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On the state space geometry of the Kuramoto-Sivashinsky flow in a periodic domain

Predrag Cvitanović, Ruslan L. Davidchack, Evangelos Siminos

arXiv:0709.2944v3nlin.CD

TL;DR

The paper asks how to characterize turbulent dynamics through its state-space patterns and geometry. It studies invariant solutions and symmetry-aware visualizations in the L = 22 Kuramoto-Sivashinsky system, finding a heteroclinic cage and many unstable relative periodic orbits that may organize turbulence.

  • Problem

    The paper seeks to characterize turbulent flows as trajectories through state space, including the patterns they follow and the geometry and natural measure of their dynamics.

    Source: §1 · p. 1

  • Method

    For the L = 22 Kuramoto-Sivashinsky system, the authors triangulate invariant solutions and unstable manifolds and visualize flow using invariant coordinates and physical observables.

    Source: §1 · pp. 2–3

  • Results

    The study finds a rigid heteroclinic cage between equilibria and a preponderance of unstable relative periodic orbits, many organized near the unstable manifold of E2.

    Source: §1 · p. 3, §9 · p. 23

  • Takeaways & Limitations

    Unstable relative periodic orbits are proposed as likely skeletons underpinning spatiotemporal turbulence in systems with continuous symmetries.

    Source: §1 · p. 3

    The L = 22 system was chosen as the smallest size empirically exhibiting sustained turbulence, but much of its observed dynamics is specific to the imposed periodicity.

    Source: §9 · p. 23

Abstract

from arXiv · show

The continuous and discrete symmetries of the Kuramoto-Sivashinsky system restricted to a spatially periodic domain play a prominent role in shaping the invariant sets of its chaotic dynamics. The continuous spatial translation symmetry leads to relative equilibrium (traveling wave) and relative periodic orbit (modulated traveling wave) solutions. The discrete symmetries lead to existence of equilibrium and periodic orbit solutions, induce decomposition of state space into invariant subspaces, and enforce certain structurally stable heteroclinic connections between equilibria. We show, on the example of a particular small-cell Kuramoto-Sivashinsky system, how the geometry of its dynamical state space is organized by a rigid `cage' built by heteroclinic connections between equilibria, and demonstrate the preponderance of unstable relative periodic orbits and their likely role as the skeleton underpinning spatiotemporal turbulence in systems with continuous symmetries. We also offer novel visualizations of the high-dimensional Kuramoto-Sivashinsky state space flow through projections onto low-dimensional, PDE representation independent, dynamically invariant intrinsic coordinate frames, as well as in terms of the physical, symmetry invariant energy transfer rates.

1. Introduction.

The paper studies turbulence as motion through an infinite-dimensional state space, using invariant solutions and their unstable manifolds to describe its geometry. For the L = 22 Kuramoto-Sivashinsky system, it examines how continuous symmetries and relative periodic orbits shape that geometry.

1. Introduction.

  • The state-space approach characterizes turbulence through the geometry of states and the natural measure describing how often asymptotic dynamics visits each region.It treats turbulent evolution as a trajectory rather than a sequence of spatial snapshots.Source: §1 · p. 1
  • The dynamical-systems account is motivated by wall-bounded shear flows, where unstable coherent structures dominate and statistical assumptions fail.The authors use “chaos,” “spatiotemporal chaos,” and “turbulence” interchangeably for lack of more precise nomenclature.Source: §1 · p. 2
  • The study triangulates invariant solutions and their unstable manifolds in a PDE-representation-independent way, with state-space dimension d ∼102 chosen for solution accuracy of ∼10−5.The goal is quantitative description, not qualitative modeling by a low-dimensional model.Source: §1 · p. 2
  • With continuous symmetry, exact temporal periodicity gives way to relative spatiotemporal periodicity, and relative equilibria and relative periodic orbits take the roles of equilibria and periodic orbits.The paper reports determining 60,000 periodic orbits and relative periodic orbits, many not small modulations of other solutions.Source: §1 · pp. 2–3
  • L = 22 is small enough for detailed exploration of equilibria, manifolds, and relative periodic orbits, yet exhibits features typical of turbulence in larger KS systems.The authors note that much of the observed dynamics is specific to this unphysical, externally imposed periodicity.Source: §1 · pp. 2–3
  • The paper presents invariant, symmetry-independent coordinate projections and physical energy-transfer observables, a heteroclinic cage between equilibria, and a preponderance of unstable relative periodic orbits.These orbits are proposed as likely skeletons underpinning spatiotemporal turbulence in systems with continuous symmetries.Source: §1 · p. 3

