2.1 Regularized sytem

Let us consider a class of $\nu $ -regularized systems

(2.3) $$ \begin{align} \displaystyle \frac{\mathrm{d} { \mathbf{x} }}{\mathrm{d} t} = {\mathbf{f}}^\nu({ \mathbf{x} }) ,\quad {\mathbf{f}}^\nu({ \mathbf{x} }):= \begin{cases} \displaystyle |{ \mathbf{x} }|^\alpha\mathbf{F}({ \mathbf{x} }/|{ \mathbf{x} }|), & { \mathbf{x} } \notin B_\nu,\\[3pt] \nu^\alpha \mathbf{H}({ \mathbf{x} }/\nu), & { \mathbf{x} } \in B_\nu, \end{cases} \end{align} $$

where $\nu> 0$ is the regularization parameter and $B_\nu = \{{ \mathbf {x} } \in \mathbb {R}^d: |{ \mathbf {x} }| \le \nu \}$ is the ball of radius $\nu $ ; recall that $\alpha < 1$ . Here, $\mathbf {H}: B_1 \mapsto \mathbb {R}^d$ is a $C^1$ -function in the unit ball such that $H(x)=F(x)$ for $|x|=1$ . Then ${\mathbf {f}}^\nu $ is $C^1(\mathbb {R}^d)$ for all $\nu>0$ . Note that the described choice of regularization leaves large freedom due to its dependence on the function $\mathbf {H}$ . The regularized field ${\mathbf {f}}^\nu $ recovers the original singular system in equation (1.2) by taking the limit $\nu \searrow 0$ . Motivated by the conceptual similarity with the viscous regularization acting at small scales in fluid dynamics [Reference Frisch27], we call $\nu $ the viscous parameter and the limit $\nu \searrow 0$ the inviscid limit.

The scaling symmetry in equation (1.4) extends to the system in equation (2.3) as follows. Let us denote the flow of the regularized system in equation (2.3) by ${\Phi }^t_\nu : \mathbb {R}^d \mapsto \mathbb {R}^d$ ; it is defined for $t \ge 0$ , $\nu> 0$ and $\alpha < 1$ . The regularized flows for arbitrary $\nu> 0$ and $\nu = 1$ are related by

(2.4) $$ \begin{align} {\Phi}^t_{\nu}({ \mathbf{x} }) = \nu\, {\Phi}^{t/\nu^{1-\alpha}}_1\bigg(\frac{{ \mathbf{x} }}{\nu}\bigg). \end{align} $$

Using this map for a deterministically or randomly chosen function $\mathbf {H}$ , we now introduce the two types of regularizations: deterministic and stochastic.

2.2 Deterministic regularization of type $\mathcal {A}_- \to \mathcal {A}_+$

Consider any initial condition ${ \mathbf {x} }_0 \in \mathcal {D}(\mathcal {A}_-)$ in the domain of the focusing attractor. The corresponding solution ${ \mathbf {x} }(t)$ of the system in equation (1.2) reaches the origin in finite time $t_b$ ; see Proposition 2.2. Let us consider the solution ${ \mathbf {x} }^\nu (t)$ of the regularized system in equation (2.3) with the same initial condition for a given viscous parameter $\nu> 0$ , provided $\nu $ is small enough so that the initial data are outside $B_\nu (\mathbf {0})$ . This solution exists and is unique globally in time. The two solutions ${ \mathbf {x} }(t)$ and ${ \mathbf {x} }^\nu (t)$ coincide up until the first time when the solution enters the ball $B_\nu $ ; see Figure 3(a). We denote this entry time by $t_{\mathsf e\mathsf n\mathsf t}^\nu $ , which has the properties

(2.5) $$ \begin{align} t_{\mathsf e\mathsf n\mathsf t}^{\nu} < t_b,\quad \lim_{\nu \searrow 0} t_{\mathsf e\mathsf n\mathsf t}^{\nu} = t_b. \end{align} $$

We introduce an escape time $t^{\nu }_{\mathsf e\mathsf s\mathsf c}> t_{\mathsf e\mathsf n\mathsf t}^\nu $ as

(2.6) $$ \begin{align} t^{\nu}_{\mathsf e\mathsf s\mathsf c} := \sup_{t>t_{\mathsf e\mathsf n\mathsf t}^\nu} \{ t : { \mathbf{x} }^\nu(t) \in B_\nu\}. \end{align} $$

Observe that the entering orbit need not necessarily escape the regularized region, e.g. it may be that $t_{\mathsf e\mathsf s\mathsf c}^\nu =+\infty $ . However, we will give conditions under which finite $t_{\mathsf e\mathsf s\mathsf c}^\nu $ exist. The corresponding entry and escape points are denoted by

(2.7) $$ \begin{align} { \mathbf{x} }_{\mathsf e\mathsf n\mathsf t}^\nu = { \mathbf{x} }(t_{\mathsf e\mathsf n\mathsf t}^\nu),\quad { \mathbf{x} }_{\mathsf e\mathsf s\mathsf c}^\nu = { \mathbf{x} }(t_{\mathsf e\mathsf s\mathsf c}^\nu), \end{align} $$

and have $|{ \mathbf {x} }_{\mathsf e\mathsf n\mathsf t}^\nu | = \nu $ and $|{ \mathbf {x} }_{\mathsf e\mathsf s\mathsf c}^\nu |> \nu $ ; see Figure 3(a). The following definition of a deterministic regularization ensures the existence of escape times.

