Revealing the phase space structure of Hamiltonian systems using the action
Francisco Gonzalez Montoya, Makrina Agaoglou, Matthaios Katsanikas
RRevealing the phase space structure of Hamiltoniansystems using the action
Francisco Gonzalez Montoya, Makrina Agaoglou, and Matthaios KatsanikasSchool of Mathematics, University of Bristol,Fry Building, Woodland Road, Bristol, BS8 1UG, United Kingdom.
Abstract
In this work, we analyse the properties of the Maupertuis’ action as a tool to revealthe phase space structure for Hamiltonian systems. We construct a scalar field with theaction’s values along the trajectories in the phase space. The different behaviour of thetrajectories around important phase space objects like unstable periodic orbits, their stableand unstable manifolds, and KAM islands generate characteristic patterns on the scalarfield constructed with the action. Using these different patterns is possible to identifythe skeleton of the phase space and understand the dynamics. Also, we present a simpleargument based on the conservation of the energy and the behaviour of the trajectoriesto understand the values of their actions. In order to show how this tool reveals the phasespace structures and its effectiveness, we compare the scalar field constructed with theactions with Poincare maps for the same set of initial conditions in the phase space of anopen Hamiltonian system with 2 degrees of freedom.
The study of phase space structure is a fundamental problem in dynamical systems. Itis essential to understand the trajectories’ behaviour and properties from a theoretical andpractical perspective. The traditional tools to visualise the phase space structure like theprojection of the trajectories in a plane or Poincare maps are important tools to understandmany properties of ODE systems’ phase space with 3 dimensions. However, the study of thephase space structure of multidimensional systems remains an open and challenging problem.New tools have been developed to study the phase space structure of multidimensional sys-tems like Fast Lyapunov Exponets (FLE) [1], Mean Exponential Growth Factor of Nearby Or-bits (MEGNO) [2], Smaller Alignment Indices (SALI), Generalized Alignment Indices (GALI)[3] and Determinant of Scattering Functions [4, 5]. The phase space structure indicators arescalar fields constructed with the system’s trajectories. The differences in the scalar field’svalues give us information of the phase spaces objects that intersect the set of trajectoriesconsidered.A family of phase space structure indicators that it has been developed recently is the La-grangian descriptors (LDs) [6–8]. Examples of systems that have been analysed, using thismethod, can be found in [9–17]. The most intuitive Lagrangian descriptor is based on tra-jectories’ arc length. The differences in the arc length of the trajectories with nearby initialconditions give us information about the phase space around those initial points. In this work,we take advantage of the Maupertuis’ action S that defines a natural metric in the phasespace of a common class of Hamiltonian systems. With the action, it is possible to construct aLagrangian descriptor to reveal the phase space structure.In Section 1, we explain in more detail the principle that underpins the detection of phasespace objects in the phase space using the nearby trajectories. In section 2, we study the1 a r X i v : . [ n li n . C D ] F e b agrangian descriptor analytically and its behaviour when the trajectories are close to thehyperbolic periodic orbit of the quadratic normal form Hamiltonian with 2 degrees of freedom(DoF). We also explain this result using an intuitive argument based on the conservation of theenergy and the behaviour or the trajectories around the unstable hyperbolic periodic orbit. Insection 3, we apply the Lagrangian descriptor based on the classical action to reveal the phasestructures in a non-integrable Hamiltonian system with 2 DoF with unbounded phase space.Finally, in section 4 we summarise our conclusions and remarks. The Lagrangian descriptors, like other chaotic indicators, are scalar fields evaluated in a set ofinitial conditions in the phase space. The scalar field’s values