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Nonlinear Dyn (2013) 73:689–704DOI 10.1007/s11071-013-0823-x

O R I G I NA L PA P E R

Detecting invariant manifolds, attractors, and generalizedKAM tori in aperiodically forced mechanical systems

Alireza Hadjighasem · Mohammad Farazmand ·George Haller

Received: 26 October 2012 / Accepted: 27 January 2013 / Published online: 22 February 2013© Springer Science+Business Media Dordrecht 2013

Abstract We show how the recently developed the-ory of geodesic transport barriers for fluid flows canbe used to uncover key invariant manifolds in exter-nally forced, one-degree-of-freedom mechanical sys-tems. Specifically, invariant sets in such systems turnout to be shadowed by least-stretching geodesics of theCauchy–Green strain tensor computed from the flowmap of the forced mechanical system. This approachenables the finite-time visualization of generalized sta-ble and unstable manifolds, attractors and generalizedKAM curves under arbitrary forcing, when Poincarémaps are not available. We illustrate these results bydetailed visualizations of the key finite-time invariantsets of conservatively and dissipatively forced Duffingoscillators.

A. HadjighasemDepartment of Mechanical Engineering, McGillUniversity, 817 Sherbrooke Ave. West, Montreal,Quebec H3A 2K6, Canadae-mail: [email protected]

M. Farazmand · G. Haller (!)Institute for Mechanical Systems, ETH Zürich,Tannenstrasse 3, 8092 Zürich, Switzerlande-mail: [email protected]

M. FarazmandDepartment of Mathematics, ETH Zürich, Rämistrasse101, 8092 Zürich, Switzerlande-mail: [email protected]

Keywords Stability of mechanical systems ·Non-autonomous dynamical systems · Invariantmanifolds · Coherent structures

1 Introduction

A number of numerical and analytical techniques areavailable to analyze externally forced nonlinear me-chanical systems. Indeed, perturbation methods, Lya-punov exponents, Poincaré maps, phase space embed-dings and other tools have been become broadly usedin mechanics [1, 2]. Still, most of these techniques,are only applicable to nonlinear systems subject to au-tonomous (time-independent), time-periodic, or time-quasiperiodic forcing.

These recurrent types of forcing allow for the anal-ysis of asymptotic features based on a finite-time sam-ple of the underlying flow map—the mapping thattakes initial conditions to their later states. Indeed,to understand the phase space dynamics of an au-tonomous system, knowing the flow map over an ar-bitrary short (but finite) time interval is enough, asall trends can be reproduced by the repeated appli-cations of this short-time map. Similarly, the periodmap of a time-periodic system (or a one-parameterfamily of flow maps for a time-quasiperiodic sys-tem) renders asymptotic conclusions about recur-rent features, such as periodic and quasiperiodic or-bits, their stable and unstable manifolds, attractors,etc.

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690 A. Hadjighasem et al.

By contrast, the identification of key features in theresponse of a nonlinear system under time-aperiodicforcing has remained an open problem. Mathemat-ically, the lack of precise temporal recurrence insuch systems prevents the use of a compact extendedphase space on which the forced system would be au-tonomous. This lack of compactness, in turn, rendersmost techniques of nonlinear dynamics inapplicable.Even more importantly, a finite-time understandingof the flow map can no longer be used to gain a fullunderstanding of a (potentially ever-changing) non-autonomous system.

Why would one want to develop an understand-ing of mechanical systems under aperiodic, finite-timeforcing conditions? The most important reason is thatmost realistic forms of forcing will take time to buildup, and hence will be transient in nature, at least ini-tially. Even if the forcing is time-independent, thefinite-time transient response of a mechanical systemis often crucial to its design, as the largest stresses andstrains invariably occur during this period.

Similar challenges arise in fluid dynamics, wheretemporally aperiodic unsteady flows are the rule ratherthan the exception. Observational or numerical datafor such fluid flows are only available for a limitedtime interval, and some key features of the flow mayonly be present for an even shorter time. For instance,the conditions creating a hurricane in the atmosphereare transient, rather than periodic, in nature, and thehurricane itself will generally only exist for less thantwo weeks [3]. As a result, available asymptotic meth-ods are clearly inapplicable to its study, even thoughthere is great interest in uncovering its internal struc-ture and overall dynamics.