Figure 3 Schematic representation of the regularization procedure in the phase space ${ \mathbf {x} } \in \mathbb {R}^d$ . (a) The blowup solution ${ \mathbf {x} }(t)$ (black curve) starts at ${ \mathbf {x} }_0 = { \mathbf {x} }(0)$ and reaches the singularity at ${ \mathbf {x} }(t_b) = \mathbf {0}$ in finite time. The regularized solution ${ \mathbf {x} }^\nu (t)$ (thick green curve) is given by the dynamical system modified in a small ball $B_\nu $ centered at the singularity. The solutions ${ \mathbf {x} }(t)$ and ${ \mathbf {x} }^\nu (t)$ coincide until and differ after the point ${ \mathbf {x} }_{\mathsf e\mathsf n\mathsf t}^\nu $ . (b) This regularization procedure is formalized by considering the two segments: the original solution ${ \mathbf {x} }(t)$ until the entry point ${ \mathbf {x} }_{\mathsf e\mathsf n\mathsf t}^\nu $ , and the regularized solution ${ \mathbf {x} }^\nu (t)$ after the escape point ${ \mathbf {x} }_{\mathsf e\mathsf s\mathsf c}^\nu $ . The two points ${ \mathbf {x} }_{\mathsf e\mathsf n\mathsf t}^\nu $ and ${ \mathbf {x} }_{\mathsf e\mathsf s\mathsf c}^\nu $ are related via the regularization map ${\Psi }_{{\mathsf D}}$ represented by the bold dashed arrow. For the regularization of type $\mathcal {A}_- \to \mathcal {A}_+$ , the first segment belongs to the cone $\mathcal {D}(\mathcal {A}_-)$ , while the second segment belongs to the cone $\mathcal {D}(\mathcal {A}_+)$ .

Definition 1. (Deterministic regularization)

Let $\mathcal {U}_-$ be a neighborhood around $\mathcal {A}_-$ in $\mathbb {S}^{d-1}$ so that the following holds. Suppose that there exists a constant $T>0$ so that, with respect to the map ${\Phi }_1^{T}$ , we have (i) $|{\Phi }_1^{T}(\mathbf {y})|> 1$ for all $\mathbf {y} \in \mathcal {U}_-$ and (ii) ${\Phi }_1^{T}(\mathcal {U}_-)\subset \mathcal {D}(\mathcal {A}_+)$ . Then the continuous map

(2.8) $$ \begin{align} {\Phi}_{1}^T:\ \mathcal{U}_- \mapsto \mathcal{D}(\mathcal{A}_+) \end{align} $$

will be called a regularization of type $\mathcal {A}_- \to \mathcal {A}_+$ .

We remark that it is simple to construct families of vector fields $\mathbf {H}$ such that deterministic regularizations of the form in equation (2.3) are of type $\mathcal {A}_- \to \mathcal {A}_+$ , in particular, when the vector field $\mathbf {F}$ satisfies our hypotheses (a) and (b). Assume now we have a regularization of type $\mathcal {A}_- \to \mathcal {A}_+$ given by ${\Phi }_{1}^T$ . Consider the initial condition ${ \mathbf {x} }_0 \in \mathcal {D}(\mathcal {A}_-)$ , a fixed $\nu>0$ (small enough) and an entry time $t_{\mathsf e\mathsf n\mathsf t}^\nu $ into the ball $B_\nu $ . Observe that by Proposition 2.2, $|{ \mathbf {x} }^\nu _{\mathsf e\mathsf n\mathsf t}/\nu -\mathbf {y}_-| \to 0$ as $\nu \searrow 0$ for the fixed-point attractor $\mathcal {A}_- = \{\mathbf {y}_-\}$ . Then for $\nu $ small enough, ${ \mathbf {x} }^\nu _{\mathsf e\mathsf n\mathsf t}/\nu \in \mathcal {U}_-$ . We will argue that an upper bound for the the escape time $t^{\nu ,*}_{\mathsf e\mathsf s\mathsf c}$ is given by $t\nu _{\mathsf e\mathsf n\mathsf t}+\nu ^{1-\alpha } T$ . Using equation (2.4), we have

