichthe saddle point P is embedded. Each crossing ofW u(P ) with W s(P ) is a homoclinic point, and we

see that there will be an infinite number of suchintersections, proving the existence of a topologi-cal horseshoe underlying chaotic dynamics in theattractor shown in Fig. 16(b).

For different parameter values we have found

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Fig. 17. Stable and unstable manifolds arising from an unstable fixed point P embedded in the chaotic attractor depictedin Fig. 16(b). The homoclinic intersections between these manifolds generate a horseshoe structure and consequently chaoticdynamics.

a more complex behavior, as illustrated by the bi-furcation diagram depicted in Fig. 18, where thevertical position in the stroboscopic map is plottedversus the damping coefficient. There is a signifi-cant range for which there appears to be a chaoticregion in coexistence with a period-2 attracting or-bit. The sudden appearance of the chaotic attractorfor δCR ≈ 0.040 is due to a boundary crises. We havechecked this observation by numerically plotting inFig. 19 the behavior of a trajectory just before thecrisis value, at δ = 0.039 < δCR. In Fig. 19(a) wesee a time series for the y-values of a trajectoryarising from an initial condition next to the chaoticattractor originated for δ > δCR. An initially longchaotic transient decays, after circa 1200 iterationsof the stroboscopic map, to the coexisting period-2stable focus F2. In Fig. 19(b) we show, in the phasespace, that during the transient the orbit wandersalong a set very similar to the post-critical chaoticattractor, and eventually asymptotes to the period-2 attractor.

These features are typical of a boundary crisis,for which the chaotic attractor existent for δ & δCR

collides with its basin boundary. For δ . δCR thechaotic attractor no longer exist, being replaced bya chaotic nonattracting saddle which resembles theattractor, in the sense that a trajectory wandersthrough the attractor remnant until it is eventuallyrepelled from it and goes to the period-2 attractor[Grebogi et al., 1983]. A chaotic saddle is comprisedby an infinite number of intersecting stable and un-stable manifolds. A point placed exactly on some ofthese manifolds would generate a trajectory stay-ing on the manifold at all times. Hence, if an initialcondition is chosen very close to some of these man-ifolds it will stay longer in the attractor remnant,compared to initial conditions farther from the sad-dle, resulting in larger transient times τ . We cancope with this diversity by considering a large num-ber of randomly chosen initial conditions in someregions containing the interesting part of the basinboundary structure, and numerically computing the

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Fig. 18. Bifurcation diagram for the stroboscopic map, showing the vertical position y(t = nT ) at discrete times versus thedamping coefficient δ. The remaining parameters are the same as in Fig. 16. The arrow indicates the occurrence of a boundarycrisis.

transient duration τ for each of them, with average〈τ〉. The transient times are expected to follow aPoisson distribution

P (τ) =1

〈τ〉 exp

(

− τ

〈τ〉

)

. (23)

There are strong theoretical arguments sup-porting this statistics [Romeiras et al., 1990], but itis also possible to justify its use through a heuristicargument. When considering the post-critical dy-namics there are just two possible outcomes for atrajectory from an initial condition near the chaoticsaddle: it will either escape or remain in the imme-diate vicinity of the chaotic saddle for a long time.However, since the saddle has zero Lebesgue mea-sure in phase space, the probability of escape to

another attractor is much larger than of not do-

ing so, for a large yet finite time τ . From the sta-

tistical point of view the problem would be cast

into a binomial probability distribution, but sincethe probability of one event is very small com-

pared to the other, it will reduce to a Poissonian

one. We have numerically verified this fact by plot-

ting in Fig. 20 a numerically obtained distribu-tion for 6000 randomly chosen initial conditions,

and a fixed value of δ = 0.040046 near the cri-

sis value δCR. The average transient duration is〈τ〉 = 49629, and a least squares fit compares well

with the distribution (23). On the other hand, the

average transient duration scales with the difference

δ−δCR in a power-law fashion 〈τ〉 ∼ (δ − δCR)−γ , as

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(a)

(b)

Fig. 19. (a) Time series for the vertical position for a chaotic transient which decays to a period-2 orbit for δ = 0.039, in thevicinity of a crisis. (b) Phase portrait for a large number of initial conditions corresponding to this situation, showing that thechaotic transient resembles the post-critical chaotic attractor. The remaining parameters are the same as in Fig. 18.

illustrated in Fig. 21, with a numerically determinedslope γ = 1.387 ± 0.056. A number of 6000 initialconditions were randomly chosen in the region oc-cupied by the chaotic attractor remnant.

