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Lagrangian Descriptors: A Method for Revealing Phase Space Structures of General Time Dependent Dynamical Systems

Ana M. Mancho, Stephen Wiggins, Jezabel Curbelo, Carolina Mendoza

arXiv:1106.1306v4nlin.CDmath.DSphysics.flu-dyn

TL;DR

The paper addresses how to reveal phase-space geometry in aperiodically time-dependent dynamical systems. It constructs Lagrangian descriptors from finite-time integrals along trajectories, and reports more accurate, faster-converging performance than FTLEs and time averages across the considered examples. The method is subject to norm, integration-time, and dynamical-system scope conditions.

  • Problem

    The paper addresses the limited availability of methods for revealing geometrical structures in aperiodically time-dependent dynamical systems.

  • Method

    Lagrangian descriptors integrate bounded, positive intrinsic geometrical or physical properties along trajectories over finite time, using forward and backward integration.

  • Results

    Across the considered examples, Lagrangian descriptors more accurately reveal geometrical structures and require less time to converge than FTLEs and finite-time averages.

  • Takeaways & Limitations

    The method provides a general approach for revealing stable and unstable manifolds and other phase-space structures in aperiodically time-dependent flows.

  • Takeaways & Limitations

    The approach requires an appropriately long integration time, and norms with γ>1 are not useful as Lagrangian descriptors for revealing significant flow structures.

Abstract

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In this paper we develop new techniques for revealing geometrical structures in phase space that are valid for aperiodically time dependent dynamical systems, which we refer to as Lagrangian descriptors. These quantities are based on the integration, for a finite time, along trajectories of an intrinsic bounded, positive geometrical and/or physical property of the trajectory itself. We discuss a general methodology for constructing Lagrangian descriptors, and we discuss a "heuristic argument" that explains why this method is successful for revealing geometrical structures in the phase space of a dynamical system. We support this argument by explicit calculations on a benchmark problem having a hyperbolic fixed point with stable and unstable manifolds that are known analytically. Several other benchmark examples are considered that allow us the assess the performance of Lagrangian descriptors in revealing invariant tori and regions of shear. Throughout the paper "side-by-side" comparisons of the performance of Lagrangian descriptors with both finite time Lyapunov exponents (FTLEs) and finite time averages of certain components of the vector field ("time averages") are carried out and discussed. In all cases Lagrangian descriptors are shown to be both more accurate and computationally efficient than these methods. We also perform computations for an explicitly three dimensional, aperiodically time-dependent vector field and an aperiodically time dependent vector field defined as a data set. Comparisons with FTLEs and time averages for these examples are also carried out, with similar conclusions as for the benchmark examples.

1 Introduction

The paper develops Lagrangian descriptors to reveal phase-space structures in aperiodically time-dependent dynamical systems and compares them with FTLEs and time averages. Across the discussed examples, the method is presented as more accurate and computationally efficient, with faster convergence to relevant structures.

  • Method: Lagrangian descriptors extend trajectory arc-length methods by integrating bounded, positive intrinsic geometrical or physical properties over finite time.The construction is intended for aperiodically time-dependent flows.
  • Evaluation: The paper compares Lagrangian descriptors side by side with FTLEs and finite-time averages of vector-field components.The comparisons use benchmark systems and a wind-driven three-layer quasigeostrophic double-gyre simulation.
  • Benchmarks: For a linear saddle with analytically known stable and unstable manifolds, the paper uses exact structures to assess integration-time effects and compare competing methods.The benchmark also supports analytical discussion of singular contours and norm choices.
  • Results: In the double-gyre example, Lagrangian descriptors converge to stable and unstable structures more quickly than FTLEs.The comparison is made against the stable and unstable subspaces of a distinguished hyperbolic trajectory.

2 Lagrangian Descriptors: Definitions, Heuristics, and Analytical Results

Lagrangian descriptors are finite-time trajectory integrals of bounded, positive intrinsic properties, designed to expose abrupt changes associated with phase-space structures. Their interpretation depends on appropriate norms, integration times, and assumptions about the dynamical system and the regions being characterized.

