Lyapunov Exponents for Random Perturbations of Coupled Standard Maps

Abstract

In this paper, we give a quantitative estimate for the first N Lyapunov exponents for random perturbations of a natural class of 2N-dimensional volume-preserving systems exhibiting strong hyperbolicity on a large but noninvariant subset of the phase space. Concrete models covered by our setting include systems of coupled standard maps, in both ‘weak’ and ‘strong’ coupling regimes.

Appendices

Appendix A: Version of the Singular Value Decomposition

Lyapunov exponents are asymptotic exponential growth rates of singular values. For this reason, we recall here some basic facts about the Singular Value Decomposition and related results used in this paper. Below, $d \ge 1$ and A is a $d \times d$ matrix. The singular values $\sigma _1(A) \ge \cdots \ge \sigma _d(A)$ are defined to be the eigenvalues of $\sqrt{A^\top A}$, listed in decreasing order and counted with multiplicity.

Theorem 29

(Singular Value Decomposition, Theorem 3.1.1 of [16]). There exist orthonormal bases $\{e_1, \cdots , e_d\}$ and $\{e_1', \cdots , e_d'\}$ of $\mathbb {R}^d$ with the property that

$$\begin{aligned} A e_i = \sigma _i(A) e_i'. \end{aligned}$$

These bases are unique up to changes of sign and rearrangements of indices in case of repeated singular values (i.e., $\sigma _i(A) = \sigma _j(A)$ for some $i \ne j$).

Recall that $\{ e_i\}$ is an (orthonormal) eigenbasis for $A^\top A$, while $\{ e_i'\}$ is an appropriate ordering of an (orthonormal) eigenbasis for $A A^\top$.

The following characterization of singular values is also used.

Lemma 30

(Min-max Principles for Singular Values, Theorem 3.1.2 of [16]). For all $1 \le i \le d$, we have that

$$\begin{aligned} \prod _{j =1}^k \sigma _j(A) = \max _{\begin{array}{c} E \subset \mathbb {R}^d \\ \dim E = i \end{array}} |\det (A|_E)|, \end{aligned}$$

where $A|_E$ is regarded as a linear mapping $E \rightarrow A(E)$.

The characterization in Lemma 30 is central to the approach taken in this paper: it directly implies that to control $\prod _1^k \sigma _j(A)$, it suffices to control the volume growth of A along k-dimensional subspaces. Since Lyapunov exponents are asymptotic exponential growth rates of singular values, this motivates why we can control sums of the top-k Lyapunov exponents by studying the ‘typical’ rate at which k-dimensional volumes grow.

Lastly, we state the following corollary of Theorem 29, which we use in Lemma 22 to estimate singular directions. Below, $E_0$ is a k-dimensional subspace, $\alpha > 0$ is fixed, $\Pi _0 : \mathbb {R}^d \rightarrow E_0$ is orthogonal projection to $E_0$, and

$$\begin{aligned} \mathcal {C}_0 := \{ v \in \mathbb {R}^d : \Vert (I - \Pi _0) v\Vert \le \alpha \Vert \Pi _0 v\Vert \} \end{aligned}$$

is a cone of vectors roughly parallel to $E_0$.

Lemma 31

Assume A is invertible, and has the property that for any $\ell$-dimensional subspace $E \subset \mathcal {C}_0$, $\ell \le k$, we have that $A(E) \subset \mathcal {C}_0$ and $A^\top (E) \subset \mathcal {C}_0$. Then, $\exists 1 \le i_1< \cdots < i_k \le d$ such that $e_{i_j}, e_{i_j}' \in \mathcal {C}_0$ for $1 \le j \le k$.

Proof

Let $\mathcal {E}_0 := \{ E \in {\text {Gr}}_k(\mathbb {R}^m) : E \subset \mathcal {C}_0\}$ and observe that $\mathcal {E}_0$ is invariant under $B := A^\top A$, which we view as a (continuous) mapping ${\text {Gr}}_k(\mathbb {R}^m) \rightarrow {\text {Gr}}_k(\mathbb {R}^m)$. In the chart $\mathcal {U}_{E_0}$ (see Sect. 5.1), the set $\mathcal {E}_0$ is convex. Since $\mathcal {E}_0$ is also compact, we see by the Brouwer fixed point theorem that B must have a fixed point in $\mathcal {E}_0$, i.e., a k-dimensional subspace E for which $B (E) = E$. Since B is self-adjoint, it follows that B has k linearly independent, orthogonal eigenvectors in E, from which we conclude $\exists i_j, 1 \le j \le k$, for which $e_{i_j} \in \mathcal {C}_0$. Applying the hypothesis to the span of $e_{i_j}$ ($\ell = 1$), we conclude that $A e_{i_j} = \sigma _{i_j}(A) e_{i_j}' \in \mathcal {C}_0$, from which it follows immediately that $e_{i_j}' \in \mathcal {C}_0$ as well (note $\sigma _i(A) \ne 0$ for all $1 \le i \le d$ if A is invertible). $\square$