are determined by the behaviourof the trajectories that cross the set of initial conditions. To visualise the principle behindthe detection of objects in the phase space we consider a 2 DoF Hamiltonian system with anunstable hyperbolic periodic orbit Γ. There are two remarkable invariant surfaces that intersectat this hyperbolic periodic orbit [18, 19]. In this case, the invariant property means that thetrajectories that start on an invariant surface are always contained in the same surface. Thesetwo surfaces are called stable and unstable manifold of the unstable hyperbolic periodic orbitΓ. The definition of the stable and unstable manifolds W s/u (Γ) is the following, W s/u (Γ) = { x | x ( t ) → Γ , t → ±∞} . (1)This means that the stable manifold W s (Γ) is the union of all the trajectories that converge tothe periodic orbit Γ as the time t goes to + ∞ . The definition of the unstable manifold W u (Γ)is similar. The unstable manifold is the set of trajectories converging to the periodic orbit asthe time t goes to −∞ .The phase space of a 2 DoF Hamiltonian system has 4 dimensions. For each value of theenergy, we can represent the dynamics in a 3 dimensional constant energy manifold. The stableand unstable manifolds W s/u (Γ) have two dimensions and form impenetrable barriers thatdivide the constant energy manifold [20, 21]. Another important property of the stable andunstable manifolds related to the chaotic dynamics is that, if a stable manifold and an unstablemanifold intersect transversely at one point, then an infinite number of transversal intersectionsexist between them. The structure generated by the stable and unstable manifolds is calledtangle and defines a set of tubes that direct the phase space dynamics. The trajectories ina tube never cross the boundaries of a tube. This fact is a consequence of the uniqueness ofordinary differential equations’ solution and the stable and unstable manifold dimension.The trajectories nearby the stable manifold W s (Γ) have similar behaviour just for some timeinterval. However, those trajectories diverge from the hyperbolic periodic orbit Γ after a while.This is a characteristic property of the trajectories in an unstable hyperbolic periodic orbitneighbourhood. Intuitively, the arc length of the trajectories on the stable manifold W s (Γ)grows like the periodic orbit’s arc length when the trajectories are close to Γ. For the othertrajectories near W s (Γ), the arc length grows similar only when the trajectories approach Γ.After the transit through the hyperbolic periodic orbit neighbourhood, the arc length growsdifferently. This difference makes possible the detection of phase space objects like stable andunstable manifolds of unstable hyperbolic orbits.Now, we give the general definition of the Lagrangian descriptor. Let us consider a systemof ODEs d x ( t ) dt = v ( x ( t )) , x ∈ R n , t ∈ R (2)2here the vector field v ( x ) ∈ C r ( r ≥
1) in a neighbourhood of the point x . The values ofthe Lagrangian descriptor depends on the initial condition x = x ( t ) and on the time interval[ t + τ − , t + τ + ]. The Lagrangian descriptor M is defined as, M ( x , t , τ + , τ − ) = M + ( x , t , τ + ) + M − ( x , t , τ − )= (cid:90) t + τ + t F ( x ( t )) dt + (cid:90) t t + τ − F ( x ( t )) dt, (3)where the function F is any positive function evaluated on the solutions x ( t ), x ( t ) = x , andthe extremes of the interval of integration τ + (cid:62) τ − (cid:54) F is chosen as positive to accumulate the effects of the trajectories’ behaviour.A natural choice is the infinitesimal arc length of the trajectories on the phase space [8]. For thedetection of phase space objects like stable and unstable manifolds of hyperbolic periodic orbits,in principle, it is possible to use any scalar field generated by the trajectories of the systemlike the final points of the trajectories in phase space or other quantities related [4, 5, 22].Nevertheless, it is not always trivial to interpret the results systematically.Notice that the first integral in the Lagrangian descriptor’s definition is calculated withtrajectories forward in time. Then, it reveals the presence of the phase space objects in the setof initial