  • Assumptions: The analysis assumes v(x,t) is C^r in x and continuous in t, with solutions existing long enough for the derived integral expressions to apply.Validity is subsequently checked in specific examples.
  • Definitions: The original descriptor M1 is the Euclidean arc length of a trajectory in phase space and depends on the initial point, time interval, and vector field.It is the first descriptor introduced before additional choices of integrand and norm.
  • Heuristic: The heuristic identifies boundaries between qualitatively different trajectory behavior through abrupt transverse changes in the descriptor, which can correspond to stable and unstable manifolds.These changes appear as singular curves in descriptor contour plots.
  • General construction: Lagrangian descriptors integrate a bounded, positive intrinsic physical or geometrical property along trajectories over a finite interval around a reference time.The trajectory starts at x* at t=t*, and the interval is (t*−τ,t*+τ).
  • Norm selection: Lγ norms with γ>1 are reported as ineffective for revealing significant underlying flow structures, unlike the successful γ≤1 choices discussed for the benchmark.The paper explores the reason for this limitation analytically.
  • Design choices: Positive accumulation along trajectories is essential to the heuristic, whereas integrals of sign-changing quantities are not covered by the argument.The paper summarizes several descriptor choices using different positive integrands and norms.
  • Scope and terminology: Interpreting hyperbolic and elliptic regions is difficult for general systems because stability types are infinite-time properties requiring invariant regions.The paper therefore mainly uses these phrases for linear and integrable systems, where the issue is less problematic.

2.1 Lagrangian Descriptors in Hyperbolic Regions

In hyperbolic regions, Lagrangian descriptors reveal stable and unstable manifolds through contour singularities whose visibility depends on integration time and norm choice. Analytical calculations for a linear saddle explain these features and their limitations.

  • Lagrangian descriptors diagnose hyperbolic regions by revealing trajectories and stable and unstable manifolds, alongside FTLEs and time averages.
  • 2.1.1 The Linear Saddle Point: For the linear saddle, M1 contours evolve from smooth patterns at τ = 0.5 toward derivative discontinuities along the manifolds as τ increases to 10.The contours at τ = 2 provide an intermediate pattern before convergence toward the manifold structure.
  • 2.1.1 The Linear Saddle Point: For λ = 1, M1 = M2 = M4, while M3 with γ = 2 loses the invariant-manifold structure and M5 clearly reveals the manifolds.The paper attributes M5's performance to the zero curvature of the stable and unstable manifold trajectories.
  • 2.1.1 The Linear Saddle Point: Constant Lyapunov-exponent contours reveal no structure in the linear saddle because the exponents are ±λ for almost every trajectory.Trajectories on either manifold are the exception, with one Lyapunov exponent equal to zero.
  • 2.1.2 Analytical Justification: The analytical justification explains why Lγ norms with γ > 1 do not produce singular contour derivatives, whereas γ < 1 produces singular derivatives across the stable manifold.For γ = 1, the construction corresponds to the arc-length descriptor M1.
  • 2.1.2 Analytical Justification: For small τ, the M1 expansion is smooth near the stable manifold and accurately matches the numerical evaluation over x0 ∈[−0.01, 0.01] when y0 = 0.5 and τ = 2.The comparison uses the numerical expression (12) and its leading-order terms (13).
  • 2.1.2 Analytical Justification: For large τ, leading-order terms accurately approximate M1 over x0 ∈[−0.1, 0.1], while the derivative appears discontinuous on the stable manifold.At finite τ, M remains regular, but its asymptotic approximation is accurate only in an exponentially narrowing gap around the manifold.

2.2 Lagrangian Descriptors in Elliptic Regions

In elliptic regions, Lagrangian descriptors represent invariant circles and shear, with their behavior depending on the trajectory property used in the integrand. For the integrable Hamiltonian example, M1 detects fixed-point circles and twistless tori, whereas acceleration-based M3 becomes uninformative when acceleration vanishes.

