Resonances in a Chaotic Attractor Crisis of the Lorenz Flow

Abstract

Local bifurcations of stationary points and limit cycles have successfully been characterized in terms of the critical exponents of these solutions. Lyapunov exponents and their associated covariant Lyapunov vectors have been proposed as tools for supporting the understanding of critical transitions in chaotic dynamical systems. However, it is in general not clear how the statistical properties of dynamical systems change across a boundary crisis during which a chaotic attractor collides with a saddle. This behavior is investigated here for a boundary crisis in the Lorenz flow, for which neither the Lyapunov exponents nor the covariant Lyapunov vectors provide a criterion for the crisis. Instead, the convergence of the time evolution of probability densities to the invariant measure, governed by the semigroup of transfer operators, is expected to slow down at the approach of the crisis. Such convergence is described by the eigenvalues of the generator of this semigroup, which can be divided into two families, referred to as the stable and unstable Ruelle–Pollicott resonances, respectively. The former describes the convergence of densities to the attractor (or escape from a repeller) and is estimated from many short time series sampling the state space. The latter is responsible for the decay of correlations, or mixing, and can be estimated from a long times series, invoking ergodicity. It is found numerically for the Lorenz flow that the stable resonances do approach the imaginary axis during the crisis, as is indicative of the loss of global stability of the attractor. On the other hand, the unstable resonances, and a fortiori the decay of correlations, do not flag the proximity of the crisis, thus questioning the usual design of early warning indicators of boundary crises of chaotic attractors and the applicability of response theory close to such crises.

Appendices

Appendix A: Transfer Operators, Resonances and Decay of Correlations

Let us shortly review the spectral theory of transfer operators relevant for this study. A more detailed exposition can be found in [43] and references therein.

1.1 Ergodicity, Mixing and Correlations

A key concept in ergodic theory is that of an invariant measure for the flow $\varPhi _t$, that is, a probability measure $\mu$ such that

$$\begin{aligned} \mu (\varPhi _t^{-1} A) = \mu (A), \quad \mathrm {for~any~} A \in \mathcal {B}. \end{aligned}$$

(A.1)

In other words, a measure is invariant if, according to this measure, the probability of a state to be in some set does not change as this state is propagated by the flow. It follows then that the average with respect to the invariant measure $\mu$ of any integrable observable g is also invariant with time, i.e.

$$\begin{aligned} \int _X g(\varPhi _t x) \mu (d x) = \int _X g(x) \mu (d x) := \langle g\rangle _\mu , \end{aligned}$$

(A.2)

so that the invariant measure gives a statistical steady-state.

A flow $\varPhi _t$ with an invariant measure $\mu$ has interesting statistical properties when $\mu$ is ergodic, that is, when the sets A which are invariant, i.e. $\varPhi _t^{-1} A = A$, are either of measure 0 or 1.¹¹ Then, by the celebrated individual ergodic theorem of Birkhoff [24, Chap. 7.3], the average of any $\mu$-integrable observable g is such that

$$\begin{aligned} \langle g\rangle _\mu = \lim _{T \rightarrow \infty } \frac{1}{T} \int _0^T g(\varPhi _t x) dt := \bar{g}, \quad \mathrm {for ~} \mu \mathrm {-almost~every~} x. \end{aligned}$$

(A.3)

Thus, when $\mu$ is ergodic, the time mean is independent of the initial state x except for a set of null measure.

Remark 2

There may exist many ergodic measures. This is actually the case for the Lorenz flow for $\rho _{\mathrm {Homo}}< \rho < \rho _A$, where, as seen in Sect. 2, three attractors coexist. Each attractor then supports at least one invariant measure [65, Chap. 4]. In this case, the equality (A.3) between ensemble averages and time averages may hold only for initial states belonging to the attractor supporting the measure, while time averages for two initial conditions in different basins of attraction will not coincide in general. More useful for experiments is then the eventual physical property of the measure. The latter ensures that the equality (A.3) between ensemble averages and time averages holds not only for initial states in a set of positive measure $\mu$, but also for states in any set of positive Lebesgue measure m in the basin of attraction of a given attractor [12].