conditions like stable manifolds. Meanwhile, the second integral is calculated with thebackward time and reveals objects like unstable manifolds.The construction of the Lagrangian descriptor based on the action is as follows. For sim-plicity, let us consider a Hamiltonian function on Cartesian coordinates H ( q , . . . , q n , p , . . . , p n ) = T ( p , . . . , p n ) + V ( q , . . . , q n ) , ( q , . . . , q n , p , . . . , p n ) ∈ R n = n (cid:88) i =1 p i m i + V ( q , . . . , q n ) , (4)where T is the kinetic energy and V is the potential energy. The Maupertuis’ action or vivaaction S for a Hamiltonian system is defined as S = (cid:90) n (cid:88) i =1 p i dq i , p i ≡ ∂L∂ ˙ q i = m i dq i dt , i ∈ { , . . . , n } . (5)where L is the Lagrangian of the system and defines the momentum. Applying the chain ruleand the definition of the momentum, we obtain that n (cid:88) i =1 p i dq i = n (cid:88) i =1 m i dq i dt dq i dt dt = n (cid:88) i =1 p i dq i dt dt = n (cid:88) i =1 p i m i dt. (6)Using this identity and Hamilton’s equations is possible to write S as S = (cid:90) n (cid:88) i =1 p i dq i = (cid:90) n (cid:88) i =1 p i m i dt = 2 (cid:90) T dt = 2 (cid:90) ( H − V ) dt. (7)Taking the integral of the kinetic energy T with respect to the time, the simplest definitionfor the Lagrangian descriptor based on the action S is3 S ( x , t , τ + , τ − ) = 2 (cid:90) t + τ + t T dt + 2 (cid:90) t t + τ − T dt. (8)This result can be generalised to Hamiltonian where the T is a quadratic function of the gen-eralised velocities ( ˙ q i , . . . , ˙ q n ) and V is only a function of the generalised coordinates ( q i , . . . , q n ).The Lagrangian descriptor based on action M S is defined by the metric ds = (cid:80) ni =1 m i dq i dq i ,see equation 6. More details about this result and an example where M S has been used arein [23] and [24]. Analytical estimation for a Lagrangian descriptor family based on differentvariants of “arc length” in phase space has been studied in detail in [25]. The Lagrangiandescriptor based on action M S has a similar structure that the elements of this family of La-grangian descriptors. An elementary exposition about Lagrangian descriptors in Hamiltoniansystems is in [26] and more examples are in [27]. M S An important question for the phase space study is the detection of hyperbolic periodic orbitsand their invariant stable and unstable manifolds. This section studies the Lagrangian descrip-tor’s behaviour evaluated on a set of initial conditions that intersect the stable and unstablemanifolds of a hyperbolic periodic orbit. The next calculations are similar to the calculationsin [25]. Let us consider the most simple integrable 2 DoF Hamiltonian system as an initialcase for the analysis of the Lagrangian descriptor based on action M S . The 2 DoF quadraticnormal form Hamiltonian is H ( x, y, p x , p y ) = T ( p x , p y ) + V ( x, y ) = ω p x + x ) + λ p y − y ) . (9)The potential energy surface V ( x, y ) is saddle with an unstable equilibrium point at theorigin, see figures 1 and 2.The Hamilton’s equations of motion are˙ x = ∂H∂p x = ωp x , (10)˙ p x = − ∂H∂x = − ωx, ˙ y = ∂H∂p y = λp y , ˙ p y = − ∂H∂y = λy. The degrees of freedom x and y are uncoupled. The motion in the x component correspondsto a harmonic oscillator and the motion y component to an inverted harmonic oscillator. Forthis 2 DoF Hamiltonian system exist only one unstable periodic orbit Γ that oscillates on the x –direction on the line y = 0 for each value of the energy E >
0. The orbit Γ is a normallyhyperbolic invariant manifold and has stable and unstable manifolds. This hyperbolic periodicorbit is given by Γ = { ( x, y, p x , p y ) ∈ R | y = p y = 0 , E = ω p x + x ) } . (11)4igure 1: Potential energy surface V ( x, y ) corresponding to quadratic normal form Hamiltonian H ( x, y, p x , p y ) with parameters ω = 1 and λ = 1. The potential energy surface has index onesaddle point at the origin.The unstable and stable invariant manifolds of Γ are W u (Γ) = { ( x, y, p x , p y ) ∈ R | y = p y , E = ω p x + x ) } ,W s (Γ) = { ( x, y, p x , p y ) ∈ R | y = − p y , E = ω p x + x ) } . (12)The Lagrangian descriptor based on the action M S for this system is M S ( x , y , p x , p y , t , τ + , τ − ) = M eS ( x , p x , t , τ + , τ − ) + M hS ( y , p y , t , τ + , τ − ) (13)= (cid:90) t + τ − t + τ + λ p x ( t, x , p x ) dt + (cid:90) t + τ − t + τ ω p y ( t, y , p y ) dt, where the terms M hS and M eS are the hyperbolic and the elliptic parts respectively. In thenext calculations, we take t = 0 and τ − = 0 to simplify the analysis.Starting from the hyperbolic part, the solutions of the equations of motion for y and p y are y ( t ) = p y e λt + e − λt ) + y e λt − e − λt ) ,p y ( t ) = p y e λt − e − λt ) + y e λt + e − λt ) . (14)In this example, the integral corresponding to the hyperbolic part is5igure 2: Projection of the unstable hyperbolic periodic orbit Γ in the configuration space.The potential energy V ( x, y ) is in colour scale on the background with some equipotential lineson black colour. The red trajectory γ is close to the unstable hyperbolic periodic orbit Γ for aninterval of time before to escape through the region with negative values of V ( x, y ) to infinity. M hS ( y , p y , τ + ) = (cid:90) τ + λ p y ( t, y , p y ) dt = (cid:90) τ + λ p y e λt − e − λt ) + y e λt + e − λt )) dt = ( t (cid:18) − λ p y λ y (cid:19) + 164 (cid:0) − λp y + 16 λp y y − λy (cid:1) e − λt + 164 (cid:0) λp y + 16 λp y y + 8 λy (cid:1) e λt ) | τ + . (15)This integral grows exponentially with time. M hS has a minimum [26] and in the limit τ + → ∞ , the minimum converge to the initial conditions on the line y = − p y contained inthe stable manifold, see equation 12. Analogous result follows for the integration backwards ontime and initial condition on the line y = p y .Now we calculate the elliptic part of the Lagrangian descriptor M eS ( x , p x , τ + ). The solu-tions of the equations of motion for harmonic oscillator are x ( t ) = p x sin( ωt ) + x cos( ωt ) ,p x ( t ) = p x cos( ωt ) − x sin( ωt ) . (16)Then, we obtain M eS ( x , p x , τ + ) = (cid:90) τ + ω p x ( t, x , p x ) dt = (cid:90) τ + ω p x cos( ωt ) − x sin( ωt )) dt. (17)If we consider that p x has period P = 2 π/ω , then τ + = N P + r , where N is an integer and r ∈ [0 , P ]. We calculate the integral starting on the initial condition p x = 0 and x = (cid:112) E/ω without loss of generality. This gives us 6 eS ( x , p x , τ ) = EN (cid:90) π sin u du + (cid:90) r ω p x dt = (18)2 πEN + (cid:90) r ω p x dt = 2 πEN + M eS ( x , p x , r ) . From the above equation, we see that the elliptic part M eS in every oscillation accumulatesthe same value of action, in contrast to the hyperbolic component M hS that grows exponentially.Considering the results for the hyperbolic and elliptic components of the Lagrangian de-scriptor based on the action, we can conclude that M S is minimum on the stable and unstablemanifolds W s/u (Γ) of the hyperbolic periodic orbit Γ. Therefore M S attain a global minimumon the periodic orbit Γ. These results can be extended to the local neighbourhood of index onesaddles of nonlinear systems due to the Moser’s theorem [28].Intuitively, it is easy to understand this result considering the shape of the potential en-ergy surface V ( x, y ) around the index one saddle point and the trajectories around the stablemanifold W s (Γ). For energies E >
0, the unstable hyperbolic periodic orbit Γ oscillates, andits kinetic energy T is a periodic function of the time. Almost all the trajectories in a neigh-bourhood of the unstable hyperbolic periodic orbit Γ separate from it and its kinetic energygrows due to the shape of V ( x, y ) in the y -direction and the conservation of E , see figures 2 and5. However, only the trajectories in W s (Γ) converge to Γ and remain bounded. Then, theirkinetic energy T converge to a periodic function and the integral respect to the time of 2 T , theaction S , is minimum for the trajectories on W s (Γ). As a result, the Lagrangian descriptorhas a minimum in the stable and unstable invariant manifolds, and the global minimum revealsthe position of their intersection in Γ, see figure 3.It is possible to generalise this result for multidimensional systems with index one saddlepoint in the potential energy using analogous arguments. In this case, the system has a NormallyHyperbolic Invariant Manifold (NHIM) associated with the saddle point in the multidimensionalpotential energy [21, 29] . The Lagrangian descriptor based on the action is minimum inthe invariant stable and unstable manifolds of the NHIM, and the global minimum in theintersection between the stable and unstable manifolds, give us the position of the NHIM inthe phase space [25]. The proof of this result is direct from the previous considerations; weneed to consider more oscillatory degrees of freedom in the argumentation.7igure 3: Lagrangian descriptor M S evaluated on the set of initial conditions on plane y – p y and x = 0 for τ + = τ − = τ = 10. The region on white colour is a the forbidden regionfor the trajectories with E = 1. The lines with the minimum value are W s (Γ) ( p y = y ) and W u (Γ) ( p y = − y ). To avoid large values of kinetic energy T and the action S , the calculationof the trajectories stop when integration time is completed or when the particle rich the thecircumference in the configuration space with radius r = 10 with centre at the origin.Figure 4: Lagrangian descriptor M S evaluated on the set of initial conditions on a line p y = 2and x = 0 for different times τ . The minimum value are intersection of the W u (Γ) with the setof initial conditions. To avoid large values of kinetic energy T and the action S , the calculationof the trajectories stop when integration time is τ or the particle rich the the circumference inthe configuration space with radius r = 10 with centre at the origin. For τ > T ( t ) as a function of time for the the unstable hyperbolic periodicorbit Γ and a trajectory γ with slightly different initial conditions. The kinetic energy for Γis a periodic function meanwhile the kinetic energy for γ grows unbounded. The values of theaction S for each trajectory is twice the area under their corresponding T ( t ) curve. M S In this section, we illustrate the Lagrangian descriptor’s capabilities M S to obtain the phasespace structure of a chaotic system. We compare the Lagrangian descriptor plots with thecorresponding Poincare maps with the same initial conditions. Let us consider a chaotic 2-DoFHamiltonian proposed to qualitatively study a special kind of chemical reaction dynamics [30].The model has been analysed in detail using the Poincare map in [31, 32].The Hamiltonian function of the system is H ( x, y, p x , p y ) = T ( p x , p y ) + V ( x, y ) = p x m + p y m + c ( x + y ) + c y − c ( x + y − x y ) (19)where the mass of the particle is m = 1 and the potential parameters are c = 5, c = 3,and c = − /
10. The potential energy V ( x, y ) is symmetric with respect to the y -axis. Thispotential has a central minimum and four index one saddles around it, see figures 7 and 6. Theshape of this potential energy surface is similar to a volcano’s caldera. More information aboutthis potential and the chemical reaction dynamics is in [30] and references therein.Let us consider two values for the energy E such that the phase space of the system isunbounded and some KAM islands are present. For the first value of the energy, E = 17, wetake the canonical conjugate plane x – p x with y = − . p y > V ( x, y ) surface. The surface has a minimum has a centralminimum and four index one saddle points at the corners.Figure 7: Potential energy V ( x, y ) in colour scale. The black lines are equipotentials.10n open Hamiltonian systems, some examples of transient chaotic systems with 2 and 3 DoFare in [22, 33–36]. The curves on green and red are the intersections of the unstable manifold W s (Γ h ) and stable manifold W u (Γ h ) of the hyperbolic periodic orbit Γ h and its symmetricperiodic orbit (with respect to the y -axis) with the Poincare plane. This periodic orbit and itssymmetric periodic orbit are Lyapunov orbits associated with the low right and low left indexone saddle points in the potential energy surface. The projection of orbit Γ h in the configurationspace is in figure 9. In the panel D) of figure 8 there is a magnification of the region where Γ h intersects with the Poincare plane at the black point around ( x, p x ) = (2 . , . h and its symmetric periodic orbit.Let us consider the origin of the signature of phase space objects in the Lagrangian