  • 2.2 Lagrangian Descriptors in Elliptic Regions: The linear elliptic example introduces the study of elliptic-region behavior, while the integrable Hamiltonian example provides insights into invariant tori and shear.The latter is presented as a simple nonlinear system with entirely elliptic behavior.
  • 2.2.1 The Linear Elliptic Point: At the linear elliptic fixed point, M1 has smooth circular contours surrounding the origin for every fixed τ, except at the origin.For this example, M1 and M2 are equal.
  • 2.2.2 An Integrable Hamiltonian System: In the integrable Hamiltonian system, constant action I defines invariant circles, whose frequencies and shear vary across tori.The exceptional circles I = 0 and I = ±1 are fixed-point circles because their frequency is zero; twistless circles have zero shear.
  • 2.2.2 An Integrable Hamiltonian System: For τ = 25, M1 detects three fixed-point circles and high-shear twistless tori; τ = 75 provides essentially the same information.The fixed-point circles appear dark blue, while twistless tori appear in reddish colors.
  • 2.2.2 An Integrable Hamiltonian System: Because all trajectories in this system have constant velocity and zero acceleration, acceleration-based M3 is identically zero and shows a flat structure for every τ > 0.Consequently, M3 has no convergence threshold in τ for this example.

2.3 Remarks on the Application of Finite Time Lyapunov Exponents (FTLEs) and Time Averages for These Examples

The paper compares Lagrangian descriptors with FTLEs and finite-time averages for revealing phase-space structures. FTLEs face finite-time, discretization, and manifold-identification issues, while finite-time averages lack a clear general interpretation and can obscure material curves.

  • Finite Time Lyapunov Exponents (FTLEs): FTLEs are computed by integrating a spatial grid forward or backward for a fixed time and assigning each initial point its maximal finite-time Lyapunov exponent.
  • Finite Time Lyapunov Exponents (FTLEs): FTLE analysis lacks theoretical guidance for choosing the finite time or determining how closely finite-time values approximate asymptotic exponents.Convergence may be relatively slow, and trajectories can change their finite-time stability type during evolution.
  • Finite Time Lyapunov Exponents (FTLEs): Numerical FTLE fields can be noisy, so obtaining smooth level curves generally requires subjective filtering and smoothing.
  • Finite Time Lyapunov Exponents (FTLEs): Locally maximal FTLE level curves are not generally material curves; precisely defined ridge curves may approximate them but can have small, nonzero flux.The literature also contains ambiguity between ridge curves and locally maximal level curves.
  • Time Averages: Finite-time averages integrate selected functions along trajectories, but their phase-space meaning is not a priori clear because no rigorous general framework is available.The approach is related to ergodic decomposition, whose Birkhoff-theorem foundation has not been proven for general aperiodic time dependence.
  • Time Averages: In the benchmark Hamiltonian system, the finite-time horizontal-velocity average changes sign across the stable manifold but remains smooth there and reveals neither the manifold nor invariant sets.Oscillating integrands can distort the output; nonnegative integrands are an essential requirement for Lagrangian descriptors.

2.4 Application to the Forced Duffing Oscillator

The forced Duffing examples test Lagrangian descriptors across integrable, periodically forced, and aperiodically forced dynamics. Their singular contours reveal hyperbolic manifolds, while smooth and complex contour regions distinguish elliptic structure and mixing more clearly than comparison methods.