A particularly important quantity in ergodic theory is the correlation function,

$$\begin{aligned} C_{f,g}(t) := \int _X f(x) g(\varPhi _t x) \mu (dx) - \langle f\rangle _\mu \langle g\rangle _\mu , \quad t \ge 0, \end{aligned}$$

(A.4)

between any observables $f, g \in L^2_\mu (X)$. It gives a measure of the relationship between the two observables as one evolves with time. The decay of correlation functions with time is a macroscopic manifestation of chaos. Indeed, this decay is equivalent to the mixing property (see e.g. [24, Chap. 4] and [65, Chap. 4]),

$$\begin{aligned} \lim _{t\rightarrow \infty } \mu (A \cap \varPhi _t^{-1} B) = \mu (A) \mu (B), \quad \mathrm {for~any~} A, B \in \mathcal {B}, \end{aligned}$$

(A.5)

of the invariant measure $\mu$, which is stronger than ergodicity. In other words, the probability for some state to be in any set B after some time t is independent of the probability of this state to be in any set A initially. Any information about the initial state of an ensemble is thus gradually forgotten.

Remark 3

For a physical measure $\mu$ supported by an attractor $\varLambda$, the correlation function can be estimated by the time mean

$$\begin{aligned} C_{f,g}(t) = \lim _{T \rightarrow \infty } \frac{1}{T} \int _0^T (f(x) - \bar{f})(g(\varPhi _t x) - \bar{g}) ~ dt, \end{aligned}$$

(A.6)

from a single time initialized in the basin of attraction of $\varLambda$. For this reason, we will see in Sect. 3.3.2 that complete information on $\mathcal {P}^\mu _t, t \ge 0$ can be obtained from from observations on the corresponding attractor alone. Unfortunately, for the forced and dissipative systems considered here, volumes contract on average [101, Chap. 2.8] so that the invariant measure $\mu$ is supported by an attractor with zero Lebesgue measure m. It follows that, as opposed to the semigroup $\mathcal {P}^m_t, t \ge 0$, no information on the dynamics away from the attractor is carried by $\mathcal {P}^\mu _t$.

1.2 Spectral Theory of Ergodicity and Mixing

One can see from the definition (A.4) that the correlation function is fully determined by the family of transfer (Koopman) operators $\mathcal {P}_t^\mu$ ($\mathcal {U}_t^\mu$) on $L^2_\mu (X)$, i.e. with respect to $\mu$. In fact, classical results relate the ergodic properties of measure-preserving dynamical systems to the behaviour of the semigroups [24,25,26]. Note first, that the invariance of the measure $\mu$ together with the invertibility of the flow ensure that the semigroups are isometries and constitute a unitary group on $L^2_\mu (X)$. As a consequence, the spectrum of the operators is contained in the unit circle $|z| = 1, z \in \mathbb {C}$ and $\mathcal {P}_t^\mu$ and $\mathcal {U}_t^\mu$ have the same eigenvalues and eigenfunctions, for any real t. Moreover, the following results hold.

Theorem 1

If the measure-preserving dynamical system $(X, \mathcal {B}, (\varPhi _t)_{t\in \mathbb {R}}, \mu )$ is

• (i) ergodic, then 1 is a simple eigenvalue of $\mathcal {P}_t^\mu , t\in \mathbb {R}$, and conversely.

• (ii) mixing, then the only one eigenvalue is one.

The first item of this theorem can be understood from the fact that ergodic systems have only one stationary density with respect to the invariant measure. The second is due to the fact that eigenfunctions associated with eigenvalues on the unit circle do not decay and thus prevent mixing for general observables.