(2.9) $$ \begin{align} { \mathbf{x} }^\nu(t^{\nu,*}_{\mathsf e\mathsf s\mathsf c})={\Phi}_{\nu}^{(t^{\nu,*}_{\mathsf e\mathsf s\mathsf c}-t^\nu_{\mathsf e\mathsf n\mathsf t})}({ \mathbf{x} }_{\mathsf e\mathsf n\mathsf t}^\nu) = \nu{\Phi}_{1}^T \bigg(\frac{{ \mathbf{x} }^\nu_{\mathsf e\mathsf n\mathsf t}}{\nu}\bigg). \end{align} $$

Since ${{ \mathbf {x} }^\nu _{\mathsf e\mathsf n\mathsf t}}/{\nu } \in \mathcal {U}_-$ , then by the definition of the deterministic regularization ${\Phi }_{1}^T({{ \mathbf {x} }^\nu _{\mathsf e\mathsf n\mathsf t}}/{\nu }) \in \mathcal {D}(\mathcal {A}_+)$ with norm bigger than one. By Proposition 2.2, ${\Phi }_{1}^t({{ \mathbf {x} }^\nu _{\mathsf e\mathsf n\mathsf t}}/{\nu })$ will stay in $\mathcal {D}(\mathcal {A}_+)$ for all $t\geq T$ and still will have norm bigger than one. Therefore, going back to equation (2.9), we may conclude that

(2.10) $$ \begin{align} t^{\nu}_{\mathsf e\mathsf s\mathsf c} < t^\nu_{\mathsf e\mathsf n\mathsf t}+\nu^{1-\alpha} T, \end{align} $$

bounding above the escape time thereby ensuring it is finite. Having the escape point and time, one defines the regularized solution

(2.11) $$ \begin{align} { \mathbf{x} }^\nu(t) = {\Phi}^{t-t_{\mathsf e\mathsf s\mathsf c}^\nu}({ \mathbf{x} }^\nu_{\mathsf e\mathsf s\mathsf c}), \quad t \ge t^{\nu}_{\mathsf e\mathsf s\mathsf c}, \end{align} $$

where ${\Phi }^t$ is the flow of the original singular system in equations (1.1) and (1.2). In the limit $\nu \searrow 0$ , we will not be interested in the solution inside the vanishing interval $t \in (t^\nu _{\mathsf e\mathsf n\mathsf t},\, t^{\nu }_{\mathsf e\mathsf s\mathsf c})$ , see Figure 3(b). Therefore, for our purposes, the regularization process is conveniently represented by the single map ${\Phi }_{1}^T$ in the Definition 1 and hence we do not need to explicitly specify the regularizing field $\mathbf {H}$ which generated this map.

2.3 Stochastic regularization

It is known that, in general, solutions ${ \mathbf {x} }^\nu (t)$ with deterministic regularization do not converge in the inviscid limit $\nu \searrow 0$ [Reference Drivas and Mailybaev20]. The limits may exist along some subsequences $\nu _n \searrow 0$ but need not be unique. We now introduce a different type of regularization by assuming that escape points are known up to some random uncertainty; see Figure 4.

Figure 4 Schematic representation of the stochastic regularization procedure in the phase space ${ \mathbf {x} } \in \mathbb {R}^d$ . The solution contains two segments: the original deterministic solution ${ \mathbf {x} }(t)$ until the entry point ${ \mathbf {x} }_{\mathsf e\mathsf n\mathsf t}^\nu $ , and the regularized solution ${ \mathbf {x} }^\nu (t)$ emanating from the random escape point ${ \mathbf {x} }_{\mathsf e\mathsf s\mathsf c}^\nu $ . The probability distribution of ${ \mathbf {x} }_{\mathsf e\mathsf s\mathsf c}^\nu $ is related to the entry point ${ \mathbf {x} }_{\mathsf e\mathsf n\mathsf t}^\nu $ via the regularization map ${\Psi }_{\mathsf R}$ . For the regularization of type $\mathcal {A}_- \to \mathcal {A}_+$ , the first segment belongs to the domain $\mathcal {D}(\mathcal {A}_-)$ and the second to $\mathcal {D}(\mathcal {A}_+)$ .

For this purpose, one may consider a family of regularized systems in equation (2.3) with the field $\mathbf {H}_{ \mathbf {a}}$ depending on a vector of parameters ${ \mathbf {a}}\in \mathbb {R}^N$ . Specifically, we consider

(2.12) $$ \begin{align} \displaystyle \frac{\mathrm{d} { \mathbf{x} }_{ \mathbf{a}}}{\mathrm{d} t} = {\mathbf{f}}^\nu({ \mathbf{x} }_{ \mathbf{a}}) ,\quad {\mathbf{f}}^\nu({ \mathbf{x} }_{ \mathbf{a}}):= \begin{cases} \displaystyle |{ \mathbf{x} }|^\alpha\mathbf{F}({ \mathbf{x} }/|{ \mathbf{x} }|), & { \mathbf{x} } \notin B_\nu,\\[3pt] \nu^\alpha \mathbf{H}_{ \mathbf{a}}({ \mathbf{x} }/\nu), & { \mathbf{x} } \in B_\nu. \end{cases} \end{align} $$

