Experimental validation of phase space conduits of transition between potential wells

Shane D. Ross,1 Amir E. BozorgMagham,1 Shibabrat Naik,1, ∗ and Lawrence N. Virgin2

1Engineering Mechanics Program, Virginia Tech, Blacksburg, Virginia 24061, USA2Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708, USA

A phase space boundary between transition and non-transition, similar to those observed in chem-ical reaction dynamics, is shown experimentally in a macroscopic system. We present a validationof the phase space flux across rank one saddles connecting adjacent potential wells and confirm theunderlying phase space conduits that mediate the transition. Experimental regions of transition arefound to agree with the theory to within 1%, suggesting the robustness of phase space conduitsof transition in a broad array of two or more degree of freedom experimental systems, despite thepresence of small dissipation.

Introduction.— Prediction of transition events andthe determination of governing criteria has significancein many physical, chemical, and engineering systemswhere rank-1 saddles are present. To name but a few,ionization of a hydrogen atom under electromagneticfield in atomic physics [1], transport of defects in solidstate and semiconductor physics [2], isomerization ofclusters [3], reaction rates in chemical physics [4, 5],buckling modes in structural mechanics [6, 7], ship mo-tion and capsize [8–10], escape and recapture of cometsand asteroids in celestial mechanics [11–13], and es-cape into inflation or re-collapse to singularity in cos-mology [14]. The theoretical criteria of transition andits agreement with laboratory experiment have beenshown for 1 degree of freedom (DOF) systems [15–17]. Detailed experimental validation of the geomet-rical framework for predicting transition in higher di-mensional phase space (> 4, that is for 2 or moreDOF systems) is still lacking. The geometric frameworkof phase space conduits in such systems, termed tubedynamics[11, 12, 18, 19], has not before been demon-strated in a laboratory experiment. It is noted thatsimilar notions of transition were developed for ide-alized microscopic systems, particularly chemical re-actions [1, 20–22] under the names of transition stateand reactive island theory. However, investigations ofthe predicted phase space conduits of transition be-tween wells in multi-well system have stayed withinthe confines of numerical simulations. In this Letter,we present a direct experimental validation of the ac-curacy of the phase space conduits, as well as the tran-sition fraction obtained as a function of energy, in a 4

dimensional phase space using a controlled laboratoryexperiment of a macroscopic system.

In [23–25], experimental validation of global charac-teristics of 1 DOF Hamiltonian dynamics of scalar trans-port has been accomplished using direct measurementof the Poincare stroboscopic sections using dye visu-alization of the fluid flow. In [23, 24], the experimen-tal and computational results of chaotic mixing werecompared by measuring the observed and simulateddistribution of particles, thus confirming the theory of

chaotic transport in Hamiltonian systems for such sys-tems. Our objective is to validate theoretical predictionsof transition between potential wells in an exemplar ex-perimental 2 DOF system, where qualitatively differ-ent global dynamics can occur. Our setup consists ofa mass rolling on a multi-well surface that is represen-tative of potential energy underlying systems that ex-hibit transition/escape behavior. The archetypal poten-tial energy surface chosen has implications in transition,escape, and recapture phenomena in many of the afore-mentioned physical systems. In some of these systems,transition in the conservative case has been understoodin terms of trajectories of a given energy crossing a hy-persurface or transition state (bounded by a normallyhyperbolic invariant manifold of geometry S2N−3 in NDOF). In this Letter, for N = 2, trajectories pass inside atube-like separatrix, which has the advantage of accom-modating the inclusion of non-conservative forces suchas stochasticity and damping [7, 10]. Based on the illus-trative nature of our laboratory experiment of a 2 DOFmechanical system, and the generality of the frameworkto higher degrees of freedom [5, 19, 26], we envision thegeometric approach demonstrated here can apply to ex-periments regarding transition across rank-1 saddles in3 or more DOF systems in many physical contexts.

