An Extension of Discrete Lagrangian Descriptors for Unbounded Maps
AAn Extension of Discrete Lagrangian Descriptorsfor Unbounded Maps
V´ıctor J. Garc´ıa-Garrido Departamento de F´ısica y Matem´aticas,Universidad de Alcal´a, 28871, Alcal´a de Henares, [email protected] 19, 2019
Abstract
In this paper we provide an extension for the method of Discrete Lagrangian De-scriptors with the purpose of exploring the phase space of unbounded maps. The keyidea is to construct a working definition, that builds on the original approach intro-duced in Lopesino et al. (2015b), and which relies on stopping the iteration of initialconditions when their orbits leave a certain region in the plane. This criterion is partlyinspired by the classical analysis used in Dynamical Systems Theory to study the dy-namics of maps by means of escape time plots. We illustrate the capability of thistechnique to reveal the geometrical template of stable and unstable invariant manifoldsin phase space, and also the intricate structure of chaotic sets and strange attractors, byapplying it to unveil the phase space of a well-known discrete time system, the H´enonmap.
Keywords:
Maps, Phase space structure, Lagrangian descriptors, Stable and unstablemanifolds, Chaotic sets.
The method of Lagrangian Descriptors (LDs) is a trajectory-based scalar diagnostic orig-inally developed in the context of Geophysics Mendoza and Mancho (2010) to study La-grangian transport and mixing processes that take place in the ocean or the atmosphere.This tool was first introduced a decade ago in the work by Madrid and Mancho (2009) asa tool to detect
Distinguished Hyperbolic Trajectories (or ’moving saddles’) in dynamicalsystems with general time dependence, and in recent years, it has been found to be an excel-lent technique for exploring and revealing the geometrical template of invariant manifoldsMancho et al. (2013); Lopesino et al. (2017); Naik et al. (2019) that characterizes phasespace regions with qualitatively distinct dyamical behavior, a long-sought goal envisionedby Henri Poincar´e in his works on the three-body problem in Celestial Mechanics Poincar´e(1890).Lagrangian Descriptors have been widely applied in the literature to problems that arisein many scientific disciplines. For instance, it has been used in Oceanography for the as-sessment of marine oil spills (Garcia-Garrido et al. , 2016), and also to plan autonomousunderwater vehicle transoceanic missions Ramos et al. (2018). Another field where LDshave received recognition is in Chemistry Craven and Hernandez (2015); Junginger andHernandez (2015). In Transition State Theory, which deals with the study of chemicalreaction dynamics, the knowledge of phase space structures is crucial to analyze reactionrates. In this framework, the tool has proved to successfully identify normally hyperbolicinvariant manifolds of Hamiltonian systems Demian and Wiggins (2017); Naik et al. (2019);Naik and Wiggins (2019), which serve as a scaffolding for the construction of the dividing1 a r X i v : . [ n li n . C D ] A ug urface to measure reation rates Craven and Hernandez (2016, 2015); Craven et al. (2017);Junginger et al. (2017); Feldmaier et al. (2017). In all these contexts, the dynamical systemthat governs the long-term evolution of particles is continuous in time, and the vector fieldthat defines it is given either by an analytical expression or by a dataset generated from anumerical model.In the discrete-time setting, LDs have also been defined for maps (Lopesino et al. , 2015b),where a mathematical connection was established between the invariant geometrical struc-tures present in a map’s phase space and the output obtained from the method. In fact,Discrete Lagrangian Descriptors (DLDs) have provided insightful results when used to de-scribe 2D chaotic maps such as the H´enon, Lozi, and Arnold’s cat maps (Lopesino et al. ,2015b,a; Garc´ıa-Garrido et al. , 2018). In all these examples which display a strong mixing ina given area of their domain, DLDs are capable of uncovering the chaotic saddles responsiblefor the chaotic dynamics.However, when Lagrangian Descriptors (both in their continous- or discrete-time ver-sions) are used to explore the phase space structures of unbounded dynamical systems,issues might arise when trajectories of initial conditions escape to infinity in finite time orat an increasing rate. This behavior will result in NaN values in the LD scalar field, obscur-ing the detection of invariant manifolds. Recent studies Junginger et al. (2017); Naik andWiggins (2019); Garc´ıa-Garrido et al. (2019) have revealed this concern