We initialize this system at some deterministic initial condition ${ \mathbf {x} }_0\in \mathcal {D}(\mathcal {A}_-)$ . For $\nu>0$ and a fixed time $t>0$ , we call the corresponding flowmap $\Phi ^t_\nu ({ \mathbf {x} }_0;{ \mathbf {a}}): \mathbb {R}^d \times \mathbb {R}^N\to \mathbb {R}^d$ , which now also depends on the parameter ${ \mathbf {a}}$ . Let us impose a probability distribution $\mu $ on values of these parameters ${ \mathbf {a}}\in \mathbb {R}^N$ for $N\geq d+1$ . Then the measure $\mu $ can be used to define a measure $\mu _{(t,{ \mathbf {x} }_0)}^\nu $ on $\mathbb {R}^d$ via a pushforward by $\Phi ^t_\nu ({ \mathbf {x} }_0;\cdot )$ with fixed t, $\nu $ , and ${ \mathbf {x} }_0$ as

(2.13) $$ \begin{align} \mu_{(t, { \mathbf{x} }_0)}^\nu = [ \Phi^t_\nu({ \mathbf{x} }_0;\cdot) ]_* \mu. \end{align} $$

To define the stochastic regularization we assume the following.

1. (1) For each ${ \mathbf {a}}\in \mathbb {R}^N$ , $\mathbf {H}_{ \mathbf {a}}$ is a regularization of type $\mathcal {A}_- \to \mathcal {A}_+$ in the sense of Definition 1. The neighborhood $\mathcal {U}_-$ and the time $T>0$ do not depend on ${ \mathbf {a}}$ .

2. (2) For $\nu =1$ and any point ${ \mathbf {x} }_0\in \mathcal {U}_-$ , the measure $\mu _{(T,{ \mathbf {x} }_0)}^1$ has an absolutely continuous (with respect to Lebesgue) density $f(1, T,{ \mathbf {x} }_0;y)$ depending on the variable y and supported in $\mathcal {D}(\mathcal {A}_+)\cap B_1^c$ .

The above hypotheses allow to define the function

(2.14) $$ \begin{align} \begin{aligned} {\Psi}_{\mathsf R}:\ \mathcal{U}_- \longrightarrow L^1(\mathcal{D}(\mathcal{A}_+)), \end{aligned} \end{align} $$

where a point ${ \mathbf {x} }_0$ is mapped to the function $f(1,T,{ \mathbf {x} }_0;y)$ . Then adding a continuity condition, we propose the following definition.

Now for ${ \mathbf {x} }_0 \in \mathcal {D}(\mathcal {A}_-)$ , consider a sufficiently small $\nu> 0$ so that the entry point ${ \mathbf {x} }^\nu _{\mathsf e\mathsf n\mathsf t}/\nu \in \mathcal {U}_-$ . The entry point is independent of the parameter ${ \mathbf {a}}$ by assumptions and we have, by our assumptions, a uniform bound on the escape time $t^{\nu }_{\mathsf e\mathsf s\mathsf c} < t^\nu _{\mathsf e\mathsf n\mathsf t}+\nu ^{1-\alpha }T$ . We shall denote

(2.15) $$ \begin{align} t^{\nu,*}_{\mathsf e\mathsf s\mathsf c}:= t^\nu_{\mathsf e\mathsf n\mathsf t}+\nu^{1-\alpha}T, \end{align} $$

which serves as a time by which all orbits have left the regularized region. The measure $\mu _{(t,{ \mathbf {x} }_0)}^\nu $ also has an absolutely continuous density, which we call $f(\nu , t,{ \mathbf {x} }_0;y)$ . Using equation (2.4) and also equation (2.9), but having in mind the dependence on the parameter ${ \mathbf {a}}$ , we obtain

(2.16) $$ \begin{align} { \mathbf{x} }_{\mathsf e\mathsf s\mathsf c}^{\nu,{ \mathbf{a}}}={\Phi}_{\nu}^{(t^{\nu,*}_{\mathsf e\mathsf s\mathsf c}-t^\nu_{\mathsf e\mathsf n\mathsf t})}({ \mathbf{x} }_{\mathsf e\mathsf n\mathsf t}^{\nu};{ \mathbf{a}}) = \nu{\Phi}_{1}^{T} \bigg(\frac{{ \mathbf{x} }^{\nu}_{\mathsf e\mathsf n\mathsf t}}{\nu};{ \mathbf{a}}\bigg). \end{align} $$

With respect to the density, $f(\nu , t,{ \mathbf {x} }_0;y)$ , we can identify the variable y with ${ \mathbf {x} }_{\mathsf e\mathsf s\mathsf c}^{\nu ,{ \mathbf {a}}}$ , and then equation (2.16) implies that