  • Cases: The study compares Lagrangian descriptors with FTLEs and velocity-component time averages for three Duffing time dependencies.The cases are integrable, periodically forced with ε = 0.1 and f(t) = sin(t), and aperiodically forced with ε = 0.15.
  • Hyperbolic regions: Abrupt changes in M1 occur at manifold locations, where the derivative transverse to the manifolds is discontinuous.Figure 11 uses the derivative of M1 to quantify these abrupt changes.
  • Hyperbolic regions: For the aperiodic case, M1 contours computed for τ = 10 align well with stable and unstable manifolds near the origin.The same correspondence is reported for the hyperbolic trajectory in Figure 12.
  • Comparison with FTLE: M1 singular contours approximate Duffing manifolds better than forward FTLE contours, which capture only the stable manifold and include non-Lagrangian ridges.The comparison covers the integrable, periodically forced, and aperiodically forced cases at τ = 10.
  • Elliptic regions: In the periodically forced elliptic region, M5 and M2 are smooth near the centre, while contours outside it become more complex and are associated with strong mixing.The contours are computed for τ = 70; the smooth centre is consistent with preserved two-frequency KAM tori.
  • Elliptic regions: In the aperiodic case, the trapped elliptic region is invaded by segments of stable and unstable manifolds near the origin.This behavior is shown using M3 and M4, with additional detail visible in M3 contours.

3 Applications to time dependent 3D flows

The paper evaluates Lagrangian descriptors in a perturbed Hill’s spherical vortex, an explicitly three-dimensional, aperiodically time-dependent flow. Descriptor cross-sections agree excellently with unstable manifolds and produce cleaner structure than FTLEs and velocity averages.

  • Benchmark flow: The perturbed Hill’s spherical vortex is used as a three-dimensional benchmark with aperiodic time dependence.The section assesses whether Lagrangian descriptors accurately reveal stable and unstable manifolds of hyperbolic trajectories in 3D flows.
  • Unstable manifold: M2 cross-sections at y = −0.3 and z = 0 agree excellently with intersections of the unstable manifold.Both cross-sections are computed at t = 5.3 for τ = 10.
  • Descriptor fields: M2 reveals detailed, clean, and sharp Lagrangian structure, while M1 gives similar output with lower contrast.The comparison is made on the vertical section at t = 5.3 and τ = 10.
  • Comparison methods: Forward and backward FTLEs introduce numerous structures without Lagrangian interpretation, while velocity averages add spurious structures and blur the true information.These comparisons are shown on the same vertical plane used for the descriptor fields.

4 Application of Lagrangian Descriptors to Velocity Fields Defined as Data Sets

The data-set application uses an aperiodic three-layer quasigeostrophic double-gyre flow with previously identified distinguished hyperbolic trajectories. Lagrangian descriptors reveal their manifolds and subspaces without prior trajectory identification, with M3 converging fastest in the tested cases.

  • Data-set flow: The application uses finite-time velocity data from a wind-driven, three-layer quasigeostrophic model with an unsteady double-gyre circulation.The data set is aperiodic and supports comparisons with previously computed DHTs and invariant manifolds.
  • Manifold comparison: Contour plots of M3, M1, M5, and M2 are compared with directly computed stable and unstable manifolds at days 290 and 320.The Lagrangian descriptors use τ = 150, and axes are measured in thousands of kilometers.
  • Methodological distinction: Lagrangian descriptors identify stable and unstable manifolds for all DHTs present during a chosen interval without requiring prior DHT identification.Direct manifold computation instead requires the DHTs to be identified and computed first.
  • Subspace convergence: Among the tested descriptors, M3 indicates the stable and unstable subspaces for the smallest τ, followed by M1, M2, and M5; M4 requires longer intervals.The ordering is reported for the situations considered in this velocity field.
  • Subspace convergence: M3 develops sharp features at the stable and unstable subspace angles for τ = 45, while FTLEs provide the comparison fields.Figure 22 marks the reference subspace angles with vertical lines and plots M3 against forward and backward FTLEs.

5 Computational performance

Lagrangian descriptors are computationally simpler than FTLEs because they require direct trajectory-property integration rather than flow-map gradients and neighboring trajectories. They also show stable and unstable manifolds simultaneously without the same post-processing burden.

