An ergodic-averaging method to differentiate covariant Lyapunov vectors

Abstract

Covariant Lyapunov vectors or CLVs span the expanding and contracting directions of perturbations along trajectories in a chaotic dynamical system. Due to efficient algorithms to compute them that only utilize trajectory information, they have been widely applied across scientific disciplines, principally for sensitivity analysis and predictions under uncertainty. In this paper, we develop a numerical method to compute the directional derivatives of the first CLV along its own direction; the norm of this derivative is also the curvature of one-dimensional unstable manifolds. Similar to the computation of CLVs, the present method for their derivatives is iterative and analogously uses the second-order derivative of the chaotic map along trajectories, in addition to the Jacobian. We validate the new method on a super-contracting Smale–Williams Solenoid attractor. We also demonstrate the algorithm on several other examples including smoothly perturbed Arnold Cat maps, and the Lorenz’63 attractor, obtaining visualizations of the curvature of each attractor. Furthermore, we reveal a fundamental connection of the derivation of the CLV self-derivative computation with an efficient computation of linear response of chaotic systems.

Appendix

1.1 A The lack of differentiability of CLVs

In general, we say that a subspace E is Hölder continuous on ${\mathbb {M}}$ if there exist constants $K,\delta > 0$ and $\beta \in (0,1]$ such that $\left\Vert E_x - E_y \right\Vert _* \le K \left\Vert x - y \right\Vert ^\beta ,$ whenever $x,y \in {\mathbb {M}}$ are such that $\left\Vert x-y \right\Vert \le \beta .$ As mentioned in Sect. 2.4, the subspaces $E^u$, $E^s$ are Hölder continuous spaces with an $\beta$ that is rarely equal to 1. The reader is referred to classical texts such as [19] (Chapter 19) or [16] for a detailed exposition on Hölder structures on hyperbolic sets.

There, the norm $\left\Vert \cdot \right\Vert _*$ uses an adapted coordinate system such as the one introduced in Sect. 3.1. The set of Hölder continuous functions themselves is independent of the coordinate system, however. The norm $\left\Vert \cdot \right\Vert _*$ used in the above references (e.g., in Theorem 19.1.6 of [19]), for our particular choice of adapted coordinates introduced in 3.1, results in the following definitions, which are exactly what one might expect. Suppose $\left\Vert x-y \right\Vert \le \delta$, and $Q_x, Q_y$ are matrix representations of the CLV basis whose ith columns, respectively, are $V^i_x, V^i_y$. Then, $\left\Vert E^u_x - E^u_y \right\Vert _* := \left\Vert Q_x[:,1:d_u] - Q_y[:,1:d_u] \right\Vert$ where the norm on the right-hand side is a matrix norm on ${\mathbb {R}}^{m\times d_u},$ say the induced 2-norm. Here, we have again used programmatic notation: given a matrix A, A[ : , i : j] refers to the columns of A from i to j, limits included. Similarly, for $E^s$, $\left\Vert E^s_x - E^s_y \right\Vert _* := \left\Vert Q_x[:,d_u+1:d] - Q_y[:,d_u+1:d] \right\Vert .$ Consistent with these definitions, for a one-dimensional $E^i$, we have $\left\Vert E^i_x - E^i_y \right\Vert := \left\Vert V^i_x - V^i_y \right\Vert ,$ which is simply the 2-norm on ${\mathbb {R}}^m.$

1.2 B Computations on the super-contracting Solenoid attractor

The super-contracting Solenoid attractor is the curve $\gamma :[0,2\pi ]\rightarrow {\mathbb {R}}^3$ (defined in Eq. 21) parameterized by a single parameter t. Since we have a closed form expression for the one-dimensional attractor, we can compute its tangent vector field, as:

$$\begin{aligned} \dfrac{d\gamma }{dt} = \begin{bmatrix} -2 r_1(t) \sin 2t -(\sin t \cos 2t)/2 \\ 2 r_1(t) \cos 2t - (\sin t\sin 2t)/2 \\ \dfrac{\cos t}{2} \end{bmatrix}, \end{aligned}$$