2. Kuramoto-Sivashinsky equation.

For periodic Kuramoto–Sivashinsky flow, continuous and discrete symmetries shape the invariant solutions and subspaces. The section describes these symmetries and the equilibria, relative equilibria, and periodic orbits they permit.

2. Kuramoto-Sivashinsky equation.

  • At L=22, truncations N=32, 64, and 128 preserve the instability of all studied equilibria and relative equilibria, while nearly all found relative periodic orbits persist.The paper selects this system size for its structurally stable chaotic attractor; long-time solutions are accurately represented using 16–128 Fourier modes, though asymptotic dynamics can change with truncation or system size.Source: §2 · pp. 3–4

2.1. Symmetries of Kuramoto-Sivashinsky equation.

  • Translations and reflections generate continuous and discrete symmetries; discrete symmetries split the state space into invariant subspaces, including the half-cell-shift-symmetric U_S.The KS flow remains within U_S = U(1) + U(2), although the nonlinear term generally mixes the irreducible subspaces.Source: §2.1 · pp. 4, 6

2.2. Equilibria and relative equilibria.

  • At the studied size L=22, the equilibria are E0, E1, E2, and E3, alongside counter-traveling pairs TW±1 and TW±2.As system size increases, n-cell equilibria emerge through pitchfork bifurcations from the zero solution; the equilibria form invariant circles under translation in the full state space.Source: §2.2 · p. 7

2.3. Relative periodic orbits, symmetries and periodic orbits.

  • A relative periodic orbit returns to its initial profile after a time period and a spatial shift, so it is periodic in a co-moving frame but quasiperiodic in the stationary frame.Reflection symmetry gives each such orbit a partner with the opposite shift.Source: §2.3 · pp. 8–9, §2.2 · p. 8
  • At L=22, no periodic orbits were found inside the Dn- or Cn-invariant subspaces because U+ dynamics is dominated by attracting heteroclinic connections.The cited earlier periodic orbits lie in U+, while this system's subspace dynamics instead follows those heteroclinic connections.Source: §2.3 · p. 9
  • All periodic orbits found at L=22 are reflection-type pre-periodic orbits: their shift over the first period is canceled by the opposite shift over the second.They return to their initial state after 2Tp.Source: §2.3 · p. 9

3. Energy transfer rates.

The section uses physical observables that are invariant under translations and reflections to describe KS dynamics independently of its ODE representation. Energy transfer rates balance on average, and averages of invariant solutions and orbits can be compared in these coordinates.

Symmetry-invariant observables

  • Translation- and reflection-invariant physical observables provide a symmetry-quotiented representation for visualizing KS solutions.The energy is intrinsic to the flow and independent of the ODE basis used to represent the PDE.Source: §3 · pp. 10–11

Energy transfer rates

  • The energy variation balances power pumped by anti-diffusion against dissipation by hyper-viscosity.The energy density on a typical orbit approaches a constant, so mean drive and dissipation balance.Source: §3 · p. 11

Orbit averages

  • Equilibria, relative equilibria, and periodic- or relative-periodic-orbit time averages fall on the diagonal in Figure 8.1(a).The passage identifies the plotted solution types and their shared diagonal relation.Source: §3 · p. 11

Energy in Fourier space

  • Large-wavenumber convergence of the energy series is insensitive to system size, while deviations at small wavenumbers determine active modes.The best current long-time energy bound cited in the passage scales as E ∝ L^2.Source: §3 · p. 11

4. Geometry of state space with L = 22.

For the L = 22 KS system, competition between wavenumbers 2 and 3 accompanies a chaotic attractor, while the state-space analysis examines invariant solutions and their manifolds. The dynamics is largely confined near an eight-dimensional manifold, but sustained chaos cannot be established rigorously for all time.