(2.17) $$ \begin{align} f(\nu,(t^{\nu,*}_{\mathsf e\mathsf s\mathsf c}-t^\nu_{\mathsf e\mathsf n\mathsf t}),{ \mathbf{x} }^{\nu}_{\mathsf e\mathsf n\mathsf t}; { \mathbf{x} }_{\mathsf e\mathsf s\mathsf c}^{\nu,{ \mathbf{a}}})= f\bigg(1,T,\frac{{ \mathbf{x} }_{\mathsf e\mathsf n\mathsf t}^{\nu}}{\nu}; \frac{{ \mathbf{x} }_{\mathsf e\mathsf s\mathsf c}^{\nu,{ \mathbf{a}}}}{\nu}\bigg). \end{align} $$

For simplicity, we will denote the measure $\mu _{(t_{\mathsf e\mathsf s\mathsf c}^\nu , { \mathbf {x} }_0)}^\nu $ as the escape measure $\mu ^\nu _{{\mathsf e\mathsf s\mathsf c}}$ . The stochastic regularization map ${\Psi }_{\mathsf R}$ defines a probability density function we call the escape density $f^\nu _{\mathsf e\mathsf s\mathsf c}$ ,

(2.18) $$ \begin{align} f^\nu_{\mathsf e\mathsf s\mathsf c} := {\Psi}_{\mathsf R}\bigg(\frac{{ \mathbf{x} }^\nu_{\mathsf e\mathsf n\mathsf t}}{\nu}\bigg)= f\bigg(1,T,\frac{{ \mathbf{x} }_{\mathsf e\mathsf n\mathsf t}^{\nu}}{\nu}; y\bigg) \in L^1(\mathcal{D}(\mathcal{A}_+)). \end{align} $$

Using equation (2.17) and a change of variables $\mathbf {y}={{ \mathbf {x} }}/{\nu }$ in $\mathbb {R}^d$ , we can conclude that the density of $\mu ^\nu _{{\mathsf e\mathsf s\mathsf c}}$ is given by

(2.19) $$ \begin{align} \mathrm{d} \mu^\nu_{\mathsf e\mathsf s\mathsf c}({ \mathbf{x} }) = f^\nu_{\mathsf e\mathsf s\mathsf c}\bigg(\frac{{ \mathbf{x} }}{\nu}\bigg) \frac{\mathrm{d} { \mathbf{x} }}{\nu^d}. \end{align} $$

We define the measure-valued stochastically regularized solution ${ \mathbf {x} }^\nu (t) \sim \mu ^\nu _t$ as

(2.20) $$ \begin{align} \mu^\nu_t = ({\Phi}^{t-t^{\nu,*}_{\mathsf e\mathsf s\mathsf c}})_* \mu^\nu_{\mathsf e\mathsf s\mathsf c}, \quad t \ge t^{\nu,*}_{\mathsf e\mathsf s\mathsf c}, \end{align} $$

where the asterisk denotes the push-forward of measure $\mu ^\nu _{\mathsf e\mathsf s\mathsf c}$ by the flow ${\Phi }^t$ of the original singular system in equations (1.1) and (1.2). Similarly to equation (2.11), the solution is now defined at all times except for a short interval $(t_{\mathsf e\mathsf n\mathsf t}^\nu ,t^{\nu ,*}_{\mathsf e\mathsf s\mathsf c})$ vanishing as $\nu \searrow 0$ .

Definition 2 completes the formulation of our main result in Theorem 1.1. This theorem states that when the randomness of regularization is removed in the limit $\nu \searrow 0$ , the limiting solution exists. This limit is independent of regularization and intrinsically random (spontaneously stochastic): different solutions are selected randomly at times $t> t_b$ with the uniquely defined probability distribution.

5.1 Proof of Proposition 1.1

By the assumptions, the system in equation (5.2) has the attractor $\mathcal {A}_+$ . Therefore, we need to understand the dynamics of the second equation (5.3). The function $F_{\mathsf r}:\mathbb {S}^{d-1}\to \mathbb {R}$ is continuous and therefore has an upper bound, $F_{\mathsf r}(\mathbf {y}) < F_M$ . Recall that the attractor $\mathcal {A}_+$ is a compact set with the defocusing property, $F_{\mathsf r}(\mathbf {y})> 0$ for any $\mathbf {y} \in \mathcal {A}_+$ . Hence, we can choose a trapping neighborhood $\mathcal {U}_+$ of $\mathcal {A}_+$ (recall this is a neighborhood in the sphere) and a positive constant $F_m$ such that

(5.4) $$ \begin{align} 0 < F_m < F_{\mathsf r}(\mathbf{y}) < F_M \quad \textrm{for } \mathbf{y} \in \mathcal{U}_+. \end{align} $$