Appendix B: Proofs of Grassmanian Geometric Lemmas 19 and 21

Proof of Lemma 19

Given $E \in {\text {Gr}}_k(\mathbb {R}^m)$, write $V = E^\perp$ and define $\mathcal {V}= (\mathcal {U}_E)^c$, which by point (A) at the beginning of Sect. 3 is the set of k-dimensional subspaces intersecting V nontransversally. We will describe $\mathcal {V}$ as the image of a fiber bundle $\mathcal {E}$, to be defined below, via a smooth mapping $\Phi : \mathcal {E}\rightarrow {\text {Gr}}_k(\mathbb {R}^m)$. As we will show, $\dim \mathcal {E}< k (m-k) = \dim {\text {Gr}}_k(\mathbb {R}^m)$, hence $\mathcal {V}$ can be covered by embedded submanifolds of dimension $< k (m-k)$.

To define $\Phi$ and $\mathcal {E}$, we first introduce some notation. Given $v \in \mathbb {R}^m$ let $I_v = \{ S \in {\text {Gr}}_k(\mathbb {R}^m) : v \in S\}$. Then, each $S \in I_v$ is uniquely specified by a corresponding $k-1$-dimensional subspace $S_v := S \cap \langle v \rangle ^\perp = (I - \Pi _v) (S)$, where $\Pi _v : \mathbb {R}^m \rightarrow \langle v \rangle$ is the orthogonal projection. So, we can (canonically) identify $I_v \cong {\text {Gr}}_{k-1}(\langle v \rangle ^\perp )$. The latter is essentially ${\text {Gr}}_{k-1}(\mathbb {R}^{m-1})$ and has dimension $(k-1)(m - 1 - (k-1)) = (k-1)(m -k)$.

Let $\pi : \mathcal {E} \rightarrow {\text {Gr}}_1(V)$ denote the fiber bundle over ${\text {Gr}}_1(V)$ with fibers ${\text {Gr}}_{k-1}(\langle v \rangle ^\perp )$. Write elements of $\mathcal {E}$ as $(v, \hat{S})$, where $v \in V, \hat{S} \in {\text {Gr}}_{k-1}(\langle v \rangle ^\perp )$. We define $\Phi : \mathcal {E} \rightarrow {\text {Gr}}_m(\mathbb {R}^k)$ to be the sum of subspaces

$$\begin{aligned} \Phi (v, \hat{S}) = \langle v \rangle + \hat{S} \end{aligned}$$

in $\mathbb {R}^m$. Evidently, the image of $\Phi$ coincides with $\mathcal {V}$. Since $\dim \mathcal {E}= (k-1) + (k-1)(m - k) = (k-1)(m - (k - 1))$, it follows that $\mathcal {V}$ can be covered by finitely many closed submanifolds of dimension $\le (k-1)(m - (k-1)) < k (m - k)$. $\square$

In what follows, given a subspace $E \subset \mathbb {R}^m$, we write $\Pi _E : \mathbb {R}^m \rightarrow E$ for its corresponding orthogonal projection.

Proof of Lemma 21

Let $E, E' \in {\text {Gr}}_k(\mathbb {R}^m)$. Then, $d_{geo}(E, E') = \sqrt{ \psi _1^2 + \cdots + \psi _k^2}$ where each $\psi _i = \psi _i(E, E') \in [0,\pi /2]$ is the i-th Jordan angle between $E, E'$, defined by, e.g.,

$$\begin{aligned} \cos \psi _i = \min _{\begin{array}{c} P \subset E \\ \dim P = i \end{array}} \max _{\begin{array}{c} v \in P \\ \Vert v \Vert = 1 \end{array}} \max _{\begin{array}{c} w \in E' \\ \Vert w \Vert = 1 \end{array}} \langle v, w\rangle \end{aligned}$$

(24)

(see Proposition 3(b) of [22]). We have $\psi _1 \le \psi _2 \le \cdots \le \psi _k$, hence $\psi _k \le d_{geo}(E, E') \le k \psi _k$.