for some examplesof unbounded continuous-time dynamical systems, where the approach of calculating LDsby integrating initial conditions on a phase space grid for the same time, known as fixed-time LDs, does not work well because of escaping trajectories. In order to circumvent thisissue, an alternative definition of LDs was introduced for continuous-time systems, wherethe value of LDs for any initial condition is calculated by evolving its trajectory for a chosenintegration time or until it leaves a sufficiently large domain in phase space. The goal ofthis paper is to adapt this definition for discrete-time systems, and demonstrate that byconsidering variable iteration number Discrete Lagrangian Descriptors (VIN-DLD) one cananalyze in full detail the phase space structures that govern the dynamics of two-dimensionalunbounded maps.This paper is organized as follows. In Section 2 we briefly describe the original definitionof the method of Discrete Lagrangian Descriptors (DLD) and propose a modified version ofit to make it work for unbounded maps. The idea behind this alternative definition is tostop the iteration of the orbits of initial conditions that leave a certain fixed phase spaceregion. This implies, in particular, that the orbits of initial conditions might be iterated fora different number of iterations. Section 3 is devoted to discussing the results of this work.First, we numerically show how the VIN-DLD succeeds in revealing the geometrical templateof invariant manifolds in the phase space for the H´enon map. Moreover, we illustrate thatthis technique is also capable of detecting KAM tori, and, as a validation, we comparethis diagnostic with the classical average-exit time distribution approach Meiss (1997). Tofinish, we demonstrate that VIN-DLD have the capability of unveiling the strange attractorthat appears in the non area-preserving H´enon map. Finally, in Section 4 we present theconclusions of this work. We begin this section by briefly explaining the basic setup for the method of Discrete La-grangian Descriptors, that was first introduced in Lopesino et al. (2015b) for 2D area-preserving maps. Consider a domain D ⊂ R and a function f that defines the discrete-timedynamical system, x i +1 = f ( x i ) , with x i = ( x i , y i ) ∈ D , ∀ i ∈ N ∪ { } . (1)We assume that this mapping is invertible so that, x i = f − ( x i +1 ) , with x i = ( x i , y i ) ∈ D , ∀ i ∈ Z − (2)2iven a fixed number of iterations N > f and f − , and take p ∈ (0 ,
1] tospecify the L p -norm that we will use, then the DLD applied to systems (1) and (2) is givenby the following function, M D p ( x , N ) = N − (cid:88) i = − N || x i +1 − x i || p = N − (cid:88) i = − N | x i +1 − x i | p + | y i +1 − y i | p (3)where x ∈ D is any initial condition. Observe that Eq. (3) can be split into two quantities, M D p ( x , N ) = M D + p ( x , N ) + M D − p ( x , N ) , (4)where, M D + p = N − (cid:88) i =0 || x i +1 − x i || p = N − (cid:88) i =0 | x i +1 − x i | p + | y i +1 − y i | p , (5)measures the forward evolution of the orbit of x and, M D − p = − (cid:88) i = − N || x i +1 − x i || p = − (cid:88) i = − N | x i +1 − x i | p + | y i +1 − y i | p , (6)accounts for the backward evolution of the orbit of x . In this framework, Lopesino et al. (2015b) provides a mathematical proof that DLDs highlight stable and unstable manifolds atpoints where the M D p field becomes non-differentiable (discontinuities or unboundedness ofits gradient). These non-differentiable points are displayed in a simple way when depictingthe M D p scalar field, and are known in the literature as ”singular features”. Therefore,DLDs are capable of recovering the geometrical template of invariant stable and unstablemanifolds present in the phase space of any map defined by Eqs. (1) and (2). Moreover,Lopesino et al. (2015b) proves for several examples that M D + p detects the stable manifoldsof the fixed points and M D − p does the same for the unstable manifolds of the fixed points.Therefore, the fixed points with respect to the map (1) will be located at the intersectionsof these structures.In the definition of DLDs, the number of iterations N considered for the computationof the orbit of any initial condition plays a crucial role for revealing the intricate structureof stable and unstable manifolds in the phase space, and hence holds the secret to unlockthe general applicability of DLDs to unbounded maps. It has been shown in Lopesino et al. (2015b) that, as the number of iterations N is increased, a richer template of geometricalphase space structures is recovered by the