Experimental setup.— We designed a surface (shownin Fig. 1(b)) that has 4 wells, one in each quadrant, withsaddles connecting the neighboring quadrants. The sur-face has 4 stable and 5 saddle (4 rank-1 and 1 rank-2) equilibrium points. Inter-well first order transitionsare defined as crossing the rank-1 saddles between thewells. On this high-precision machined surface, accu-rate to within 0.003 mm and made using stock poly-carbonate, a small rubber-coated spherical steel massreleased from rest can roll without slipping under theinfluence of gravity. The mass is released from differentlocations on the machined surface to generate experi-mental trajectories. The mass is tracked using a Prosil-ica GC640 digital camera mounted on a rigid frame at-tached to the surface as shown in Fig. 1(a), with a pixelresolution of about 0.16 cm. The tracking is done bycapturing black and white images at 50 Hz, and cal-

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Tracking camera

Machined surface

(b)(a)

30 cm 30 cm

12 0 12x (cm)

12

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12

y(c

m)

U2 U1

(c)

3

6

9

12

H(x,y

)(c

m)

12

0

12

y(c

m)

∆E< 0 (d)

12 0 12x (cm)

12

0

12y

(cm

)∆E> 0 (e)

FIG. 1. (a), (b) Experimental apparatus showing the machined sur-face, tracking camera, and the rubber coated steel ball. (c) A typi-cal experimental trajectory, shown in white, on the potential energysurface where the contours denote isoheights of the surface. Thisinstance of the trajectory was traced by the ball released from rest,marked by a red cross. (d) and (e) Show energetically accessibleregion projected on the configuration space in white for ∆E < 0:∆E = −100 (cm/s)2 and ∆E > 0: ∆E = 100 (cm/s)2, respectively.

culating the coordinates of the mass’s geometrical cen-ter. We recorded 120 experimental trajectories of about10 seconds long, only using data after waiting at leastthe Lyapunov time of ≈ 0.4 seconds [27] ensuring thatthe trajectories were well-mixed in the phase space. Toanalyze the fraction of trajectories that leave/enter awell, we obtain approximately 4000 intersections witha Poincare surface-of-section, U1, shown as a black line,for the analyzed range of energy. One such trajectoryis shown in white in Fig. 1(c). These intersections arethen sorted according to energy. The intersection pointson U1 are classified as a transition from quadrant 1 to 2

if the trajectory, followed forward in time, leaves quad-rant 1. Four hundred transition events were recorded.

Theory.— The initial mathematical model of the tran-sition behavior of a rolling ball on the surface, H(x, y),shown in Fig. 1, is described in [27]. The equa-tions of motion are obtained from the Hamiltonian,H(x, y, px, py) = T(x, y, px, py) + V(x, y), where massfactors out and where the kinetic energy (translationaland rotational for a ball rolling without slipping) is,

T =514

(1 + H2y)p2

x + (1 + H2x)p2

y − 2Hx Hy px py

1 + H2x + H2

y(1)

where H(·) = ∂H∂(·) . The potential energy is V(x, y) =

gH(x, y) where g = 981 cm/s2 is the gravitational ac-celeration and the height function is

H = α(x2 + y2)− β

(√x2 + γ +

√y2 + γ

)− ξxy + H0. (2)

transition tube from quadrant 1 to 2 periodic orbit for excess energy, ∆E(a) (b) W s

1−2,U1

FIG. 2. (a) For a fixed excess energy, ∆E, above the critical valueEe, the permissible regions (in white) are connected by a bottleneckaround the saddle equilibria. All motion from the well in quadrant 1

to quadrant 2 must occur through the interior of a stable manifold as-sociated with an unstable periodic orbit in the bottleneck between thequadrants; seen as a 2D configuration space projection of the 3D en-ergy manifold. We show the stable manifold (cyan) and the periodicorbit (black) for an excess energy of ∆E = 100 (cm/s)2. A trajectorycrossing the U−1 section inside the stable manifold will transition (red)into the quadrant 2 well, while one that is outside stays (blue) insidequadrant 1. The zoomed-in inset in the figure shows the structure ofthe manifold and how precisely the separatrix divides transition andnon-transition trajectories. (b) In the (x, y, vy) projection, the phasespace conduit for imminent transition from quadrant 1 to 2 is the sta-ble manifold (cyan) of geometry R1 × S1 (i.e., a cylinder). The sameexample trajectories (red and blue) as in (a) that exhibit transition andnon-transition behavior starting inside and outside the stable mani-fold, respectively, are shown in the 3D projection and projected onthe (x, y) configuration space. A movie of a nested sequence of thesemanifolds can be found here.

This is the analytical function for the machined sur-face shown in Fig. 1(b) and the isoheights shownin Fig. 1(c). We use parameter values (α, β, γ, ξ, H0) =(0.07, 1.017, 15.103, 0.00656, 12.065) in the appropriateunits [28].