To connect this with the Hausdorff metric, by [19] Theorem I-6.34 and some elementary arguments, we have

$$\begin{aligned} d_H(E, E')&= \Vert (I - \Pi _{E'}) \Pi _E\Vert = \sup _{v \in E} \Vert (I - \Pi _{E'}) v\Vert = \sup _{v \in E} d(v, E') \\&= \sin \angle (v, \Pi _{E'} v). \end{aligned}$$

On the other hand, by (24), we have

$$\begin{aligned} \max _{\begin{array}{c} w \in E' \\ \Vert w \Vert = 1 \end{array}} \langle v, w \rangle = \left\langle v, \frac{\Pi _{E'} v}{\Vert \Pi _{E'} v\Vert } \right\rangle = \cos \angle (v, \Pi _{E'} v), \end{aligned}$$

hence $\psi _k = \max _{v \in E, \Vert v \Vert = 1} \angle (v, \Pi _{E'} v)$. We conclude, then, that $d_H(E, E') = \sin \psi _k.$ In particular, $\frac{2}{\pi } \psi _k \le d_H(E, E') \le \psi _k$. This completes the proof. $\square$

Appendix C: Proof of Proposition 6: Explicit Noise Model $R_\omega$ Satisfying (E), (C) and (ND)

Below, we write $d = 2N$ for short. Throughout, $\{ z_i\}_{i = 1}^K$ is a set such that the balls $\{ B_{1/20}(z_i)\}$ of radius 1/20 centered at the $z_i$ form an open cover of $\mathbb {T}^d$.

Proof

For $(z, E) \in {\text {Gr}}_{d/2}(\mathbb {T}^d)$ fixed, we write

$$\begin{aligned} \Psi _{(z, E)} : \Omega _0 \rightarrow {\text {Gr}}_{d/2}(\mathbb {T}^d), \quad \Psi _{(z, E)}(\omega ) := (R_{\omega }z, D_z R_{ \omega }(E)). \end{aligned}$$

Here and throughout, elements $\omega \in \Omega _0$ are written $\omega = (v, (U^{(i)}))$. Observe that $\Psi = \Psi _{(z, E)}$ is a continuous mapping sending the origin to (z, E). From this, we see that (C) is a straightforward consequence of hypothesis (i). Condition (E) follows from Lemma 9, the hypotheses of which are guaranteed by (iii) and the fact $(z, E) \mapsto \Psi _{(z, E)}$ is continuous as a mapping from ${\text {Gr}}_{d/2}(\mathbb {T}^d)$ into the space of continuous mappings $\Omega _0 \rightarrow {\text {Gr}}_{d/2}(\mathbb {T}^d)$ in the compact-open topology.

It remains to check condition (ND). For this, by a compactness argument and the Constant Rank Theorem, it suffices to show that for $(z, E) \in {\text {Gr}}_{d/2}(\mathbb {T}^d)$ fixed, the mapping $\Psi = \Psi _{(z, E)}$ is a submersion. To simplify the argument, we begin with the following observation regarding the ‘upper triangular’ structure of $D \Psi$: writing $v = (v_1, \cdots , v_d) \in \mathbb {R}^d$ and $z = (z_1, \cdots , z_d)$,² we have that

$$\begin{aligned} D_{(t, (U^{(i)}))} \Psi \left( \frac{\partial }{\partial v_i} \right) = \frac{\partial }{\partial z_i}. \end{aligned}$$

That is, varying v does not change at all the ${\text {Gr}}_{d/2}(\mathbb {R}^d)$ coordinate in the image. Therefore, it suffices to show that when v is held fixed, we have that

$$\begin{aligned} (U^{(i)}) \mapsto D_z R_{(v, (U^{(i)}))}(E) \end{aligned}$$

is a submersion ${\text {Skew}}(d)^K \rightarrow {\text {Gr}}_{d/2}(\mathbb {R}^d)$.

In fact, we will show that for each z, it suffices to consider tangent directions corresponding to a single $U^{(i)}$. To see this, we make the following claim.

Claim 32

There exists $c = c_{K, d} > 0$ with the following property. For any $z \in \mathbb {T}^d$ there exists $1 \le j \le K$ such that for any $(U^{(i)}) \in {\text {Skew}}(d)^K$, $\Vert (U^{(i)})\Vert \le c$, we have that

$$\begin{aligned} d(\Phi ^{(j-1)}_{U^{(j-1)}} \circ \cdots \circ \Phi ^{(1)}_{U^{(1)}} z, z_j) \le \frac{1}{10}. \end{aligned}$$

Indeed, the claim holds with j any index for which $d(z, z_j) \le 1/20$ (see (9)), assuming $c = c_{K, d}$ is taken small enough.