The rate of decay of correlations, or mixing, characterizes the (weak) convergence of ensembles to a statistical steady-state and has been the subject of intense research these past decades. Once again, the latter can be studied from the spectral properties of the semigroups [31]. However, the eigenvalues responsible for such decay, the Ruelle–Pollicott resonances [29, 30], lie inside the unit disk and are thus not accessible from operators on regular functional spaces (here, the resonances refer to the eigenvalues of the finite time operators rather than there infinitesimal generator). Anisotropic Banach spaces of distributions capturing the dynamics of contraction and expansion of the system and on which the semigroups are contracting should instead be considered [31,32,33,34,35,36].¹²

A certain degree of hyperbolicity is necessary to adapt these spaces to the dynamics, and the robustness to perturbations of the spectrum or the convergence of numerical algorithms is not guarantied in general (see [33, 105] and [106, 107], respectively, for results in this direction). In the present study, it is found, however, that the discretization scheme used in Sect. 3.3 yields reasonably good approximations of the resonances.

The spectral properties of the semigroups with respect to an invariant measure $\mu$ thus allows for the study of the ergodic properties of this measure. On the other hand, for the study of the global stability of this set, operators acting on a neighborhood of positive Lebesgue measure of a chaotic set should be considered.

1.3 Spectral Theory of Global Stability

In Sect. 1, we have advocated on heuristic grounds that the global stability of invariant sets could be studied from the evolution of densities and observables. In this case, however, the transfer (Koopman) operators $\mathcal {P}_t^m$ ($\mathcal {U}_t^m$) with respect to the Lebesgue measure m should be considered. Indeed, this should allow one not only to study the mixing dynamics of an invariant measure, but also the contraction to/escape from such a measure. This has allows for new developments in the theory of the global stability of stationary points and periodic orbits from the behavior of $\mathcal {P}_t^m$ [41] and $\mathcal {U}_t^m$ [42]. This approach should also be amenable to chaotic invariant sets. However, the same difficulties as mentioned in the previous “Spectral Theory of Ergodicity and Mixing” section of Appendix A, regarding the functional analytic framework appropriate to capture the resonances inside the unit disk, are encountered. In order to better interpret the numerical results obtained in Sect. 4, let us however give a few comments regarding the eigenfunctions of the Koopman operators $\mathcal {U}^X_t$ and $\mathcal {U}^\varLambda _t$ on the space C(X) of continuous functions on the compact metric space (X, d) (e.g. $X=\mathbb {R}^n$ with d the distance induced by the Euclidean norm) and its restriction to the support $\varLambda$ of some invariant measure $\mu$, respectively. Since (X, d) is a normal space and $\varLambda$ is a closed subset of X, the Tietze–Urysohn Extension Theorem (e.g. [108]) ensures that for any continuous function $g^\varLambda$ on $\varLambda$, there exists a continuous function $g^X$ on X whose restriction to $\varLambda$ is $g^\varLambda$. The restriction $\mathcal {U}^\varLambda _t$ of $\mathcal {U}^X_t$ on $C(\varLambda )$ is thus defined such that $\mathcal {U}^\varLambda _t g^\varLambda = \mathcal {U}^X_t g^X$.

1.3.1 Multiple Attractors

We first discuss the coexistence of multiple attractors with disjoint basins of attraction, (such as in the Lorenz flow for $\rho _{\mathrm {Homo}}< \rho < \rho _A$), in terms of multiplicity of the eigenvalue 1. When the state space X is forward invariant, i.e. such that $X \subset \varPhi _t^{-1} X$ then one has that

$$\begin{aligned} \mathcal {U}^X_t \mathbf {1}_X = \mathbf {1}_{\phi _t^{-1} X \cap X} = \mathbf {1}_{X}, \quad t \ge 0. \end{aligned}$$

(A.7)

Thus, the forward invariance of X implies that the function $\mathbf {1}_X$, constant on X, is an eigenfunction of the Koopman semigroup associated with the eigenvalue 1. This is for example the case when the flow has a single globally asymptotically stable attractor on X. If instead, there is a single repeller and $\varPhi _t^{-1} X \subset X$, then $\mathcal {U}^X_t \mathbf {1}_X = \mathbf {1}_{\phi _t^{-1} X \cap X} = \mathbf {1}_{\phi _t^{-1} X}$ so that $\mathbf {1}_X$ is no longer an eigenfunction associated with 1.