We define the two quantities

(5.5) $$ \begin{align} w_m = \frac{1}{(1-\alpha)F_M}> 0, \quad w_M = \frac{1}{(1-\alpha)F_m} > w_m. \end{align} $$

For any $\mathbf {y} \in \mathcal {U}_+$ , the derivative in equation (5.3) satisfies the inequalities $dw/ds> 0$ for $0 < w \le w_m$ and $dw/ds < 0$ for $w \ge w_M$ . Thus, the region

(5.6) $$ \begin{align} \mathcal{U}^{\prime}_+ = \{ (\mathbf{y},w): \ \mathbf{y} \in \mathcal{U}_+, \ w \in (w_m,w_M) \} \end{align} $$

is trapping for the system in equations (5.2) and (5.3), and it attracts any solution starting in $\mathcal {B}(\mathcal {A}_+) \times \mathbb {R}^+$ .

Lemma 5.1. The function

(5.7) $$ \begin{align} G(\mathbf{y}) = \int_0^{+\infty} \exp\bigg[(\alpha-1)\int_0^{s_1} F_{\mathsf r}(X^{-s_2}(\mathbf{y}))\, \mathrm{d} s_2 \bigg]\,\mathrm{d} s_1 \end{align} $$

is continuous on the attractor $\mathcal {A}_+$ .

Proof. Convergence of the integral in equation (5.7) follows from the existence of positive lower bound $F_m$ in equation (5.4) and the condition $\alpha < 1$ . For $p>0$ , we split the integral in equation (5.7) into two segments for $s_1 \in [0,p]$ and $s_1 \in [p,+\infty )$ with an arbitrary parameter $p> 0$ . This yields

(5.8) $$ \begin{align} G(\mathbf{y}) = G_{p}(\mathbf{y})+R_{p}(\mathbf{y}), \end{align} $$

where

(5.9) $$ \begin{align} G_{p}(\mathbf{y}) = \int_0^{p} \exp\bigg[(\alpha-1) \int^{s_1}_0 F_{\mathsf r}(X^{-s_2}(\mathbf{y}))\, \mathrm{d} s_2\bigg]\,\mathrm{d} s_1, \end{align} $$

(5.10) $$ \begin{align} R_{p}(\mathbf{y}) = \int_{p}^{+\infty} \exp\bigg[(\alpha-1)\int^{s_1}_0 F_{\mathsf r}(X^{-s_2}(\mathbf{y}))\, \mathrm{d} s_2\bigg]\,\mathrm{d} s_1. \end{align} $$

The positive function $R_p$ can be bounded using the property $F_{\mathsf r}(\mathbf {y})> F_m > 0$ from equation (5.4) as

(5.11) $$ \begin{align} R_{p}(\mathbf{y}) < \int^{+\infty}_{p} \exp[(\alpha-1)F_m s_1]\,\mathrm{d} s_1 = \frac{\exp[(\alpha-1)F_m p]}{(1-\alpha)F_m}. \end{align} $$

By choosing p sufficiently large, we have that $R_{p}(\mathbf {y}) < {\varepsilon }/{4}$ and this bound is valid for any $\mathbf {y} \in \mathcal {A}_+$ . Then

(5.12) $$ \begin{align} | G(\mathbf{y}')-G(\mathbf{y}) | < | G_{p}(\mathbf{y}')-G_{p}(\mathbf{y})| +\frac{\varepsilon}{2}. \end{align} $$

The function $G_{p}(\mathbf {y})$ in equation (5.9) contains integration over finite intervals and, therefore, it is a continuous function defined for any $\mathbf {y} \in \mathbb {S}^{d-1} $ . One can choose $\delta> 0$ such that $| G_{p}(\mathbf {y}')-G_{p}(\mathbf {y})| < \varepsilon /2$ for any $\mathbf {y}$ and $\mathbf {y}' \in \mathbb {S}^{d-1} $ with $|\mathbf {y}'-\mathbf {y}| < \delta $ . This yields the desired property as the consequence of equation (5.12).

Let $X^s: \mathbb {S}^{d-1} \mapsto \mathbb {S}^{d-1}$ denote the flow of the system in equation (5.2) and the pair ( $X^s $ , $X_w^s$ ) with $X_w^s: \mathbb {S}^{d-1} \times \mathbb {R}^+ \mapsto \mathbb {R}^+$ denote the flow of the system in equations (5.2) and (5.3). We will show the following properties, observing that the first expression in equation (5.13) is that of generalized synchronization whereas $w(s)$ gets synchronized with the evolution of $\mathbf {y}(s)$ [Reference Kocarev and Parlitz29].