  • Computational requirements: FTLE computation requires the gradient of the flow map, whereas Lagrangian descriptors integrate the selected quantity along trajectories.In two-dimensional flows, the FTLE gradient calculation uses neighboring points and requires tuning grid spacing against integration time.
  • Computational requirements: Lagrangian descriptors avoid the FTLE method’s grid-spacing and integration-time tuning needed to maintain accurate flow-map gradients.The FTLE calculation also has greater programming complexity when neighboring trajectories are stored and reused.
  • Computational requirements: Four integrations are required per point for the FTLE approach described, compared with one integration for a Lagrangian descriptor.This difference is especially important for geophysical data sets, where trajectory integration requires expensive interpolation.
  • Output interpretation: Lagrangian descriptors provide stable and unstable manifolds simultaneously, whereas FTLEs require post-processing into one image.FTLE post-processing commonly filters field values with a threshold, potentially removing spurious structures.

6 Conclusions and Outlook

The paper proposes Lagrangian descriptors as finite-time integrals of positive, bounded intrinsic properties for revealing phase-space geometry in aperiodically time-dependent systems. Across examples, they reveal manifolds more accurately and converge faster than FTLEs and finite-time averages.

  • Lagrangian descriptors use finite-time integrals of positive, bounded intrinsic geometrical or physical properties to reveal phase-space structures.
  • The method can reveal stable and unstable manifolds in the same calculation.
  • A sufficiently long integration time τ is required; beyond a system-dependent threshold, increasing τ reveals increasingly detailed geometrical structure.
  • Compared with FTLEs and finite-time component averages, Lagrangian descriptors more accurately reveal structures and require shorter convergence times across the examples considered.

A Numerical computation of Lagrangian descriptors

The numerical construction integrates a chosen positive bounded scalar along trajectories over a finite time interval. The appendix describes trajectory-based evaluation, derivative construction, numerical integration, and finite-difference implementations for analytic fields and data sets.

  • A general descriptor integrates a positive bounded scalar F(x(t)) representing an intrinsic geometrical or physical property along a trajectory.
  • M1 is computed by integrating the velocity norm ∥v(x(t), t)∥ from t*−τ to t*+τ along a trajectory initialized at x(t*) = x*.
  • Geometrically, M1 equals the area under the graph of the velocity norm over the specified time interval.
  • The area can be obtained from an auxiliary variable Y as Y(t*+τ) − Y(t*−τ).
  • Velocity, acceleration, and the time derivative of acceleration are obtained separately before forming combinations for specific descriptors.

B Computation of FTLE’s

The FTLE appendix computes stretching from the flow map and its gradient. The resulting matrix construction supplies the maximum eigenvalue used to define the finite-time Lyapunov exponent.

  • The FTLE field στ is computed from the maximum eigenvalue λmax of a matrix Δ.
  • The flow map advects an initial point x* from time t* to t*+τ, providing the basis for constructing Δ.
  • The gradient of the flow map defines the matrix N used in the FTLE calculation.
  • Using N^T as the transpose of N, Δ is formed as a symmetric matrix.
  • In two dimensions, the flow-map gradient at a grid point can be computed with central differences.

C Regularity of Lagrangian descriptors

For velocity fields supplied as data sets, interpolation regularity affects descriptor quality. Velocity-based descriptors are the most regular and are preferred over higher-derivative descriptors when interpolation is not sufficiently smooth.

  • Discrete velocity data must be interpolated in space and time to compute particle trajectories.
  • The described interpolation uses bicubic spatial interpolation and third-order Lagrange temporal polynomials, yielding C1 spatial and C0 temporal continuity.
  • Descriptors are sensitive to interpolation quality, especially those involving the time derivative of acceleration because they require second-order spatial velocity derivatives.
  • M2 is more regular than the time-derivative-of-acceleration descriptor because its regularity follows that of acceleration.
  • FTLE computations gain regularity through integration of the linearized velocity field, but their performance is similar to acceleration-based descriptors.
  • For data sets with insufficiently regular interpolation, velocity-based Lagrangian descriptors are preferred, including over FTLEs.
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