In response to these challenges in fluid dynam-ics, a number of diagnostic tools have been devel-oped [4, 5]. Only very recently, however, has a rig-orous mathematical theory emerged for dynamicalstructures in finite-time aperiodic flow data [6]. Thistheory finds that finite-time invariant structures in adynamical system are governed by intrinsic, metricproperties of the finite-time flow map. Specifically, intwo-dimensional unsteady flows, structures acting astransport barriers can be uncovered with the help ofgeodesics of the Cauchy–Green strain tensor used incontinuum mechanics [7]. This approach generalizesand extends earlier work on hyperbolic LagrangianCoherent Structures (LCS), which are locally most re-pelling or attracting material lines in the flow [8–11].

In this paper, we review the geodesic transport the-ory developed in [6] in the context of one-degree-of-freedom, aperiodically forced mechanical systems.We then show how this theory uncovers key in-variant sets under both conservative and dissipativeforcing in cases where classic techniques, such asPoincaré maps, are not available. Remarkably, thesefinite-time invariant sets can be explicitly identifiedas parametrized curves, as opposed to plots requiringpost-processing or feature extraction.

The organization of this paper is as follows. Sec-tion 2 is divided into two subsections: Sect. 2.1 pro-vides the necessary background for the geodesic the-ory of transport barriers developed in [6]. In Sect. 2.2,we describe a numerical implementation of this the-ory that detects finite-time invariant sets as trans-port barrier. Section 3 presents results from the ap-plication of this numerical algorithm to one degree-of-freedom mechanical systems. First, as a proof ofconcept, Sect. 3.1 considers conservative and dissipa-tive time-periodic Duffing oscillators, comparing theirgeodesically extracted invariant sets with those ob-tained form Poincaré maps. Next, Sect. 3.2 deals withinvariant sets in aperiodically forced Duffing oscilla-tors, for which Poincaré maps or other rigorous extrac-tion methods are not available. We conclude the paperwith a summary and outlook.

2 Set-up

The key invariant sets of autonomous and time-periodic dynamical systems–such as fixed points, pe-riodic and quasiperiodic motions, their stable and un-stable manifolds, and attractors–are typically distin-guished by their asymptotic properties. In contrast,invariant sets in finite-time, aperiodic dynamical sys-tems solely distinguish themselves by their observedimpact on trajectory patterns over the finite time in-terval of their definition. This observed impact is apronounced lack of trajectory exchange (or transport)across the invariant set, which remains coherent intime, i.e., only undergoes minor deformation. Well-understood, classic examples of such transport barri-ers include local stable manifolds of saddles, parallelshear jets, and KAM tori of time-periodic conservativesystems. Until recently, a common dynamical featureof these barriers has not been identified, hindering theunified detection of transport barriers in general non-autonomous dynamical systems.

Manifolds, attractors and KAM surfaces in aperiodic systems 691

As noted recently in [6], however, a common fea-ture of all canonical transport barriers in two dimen-sions is that they stretch less under the flow than neigh-boring curves of initial conditions do. This observationleads to a nonstandard calculus of variations problemwith unknown endpoints and a singular Lagrangian.Below we recall the solution of this problem from[6], with a notation and terminology adapted to one-degree-of-freedom mechanical oscillators.

A one-degree-of-freedom forced nonlinear oscilla-tor can generally be written as a two-dimensional dy-namical system

x = v(x, t), x ! U " R2, t ! [t0, t1], (1)

with U denoting an open set in the state space, wherethe vector x labels tuples of positions and velocities.The vector v(x, t), assumed twice continuously differ-entiable, contains the velocity and acceleration of thesystem at state x and at time t .

Let x(t1; t0, x0) denote the final state of system (1)at time t1, given its state x0 at an initial time t0. Theflow map associated with (1) over this time interval isdefined as

Ft1t0

: x0 #$% x(t1; t0, x0), (2)

which maps initial states to final states at t1. TheCauchy–Green (CG) strain tensor associated with theflow map (2) is defined as

Ct1t0

(x0) =!DF

t1t0

(x0)"&

DFt1t0

(x0), (3)

where DFt1t0

denotes the gradient of the flow map (2),and the symbol & refers to matrix transposition.

Note that the CG tensor is symmetric and positivedefinite. As a result, it has two positive eigenvalues0 < !1 ' !2 and an orthonormal eigenbasis {"1, "2}.We fix this eigenbasis so that

Ct1t0

(x0)"i (x0) = !i (x0)"i (x0),##"i (x0)

## = 1, i ! {1,2}, (4)

"2(x0) =#"1(x0), # =$

0 $11 0

%.

We suppress the dependence of !i and "i on t0 and t1for notational simplicity.

2.1 Geodesic transport barriers in phase space

A material line $t = F tt0($t0) is an evolving curve of

initial conditions $t0 under the flow map F tt0

. As shownin [6], for such a material line to be a locally least-stretching curve over [t0, t1], it must be a hyperbolic,a parabolic or an elliptic line (see Fig. 1).