(34)

where

$$\begin{aligned} r_1(t) = \left( s_0 + \dfrac{\cos t}{2}\right) . \end{aligned}$$

As explained in Sect. 4.1, $V^1(t) = \gamma '(t)/\left\Vert \gamma '(t) \right\Vert .$ Further, we analytically calculate that

$$\begin{aligned}&\partial _{\gamma '(t)/\Vert \gamma '(t)\Vert } \left( \gamma '(t)/\Vert \gamma '(t)\Vert \right) \nonumber \\&\quad =\dfrac{1}{2} \begin{bmatrix} -(193 \cos t + 392 \cos 2t + 267 \cos 3t + 68 \cos 4t + 6 \cos 5t + 36)/c_1 \\ -(189 \sin t + 392 \sin 2t + 267 \sin 3t + 68 \sin 4t + 6 \sin 5t)/c_1 \\ -(19 \sin t + 8\sin t \cos t + 2 \sin t \cos 2t - 2 \sin 2t \cos t)/(c_1/2) \end{bmatrix} \nonumber \\&\quad \begin{bmatrix} -\sin 2t/r_1&\cos 2t/r_1&0 \end{bmatrix} \; \dfrac{\gamma '(t)}{\left\Vert \gamma '(t) \right\Vert } \end{aligned}$$

(35)

where

$$\begin{aligned} c_1 := 2(16 \cos t + 2 \cos 2t + 19)^{3/2}. \end{aligned}$$

In Figs. 2and 3, we observe that the vector field $W^1$ computed using the differential CLV method (Eq. 19), matches almost exactly against the above expression in Eq. 35.

1.3 C Convergence of the differential CLV method

In this section, we show that convergence of Eq. 19 is guaranteed when $i=1$. Moreover, the asymptotic convergence is exponentially fast. Fix a reference trajectory $q, q_1, \ldots ,$, and use the notation $f_n$ to denote $f(x_n)$. Let $W^i, W_1^i,\cdots$ and ${{\tilde{W}}}^i,{{\tilde{W}}}_1^i,\ldots$ be two sequences of vectors generated by iterating Eq. 19. Then, from Eq. 19,

$$\begin{aligned}&\left\Vert W_n^i - {{\tilde{W}}}_n^i \right\Vert = \dfrac{1}{\prod _{m=0}^{n-1}z_{m,i}^2}\nonumber \\&\quad \left\Vert \prod _{m=0}^{n-1} \Big ( (I - V^i_{m+1} (V^i_{m+1})^T)\, (d\varphi )_m \Big ) (W^i - {{\tilde{W}}}^i) \right\Vert . \end{aligned}$$

(36)

We can apply Oseledets MET to the cocycle $\mathrm{Coc}(x_m,n) = \prod _{k=0}^{n-1}(I - V^i_{m+k+1} (V^i_{m+k+1})^T)(d\varphi )_{m+k},$ and to the Jacobian cocycle to obtain the following asymptotic inequality. In particular, using the relationship Eq. 6, we get that for every $\epsilon > 0$, there exists an $N \in {\mathbb {N}}$ such that for all $n \ge N$,

$$\begin{aligned}&\left\Vert W_{n}^i - {{\tilde{W}}}_{n}^i \right\Vert =\nonumber \\ \dfrac{1}{\prod _{m=0}^{n-1}(z^i_m)^2}&\qquad \left\Vert \prod _{m=0}^{n-1} \Big ( (I - V^i_{m+1} (V^i_{m+1})^T)(d\varphi )_m \Big ) (W^i - {{\tilde{W}}}^i) \right\Vert \nonumber \\&\quad \le e^{-2n(\lambda _i - \epsilon )}\; e^{n(\omega _i + \epsilon )} \;\left\Vert W^i - {{\tilde{W}}}^i \right\Vert . \end{aligned}$$

(37)

In the above inequality 37, $\omega _i := \max _{j\ne i,1\le j\le d_u} \lambda _j.$ Thus, asymptotic exponential convergence is guaranteed whenever $2\lambda _i \ge \omega _i,$ which is of course true when $i=1.$