method, since we are including into the analysismore information about the past and future history of the orbits of the map. The issuearises when we deal with an unbounded map for which orbits can escape to infinity at anincreasing rate. This happens for instance in the H´enon map, which is the example wewill use to illustrate how the definition of DLDs has to be modified in order to make themethod work for general open maps. Notice that in the original definition of DLDs givenin Eq. (3), all the initial conditions chosen in the domain D ⊂ R are iterated using f andalso f − for the same number of iterations N . Consequently, for an unbounded map, someinitial conditions might escape to infinity very fast, yielding extremely large and N aN valuesin the computation of
M D p , and this issue obscures completely the detection of invariantmanifolds in some regions of phase space.The modification that we propose for the definition of DLDs in order to avoid the problemof orbits escaping to infinity in unbounded maps, which would end up rendering the M D p scalar field values useless, is the following. Start by fixing a planar region R ∈ R in thephase space, that we call the interaction region , and take a fixed number of iterations N > x ∈ D , we calculate M D p along its orbit as we iterate it with f and f − , and we do so until we reach the maximum number of iterations N or the orbitleaves the interaction region R . Therefore, we define the numbers, N ± x = max k =1 ,...,N { k | f ± k ( x ) ∈ R} (7)3here N + x and N − x are, respectively, the number of forward ( f ) and backward ( f − ) it-erations of the orbit O ( x ). As a result, initial conditions might be iterated a differentnumber of times depending on whether they remain inside the interaction domain R or theyeventually escape it. We use this idea to define what we call Variable Iteration NumberDiscrete Lagrangian Descriptor (VIN-DLD). This modified version of DLDs is given by theformula, D p ( x , N ) = N + x − (cid:88) i = − N − x || x i +1 − x i || p = N + x − (cid:88) i = − N − x | x i +1 − x i | p + | y i +1 − y i | p . (8)Notice that if one takes a large interaction region R of the planar phase space, the expectedresult is that this new variable iteration number definition of DLD approaches the values ofthe original DLD in Eq. (3). Therefore, the same mathematical properties that were provedfor the DLD in Lopesino et al. (2015b) also hold in this case, that is, the modified VIN-DLDalso has the capability of detecting the stable and unstable manifolds of fixed points andperiodic orbits at locations where D p is non-differentiable.To finish, we would also like to remark that, since the interaction region R has to bechosen large enough so that the values of the VIN-DLD approach those obtained with theFixed Iteration Number Discrete Lagrangian Descriptor (FIN-DLD) in Eq. (3), the shapeof R is not important in order for the method to work. For the purpose of simplifiyng theanalysis, and without loss of generality, it is convenient to choose R as a circle centeredabout the origin, but a square would work perfectly fine too. In particular, to analyze thephase space structures present in the H´enon that we discuss next we will use a circle ofradius r = 100. In this section we present the results of applying the VIN-DLD defined in Eq. (8) to analyzethe phase space structures of a well-known example with unbounded behavior, the H´enonmap H´enon (1976), which is a hallmark for studying dynamical properties of two-dimensionaldiscrete-time systems. This invertible map is given by the system, (cid:40) x n +1 = A + By n − x n y n +1 = x n , ∀ n ∈ N ∪ { } (9)where A, B ∈ R are the model parameters, and the inverse mapping is: x n = y n +1 y n = x n +1 − A + y n +1 B , ∀ n ∈ Z − ∪ { } (10)The H´enon map is an area preserving map when | B | = 1, and is orientation-preserving for B = −
1. Moreover, in Devaney and Nitecki (1979) it is shown that, if
A > (cid:0) √ (cid:1) (1 + | B | ) / A = 9 . B = − A = 0 .
298 and B = 1, and for this case we show howDLDs detect KAM tori and also the stable and unstable manifolds. Finally, we demosntratethat the method succeeds in revealing the intricate strucutre of the strange attractor thatappears in the system for the classical value of the parameters A = 1 . B = 0 .
3, whichwas first discovered by H´enon in H´enon (1976).We begin our analysis by applying the FIN-DLD introduced in Eq.(3) to the map withparameters A = 9 . B = −
1. This computation was already carried out in Lopesino4 t al. (2015b), but the p -norm chosen to reveal the chaotic saddle was p = 0 .