Finally, in Fig. 22 we show a bifurcation dia-gram for the stroboscopic map in which the vary-ing parameter is the forcing amplitude B. In thiscase the multistable behavior includes the coexis-tence of chaotic and periodic attractors. A period-2orbit is born at BSN = 1.2 through a saddle-nodebifurcation, coexisting with a stable period-1 orbitwhich undergoes a period-doubling bifurcation cas-

cade beginning at BBIFF = 1.8 and accumulating atBACC = 2.7. The chaotic attractor generated imme-diately after this parameter value has a rather largelifetime, when compared with the results for whichthe damping coefficient was varied, and the chaoticattractor dies at BCR = 3.2 following an interiorcrisis, probably due to a collision with the unsta-ble period-2 orbit generated at the saddle-node bi-furcation that occurred when B = BSN. A similarcollision with an unstable period-1 orbit probablyaccounts for the second crises observed at BCR.Moreover, a period-3 orbit is born at B = 1.5

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Fig. 20. Frequency histogram for the chaotic transient duration, using 6000 initial conditions randomly chosen andδ = 0.040046, just before the crisis value. The dashed line is a least squares fit with slope 1/〈τ 〉 = 1/49629.

Fig. 21. Average chaotic transient duration versus the normalized difference between δ and its crisis value, using 6000 initialconditions randomly chosen inside the region occupied by the chaotic attractor remnant. The dashed line is a least squares fitwith slope 1.387 ± 0.056, and the error bars refer to six sets of 1000 initial conditions each.

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Fig. 22. Bifurcation diagram for the stroboscopic map, showing the vertical position y(t = nT ) at discrete times versus theforcing amplitude B. The remaining parameters are the same as in Fig. 16. Coexisting period-N orbits are represented bydifferent colors, and indicated as pN in the figure.

through a saddle-node bifurcation and dies atB = 2.0, after undergoing a period-doubling bifur-cation cascade.

7. Conclusions

The behavior of a suspension bridge due to exter-nal periodic forcing caused by impinging wind isa complex subject from the point of view of dy-namical systems. We have considered a simplifiedmodel which takes into account the response of theroadbed as an elastic beam, connected by one-sidedsprings to the main cable. This asymmetric responsecauses the stiffness of the combined system to bepiecewise linear. Instead of solving the partial dif-ferential equation for the beam oscillations, we haveproceeded by choosing the most interesting vibra-tion mode, which appears to be the first transver-sal harmonic, and studying its evolution using asecond-order nonautonomous ordinary differentialequation.

The dynamics related to this mode reducesto a one-degree of freedom weakly dissipativesystem with forcing. We considered the periodic(harmonic, sub and super-harmonic) and quasiperi-odic responses, and relate these to the ratio betweenthe unperturbed and external forcing frequencies.In this way, we were able to explain, at least qual-itatively, the main features observed in numericalsimulation, like the interaction of resonances andchaotic motion, by a reasoning based on KAM the-ory and some ideas from canonical perturbationmethods.

When a small dissipative term is added tothe formerly Hamiltonian system, the main conse-quences can be predicted in rather general grounds.The resonances of the conservative system, whichcorrespond to nonhyperbolic periodic orbits of thePoincare map, become attractors of the stable focitype, and the quasiperiodic tori around them dis-appear to give way to the basins of the correspond-ing attractors displayed by the weakly dissipative

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system. The chaotic trajectories of the Hamiltoniansystem are replaced by chaotic transients (in theweakly dissipative case) which asymptote to the at-tractors. Hence the main feature of this case is amultistable behavior with a dominance of periodicattractors, although we were able to show the exis-tence of chaotic attractors as well.