descriptorplots using the conservation of the energy and the behaviour of the trajectories. The trappedregions inside the KAM islands have large values of M S . The trapped trajectories in theKAM island remain all the time in a region nearby the global minimum of the potential energy V , see figures 6 and 7. That means that for large values of time t the kinetic energy T ( t )of trapped trajectories is larger than the kinetic energy of the trajectories that spend only alapse close to the minimum of V and then reach an exit of the caldera. Thus, the Lagrangiandescriptor’s values M S for the trapped trajectories in the island are larger than the values forthe trajectories that reach an exit.The stable and unstable manifolds of the hyperbolic periodic orbit Γ h generate abruptchanges in the Lagrangian descriptor’s values. To see this property, let us consider a line ofinitial conditions that intersect W s (Γ h ). The line is contained on the domain of panel C) offigure 8 and has p x = 0 .
62. The results for different integration times are in figure 10. Wecan see how the jumps are generated when τ is increased. The first jump on the right sidecorresponds to the intersection of the stable manifold W (Γ h ) s with the set of initial conditions.We see a jump and not a soft minimum like in the example in the previous section due tothe large integration time and the stop condition for trajectories that reach the circumferenceoutside the caldera. The trajectories on the right side of the W (Γ h ) s reach the circumferenceand calculation of their M S stops.As a second example, let us consider another value of the energy where the phase spacehas a different structure. The figures in panels B) and D) of figure 11 show the Poincare mapcorresponding to the canonical conjugate plane x – p x with y = 0, p y > E = 23. Thecentral black point corresponds to the intersection between the Poincare plane and an inversehyperbolic periodic orbit Γ i . Figure 12 shows the projection of Γ i in the configuration space.This periodic orbit belongs to the family of periodic orbits of the central minimum [31]. Thered and green lines on the Poincare maps of figure 11 are the stable and unstable manifolds W s (Γ i ) and W u (Γ i ).In this case, the results are different from the results of the integrable example wherethe Lagrangian descriptor has a minimum at hyperbolic periodic orbit Γ. For the inversehyperbolic periodic orbit Γ i , the Lagrangian descriptor M S have a maximum at the intersectionbetween the inverse hyperbolic periodic orbit and the set of initial conditions. Figure 13 shows M S evaluated on a line of initial conditions that intersect the Γ i for different values of theintegration time τ . To explain this different behaviour, with respect to the behaviour of M S for the hyperbolic periodic orbit Γ associated with the saddle in the potential energy surface,we consider the conservation of the energy and the shape of the potential energy surface aroundthe projection of Γ i again. The projection of Γ i in the configuration space is at the minimumof V ( x, y ) with respect to the x -direction. The trajectories in the neighbourhood of Γ i have11) B)C) D)Figure 8: Lagrangian descriptor M S and Poincare map with mixed phase space for E = 17.The Poincare section is the canonical conjugate plane x – px with y = − . p y > M S stop when the integration time is τ = 20 or the particlesreach the a circumference in the configuration space with radios r = 3 and centre at the origin.A)Figure 9: Projection in configuration space of the periodic orbit Γ h associated to a saddlepoint in the potential energy surface and a trajectory γ with slightly different initial con-ditions. The potential energy V ( x, y ) is on the background. The periodic orbit Γ h inter-sects the Poincare surface x – p x with y = − . x, p x ) = (2 . , . M S for different times τ = 18 , ,