Geometry of state space with L = 22

  • The L = 22 system’s wavenumber-2/3 competition accompanies a chaotic attractor, though sustained chaos cannot be established for all time.The attractor could instead be a long transient on the way to an attracting periodic state; small-system attractor structure is sensitive to parameter variation.Source: §4 · p. 11
  • The L = 22 chaotic dynamics takes place mostly near an eight-dimensional manifold, with strong contraction in other dimensions.Two zero Lyapunov exponents correspond to time and space translation symmetries; 62–126 real Fourier modes suffice for invariant solutions accurate within 10^-5.Source: §4 · p. 12
  • The investigation determines a large set of short-period relative periodic orbits and examines equilibria and relative equilibria for L = 22.The study also considers unstable solutions and associated Smale horseshoe structures as sources of insight into observed unstable coherent structures.Source: §4 · p. 12

5. Equilibria and relative equilibria for L = 22.

For L = 22, the paper identifies equilibria and traveling waves, then traces their unstable manifolds to reveal heteroclinic and homoclinic connections that organize the flow.

5. Equilibria and relative equilibria for L = 22.

  • E1, E2, and E3 are reflection-symmetric equilibria; E2 and E3 also have L/2 and L/3 translation symmetries, respectively.Their dominant wavenumbers are 1, 2, and 3; E0 is the zero solution.Source: §5 · p. 12
  • At L = 22, E0 has three pairs of unstable eigenvalues, ordered by modes k = 2, 3, and 1; eigenvectors provide flow-intrinsic coordinates for visualizing manifolds.The leading eigenvalues and eigenvector symmetries are tabulated for equilibria and relative equilibria.Source: §5 · p. 13, Table 5.1 · p. 15
  • Two pairs of traveling waves have velocities c = ±0.73699 and ±0.34954, and slightly perturbed profiles decay into the chaotic attractor.These relative equilibria are labeled TW±1 and TW±2.Source: §5 · p. 13, Figure 5.3 · p. 16

5.1. Unstable manifolds of equilibria and their heteroclinic connec-

  • E1’s first unstable plane appears to enter the chaotic attractor, while almost all trajectories in its second plane converge to E2, including a connection from E1 to E3.E1 has two spiraling-out unstable planes; its larger unstable subspace is expected to give it less influence on long-time dynamics than E2 and E3.Source: §5.1 · pp. 14–16
  • Except for connections to E3, trajectories in E2’s two-dimensional unstable manifold converge to E2 shifted by L/4, forming a homoclinic connection.The E2-to-E3 heteroclinic connections are the exception to this convergence.Source: §5.1 · p. 16
  • E3’s unstable manifold connects to three translated E2 equilibria, while symmetry-invariant subspaces support heteroclinic loops between L/4-shifted E2 states.The shared invariant subspaces make τ1/4E2 a sink on U+ and explain connections from E1, E2, and E3 to it.Source: §5.1 · pp. 17–19

6. Relative periodic orbits for L = 22.

The L = 22 search finds many relative periodic orbits and relates their periods and shifts to the geometry of unstable manifolds. Shorter orbits capture coarse attractor structure, while the equilibrium cage helps organize the dynamics.

6. Relative periodic orbits for L = 22.

  • Over 30,000 relative periodic orbits with ℓp > 0 were found for Tp < 200, and each has a reflection-symmetric partner with shift −ℓp.The partner is up(x) → −up(−x).Source: §6 · p. 20
  • Short-period relative periodic orbits outline the attractor’s coarse structure, while longer-period orbits resolve finer dynamics.The shortest orbit, Tp = 16.4, is the most unstable, with one positive Floquet exponent of 0.328.Source: §6 · p. 20
  • Relative periodic orbits near E2’s unstable manifold have shifts ℓp ≈ L/4, supporting the role of the equilibrium cage in organizing chaotic dynamics.Orbits within E2’s unstable manifold also have this shift.Source: §6 · pp. 20–21

7. Pre-periodic orbits.

The paper distinguishes pre-periodic orbits that return after a discrete symmetry from relative periodic orbits with spatial shifts. For L = 22, the search finds many reflection-symmetric pre-periodic solutions.