The multistable structure also shows a com-plicated basin boundary structure because of thelarge number of coexisting periodic (and perhapschaotic) attractors. However, this does not neces-sarily mean that the basin boundaries are fractal.Due to a logarithmic accumulation of basin bandstowards the stable manifold of a fixed point belong-ing to the basin boundary, the uncertain fractionof phase space sometimes appears to obey a power-law scaling in spite of the nonfractal character ofthe basin boundary, calling for a correct interpre-tation of the final state sensitivity concept for suchsystems. Nevertheless, from the practical point ofview, the coexistence of a large number of predomi-nantly periodic attractors with a complicated basinboundary structure is already important, since ex-ternal noise is able to drive the system off a givenbasin and jump to another one, causing damage dueto the resulting amplitude jumps.

Acknowledgments

This work was made possible by partial financialsupport from the following Brazilian governmentagencies: CNPq, CAPES, Fundacao Araucaria,FUNPAR-UFPR, and FAPESP. We are grateful toS. R. Lopes, E. Macau and J. R. Balthazar for usefulcomments and suggestions.

References

Aidanpaa, J. O., Shen, H. H. & Gupta, R. B. [1994]“Stability and bifurcations of a stationary state foran impact oscillator,” Chaos 4, 621–630.

Amann, O. H., von Karman, T. & Woodruff, G. B. [1941]The Failure of the Tacoma Narrows Bridge (FederalWorks Agency, Washington).

Billah, K. Y. & Scanlan, R. H. [1991] “Resonance,Tacoma narrows bridge failure, and undergraduatephysics textbooks,” Am. J. Phys. 59, 118–124.

Blazejczyk-Okulewska, B., Czolczynski, K., Kapitaniak,T. & Wojewoda, J. [1999] Chaotic Mechanics in Sys-

tems with Friction and Impacts (World Scientific,Singapore).

Blazejczyk-Okulewska, B. [2000] “Study of the impactoscillator with elastic coupling of masses,” Chaos

Solit. Fract. 11, 2487–2492.

Blevins, R. D. [1997] Flow-Induced Vibration (VanNos-trand Reinhold, NY).

Cao, Q., Xu, L., Djidjeli, K., Price, W. G. & Twizell, E.H. [2001] “Analysis of period-doubling and chaos ofa non-symmetric oscillator with piecewise-linearity,”Chaos Solit. Fract. 12, 1917–1927.

Chin, W., Ott, E., Nusse, H. E. & Grebogi C. [1994]“Grazing bifurcations in forced impact oscillators,”Phys. Rev. E50, p. 4427.

Chirikov, B. [1979] “A universal instability of many-dimensional oscillator systems,” Phys. Rep. 52,263–379.

Di Bernardo, M., Budd, C. J. & Champneys, A. R.[2001] “Normal form maps for grazing bifurcations inn-dimensional piecewise-smooth dynamical systems,”Physica D160, 222–254.

Doole, S. H. & Hogan, S. J. [1996] “A piecewise linearsuspension bridge model: Nonlinear dynamics and or-bit continuation,” Dyn. Stab. Syst. 11, 19–47.

Doole, S. H. & Hogan, S. J. [2000] “Nonlinear dynam-ics of the extended Lazer–McKenna bridge oscillationmodel,” Dyn. Stab. Syst. 15, 43–58.

Feudel, U., Grebogi, C., Hunt, B. R. & Yorke, J. A. [1996]“Map with more than 100 coexisting low-period peri-odic attractors,” Phys. Rev. E54, 71–81.

Feudel, U. & Grebogi, C. [1997] “Multistability and thecontrol of complexity,” Chaos 7, 597–604.

Feudel, U., Grebogi, C., Poon, L. & Yorke, J. A. [1998]“Dynamical properties of a simple mechanical systemwith a large number of coexisting periodic attractors,”Chaos Solit. Fract. 9, 171–180.

Grebogi, C., Ott, E. & Yorke, J. A. [1982] “Chaotic at-tractors in crisis,” Phys. Rev. Lett. 48, 1507–1510.

Grebogi, C., Ott, E. & Yorke, J. A. [1983] “Crises,sudden changes in chaotic attractors, and transientchaos,” Physica D7, 181–200.

Grebogi, C., Ott, E. & Yorke, J. A. [1988] “Unstable pe-riodic orbits and the dimension of multifractal chaoticattractors,” Phys. Rev. E37, 1711–1724.

Guckenheimer, J. & Holmes, P. [1983] Nonlinear

Oscillations, Dynamical Systems and Bifurcation of

Vector Fields (Springer, NY).Hartog, D. & Pieter, J. [1987] Advanced Strength of

Materials (Dover, NY).Heertjes, M. F. & Van De Molengraft, M. J. G.