Let M(E) be the energy manifold in the 4D phasespace given by setting the total energy equal to aconstant, E, i.e., M(E) = {(x, y, px, py) ⊂ R4 |H(x, y, px, py) = E}. The projection of the energy man-ifold onto the (x, y) configuration space is the region ofenergetically possible motion for a mass with energy E,and is given by M(E) = {(x, y) | V(x, y) ≤ E}. Theboundary of M(E) is the zero velocity curve and is de-fined as the locus of points in the (x, y) plane where thekinetic energy is zero. The mass is only able to move onthe side of the curve where the kinetic energy is posi-tive, shown as white regions in Fig. 1(d) and (e). Thecritical energy for transition, Ee, is the energy of therank-1 saddle points in each bottleneck, which are allequal. This energy divides the global behavior of themass into two cases, according to the sign of the excessenergy above the saddle, ∆E = E− Ee:

Case 1: ∆E < 0 —the mass is safe against transi-tion and remains inside the starting well since potentialwells are not energetically connected (Fig. 1(d)).

Case 2: ∆E > 0 —the mass can transition by cross-ing the bottlenecks that open up around the sad-dle points, permitting transition between the potentialwells (Fig. 1(e) and Fig. 2(a) show this case).

Thus, transition between wells can occur when ∆E >

3

0 and this constitutes a necessary condition. The suffi-cient condition for transition to occur is when a trajec-tory enters a codimension-1 invariant manifold associ-ated with the unstable periodic orbit in the bottleneckas shown by non-transition and transition trajectoriesin Fig.2(a) [18]. In 2 DOF systems, the periodic or-bit residing in the bottleneck has an invariant manifoldwhich is codimension-1 in the energy manifold and hastopology R1 × S1, that is a cylinder or tube [28]. Thisimplies that the transverse intersection of these mani-folds with Poincare surfaces-of-sections, U1 and U2, aretopologically S1, a closed curve [7, 10, 18]. All the tra-jectories transitioning to a different potential well (orhaving just transitioned into the well) are inside a tubemanifold, for example as shown in Fig. 2(b) [18, 19].For every ∆E > 0, the tubes in phase space (or moreprecisely, within M(E)) that lead to transition are thestable (and that lead to entry are the unstable) mani-folds associated with the unstable periodic orbit of en-ergy E. Thus, the mass’s imminent transition betweenadjacent wells can be predicted by considering whereit crosses U1 as shown in Fig. 3, relative to the inter-section of the tube manifold. Furthermore, nested en-ergy manifolds have corresponding nested stable andunstable manifolds that mediate transition. To sim-plify analysis, we focus only on the transition of tra-jectories that intersect U1 in the first quadrant. Thissurface-of-section is best described in polar coordinates(r, θ, pr, pθ); U±1 = {(r, pr) | θ = π

4 , − sign(pθ) = ±1},where + and − denote motion to the right and left ofthe section, respectively [28]. This Hamiltonian flowon U±1 defines a symplectic map with typical featuressuch as KAM tori and chaotic regions, shown in Fig. 3

for two values of excess energy.

Based on these phase space conduits that lead totransition, we would like to calculate what fraction ofthe energetically permissible trajectories will transitionfrom/into a given well. This can be answered in partby calculating the transition rate of trajectories cross-ing the rank-1 saddle in the bottleneck connecting thewells. For computing this rate—surface integral of tra-jectories crossing a bounded surface per unit time—weuse the geometry of the tube manifold cross-section onthe Poincare section. For low excess energy, this com-putation is based on the theory of flux over a rank-1saddle [29], which corresponds to the action integralaround the periodic orbit at energy ∆E. By the Poincareintegral invariant [30], this action is preserved for sym-plectic maps, such as P± : U±1 → U±1 , and is equivalentto computing the area of the tube manifold’s intersec-tion with the surface-of-section. The transition fractionat each energy, ptrans(∆E), is calculated by the fractionof energetically permissible trajectories at a given ex-cess energy, ∆E, that will transition. This is given bythe ratio of the cross-sections on U1 of the tube to theenergy surface. The transition area, to leading order in∆E [29], is given by Atrans = Tpo∆E, where Tpo = 2π/ωis the period of the periodic orbits of small energy in the

0 4 8 12 16r (cm)

80

40

0

40

80

pr(c

m/s

)

U −1∆E= 100 (cm/s)2(a)

W s1− 2,U−

1int(W s

1− 2)

0 4 8 12 16r (cm)

80

40

0

40

80

pr(c

m/s

)

U −1∆E= 500 (cm/s)2(b)

W s1− 2,U−

1int(W s

1− 2)