20. The set of initialconditions is a line segment on the domain of the figure 8 C). The first abrupt jump on theright side corresponds to the intersection of the stable manifold W s (Γ) with the set of initialconditions. All the trajectories with initial conditions on the right reach the circumference with r = 3. For this reason we only see an abrupt jump on M S .A) B)C) D)Figure 11: Lagrangian descriptor M S and Poincare map with mixed phase space for E = 23.The Poincare surface of section is the canonical conjugate plane x – p x with y = 0 and p y > τ = 20.13igure 12: Projection on configuration space of the periodic orbit Γ i corresponding to the pointin the centre of the figures in 11 and a trajectory γ with slightly different initial conditions.The potential energy V ( x, y ) is the background in colour scale with some equipotential lines onblack. The periodic orbit Γ i oscillates on y -direction between two equipotential lines.A)Figure 13: Lagrangian descriptor M S for different times t . The set of initial conditions is aline segment on the figure 11 D) with p x = 0. The maximum corresponds to the intersectionof the inverse hyperbolic periodic orbit Γ i with the set of initial conditions.14arger values of V ( x, y ) and their kinetic energy T is smaller than the kinetic energy for Γ i .Consequently, the Lagrangian descriptor has a maximum on Γ i and its stable and unstablemanifolds W s (Γ i ) and W u (Γ i ). 15 Conclusions and remarks
We construct a natural phase space structure indicator for Hamiltonian systems based onthe Maupertuis’ action S . For this construction, it is necessary that the kinetic energy is aquadratic function of the generalised velocities, and its potential energy is only a function ofthe generalised coordinates. The simplest way to calculate the Lagrangian descriptor M S iswith the integral of the kinetic energy with respect to the time of the trajectories of the system.The Lagrangian descriptor based on the action is a convenient tool to study the phase space ofopen Hamiltonian systems. The energy conservation is essential to interpret the values of M S and link them with the potential energy V . There is a simple relationship between its valuesand the different phase space regions. The trapped regions correspond to large values of M S and the unbounded regions to smaller values of M S .With the action as a phase space structure indicator, it is possible to identify regular regions,unbounded regions, resonant islands and the transient chaotic sea around them formed byhomoclinic and heteroclinic tangles. In the regions where the dynamics is confined to a finiteregion on the phase space, the values of M S change smoothly for large integration times andis clear how to identify the central periodic orbits in the islands. However, we can alwaysfind stable periodic orbits on the KAM islands centres with the Poincare map. Therefore,the Lagrangian descriptor and the Poincare maps are complementary tools to reveal the phasespace structure.We find that M S has a minimum value on their stable and unstable manifolds of thehyperbolic periodic orbits. On the other hand, this Lagrangian descriptor has a maximal valuefor the stable and unstable manifolds for the inverse hyperbolic periodic orbits. These resultsare intuitive if we consider the conservation of the energy and the trajectories’ behaviour in theperiodic orbits’ neighbourhood. It is possible to generalise these results for NHIMs and theirstable and unstable manifolds in systems with more dimensions.In some situations with unbounded negative potentials, it is convenient to stop calculationof the Lagrangian descriptor’s trajectories. In this way, the Lagrangian descriptor reveals onlythe phase space structure in one particular region. It is important to consider the stop ofthe trajectories to interpret the Lagrangian descriptor. For the Lagrangian descriptor plot onfigures 11 and 13, the stop of the integration of the trajectories generates an abrupt jump thatreveals Γ h . We acknowledge Stephen Wiggins for discussions about the action and the support of EPSRCGrant no. EP/P021123/1.
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Physical Review E , 102:012215, 2020.[17] Francisco Gonzalez Montoya and Stephen Wiggins. Revealing roaming on the double morsepotential energy surface with lagrangian descriptors.
Journal of Physics A: Mathematicaland Theoretical , 53(23):235702, may 2020.[18] E Ott.
Chaos in Dynamical Systems . Cambrige University Press, 2002.[19] R Abraham and C Shaw.