1.4 D Regularization of Ruelle’s formula

Here, we briefly describe the derivation of Eq. 32 from Ruelle’s formula (Eq. 31). The reader is referred to Ruelle’s original papers [30, 31], or to [12] for an alternative derivation of a regularized response to unstable perturbations. In the case of one-dimensional unstable manifolds, which is the focus of this paper, we can obtain Eq. 32 by the following sequence of steps:

• Disintegration of the SRB measure on the unstable manifolds. Let $\varXi$ be a partition of ${\mathbb {M}}$ subordinate to the unstable manifold [22], and let $\rho _0$ be the conditional density of the SRB measure on elements of $\varXi$. Then, disintegration results in the following expression for the (nth term in the) linear response to the unstable perturbation $a V^1$,

  $$\begin{aligned}&\langle d(J\circ \varphi ^n) \cdot a\,V^1 , \mu _0 \rangle \nonumber \\&\quad = \int _{{\mathbb {M}}/\varXi } \int _{\varXi (x)} \Big ( d(J\circ \varphi ^n) \cdot a\, V^1 \Big )\nonumber \\&\qquad \circ {\mathcal {C}}_{x,1}(t)\, \rho _0\circ {\mathcal {C}}_{x,1}(t)\, dt \, d{\hat{\mu }}_0(x) \end{aligned}$$

  (38)
  $$\begin{aligned}&= \int _{{\mathbb {M}}/\varXi } \int _{\varXi (x)} a\circ {\mathcal {C}}_{x,1}(t) \, \dfrac{d(J\circ \varphi ^n\circ {\mathcal {C}}_{x,1})}{dt}(t)\, \rho _0\nonumber \\&\qquad \circ {\mathcal {C}}_{x,1}(t)\, dt \, d{\hat{\mu }}_0(x). \end{aligned}$$

  (39)

  In the above expression, $\varXi (x)$ is the element of $\varXi$ containing x, and the quotient measure of the SRB measure on $M/\varXi$ is denoted ${\hat{\mu }}_0$.

• Applying integration by parts on the inner integral, we obtain,

  $$\begin{aligned}&\langle d(J\circ \varphi ^n) \cdot a\,V^1 , \mu _0 \rangle \nonumber \\&= \int _{{\mathbb {M}}/\varXi } \int _{\varXi (x)} \dfrac{d\big ((a\, \rho _0\,J\circ \varphi ^n)\circ {\mathcal {C}}_{x,1}\big )}{dt}(t)\nonumber \\&\qquad dt \, d{\hat{\mu }}_0(x) \end{aligned}$$

  (40)
  $$\begin{aligned}&\quad - \int _{{\mathbb {M}}/\varXi } \int _{\varXi (x)} J\circ \varphi ^n\circ {\mathcal {C}}_{x,1}(t)\nonumber \\&\quad \Big ( \dfrac{a\circ {\mathcal {C}}_{x,1}(t)}{\rho _0\circ {\mathcal {C}}_{x,1}(t)} \dfrac{d (\rho _0\circ {\mathcal {C}}_{x,1})}{dt}(t) \end{aligned}$$

  (41)
  $$\begin{aligned}&\quad + \dfrac{d (a\circ {\mathcal {C}}_{x,1})}{dt}(t) \Big ) \rho _0\circ {\mathcal {C}}_{x,1}(t) \, dt\, d{\hat{\mu }}_0(x). \end{aligned}$$

  (42)

  The first term on the right-hand side of the above equation vanishes, as noted by Ruelle [30, 31] for arbitrary-dimensional unstable manifolds in Theorem 3.1(b). Applying the divergence theorem on the first term, we obtain integrals over boundaries of the partition elements, which incur cancellations in the outer integral.

• Using the definitions of b and g in the above equation, we obtain

  $$\begin{aligned} \langle d(J\circ \varphi ^n) \cdot a\,V^1 , \mu _0 \rangle&= - \langle J\circ \varphi ^n \Big (a \, g + b \Big ), \mu _0\rangle . \end{aligned}$$

  (43)

  Eq. 32 is now obtained when we rewrite the above ensemble average as an ergodic average.