Proof. Let us verify that equation (5.3) has the explicit solution in the form

(5.15) $$ \begin{align}w(s) = X_{w}^s(\mathbf{y}_0,w_0) &= \displaystyle w_0\exp\bigg[(\alpha-1)\int^{s}_0 F_{\mathsf r}(X^{s-s_2}(\mathbf{y}_0))\, \mathrm{d} s_2\bigg] \nonumber\\& \quad \displaystyle +\int^{s}_0 \exp\bigg[(\alpha-1)\int^{s_1}_0 F_{\mathsf r} (X^{s-s_2}(\mathbf{y}_0))\, \mathrm{d} s_2\bigg]\mathrm{d} s_1.\end{align} $$

It is easy to see that $w(0) = w_0$ . The change of integration variable $\tilde {s}_2 = s_2-s$ yields

(5.16) $$ \begin{align} \frac{\mathrm{d} }{\mathrm{d} s} \int^{s}_0 F_{\mathsf r}(X^{s-s_2}(\mathbf{y}_0))\, \mathrm{d} s_2 = F_{\mathsf r}(X^{s}(\mathbf{y}_0)),\quad \end{align} $$

(5.17) $$ \begin{align} \frac{\mathrm{d} }{\mathrm{d} s} \int^{s_1}_0 F_{\mathsf r}(X^{s-s_2}(\mathbf{y}_0))\, \mathrm{d} s_2 = F_{\mathsf r}(X^{s}(\mathbf{y}_0))-F_{\mathsf r}(X^{s-s_1}(\mathbf{y}_0)). \end{align} $$

Taking the derivative of equation (5.15) and using equations (5.16) and (5.17), we have

(5.18) $$ \begin{align} \displaystyle \frac{\mathrm{d} w}{\mathrm{d} s} &= \displaystyle w_0(\alpha-1) F_{\mathsf r}(X^{s}(\mathbf{y}_0)) \exp\bigg[(\alpha-1)\int^{s}_0 F_{\mathsf r}(X^{s-s_2}(\mathbf{y}_0))\, \mathrm{d} s_2\bigg] \nonumber\\& \quad \displaystyle + \exp\bigg[(\alpha-1)\int^{s}_0 F_{\mathsf r} (X^{s-s_2}(\mathbf{y}_0))\, \mathrm{d} s_2\bigg] \nonumber\\& \quad \displaystyle + (\alpha-1)F_{\mathsf r}(X^{s}(\mathbf{y}_0)) \int^{s}_0 \exp\bigg[(\alpha-1)\int^{s_1}_0 F_{\mathsf r} (X^{s-s_2}(\mathbf{y}_0))\, \mathrm{d} s_2\bigg]\,\mathrm{d} s_1 \nonumber\\& \quad \displaystyle - (\alpha-1) \int^{s}_0 F_{\mathsf r}(X^{s-s_1}(\mathbf{y}_0)) \exp\bigg[(\alpha-1)\int^{s_1}_0 F_{\mathsf r} (X^{s-s_2}(\mathbf{y}_0))\, \mathrm{d} s_2\bigg]\,\mathrm{d} s_1. \end{align} $$

The term in the last line is integrated explicitly with respect to $s_1$ as

(5.19) $$ \begin{align} -\exp\bigg[(\alpha-1)\int^{s_1}_0 F_{\mathsf r} (X^{s-s_2}(\mathbf{y}_0))\, \mathrm{d} s_2\bigg]\bigg|_{s_1 = 0}^{s_1 = s}= 1-\exp\bigg[(\alpha-1)\int^{s}_0 F_{\mathsf r} (X^{s-s_2}(\mathbf{y}_0))\, \mathrm{d} s_2\bigg]. \end{align} $$

Combining the expressions in equations (5.15), (5.18), and (5.19) with $\mathbf {y}(s) = X^{s}(\mathbf {y}_0)$ , one verifies that equation (5.3) is indeed satisfied.

Note that $X^s(\mathbf {y}_0) \in \mathcal {A}_+$ for any $s \in \mathbb {R}$ and initial point on the attractor, $\mathbf {y}_0 \in \mathcal {A}_+$ . Because of the positive lower bound $F_m$ in equation (5.4) and $\alpha < 1$ , the first term on the right-hand side of equation (5.15) vanishes in the limit $s \to +\infty $ uniformly for all initial points $\mathbf {y}_0 \in \mathcal {A}_+$ and $w_0 \in (w_m,w_M)$ . For the same reason, the limit $s \to +\infty $ of the last term in equation (5.15) converges uniformly in this region. Therefore, taking the limit $s \to +\infty $ in equation (5.15) with $\mathbf {y}_0 = X^{-s}(\mathbf {y})$ yields the equivalence of relations in equations (5.7) and (5.13), proving items (i) and (ii) of the lemma.