The initial position $t0 of a hyperbolic material lineis tangent to the vector field "1 at all its points. Suchmaterial lines are compressed by the flow by locallythe largest rate, while repelling all nearby materiallines at an exponential-in-time rate. The classic exam-ple of a hyperbolic material lines is the unstable man-ifold of a saddle-type fixed point.

A parabolic material line is an open material curvewhose initial position $t0 is tangent to one of the di-rections of locally largest shear. At each point of thephase space, the two directions of locally largest shearare given by

%± =& (

!2(!1 + (

!2"1 ±

& (!1(

!1 + (!2

"2, (5)

as derived in [6]. Parabolic material lines still repelmost nearby material lines (except for those parallel tothem), but only at a rate that is linear in time. Classicexamples of parabolic material lines in fluid mechan-ics are the parallel trajectories of a steady shear flow.

Finally, an elliptic material line is a closed curvewhose initial position $t0 is tangent to one of the twodirections of locally largest shear given in (5). As aresult, elliptic lines also repel nearby, nonparallel ma-terial lines at a linear rate, but they also enclose aconnected region. Classic examples of elliptic mate-rial lines are closed trajectories of a steady, circularshear flow, such as a vortex.

Initial positions of hyperbolic material lines are,by definition, strainlines, i.e., trajectories of the au-tonomous differential equation

r ) = "1(r), r ! U " R2, (6)

where r : [0,&] #% U is the parametrization of thestrainline by arclength. A hyperbolic barrier is then astrainline that is locally the closest to least-stretchinggeodesics of the CG tensor, with the latter viewed as ametric tensor on the domain U of the phase space. Thepointwise closeness of strainlines to least-stretchinggeodesics can be computed in terms of the invariantsof the CG strain tensor. Specifically, the C2 distance


Fig. 1 The three types oftransport barrier intwo-dimensional flows(Color figure online)

(difference of tangents plus difference of curvatures)of a strainline from the least-stretching geodesic of Ct

t0

through a point x0 is given by the geodesic strain de-viation

d"1g (x0) = |*+!2, "2, + 2!2'1|

2!32

, (7)

with '1(x0) denoting the curvature of the strainlinethrough x0 [6]. A hyperbolic barrier is a compactstrainline segment on which d

"1g is pointwise below a

small threshold value, and whose averaged d"1g value is

locally minimal relative to all neighboring strainlines.Similarly, initial positions of parabolic and elliptic

material lines are, by definition, shearlines, i.e., trajec-

tories of the autonomous differential equation

r ) = %±(r), r ! U " R2. (8)

A parabolic barrier is an open shearline that is close toleast-stretching geodesics of the CG tensor. The point-wise C2-closeness of shearlines to least-stretchinggeodesics is given by the geodesic shear deviation

d%±g (x0) =

(1 + !2 $ (

!1(1 + !2

+####

*+!2, "1,2!2

(1 + !2

- *+!2, "2,((

1 + !23 $ (

!25)

2!32(

1 + !23

####


- '1[(!2

5 + (1 $ !22)

(1 + !2]

!22(

1 + !2

+ '2(1 + !2

, (9)

with '2(x0) denoting the curvature of the "2 vectorfield at the point x0 [6]. The geodesic shear devia-tion should pointwise be below a small threshold levelfor an open shearline to qualify as a parabolic barrier.Similarly, a closed shearline is an elliptic barrier ifits pointwise geodesic shear deviation is smaller thansmall threshold level.

For the purposes of the present discussion, we calla mechanical system of the form (1) conservative ifit has vanishing divergence, i.e., + · v(x, t) = 0, with+ referring to differentiation with respect to x. Thisproperty implies that flow map of (1) conserves phase-space area for all times [13].

While a typical material line in such a conserva-tive system will still stretch and deform significantlyover time, the length of a shearline will always bepreserved under the area-preserving flow map F

t1t0

(cf.[6]). An elliptic barrier in a conservative system will,therefore, have the same enclosed area and arclengthat the initial time t0 and at the final time t1. These twoconservation properties imply that an elliptic barrier ina non-autonomous conservative system may only un-dergo translation, rotation and some slight deforma-tion, but will otherwise preserve its overall shape. Asa result, the interior of an elliptic barrier will not mixwith the rest of the phase-space, making elliptic barri-ers the ideal generalized KAM curves in aperiodicallyforced conservative mechanical systems.