When several attractors $\varLambda _1, \ldots , \varLambda _l$ coexist, each basin of attraction $B(\varLambda _i)$ is forward invariant and $\mathcal {U}^X_t \mathbf {1}_{B(\varLambda _i)} = \mathbf {1}_{B(\varLambda _i)}, t \ge 0$. Thus, to each attractor $\varLambda _i$ corresponds a Koopman eigenfunction $\mathbf {1}_{B(\varLambda _i)}$ associated with a generator eigenvalue zero. However, only l of the $l+1$ functions $\mathbf {1}_X, \mathbf {1}_{B(\varLambda _1)}, \ldots , \mathbf {1}_{B(\varLambda _l)}$ are independent, so that there are only l eigenvalues 1. For example, when two attractors coexist, a possibility is to have as independent eigenfunctions $\mathbf {1}_X$ and $\mathbf {1}_{B(\varLambda _1)} - \mathbf {1}_{B(\varLambda _2)}$. Adding suitably defined (e.g. Gaussian) noise leads to a unique invariant measure; reducing the intensity of the noise to zero leads in the limit to selecting a special invariant measure constructed as linear combination of all the invariant measure of the deterministic case. Note, however, that the numerical approximations of the basins of attraction may be difficult when the geometry of the boundary is convoluted. The fact that the numerical discretization of the state space leads effectively to introducing some noise in the system might explain why also in the deterministic case one can miss the presence of various coexisting and independent invariant measures.

1.3.2 Correspondence Between the Eigenvalues of $\mathcal {U}_t^\varLambda$ and $\mathcal {U}_t^X$

Let us discuss the correspondence between the eigenvalues of $\mathcal {U}_t^X$ and $\mathcal {U}_t^\varLambda$ for a hyperbolic attractor $\varLambda$. Following [45], we refer to the eigenvalues of $\mathcal {U}_t^X$ as the unstable resonances and those of the $\mathcal {U}_t^\varLambda$ which do not correspond to the latter as the stable resonances.

What we show below is strictly applicable only for eigenvalues on the unit circle, as discussed in Remark 4. Nonetheless, we believe that it may be practically relevant in the region near the unit circle, and, using more advanced mathematical tools, could be extended for the unit disk, away form the essential spectrum.

We consider the particular case where the invariant measure $\mu$ is supported by a uniformly hyperbolic attractor $\varLambda \subset X$ for the continuously differentiable flow $\varPhi _t, t \in \mathbb {R}$ on (X, d). For convenience, we will assume that X is the basin of attraction of $\varLambda$. The global stable manifold $W_s$ through the point $x \in X$ can then be characterized topologically by

$$\begin{aligned} W_s(x) = \{y \in X: \quad d(\varPhi _t(x), \varPhi _t(y)) \rightarrow 0, \quad t \rightarrow \infty \}. \end{aligned}$$

(A.8)

It follows directly that $W_s$ is invariant, i.e. $\varPhi _t W_s(x) \subset W_s(x)$ and $\cup _{x \in \varLambda } W_s(x) = X$, since X is the basin of attraction of $\varLambda$. Moreover, as a consequence of the stable manifold theorem [65, Chap. 6], $x \mapsto W_s(x)$ is continuous.