05. The reasonfor choosing such a small value for p is to balance the large values obtained in the DLD dueto the orbits of initial conditions escaping to infinity very fast in this case. We can see in Fig.1A that for N = 5 iterations, the method reveals some of the features of the chaotic saddle.However, when we increase the iterations to N = 10, see Fig. 1B, most of the orbits haveescaped to infinity, yielding NaN values when computing DLDs and obscuring completelythe phase space structure of the map.A) B)Figure 1: Phase space structures obtained for the H´enon map with model parameters A = 9 . B = − p = 0 .
05. A) N = 5 iterations. B) N = 10 iterations.For N = 5 the method captures the basic features of the chaotic saddle. However, for N = 10 iterations many orbits have already escaped to infinity, giving extremely large M D p values and rendering the diagnostic useless in this case.With this example, it becomes clear that stopping the orbits of initial conditions at somepoint along their evolution might help circumvent this issue. To do so, we fix a circularregion of radius r = 100 about the origin in the phase plane and apply the variable-iterationdefinition in Eq. (8). Therefore, given a fixed number of iterations N , we will calculate DLDsalong the orbit of an initial condition until we reach the maximum number of iterations N ,or the trajectory leaves the circular region. We can clearly see in Fig. 2A that the VIN-DLD, using p = 0 .
05 recovers for N = 5, the basic characteristics of the chaotic saddle asthe FIN-DLD does. Furthermore, if we increase the number of iterations to N = 10 weobserve, see Fig. 2B, that this time the technique succeeds in recovering more features ofthe intricate and fractal structure of the chaotic saddle. This is expected from the method asthe number of iterations gets large, since we are incorporating more information about thefuture and past history of orbits into the DLD computation and, consequently, this resultsin a detailed detection of the stable and unstable manifolds. This exapmple demonstratesthat the variable iteration number definition of DLDs overcomes the issue encountered bythe FIN-DLD when applied to unbounded maps with trajectories escaping to infinity atan increaseing rate. It is important to highlight that, once D p has been calculated, we candirectly extract from it the stable and unstable manifolds of the map by means of processingthe values of its gradient, since these invariant manifolds are mathematically connected withthe points at which the DLD scalar field is non-differentiable, as was proved in Lopesino et al. (2015b). We illustrate this capability of the method in Figs. 2C-D. As a validation, wecompare the output of D p with exit time distributions Meiss (1997), i.e. the total number ofiterations (forwards plus backwards) that the orbit of each initial condition remains insidea fixed phase space region. Therefore, the transit time distribution is given by: T x = N + x + N − x (11)where N ± x are defined in Eq. (7). Notice that, in order to compute D p we need to keeptrack of the number of iterations for which the orbit of each initial condition remains inside5 given phase space region (before escaping), thus, in the process of calculating VIN-DLDwe also obtain simultaneously the transit time distribution. The comparison between bothmethods shows an excellent agreement as displayed in Figs. 2E-F.We turn our attention next to analyze the phase space structures of the H´enon map withmodel parameters A = 0 .
298 and B = 1. In this case we know that there is dynamicaltrapping due to the stickiness of the KAM tori present in two regularity islands of the phasespace, wich are surrounded by the homoclinic and heteroclinic intersections of the stable andunstable manifolds of periodic saddles in the chaotic sea de Oliveira et al. (2019). In orderto uncover the invariant manifolds we apply D p with p = 0 . N = 15 and N = 25iteartions. We illustrate in this way how the method captures more features of the stable andunstable manifolds as the number of iterations is increased. The result of this calculationis displayed in Figs. 3A-B. As we have explained, the stable and unstable manifolds arerevealed at points of the scalar field, known in the literature as singular features, where D p is non-differentiable (the gradient becomes unbounded). We also demonstrate the potentialof extracting the invariant stable and unstable manifolds of the map from the gradient of D p in Figs. 3C-D. Moreover, observe that the islands of regularity corresponding to KAMtori are recovered by regions where the D p function is smooth. In fact, a mathematicalconnection has been established between time-averaged Lagrangian descriptors Lopesino et al. (2017) and KAM tori through Birkhoff’s Ergodic Partition Theorem Mezic and Wiggins(1999) under certain assumptions. Despite the fact that this proof has only been given forcontinuous-time dynamical systems, since Eq. (3) can be interpreted as a discretized versionof the p -norm definition of LDs inntroduced in Lopesino et al. (2017), this property will alsohold for maps. Therefore, if we define the iteration-averaged DLD as, (cid:104)D p (cid:105) = 1 N D p ( x , N ) (12)then the contours of (cid:104)D p (cid:105) when convergence has been reached as the number of iterations N → ∞ correspond to invariant sets in the phase space. With this approach, we can use D p to easily recover the KAM tori as we demonstrate in Fig. 4A, where we have representedthe contours of (cid:104)D p (cid:105) calculated for N = 500 iterations. We compare this output to thatobtained by plotting the transit time distribution, see Fig. 4B. Since orbits are trappedin the tori forever, the exit-time plot is flat all over the regularity island and thus transittimes do not give further information about the structure of the KAM tori in that region.To summarize, we have illustrated that VIN-DLD has some advantages with respect to theanalysis of transit time distributions. First, it is straightforward to extract the stable andunstable manifolds of the map directly from the computation of ||∇D p || , and second, itsimultaneously depicts the KAM tori associated to islands of regularity when the number ofiterations is chosen large enough, thanks to its connection with Birkhoff’s Ergodic PartitionTheorem.To finish, we will apply D p with p = 0 . N = 10 iterations to uncover the phase spacestructure of the H´enon map with model parameters A = 1 . B = 0 .