7. Pre-periodic orbits.

  • A relative periodic orbit is an ordinary periodic orbit when it lies in U+ or returns to a reflection or discrete rotation after one period.In the latter case, it is periodic with period mTp.Source: §7 · p. 21
  • The L = 22 flow has no periodic orbits of the U+ antisymmetric type.Prior studies investigated the dynamics in that subspace.Source: §7 · p. 21
  • The search found over 30,000 pre-periodic orbits with Tp < 200 and reflection symmetry, satisfying the type-(b) condition with γ = R ∈ D1.Some were repeats encountered during relative-periodic-orbit searches; most were found by solving the reflection pre-periodic condition.Source: §7 · p. 21

8. Energy transfer rates for L = 22.

The energy-transfer section visualizes equilibria, relative solutions, and turbulent trajectories through energy and power observables. It cautions that these low-dimensional projections can hide differences between states and that reliable averages require a hierarchy of orbits.

8. Energy transfer rates for L = 22.

  • For L = 22, the (P, D) and (E, P) plots include equilibria, relative equilibria, a relative periodic orbit, and a typical turbulent trajectory.In the (P, D) plane, equilibria and relative equilibria, as well as time averages on periodic and relative periodic orbits, lie on the diagonal.Source: §8 · p. 21
  • Three-dimensional (E, P, D) projections can mislead: clearly separated points remain separated in the full state space, but states with different topology can have similar energies.The E1 equilibrium is given as an example of a point outside the recurrent set.Source: §8 · pp. 21–22
  • A single relative periodic orbit can approximate the long-time mean power, but accurate predictions require a hierarchy of increasingly long periodic orbits.The example orbit has (Tp, ℓp) = (32.8, 10.96), and its mean power is numerically close to the turbulent time average.Source: §8 · p. 22
  • Each relative periodic orbit has its own mean-velocity frame, making multiple such orbits difficult to visualize together.In that frame, an individual relative periodic orbit becomes periodic.Source: §8 · p. 22

9. Summary.

For the L = 22 Kuramoto-Sivashinsky flow, the paper maps how continuous and discrete symmetries shape recurrent solutions and organize state-space dynamics. It identifies many unstable relative periodic orbits, proposes visualizations of the flow, and cautions that much observed dynamics is specific to the imposed periodicity.

9. Summary.

  • The L = 22 system is the smallest size where KS empirically exhibits sustained turbulence, and the paper offers a detailed geometrical portrait of its state space.The study uses KS to test dynamical-systems approaches to moderate-Reynolds-number turbulence in an infinite-dimensional PDE setting.Source: §9 · p. 22
  • The relative periodic orbit with (Tp, ℓp) = (33.5, 4.04) is shown over four periods in a stationary projection and in a co-moving mean-velocity frame.The stationary projection uses the v1, v2, and v3 axes from Figure 5.6.Source: Figure 8.3 · p. 23
  • A large number of unstable relative periodic orbits and periodic orbits were determined numerically, alongside new insights into continuous and discrete symmetries and equilibrium connections.The paper identifies continuous families of traveling waves and relative periodic orbits in the translation-invariant system.Source: §9 · pp. 22–23
  • Two visualizations project the flow onto PDE-independent, dynamically invariant frames and onto physical, symmetry-invariant energy-transfer rates.The authors describe the dynamics as low-dimensional local unstable manifolds connected by fast transient interludes.Source: §9 · p. 23
  • Many relative periodic orbits appear organized by the unstable manifold of E2, following the homoclinic loop between E2 and τ1/4E2.Relative periodic orbits require a reformulation of periodic orbit theory, an issue left for future work.Source: §9 · p. 23
  • Much of the observed L = 22 dynamics is specific to the externally imposed periodicity, leaving the nature of solutions as L → ∞ unresolved.The paper identifies the structure of the L = ∞ periodic orbit theory as an open question.Source: §9 · p. 23
  • Equilibria, relative equilibria, periodic orbits, and relative periodic orbits together form the repertoire of recurrent spatiotemporal patterns explored by turbulent dynamics.The paper presents this as embodying Hopf’s vision.Source: §9 · p. 23

Appendix A. Integrating Kuramoto-Sivashinsky equation numerically.

The appendix describes a Fourier-mode representation and truncation of the Kuramoto-Sivashinsky equation, then outlines numerical integration and tangent-space calculations used to study its flow.