Proposition 2

Let $\varPhi _t, t \in \mathbb {R}$ be a continuously differentiable flow on the compact space X. Let $\varLambda$ be a uniformly hyperbolic attractor and denote by $\mathcal {U}_t^\varLambda : C(\varLambda ) \rightarrow C(\varLambda ), t\ge 0,$ the restriction to $C(\varLambda )$ of the Koopman operator $\mathcal {U}^X_t: C(X) \rightarrow C(X), g \mapsto g \circ \varPhi _t$. Assume that $\psi ^\varLambda$ in $C(\varLambda )$ is an eigenfunction of $\mathcal {U}_t^\varLambda$ associated with the eigenvalue $\zeta \in \mathbb {C}$ for some $t > 0$, i.e. $\mathcal {U}_t^\varLambda \psi ^\varLambda = \zeta \psi ^\varLambda$. Then the function $\psi ^X$ such that

$$\begin{aligned} \psi ^X(y) = \psi ^\varLambda (x), \quad \mathrm {whenever~} y \in W_s(x) \end{aligned}$$

(A.9)

is in C(X) and is an eigenfunction for $\mathcal {U}_t^X$ associated with the eigenvalue $\zeta$.

In other words, $\psi ^X$ takes on a leaf of the global stable manifold a constant value given by that of $\psi ^\varLambda$ on this leaf.

Proof

Let us first verify that $\psi ^X$ is indeed in C(X). Note first that $\psi ^X$ is defined on X, since $W_s = X$. The continuity of $\psi ^X$ follows from that of $\psi ^\varLambda$ and $x \mapsto W_s(x)$. To see this, let $\{y_n\}$ be a sequence in X converging to y. For each $y_n$ there is an $x_n$ in $\varLambda$ such that $y_n \in W_s(x_n)$. Thus

$$\begin{aligned} \psi ^X(y_n) = \psi ^\varLambda (x_n) \end{aligned}$$

(A.10)

From the continuity of $x \mapsto W_s(x)$, the limit x of $x_n$ exists and is such that $y \in W_s(x)$ and $\psi ^X(y) = \psi ^\varLambda (x)$. From the continuity of $\psi ^\varLambda$, it follows that

$$\begin{aligned} \lim _{n \rightarrow \infty } \psi ^X(y_n) = \lim _{n \rightarrow \infty } \psi ^\varLambda (x_n) = \psi ^\varLambda (x) = \psi ^\varLambda (y), \end{aligned}$$

(A.11)

so that $\psi ^X$ is continuous on the metric space (X, d).

That $\psi ^X$ is an eigenfunction follows directly from the invariance of the global stable manifold:

$$\begin{aligned} \mathcal {U}_t^X \psi ^X(y) = \psi ^X(\varPhi _t y)&= \psi ^\varLambda (x)&\mathrm {whenever~} \varPhi _t y \in W_s(x) \end{aligned}$$

(A.12)

$$\begin{aligned}&= \psi ^\varLambda (x)&\mathrm {whenever~} y \in W_s(\varPhi _t^{-1} x) \end{aligned}$$

(A.13)

$$\begin{aligned}&= \psi ^\varLambda (\varPhi _t z)&\mathrm {whenever~} y \in W_s(z) \end{aligned}$$

(A.14)

$$\begin{aligned}&= \zeta \psi ^\varLambda (z)&\mathrm {whenever~} y \in W_s(z) \end{aligned}$$

(A.15)

$$\begin{aligned}&= \zeta \psi ^X(y). \end{aligned}$$

(A.16)

$\square$

Thus, to each eigenfunction $\mathcal {U}_t^\varLambda$ on $C(\varLambda )$ corresponds an eigenfunction of $\mathcal {U}_t^X$ on C(X) associated with the same eigenvalue, i.e. the spectrum of $\mathcal {U}_t^\varLambda$ is a subset of the spectrum of $\mathcal {U}_t^X$.