2.2 Computation of invariant sets as transportbarriers

In this section, we describe numerical algorithms forthe extraction of hyperbolic and elliptic barriers in aone-degree-of-freedom mechanical system with gen-eral time dependence. Parabolic barriers can in princi-ple also exist in mechanical systems, but they do notarise in the simple examples we study below. In con-trast, parabolic barriers are more common in geophys-ical fluid mechanics where they typically represent un-steady shear jets.

Our numerical algorithms require a careful compu-tation of the CG tensor. In most mechanical systems,

trajectories separate rapidly, resulting in an exponen-tial growth in the entries of the CG tensor. This growthnecessitates the use of a well-resolved grid, as well asthe deployment of high-end integrators in solving forthe trajectories of (1) starting form this grid. Furthercomputational challenges arise from the handling ofthe unavoidable orientational discontinuities and iso-lated singularities of the eigenvector fields "1 and "2.The reader is referred to Farazmand & Haller [10] fora detailed treatment of these computational aspects.

As a zeroth step, we fix a sufficiently dense grid G0of initial conditions in the phase-space U , then advectthe grid points from time t0 to time t1 under system (1).This gives a numerical representation of the flow mapF

t1t0

over the grid G0. The CG tensor field Ct1t0

is thenobtained by definition (3) from F

t1t0

. In computing thegradient DF

t1t0

, we use careful finite differencing overan auxiliary grid, as described in [10].

Since, at each point x0 ! G0, the tensor Ct1t0

(x0) is atwo-by-two matrix, computing its eigenvalues {!1,!2}and eigenvectors {"1, "2} is straightforward. With theCG eigenvalues and eigenvectors at hand, we locatethe hyperbolic barriers using the following algorithm.

Algorithm 1 (Locating hyperbolic barriers)

1. Fix a small positive parameter ("1 as the admissi-ble upper bound for the pointwise geodesic straindeviation of hyperbolic transport barriers.

2. Calculate strainlines by solving the ODE (6) nu-merically, with linear interpolation of the strainvector field between grid points. Truncate strain-lines to compact segments whose pointwise geo-desic strain deviation is below ("1 .

3. Locate hyperbolic barriers as strainline segments$t0 with locally minimal relative stretching, i.e.,strainline segments that locally minimize the func-tion

q($t0) = l($t1)

l($t0). (10)

Here l($t0) and l($t1) denote the length of thestrainline $t0 and the length of its advected image$t1 , respectively.

Computing the relative stretching (10) of a strainline$t0 , in principle, requires advecting the strainline totime t1. However, as shown in [6], the length of the ad-vected image satisfies l($t1) =

'$t0

(!1 ds, where the


Fig. 2 Locating closed shearlines using a Poincaré section ofthe shear vector field. Closed shearlines pass through the fixedpoints of the corresponding Poincaré map

integration is carried out along the strainline $t0 . Thisrenders the strainline advection unnecessary.

Numerical experiments have shown that a directcomputation of "1 is usually less accurate than thatof "2 due to the attracting nature of strongest eigen-vector of the CG tensor [10]. For this reason, comput-ing "1 as an orthogonal rotation of "2 is preferable.Moreover, it has been shown [12] that strainlines canbe computed more accurately as advected images ofstretchlines, i.e. curves that are everywhere tangent tothe second eigenvector of the backward-time CG ten-sor C

t0t1

. In the present paper, this approach is taken forcomputing the strainlines.

Computing elliptic barriers amounts to finding limitcycles of the ODE (8). To this end, we follow the ap-proach used in [6, 12] by first identifying candidateregions for shear limit cycles visually, then calculat-ing the Poincaré map on a one-dimensional sectiontransverse to the flow within the candidate region (seeFig. 2). Hyperbolic fixed points of this map can be lo-cated by iteration, marking limit cycles of the shearvector field (see [12] for more detail).

This process is used in the following algorithm tolocate elliptic barriers.

Algorithm 2 (Locating elliptic barriers)

1. Fix a small positive parameter )%± as the admis-sible upper bound for the average geodesic sheardeviation of elliptic transport barriers.

2. Visually locate the regions where closed shearlinesmay exist. Construct a sufficiently dense Poincarémap, as discussed above. Locate the fixed points ofthe Poincaré map by iteration.

3. Compute the full closed shearlines emanating fromthe fixed points of the Poincaré map.

4. Locate elliptic barriers as closed shearlines whoseaverage geodesic deviation *d%±

g , satisfies *d%±g , <

(%± .

In the next section, we use the above algorithms forlocating invariant sets in simple forced and dampednonlinear oscillators.