Appendix A. Integrating Kuramoto-Sivashinsky equation numerically.

  • Because the state is real, its Fourier modes satisfy the conjugate relation ˆu−k = ˆu∗k.Source: Appendix A · p. 24
  • The Fourier-mode system is truncated to N − 2 independent variables, using a discrete Fourier transform and the restrictions a0 = 0 and aN/2 = 0.The zero mode follows from Galilean invariance, and the Nyquist mode is set to zero assuming even N.Source: Appendix A · p. 24
  • A displacement vector in the tangent space is evolved alongside the state to obtain the fundamental matrix or compute Lyapunov exponents.The tangent-space equation is solved simultaneously with the state equation.Source: Appendix A · p. 24
  • Equations (A.4) and (A.6) are solved with the fourth-order exponential time-differencing Runge-Kutta method, ETDRK4.Source: Appendix A · p. 25

Appendix B. Determining stability properties of equilibria, traveling waves, and relative periodic orbits.

The appendix characterizes stability through the flow’s fundamental matrix, adapting the calculation for periodic orbits, relative periodic orbits, equilibria, and traveling waves.

Appendix B. Determining stability properties of equilibria, traveling waves, and relative periodic orbits.

  • The fundamental matrix is built from derivatives of the flow map with respect to the real and imaginary parts of the complex state vector.Source: Appendix B · p. 25
  • Periodic-orbit stability is determined by whether the eigenvalues of J(ap, Tp) lie inside or outside the unit circle.Source: Appendix B · p. 25
  • For a relative periodic orbit, translation symmetry changes the stability matrix to g(ℓp) J(ap, Tp), where g(ℓp) is the translation action.Source: Appendix B · p. 25
  • Equilibrium stability is classified by the signs of the real parts of the eigenvalues of A(aq).Source: Appendix B · p. 26
  • Traveling-wave stability is evaluated in a co-moving frame, with stability matrix A(aq) + cqL and L the Lie algebra translation generator.Source: Appendix B · p. 26

Appendix C. Levenberg–Marquardt searches for relative periodic orbits.

The appendix details a multiple-shooting search using Levenberg–Marquardt, with close returns in the chaotic attractor as seeds. The search detected over 30,000 each of relative periodic and pre-periodic orbits, while tests at higher numerical resolution rejected only a handful.

Appendix C. Levenberg–Marquardt searches for relative periodic orbits.

  • The search uses multiple shooting with the Levenberg–Marquardt algorithm implemented in MINPACK’s lmder.Source: Appendix C · p. 26
  • The authors omit added constraints because LM can solve an underdetermined system; they report numerical evidence that added constraints often increase steps or prevent convergence.Source: Appendix C · p. 26
  • The search solves N − 2 equations with N unknowns for relative periodic orbits, and uses a corresponding system for pre-periodic orbits.Source: Appendix C · p. 26
  • A starting guess comes from a close return in the chaotic attractor, with integration steps rounded so the initial step size does not exceed h0.The shooting system divides the integration into stages and adjusts the step size to reach the target period.Source: Appendix C · p. 27
  • Seeds are generated by integrating random points for a transient and minimizing the return distance over time and spatial shift within period windows from 10 to 200.The period range is divided into windows of length 10, and the minimum is searched on a grid.Source: Appendix C · pp. 28–29
  • Over 30,000 distinct prime relative periodic orbits with ℓ > 0 and over 30,000 distinct prime pre-periodic orbits with T < 200 were found.Only a handful of orbits failed a higher-resolution test using N = 64 and h = 0.1 and were excluded from the list of 60,000+ detections.Source: Appendix C · p. 29
  • The hit rate declines from about 70% for windows with tmax ≤ 80 to about 30% for windows with tmin ≥ 160.The search generated approximately equal numbers of seeds for relative periodic and pre-periodic orbits in each window.Source: Appendix C · p. 29
  • Detected relative periodic and pre-periodic orbit counts are approximately equal and grow exponentially with T up to about 130, so searches mostly miss orbits with T > 130.Fits over T ∈ [70, 120] suggest totals could exceed 10^5 each for T < 200.Source: Figure C.1 · p. 28, Appendix C · p. 29
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