Remark 4

There is, however, a major caveat to the applicability of this results. Indeed, it cannot be applied to eigenfunctions associated with eigenvalues inside the unit disk, since $\mathcal {U}_t^\varLambda$ has a roughening effect due to the contraction on the unstable manifold manifold of $\varLambda$, backward in time. For that purpose, spaces a distributions should be considered for $\psi ^\varLambda$.

Remark 5

A similar result holds for the nonuniformly hyperbolic case [66]. However, the stable foliation is then only measurable so that eigenfunctions in spaces of measurable functions should be considered.

Appendix B: Robustness of the Resonances

In this appendix, we explain and give support to the choice of parameters used to obtain the results of Sect. 4 and given in Sect. 3.4.

1.1 Numerical Integration

In this study, the time step of integration is particularly important for the position of the attractor crisis. It appears that for larger time steps, the attractor crisis occurs for values of $\rho$ smaller than $\rho _A$. A time step of $10^{-4}$ allows for the crisis to occur within $1\%$ of $\rho _A$, while keeping the numerical integration tractable.

1.2 Discretization

The choice of the grid used to approximate the transfer operators by transition matrices is the most critical step of the numerical application in this study. First, in order for probabilities to be conserved, the grid should cover a bounded set within which any trajectory remains. This condition if fulfilled by a grid covering the ball $R_o$ (see Sect. 2). A discretization of $R_o$ is then easily implemented when working in spherical coordinates $(r, \theta , \phi )$.

Second, due to the fine-grained geometry of the eigenvectors associated with chaotic sets, the generator eigenvalues are slow to converge with the grid resolution.¹³ In this study, the focus is, however, on the generator eigenvalues close to the imaginary axis, which are expected to be more robust to perturbations than eigenvalues further from the imaginary axis. The results of a test of convergence with respect to the grid resolution $n_d$-by-$n_d/2$-by-$n_d$ is given in Fig. 10. One can see that for $n_d \ge 400$, the real part of the first nonzero eigenvalue is close to convergence. To get an idea of the value to which this real part would converge for higher resolutions, the dashed orange line represents a least square fit of an exponential $a e^{b n_d},$ with $a, b \in \mathbb {R}$, to it. The quality of this fit and the fact that the fitted curve is an exponential converging to zero suggests that this eigenvalue, for $\rho \approx 24$, would converge to the imaginary axis if the resolution were to be further increased. Eventually, a grid resolution of 400-by-200-by-400 is chosen, allowing for the real part of the first nonzero generator eigenvalue to remain within $2\%$ of the corresponding value obtained for $n_d = 500$. The same grid is used to estimate $\mathbf {P}^\mu _\tau$, for which similar numerical convergence is also observed (not shown here).

Fig. 10

Convergence test of the real parts of the five leading generator eigenvalues calculated from $\mathbf {P}^m_\tau$ with respect to the grid resolution $n_d$-by-$n_d/2$-by-$n_d$, for $\rho = 24$, $\tau = 0.05$ and from $3.2 \times 10^{10}$ trajectories

1.3 Transition Time

The transition time $\tau$ for which the transition matrices $P_\tau ^m$ and $P_\tau ^\mu$ are estimated is a key parameter. In theory, the spectral mapping formula (3.8) allows to calculate the generator eigenvalues $\lambda _k$ from the eigenvalues $\zeta _k(\tau ), k \ge 0$ for the transfer operator $\mathcal {P}_\tau ^\eta$, for any transition time $\tau$. A first issue is, however, that, in taking the complex logarithm divided by $\tau$ to get the $\lambda _k$ from the $\zeta _k(\tau )$, the imaginary part of the $\lambda _k$ is only known modulo $2\pi /\tau$. An arbitrary choice of the principal part would then be valid only for true $\lambda _k$ such that $|\mathfrak {I}(\lambda _k)| \le \pi /\tau$, so that this window shortens as $\tau$ is increased (see [109, Sect. 2.4]).