3 Results

We demonstrate the implementation of the geodesictheory of transport barriers on four Duffing-type os-cillators. As a proof of concept, in the first two exam-ples (Sect. 3.1), we consider periodically forced Duff-ing oscillators for which we can explicitly verify ourresults using an appropriately defined Poincaré map.

The next two examples deal with aperiodicallyforced Duffing oscillators (Sect. 3.2). In these exam-ples, despite the absence of a Poincaré map, we stillobtain the key invariant sets as hyperbolic and ellipticbarriers.

To implement Algorithms 1 and 2 in the forthcom-ing examples, the CG tensor is computed over a uni-form grid G0 of 1000 . 1000 points. A fourth orderRunge–Kutta method with variable step-size (ODE45in MATLAB) is used to solve the first-order ODEs (1),(6), and (8) numerically. The absolute and relative tol-erances of the ODE solver are set equal to 10$4 and10$6, respectively. Off the grid points, the strain andshear vector fields are obtained by bilinear interpola-tion.

In each case, the Poincaré map of Algorithm 2 isapproximated by 500 points along the Poincaré sec-tion. The zeros of the map are located by a standardsecant method.

3.1 Proof of concept: periodically forced Duffingoscillator

Case 1: Pure periodic forcing, no damping Considerthe periodically forced Duffing oscillator

x1 = x2,


Fig. 3 Five hundred iterations of the Poincaré map for the periodically forced Duffing oscillator. Two elliptic regions of the phase-spacefilled by KAM tori are shown

x2 = x1 $ x31 + ( cos(t).

For ( = 0, the system is integrable with one hyper-bolic fixed point at (0,0), and two elliptic fixed points(1,0) and ($1,0), respectively. As is well known,there are two homoclinic orbits connected to the hy-perbolic fixed point, each enclosing an elliptic fixedpoint, which is in turn surrounded by periodic orbits.These periodic orbits appear as closed invariant curvesfor the Poincaré map P := F 2*

0 . The fixed points ofthe flow are also fixed points of P .

For 0 < ( / 1, the Kolmogorov–Arnold–Moser(KAM) theory [13] guarantees the survival of mostclosed invariant sets for P . Figure 3 shows these sur-viving invariant sets (KAM curves) of P obtained for( = 0.08. For the KAM curves to appear continuous-looking, nearly 500 iterations of P were needed, re-quiring the advection of initial conditions up to timet = 1000* . The stochastic region surrounding theKAM curves is due to chaotic dynamics arising fromthe transverse intersections of the stable and unstablemanifold of the perturbed hyperbolic fixed point of P .

The surviving KAM curves are well-known, clas-sic examples of transport barriers. We would like tocapture as many of them as possible as elliptic barriersusing the geodesic transport theory described in pre-vious sections. Note that not all KAM curves are ex-pected to prevail as locally least-stretching curves for agiven choice of the observational time interval [t0, t1];some of these curves may take longer to prevail due totheir shape and shearing properties.

We use the elliptic barrier extraction algorithm ofSect. 2.2 with )%± = 0.7. Figure 4 shows the result-ing shearlines in the KAM regions, with the closedones marked by red. Note that these shearlines wereobtained from the CG tensor computed over the timeinterval [0,8*], spanning just four iterations of thePoincaré map. Despite this low number of iterations,the highlighted elliptic barriers are practically indis-tinguishable form the KAM curves obtained from 500iterations.

Figure 5 shows the convergence of an elliptic bar-rier to a KAM curve as the integration time T = t1 $ t0increases. Note how the average geodesic deviation*d%±

g , decreases with increasing T , indicating decreas-ing deviation from nearby Cauchy–Green geodesics.

Remarkably, constructing these elliptic barriersrequires significantly shorter integration time (onlyfour forcing periods) in comparison to visualizationthrough the Poincaré map, which required 500 forc-ing periods to reveal KAM curves as continuous ob-jects. Clearly, the overall computational cost for con-structing elliptic barriers still comes out to be higher,since the CG tensor needs to be constructed on a rel-atively dense grid G0, as discussed in Sect. 2.2. Thishigh computational cost will be justified, however, inthe case of aperiodic forcing (Sect. 3.2), where noPoincaré map is available.