With (z, E) and the above value of j fixed, we now set about checking that

$$\begin{aligned} U^{(j)} \mapsto D_z R_{(t, U^{(i)})} (E) \end{aligned}$$

is a submersion. Since $T_v, \Phi ^{(i)}_{U^{(i)}}$ are all diffeomorphisms of $\mathbb {T}^d$, it suffices to check that $U^{(j)} \mapsto D_{z'} \Phi ^{(j)}_{U^{(j)}}(E')$ is a submersion ${\text {Skew}}(d) \rightarrow {\text {Gr}}_{d/2}(\mathbb {T}^d)$, where $z' = \Phi ^{(j-1)}_{U^{(j-1)}} \circ \Phi ^{(1)}_{U^{(1)}}(z)$ and $E' = D_z \Phi ^{(j-1)}_{U^{(j-1)}} \circ \Phi ^{(1)}_{U^{(1)}}(E)$. By our choice of j, Claim 32 ensures $d(z', z_j) \le 1/10$, hence $D_{z'} \Phi ^{(j)}_{U^{(j)}} = \exp (U^{(j)})$. In view of the composition

$$\begin{aligned} U^{(j)} \mapsto \exp (U^{(j)}) \mapsto \exp (U^{(j)})(E') \end{aligned}$$

and the fact that $U \mapsto \exp (U)$ is a local diffeomorphism ${\text {Skew}}(d) \rightarrow \mathcal {O}(d)$, it suffices to check that $O \mapsto O(E)$ is a submersion $\mathcal {O}(d) \mapsto {\text {Gr}}_{d/2}(\mathbb {R}^d)$. Since surjectivity of a derivative is an open property, it suffices to check that the differential of $O \mapsto O(E)$ is a submersion at the identity ${\text {Id}}\in \mathcal {O}(d)$. $\square$

Claim 33

Fix $1 \le k \le d$ and $E_0 \in {\text {Gr}}_k(\mathbb {R}^d)$. Define $\Xi : \mathcal {O}(d) \rightarrow {\text {Gr}}_k(\mathbb {R}^d)$, $\Xi (O) := O(E_0)$. Then, $D_{{\text {Id}}} \Xi : {\text {Skew}}(d) \rightarrow T_{E_0} {\text {Gr}}_k(\mathbb {R}^d)$ is surjective.

Proof of Claim

We evaluate the differential explicitly in coordinates. Recall the chart $\mathcal {U}_{E_0} \cong L(E_0, E_0^\perp )$ for ${\text {Gr}}_k(\mathbb {R}^d)$ at $E_0$. As one can check, in this chart, $\Xi (O) = O(E_0)$ is represented as

$$\begin{aligned} \Xi (O) = {\text {graph}}_{E_0} g(O), \quad g(O) := ({\text {Id}}- \Pi _{E_0}) O (\Pi _{E_0} O|_{E_0})^{-1}. \end{aligned}$$

Therefore, in these coordinates we have (writing $\Pi _{E_0} = \Pi , \Pi ^\perp = {\text {Id}}- \Pi _{E_0}$)

$$\begin{aligned} D_O g (U)= \Pi ^\perp U (\Pi O|_{E_0})^{-1} + \Pi ^\perp O (\Pi O|_{E_0})^{-1} U (\Pi O|_{E_0})^{-1} \end{aligned}$$

for $U \in T_O \mathcal {O}(d)$. Evaluating at $O = {\text {Id}}$, we see that

$$\begin{aligned} D_{{\text {Id}}} g(U) = \Pi ^\perp U|_{E_0} \end{aligned}$$

This is clearly surjective as a linear mapping ${\text {Skew}}(d) \mapsto L(E_0, E_0^\perp )$; given an arbitrary $B \in L(E_0, E_0^\perp )$, we have $D_{{\text {Id}}} g(U) = B$ for any U of the form

$$\begin{aligned} U = \begin{pmatrix} \Pi U|_{E_0} &{} \Pi U|_{E_0^\perp } \\ \Pi ^\perp U|_{E_0} &{} \Pi ^\perp U|_{E_0^\perp } \end{pmatrix} = \begin{pmatrix} * &{} - B^\top \\ B &{} * \end{pmatrix}. \end{aligned}$$

$\square$