FIG. 3. Poincare section, P− : U−1 → U−1 , of trajectories where U−1 :={(r, pr)| θ = π/4, pθ > 0}, at excess energy (a) ∆E = 100 (cm/s)2

and (b) ∆E = 500 (cm/s)2. The blue curves with cyan interior denotethe intersection of the tube manifold (stable) associated with the un-stable periodic orbit with U−1 . It is to be noted that these manifoldsact as a boundary between transition and non-transition trajectories,and may include KAM tori spanning more than one well. The inte-rior of the manifolds, int(·), denote the region of imminent transitionto the quadrant 2 from quadrant 1. A movie showing the Poincaresection for a range of excess energy can be found here.

bottleneck, where ω is the imaginary part of the com-plex conjugate pair of eigenvalues resulting from thelinearization about the saddle equilibrium point [29].The area of the energy surface projection on U1, to lead-ing order in ∆E > 0, is AE = A0 + τ∆E, where,

A0 =2∫ rmax

rmin

√145(Ee − gH(r))(1 + 4H2

r (r)) dr, (3)

and τ =∫ rmax

rmin

√145

(1 + 4H2r (r))

(Ee − gH(r))dr. (4)

The transition fraction, under the well-mixed assump-tion mentioned earlier, is given in 2 DOF by

ptrans =Atrans

AE=

Tpo

A0∆E(

1− τ

A0∆E +O(∆E2)

). (5)

For small positive excess energy, the predicted growthrate is Tpo/A0 ≈ 0.87× 10−3 (s/cm)2. For larger valuesof ∆E, the cross-sectional areas are computed numeri-cally using Green’s theorem, see Fig. 4(b).

As with any physical experiment there is dissipationpresent, but over the time-scale of interest, the motionapproximately conserves energy. We compare δE, thetypical energy lost during a transition, with the typi-cal excess energy, ∆E > 0, when transitions are possi-ble. The time-scale of interest, ttrans, corresponds to thetime between crossing U1 and transitioning across thesaddle into a neighboring well. The energy loss overttrans in terms of the measured damping ratio ζ ≈ 0.025is δE ≈ πζv2(∆E) where the squared-velocity v2(∆E) isapproximated through the total energy. For our exper-imental trajectories, all starting at ∆E > 1000 (cm/s)2,we find δE/∆E � 1, suggesting the appropriateness ofthe assumption of short-time conservative dynamics tostudy transition between wells [7, 10].

Results.— For each of the recorded trajectories, wedetect intersections with U1 and determine the instan-taneous ∆E. Grouping intersection points by energy,for example Fig. 4(a), we get an experimental transition

4

12 16-80

-40

0

40

80

0 4 8

[40, 140] (b)

FIG. 4. (a) On the Poincare section, U−1 , we show a narrow rangeof energy (∆E ∈ (40, 140) (cm/s)2) and label intersecting trajecto-ries as no transition (black) and imminent transition (red) to quad-rant 2, based on their measured behavior. The stable invariant man-ifold associated with the bottleneck periodic orbit at excess energy,∆E = 140 (cm/s)2, intersects the Poincare section, U−1 , along theblue curve. Its interior is shown in cyan and includes the experi-mental transition trajectories. The outer closed curve (magenta) is theintersection of the boundary of the energy surface M(∆E) with U−1 .(b) Transition fraction of trajectories as a function of excess energyabove the saddle. The theoretical result is shown (blue curve) and ex-perimental values are shown as filled circles (black) with error bars.For small excess energy above critical (∆E = 0), the transition fractionshows linear growth (see inset) with slope 1.0± 0.23× 10−3 (s/cm)2

and shows agreement with the analytical result (5). A movie of in-creasing transition area on the Poincare section, U−1 , can be foundhere.

fraction, Fig. 4(b), by dividing points which transitionedby the total in each energy range. Despite the experi-mental uncertainty from the image analysis, agreementbetween observed and predicted values is satisfactory.In fact, a linear fit of the experimental results for smallexcess energy gives a slope close to that predicted by(5) within the margin of error. Furthermore, the clus-tering of observed transitioning trajectories in each en-ergy range, as in Fig. 4(a), is consistent with the theoryof tube dynamics. The predicted transition regions ineach energy range account for more than 99% of theobserved transition trajectories.

Conclusions.— We considered a macroscopic 2 DOFexperimental system showing transitions between po-tential wells and a dynamical systems theory of the con-duits which mediate those transitions [7, 10, 18]. Theexperimental validation presented here confirms the ro-bustness of the phase space conduits between multi-stable regions, even in the presence of non-conservativeforces, providing a strong footing for predicting transi-tions in a wide range of physical systems.

SDR and LNV thank the NSF for partially fundingthis work through grants 1537349 and 1537425.

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