In the context of one-degree-of-freedom mechan-ical systems, the outermost elliptic barrier marks theboundary between regions of chaotic dynamics and re-gions of oscillations that are regular on a macroscopic


Fig. 4 Shearlines (black) of the periodically forced Duffing oscillator computed at t0 = 0, with integration time T = 8* . The extractedelliptic barriers with *d%±

g , ' )%± = 0.7 are shown in red (Color figure online)

Fig. 5 Convergence of an elliptic barrier (red) to a KAM curve(black) as the integration time T = t1 $ t0 increases. The grad-ually decreasing average geodesic deviation *d%±

g , confirms the

convergence to Cauchy–Green geodesics that closely shadowthe underlying KAM torus (Color figure online)

scale. To demonstrate this sharp dividing property ofelliptic barriers, we show the evolution of system (12)from three initial states, two of which are inside the el-liptic region and one of which is outside (Fig. 6a). Thesystem exhibits rapid changes in its state when startedfrom outside the elliptic region. In contrast, more regu-lar behavior is observed for trajectories starting insidethe elliptic region. This behavior is further depicted inFig. 7, which shows the evolution of the x1-coordinateof the trajectories as a function of time.

Case 2: Periodic forcing and damping Consider nowthe damped-forced Duffing oscillator

x1 = x2,(11)

x2 = x1 $ x31 $ +x2 + ( cos(t),

with + = 0.15 and ( = 0.3. This system is known tohave a chaotic attractor that appears as an invariant setof the Poincaré map P = F 2*

0 (see, e.g., [1]). Here,we show that the attractor can be very closely ap-proximated by hyperbolic barriers computed via Al-gorithm 1.

Figure 8a shows strainlines computed backward intime with t0 = 0 and integration time T = t1 $ t0 =$8* . The strainline with globally minimal relativestretching (10) is shown in Fig. 8b. Black dots markthe points where the geodesic deviation d

"1g exceeds

the admissible upper bound ("1 = 10$3. At its tail(covered by black dots), the strainline persistently de-viates from CG geodesics, and hence should be trun-


Fig. 6 (a) The outermost elliptic barrier (black curve) and threeinitial conditions: Two inside the elliptic barrier (blue and green)and one outside the elliptic barrier (red). (b) The correspondingtrajectories are shown in the extended phase space of (x1, x2, t).The closed black curves mark the elliptic barrier at t0 = 0 andt1 = 16* (Color figure online)

cated. The resulting hyperbolic barrier, as a finite-timeapproximation to the chaotic attractor, is shown inFig. 8c.

The approximate location of the attractor can alsobe revealed by applying the Poincaré map to a fewinitial conditions (tracers) released from the basin ofattraction. For long enough advection time, the ini-tial conditions converge to the attractor highlightingits position (see Figs. 9a and 9b). In Fig. 9c, the hy-perbolic barrier is superimposed on the advected trac-ers showing close agreement between the two. Fig-ure 9d shows the tracers advected for a longer time(T = 40* ) together with the hyperbolic barrier; thetwo virtually coincide. Note that the hyperbolic barrieris a smooth, parametrized curve (computed as a trajec-

Fig. 7 The x1-coordinate of the trajectories of Fig. 6

tory of (6)), while the tracers form a set of scatteredpoints.

3.2 The aperiodically forced Duffing oscillator

In the next two examples, we study aperiodicallyforced Duffing oscillators. In the presence of aperi-odic forcing, the Poincaré map P is no longer definedas the system lacks any recurrent behavior. However,KAM-type curves (i.e., closed curves, resisting signif-icant deformation) and generalized stable and unstablemanifolds (i.e., most repelling and attracting materiallines) exist in the phase-space and determine the over-all dynamics of the system.

Case 1: Purely aperiodic forcing, no damping Con-sider the Duffing oscillator

x1 = x2,(12)

x2 = x1 $ x31 + f (t),

where f (t) is an aperiodic forcing function ob-tained from a chaotic one-dimensional map (seeFig. 10).

While KAM theory is no longer applicable, onemay still expect KAM-type barriers to survive forsmall forcing amplitudes. Such barriers would nolonger be repeating themselves periodically in the ex-tended phase space. Instead, a generalized KAM bar-rier is expected to be an invariant cylinder, with crosssections showing only minor deformation. The exis-


Fig. 8 Construction of the attractor of the damped-forced Duffing oscillator as a hyperbolic transport barrier (Color figure online)

tence of such structures can, however, be no longerstudied via Poincaré maps.