Dynamics: The Geometry of Behavior . Addison Wesley LongmanPublishing, 1992.[20] Z Kov´acs and L Wiesenfeld. Topological aspects of chaotic scattering in higher dimensions.
Phys. Rev. E , 63(5):56207, apr 2001.[21] S Wiggins, L Wiesenfeld, C Jaff´e, and T Uzer. Impenetrable Barriers in Phase-Space.
Phys. Rev. Lett. , 86(24):5478–5481, jun 2001.[22] F Gonzalez and C Jung. Rainbow singularities in the doubly differential cross sectionfor scattering off a perturbed magnetic dipole.
Journal of Physics A: Mathematical andTheoretical , 45(26):265102, 2012.[23] Francisco Gonzalez Montoya and Stephen Wiggins. The phase space structure and theescape time dynamics in a van der waals model for exothermic reactions, 2020.[24] Rafael Garcia-Meseguer and Barry K. Carpenter. Re-evaluating the transition state forreactions in solution.
European Journal of Organic Chemistry , 2019(2-3):254–266, 2019.[25] Shibabrat Naik, V´ıctor J Garc´ıa-Garrido, and Stephen Wiggins. Finding nhim: Identifyinghigh dimensional phase space structures in reaction dynamics using lagrangian descriptors.
Communications in Nonlinear Science and Numerical Simulation , 79:104907, 2019.[26] Makrina Agaoglou, Broncio Aguilar-Sanjuan, V´ıctor Jos´e Garc´ıa Garrido, FranciscoGonz´alez-Montoya, Matthaios Katsanikas, Vladim´ır Krajˇn´ak, Shibabrat Naik, andStephen Wiggins. Lagrangian Descriptors: Discovery and Quantification of Phase SpaceStructure and Transport, July 2020. 10.5281/zenodo.3958985.[27] Makrina Agaoglou, Broncio Aguilar-Sanjuan, Victor Jose Garc´ıa-Garrido, Rafael Garc´ıa-Meseguer, Francisco Gonz´alez-Montoya, Matthaios Katsanikas, Vladimir Krajˇn´ak,Shibabrat Naik, and Stephen Wiggins. Chemical reactions: A journey into phase space,December 2019. 10.5281/zenodo.3568210.[28] J¨urgen Moser. On the generalization of a theorem of a. liapounoff.
Communications onPure and Applied Mathematics , 11(2):257–271, 1958.[29] S. Wiggins. The role of normally hyperbolic invariant manifolds (nhims) in the context ofthe phase space setting for chemical reaction dynamics.
Regular and Chaotic Dynamics ,21(6):621–638, 2016.[30] Peter Collins, Zeb C. Kramer, Barry K. Carpenter, Gregory S. Ezra, and Stephen Wiggins.Nonstatistical dynamics on the caldera.
The Journal of Chemical Physics , 141(3):034111,2014. 1831] Matthaios Katsanikas and Stephen Wiggins. Phase space structure and transport ina caldera potential energy surface.
International Journal of Bifurcation and Chaos ,28(13):1830042, 2018.[32] Matthaios Katsanikas and Stephen Wiggins. Phase space analysis of the nonexistence ofdynamical matching in a stretched caldera potential energy surface.
International Journalof Bifurcation and Chaos , 29(04):1950057, 2019.[33] Ying-Cheng Lai and Tam´as T´el.
Transient Chaos . Springer-Verlag New York, 2011.[34] T´amas T´el. The joy of transient chaos.
Chaos: An Interdisciplinary Journal of NonlinearScience , 25(9):97619, 2015.[35] D´aniel J´anosi and Tam´as T´el. Chaos in hamiltonian systems subjected to parameter drift.
Chaos: An Interdisciplinary Journal of Nonlinear Science , 29(12):121105, 2019.[36] F Gonzalez, G Drotos, and C Jung. The decay of a normally hyperbolic invariant manifoldto dust in a three degrees of freedom scattering system.