[2001] “Controlling the nonlinear dynamics of a beamsystem,” Chaos Solit. Fract. 12, 49–66.

Hindmarch, A. C. [1983] “ODEPACK: a systematizedcollection of ODE solvers,” in Scientific Comput-

ing, eds. Stepleman, R. S. et al. (North-Holland,Amsterdam).

Jerrelind, J. & Stensson, A. [2000] “Nonlinear dynamicsof parts in engineering systems,” Chaos Solit. Fract.

11, 2413–2418.Kapitaniak, T. [2000] Chaos for Engineers (Springer,

NY).

March 31, 2004 9:38 00963


Kim, Y. B. & Noah, S. T. [1991] “Stability and bifurca-tion analysis of oscillators with piecewise-linear char-acteristics: A general approach,” J. Appl. Mech. 58,545–553.

Kraut, S., Feudel, U. & Grebogi, C. [1999] “Preference ofattractors in noisy multistable systems,” Phys. Rev.

E59, 5253–5260.Lazer, A. C. & McKenna, P. J. [1990] “Large amplitude

periodic oscillations in suspension bridges: Some newconnections with nonlinear analysis,” SIAM Rev. 58,537–578.

Lichtenberg, A. J. & Lieberman, M. A. [1997] Regular

and Chaotic Dynamics, 2nd edition (Springer, NY).Nayfeh, A. H. & Mook, D. T. [1979] Nonlinear Oscilla-

tions (Wiley-Interscience, NY).Macau, E. E. N. & Grebogi, C. [1999] “Driving trajecto-

ries in complex systems,” Phys. Rev. E59, 4062–4070.McDonald, S. W., Grebogi, C., Ott, E. & Yorke, J. A

[1983] “Final state sensitivity: An obstruction to pre-dictability,” Phys. Lett. A99, 415–418.

McDonald, S. W., Grebogi, C., Ott, E. & Yorke, J.A. [1985] “Fractal basin boundaries,” Physica D17,125–153.

Nordmark, A. B. [1991] “Non-periodic motion causedby grazing incidence in impact oscillators,” J. Sound

Vibr. 2, 279–297.Pentek, A., Toroczkai, Z., Tel, T., Grebogi, C. & Yorke,

J. A. [1995] “Fractal boundaries in open hydrodynam-ical flows: Signatures of chaotic saddles,” Phys. Rev.

E51, 4076–4088.

Popp, K. & Stelter, P. [1990] “Stick-slip vibrations andchaos,” Phil. Trans. R. Soc. Lond. A332, 89–105.

Romeiras, F. J., Grebogi, C., Ott, E. & Dayawansa,W. P. [1990] “Controlling chaotic dynamic systems,”Physica D58, 165–192.

Schmidt, G. & Wang, B. W. [1985] “Dissipative standardmap,” Phys. Rev. A32, 2994–2999.

Shaw, S. W. & Holmes, P. J. [1983] “A periodicallyforced piecewise linear oscillator,” J. Sound Vibr. 90,129–155.

Thomspson, J. M. T., Bokaian, A. R. & Ghaffari, R.[1983] “Subharmonic resonances and chaotic motionsof a bilinear oscillator,” I.M.A. J. Appl. Math. 31,207–234.

Whiston, G. S. [1987] “The vibro-impact response of aharmonically excited and preloaded one-dimensionallinear oscillator,” J. Sound Vibr. 115, 303–319.

Wiercigroch, M. [2000] “Modelling of dynamical systemswith motion dependent discontinuities,” Chaos Solit.

Fract. 11, 2429–2442.Wiercigroch, M. & DeKraker, B. [2000] Applied Nonlin-

ear Dynamics and Chaos of Mechanical Systems with

Discontinuities (World Scientific, Singapore).Wiggins, S. [1997] Introduction to Applied Nonlinear Dy-

namics and Chaos (Springer, NY).Zavodney, L. D., Nayfeh, A. H. & Sanchez, N. E. [1990]

“Bifurcations and chaos in parametrically excitedsingle-degree-of-freedom systems,” Nonlin. Dyn. 1,1–21.

multistability, basin boundary structure, and …

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