Figure 11 confirms that generalized KAM-typecurves, obtained as elliptic barriers, do exist in thisproblem. These barriers are computed over the timeinterval [0,4*] (i.e. t0 = 0 and t1 = t0 + T = 4* ). Asdiscussed in Sect. 2.1, the arclength of an elliptic bar-rier at the initial time t0 is equal to the arclength ofits advected image under the flow map F

t1t0

at the finaltime t1. This arclength preservation is illustrated nu-merically in Fig. 12, which shows the relative stretch-

ing,

+&(t) = &($t ) $ &($0)

&($0)(13)

of the time-t image $t of an elliptic barrier $0, with &

referring to the arclength of the curve. Ideally, the rel-ative stretching of each elliptic barrier should be zeroat time t1 = 4* , i.e. +&(4*) = 0. Instead, we find thatthe relative stretching +&(4*) of the computed ellip-tic barriers is at most 1.5 %. This deviation from zeroarises from numerical errors in the computation of the


Fig. 9 (a) Attractor of system (11) obtained from four iteratesof the Poincaré map. (b) Attractor obtained from 20 iterates ofthe Poincaré map. (c) Attractor computed as a hyperbolic barrier(red), compared with the Poincaré map (blue) computed for thesame integration time (four iterates). (d) Comparison of attrac-

tor computed as a hyperbolic barrier (red) with the one obtainedfrom 20 iteration of the Poincaré map (blue). The integrationtime for locating the hyperbolic barrier is T = t1 $ t0 = $8*(Color figure online)

Fig. 10 Chaotic forcing function f (t) for (12)

CG strain tensor Ct1t0

, which in turn causes small inac-curacies in the computation of closed shearlines.

As noted earlier, the small relative stretching andthe conservation of enclosed area for an elliptic bar-rier in incompressible flow only allows for small de-formations when the barrier is advected in time. Thisis illustrated in Fig. 13, which shows the blue ellipticbarrier of Fig. 11b in the extended phase-space. Eachconstant-time slice of the figure is the advected imageof the barrier.

Finally, we point out that the stability of the trajec-tories inside elliptic barriers show a similar trend asin the case of the periodically forced Duffing equation(Figs. 6 and 7). Namely, perturbations inside the ellip-


Fig. 11 Closed shearlines for (12) computed in two elliptic regions. The figure shows the shearlines at time t0 = 0. The integrationtime is T = 4*

Fig. 12 The relative stretching +&(t) . 100 of closed shear-lines of Fig. 11. The colors correspond to those of Fig. 11. Bytheir arclength preservation property, the advected elliptic bar-riers must theoretically have the same arclength at times t0 = 0

and t1 = 4* . The numerical error in arclength conservation issmall overall, but more noticeable for oscillations with large am-plitudes (green and red curves of the right panel) (Color figureonline)

Fig. 13 GeneralizedKAM-type cylinder in theextended phase space of theaperiodically forcedundamped Duffingoscillator. The cylinder isobtained by advection ofthe closed shearline shownin blue in Fig. 11(b)


Fig. 14 (a) Strainlines computed in backward time from t0 = 30 to t1 = 10. (b) The resulting hyperbolic barrier extracted withmaximum admissible geodesic deviation of ("1 = 10$5

tic regions remain small while they grow significantlyinside the hyperbolic regions.

Case 2: Aperiodic forcing with damping In this fi-nal example, we consider the aperiodically forced,damped Duffing oscillator

x1 = x2,(14)

x2 = x1 $ x31 $ +x2 + f (t),

with damping coefficient + = 0.15. The forcing func-tion f (t) is similar to that of Case I above, but with anamplitude twice as large. As a result, none of the ellip-tic barriers survive even in the absence of damping.

Again, because of the aperiodic forcing, the behav-ior of this system is a priori unknown and cannot beexplored using Poincaré maps. In order to investigatethe existence of an attractor, strainlines (Fig. 14a) arecomputed from the backward-time CG strain tensorC

t1t0

with t0 = 30 and t1 = 10. The strainline with mini-mum relative stretching (10) is then extracted. The partof this strainline satisfying d

"1g < ("1 is considered as

the most influential hyperbolic barrier (Fig. 14b). Theadmissible upper bound ("1 for the geodesic deviationis fixed as 10$5.

In order to confirm the existence of the extracted at-tractor, we advect tracer particles in forward time, firstfrom time t1 = 10 to time t0 = 30, then from t1 = 0to time t0 = 30. Because of the fast-varying dynam-ics and weak dissipation, a relatively long advection

time is required for the tracers to converge to the at-tractor. Figure 15 shows the evolution of tracers over[t1, t0]. Note that the attractor inferred from the trac-ers is less well pronounced than the hyperbolic barrierextracted over the same length of time. This showsa clear advantage for geodesic transport theory oversimple numerical experiments with tracer advection.For a longer integration time from t0 = 0 to t = 30,the tracers eventually converge to the hyperbolic bar-rier.