3. For this case,we know that the map does not preserve area, and that it has a strange attractor that hasbeen widely studied in the literature H´enon (1976). In Fig. 5 we display the results of thiscomputation, confirming that the method successfully unveils the intricate structure of theattractor. Moreover, the structure of the strange attractor can be recovered completely ina simple way from the gradient of the backward iteration of the method which, as we haveexplained, captures the unstable manifolds of the map.
In this work we have extended the method of Discrete Lagrangian Descriptors with thepurpose of improving its capability for revealing the phase space structure of unboundedmaps, where trajectories can escape to infinity at an increasing rate. In order to achieve thisgoal, and given a fixed number of maximum iterations, we have modified the definition ofDLDs so that its value is accumulated along the orbit of each initial condition up to the fixed6) B)C) D)E) F)Figure 2: Phase space structures of the H´enon map for the model parameters A = 9 . B = −
1. The left column corrsponds to N = 5 iterations, and the right column is for N = 10 iterations. A) and B) VIN-DLD calculated with p = 0 .
05. C) and D) Stable (blue)and unstable (red) manifolds extracted from the gradient of DLD. E) and F) Transit timedistributions. 7) B)C) D)E) F)Figure 3: Phase space structures of the H´enon map for the model parameters A = 0 . B = 1. The left column corrsponds to N = 15 iterations, and the right column is for N = 25 iterations. A) and B) VIN-DLD calculated with p = 0 .
5. C) and D) Stable (blue)and unstable (red) manifolds extracted from the gradient of DLD. E) and F) Transit timedistributions. 8) B)Figure 4: Phase space KAM tori of the H´enon map for the model parameters A = 0 .
298 and B = 1. A) Contour levels of the VIN-DLD calculated with p = 0 . N = 500 iterations.B) Transit time distribution.A) B)C)Figure 5: Strange attractor of the H´enon map for the model parameters A = 1 . B = 0 .
3. A) Variable iteration number DLD calculated with p = 0 . N = 10 iterations.B) Unstable manifold extracted from the gradient of D p . C) Transit time distribution.9umber of iterations, or until the trajectory leaves a certain region in the plane. In orderto illustrate the success of this alternative variable iteration number definition to unveilthe template of geometrical phase space structures we have applied it to the well-knownH´enon map, for which the original FIN-DLD has issues in order to fully reveal the intricategeometry of the stable and unstable manifolds, and also the presence of a chaotic saddle anda strange attractor for some values of the model parameters. Finally, and as a validation,we have compared our simulations of DLDs with the classical escape time plots used in theDynamical System’s literature to uncover invariant manifolds. Our results are in excellentagreement with the well-studied properties of the H´enon map and therefore we are confidentthat this technique will surely provide the Dynamical System’s community with a valuableresource to explore the dynamics and dynamical characteristics of discrete time systems. Acknowledgments
I would like to thank Prof. Stephen Wiggins for his useful comments and suggestionsthat helped me improve the contents of this paper. The author also acknowledges thefinancial support received from the EPSRC Grant No. EP/P021123/1 and the Office ofNaval Research Grant No. N00014-01-1-0769 for his research visits over the past year atthe School of Mathematics, University of Bristol. This work is the result of many fruitfuldiscussions with Prof. Stephen Wiggins during that period.
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