Second, a compromise should be found in order to estimate correctly as many eigenvalues as possible, while only approximating those for which the eigenvectors can be resolved for a given grid. Indeed, the longer the transition time $\tau$, the smaller $|\zeta _k(\tau )| = e^{\mathfrak {R}(\lambda _k) \tau }$ for $\lambda _k$ with a small real part. Thus, in order to be able to estimate the eigenvalue $\zeta _k(\tau )$ numerically, $\tau$ should be sufficiently small for $e^{\mathfrak {R}(\lambda _k) \tau }$ to be larger than a threshold under which numerical errors become important. This threshold depends on several factors such as the sampling, the nonnormality of the transfer operators and roundoff errors [109].

On the other hand, it is not always a good strategy to take $\tau$ short to approximate as many $\lambda _k$ far from the imaginary axis as possible. Indeed, eigenvalues further from the imaginary tend to be associated with eigenvectors with more and more changes of sign. For the latter to be resolved, the grid resolution should be higher and higher. However, if for a fixed grid, eigenvalues $\zeta _k(\tau )$ associated with eigenvectors which cannot be appropriately resolved have not decayed, their imprecise approximation will also have an impact on eigenvalues closer to the imaginary axis. As a rule of thumb, the lower the grid resolution, the larger should $\tau$ be, so as to approximate only the eigenvalues for which the eigenvectors can be properly resolved at this resolution. For a grid of 400-by-200-by-400, a transition time $\tau$ of 0.05 time units is found to give of a good compromise.

Fig. 11

Convergence test of the real parts of the leading generator eigenvalues of $\mathbf {P}_\tau ^m$ (left) and $\mathbf {P}_\tau ^\mu$ (right) with respect to the number of samples $N_s$, for $\rho = 24$ and $\rho = 28$, respectively, and a grid of $400-by-200-by-400$

1.4 Number and Length of Trajectories

Ulam’s method relies on the estimation of transition probabilities from time series. The quality of these estimations depend on the sampling. To test the sampling, one strategy is to estimate confidence intervals (see e.g. [86, SI] and [88]). Another approach, followed here, is to directly test the convergence of the generator eigenvalues with respect to the number of samples $N_s$. This convergence of the transition probabilities and the eigenvalues with $N_s$ is known to occur at a rate of $\mathcal {O}(N_s^{1/2})$ [90, 109]. When estimating $\mathbf {P}_\tau ^m$ from many short time series, the number of samples is given by the number of trajectories. In this case, the robustness of the eigenvalues to $N_s$ is shown in the left panel in Fig. 11. One can see that a number $6.4 \times 10^{9}$ of trajectories is more than enough for an estimation on the chosen grid of 400-by-200-400. When estimating $\mathbf {P}_\tau ^\mu$ from a few long time series, the number of samples is given by the number of trajectories by their lengths $T_\mathrm {samp}$ divided by their sampling rate. Here, a sampling rate of 100 samples per time unit is used and 24 long trajectories are used (in order to distribute each on a computer thread). The robustness of the eigenvalues to $T_\mathrm {samp}$ is shown in the right panel in Fig. 11. One can see that a length $T_\mathrm {samp}$ of $1 \times 10^{5}$ time units is more than enough for an estimation on the chosen grid of 400-by-200-400.

1.5 Numerical Eigenvalue Problem

We have seen that the fine-grained geometry of the eigenvectors of the Lorenz flow require a large grid resolution in order to achieve numerical convergence, if only for the leading eigenvalues. While, the estimation of the transition matrices from time series is, an embarrassingly parallel problem, which can be easily distributed on several nodes of a calculator, solving the eigenproblem for such large transition matrices is more challenging, both in terms of computations and memory. On the other hand, the sparse structure of the transition matrices allows to use iterative eigensolvers. Here, we have chosen the block Krylov Schur algorithm implemented in the Anasazi package [91] of the Trilinos library. This algorithm is a common and robust choice for such problems and the Anasazi implementation makes it straightforward to distribute it on several nodes of a calculator.