Repelling hyperbolic barriers can be computedsimilarly using forward-time computations. Figure 16shows both hyperbolic barriers (stable and unstablemanifolds) at time t0 = 30. The repelling barrier iscomputed from the CG strain tensor Ct1

t0with t0 = 30

and t1 = 50.

4 Summary and conclusions

We have shown how the recently developed geodesictheory of transport barriers [6] in fluid flows can beadapted to compute finite-time invariant sets in one-degree-of-freedom mechanical systems with generalforcing. Specifically, in the presence of general timedependence, temporally aperiodic stable- and unsta-ble manifolds, attractors, as well as generalized KAMtori can be located as hyperbolic and elliptic barri-ers, respectively. The hyperbolic barriers are computedas distinguished strainlines, i.e. material lines along


Fig. 15 (a) Tracers advected over the time interval from t1 = 10to t0 = 30. (b) Tracers advected over a longer time interval fromt1 = 0 to t0 = 30. (c) The hyperbolic barrier (red) superimposed

on the tracers advected for the same time interval. (d) Compari-son of the hyperbolic barrier (red) with the tracers advected forthe longer time interval (Color figure online)

which the Lagrangian strain is locally maximized. Theelliptic barriers, on the other hand, appear as dis-tinguished shearlines, i.e. material lines along whichthe Lagrangian shear is locally maximized. The bar-riers are finally identified as strainlines and shearlinesthat are most closely approximated by least-stretchinggeodesics of the metric induced by Cauchy–Greenstrain tensor.

We have used four simple examples for illustra-tion. First, as benchmarks, we considered periodicallyforced Duffing equations for which stable and unsta-ble manifolds, attractors and KAM curves can alsobe obtained as invariant sets of an appropriately de-fined Poincaré map. We have shown that elliptic barri-

ers, computed as closed shearlines, coincide with theKAM curves. Also, stable and unstable manifolds, aswell as attractors, can be recovered as hyperbolic bar-riers. More precisely, as the integration time T = t1 $t0 of the Cauchy–Green strain tensor Ct1

t0increases,

the elliptic barriers in the periodically forced Duff-ing equations converge to KAM curves. Similarly, thechaotic attractor of the periodically forced and dampedDuffing equation is more and more closely delineatedby a hyperbolic barrier computed from the backward-time Cauchy–Green strain tensor Ct0

t1for increasing

T = t0 $ t1 where t0 > t1.In the second set of examples, we have computed

similar structures for an aperiodically forced Duff-


Fig. 16 Attracting (blue) and repelling (red) barriers at t0 = 30extracted from backward-time and forward-time computations,respectively (Color figure online)

ing oscillator with and without damping. In this case,Poincaré maps are no longer well-defined for the sys-tem, and hence we had to advect tracer particles toverify the predictions of the geodesic theory. Notably,tracer advection takes longer time to reveal the struc-tures in full detail than the geodesic theory does. Also,tracer advection is only affective as a visualization toolif it relies on a small number of particles, which inturn assumes that one already roughly knows the loca-tion of the invariant set to be visualized. Finally, un-like scattered tracer points, geodesic barriers are re-covered as parametrized smooth curves that provide asolid foundation for further analysis or highly accurateadvection.

In our examples, elliptic barriers have shown them-selves as borders of subsets of the phase-space thatbarely deform over time. In fact, as illustrated in Fig. 6,outermost elliptic barriers define the boundary be-tween chaotic and regular dynamics. Trajectories initi-ated inside elliptic barriers remain confined and robustwith respect to small perturbations. We believe thatthis property could be exploited for stabilizing me-chanical systems with general time dependence. Forinstance, formulating an optimal control problem forgenerating elliptic behavior in a desired part of thephase-space is a possible approach.

Undoubtedly, the efficient and accurate computa-tion of invariant sets as geodesic transport barriers re-quires dedicated computational resources. Smart algo-rithms reducing the computational cost are clearly ofinterest. Parallel programming (both at CPU and GPU

levels) has previously been employed for Lagrangiancoherent structure calculations and should be useful inthe present setting as well (see e.g. [14]). Other adap-tive techniques are also available to lower the numer-ical cost by reducing the computations to regions ofinterest (see e.g. [15, 16]).

In principle, invariant sets in higher-degree-of-freedom mechanical systems could also be capturedby similar techniques as locally least-stretching sur-faces. The development of the underlying multi-dimensional theory and computational platform, how-ever, is still under way.

Acknowledgements M.F. would like to thank the Departmentof Mathematics at McGill University where this research waspartially carried out. G.H. acknowledges partial support by theCanadian NSERC under grant 401839-11.

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