To prove item (iii), consider the solution in equation (5.15) with $w_0 = G(\mathbf {y}_0)$ given by equation (5.7). This yields

(5.20) $$ \begin{align} w(s) &= \displaystyle \int_0^{+\infty} \exp\bigg[(\alpha-1)\int_{-s}^{s_1} F_{\mathsf r}(X^{-s_2}(\mathbf{y}_0))\, \mathrm{d} s_2\bigg]\,\mathrm{d} s_1 \nonumber\\& \quad \displaystyle +\int^{s}_0 \exp\bigg[(\alpha-1)\int^{s_1}_0 F_{\mathsf r} (X^{s-s_2}(\mathbf{y}_0))\, \mathrm{d} s_2\bigg]\,\mathrm{d} s_1, \end{align} $$

where we combined the product of two exponents in the first term into the single one. After changing the integration variables $s_1 = s^{\prime }_1-s$ and $s_2 = s^{\prime }_2-s$ in the first integral term of equation (5.20), the full expression reduces to the simple form

(5.21) $$ \begin{align} w(s) = \int_0^{+\infty} \exp\bigg[(\alpha-1)\int_{0}^{s_1} F_{\mathsf r}(X^{s-s_2}(\mathbf{y}_0))\, \mathrm{d} s_2\bigg]\,\mathrm{d} s_1 = G(\mathbf{y}(s)), \end{align} $$

where $G(\mathbf {y})$ is given by equation (5.7) and $\mathbf {y}(s) = X^s(\mathbf {y}_0)$ .

Lemma 5.2 shows that $\mathcal {A}^{\prime }_+$ from equation (1.9) is the invariant set for the system in equations (5.2) and (5.3). This set has the same structure of orbits as the attractor $\mathcal {A}_+$ of the system in equation (5.2). We need to show that $\mathcal {A}^{\prime }_+$ is an attractor with the trapping neighborhood in equation (5.6). Since $\mathcal {A}_+$ is the attractor of the first equation (5.2), it is sufficient to prove that

(5.22) $$ \begin{align} \lim_{s \to +\infty} |w(s)-G(\mathbf{y}(s))| = 0 \end{align} $$

uniformly for all initial conditions $\mathbf {y}_0 \in \mathcal {A}_+$ and $w_0 \in (w_m,w_M)$ . Since $\mathbf {y}(s) = X^s(\mathbf {y}_0)$ and $w(s) = X_{w}^s(\mathbf {y}_0,w_0)$ , we rewrite equation (5.22) as

(5.23) $$ \begin{align} \lim_{s \to +\infty} |X_{w}^s(X^{-s}(\mathbf{y}(s)),w_0)-G(\mathbf{y}(s))| = 0. \end{align} $$

The uniform convergence in this expression follows from Lemma 5.2.

It remains to prove the relations

(5.24) $$ \begin{align} \mathrm{d} \mu^{\prime}_{\mathsf{phys}}(\mathbf{y},w) = \frac{\delta(w-G(\mathbf{y}))}{c\, G(\mathbf{y})} \,\mathrm{d} \mu_{\mathsf{phys}}(\mathbf{y})\,\mathrm{d} w, \quad c = \int \frac{\mathrm{d} \mu_{\mathsf{phys}}(\mathbf{y})}{G(\mathbf{y})}. \end{align} $$

Because of the synchronization condition in equation (5.13), the physical measure $\mu _{{\mathsf s\mathsf y\mathsf n}}$ for the attractor $\mathcal {A}^{\prime }_+$ of the system in equations (5.2) and (5.3) is obtained from the physical measure $\mu _{\mathsf {phys}}$ of attractor $\mathcal {A}_+$ as

(5.25) $$ \begin{align} \mathrm{d} \mu_{{\mathsf s\mathsf y\mathsf n}}(\mathbf{y},w) = \delta(w-G(\mathbf{y}))\,\mathrm{d} \mu_{\mathsf{phys}}(\mathbf{y})\,\mathrm{d} w. \end{align} $$

This measure corresponds to the dynamics of the system in equations (5.2) and (5.3). The time change $\mathrm {d} s = G(\mathbf {y})\mathrm {d} \tau $ following from equation (5.1) with $w = G(\mathbf {y})$ transforms equation (5.25) to the physical measure in equation (5.24) for the system ni equation (1.5); see [Reference Cornfeld, Fomin and Sinai15, Ch. 10].

5.2 Proof of Theorem 1.1

Let us consider the variables

(5.26) $$ \begin{align} w = (t-t^\nu)|{ \mathbf{x} }|^{\alpha-1},\quad \tau = \log(t-t^\nu), \end{align} $$

where the temporal shift $t^\nu $ , specified later in equation (5.33), depends on the regularization parameter $\nu> 0$ . Observe that $t^\nu $ was not present in the original definition of equation (1.6), but it does not affect the system in equation (1.5): at times $t> t^\nu $ , each non-vanishing solution ${ \mathbf {x} }(t)$ of equations (1.1) and (1.2) is uniquely related to the solution $\mathbf {y}(\tau ),~w(\tau )$ of the system in equation (1.5) through the relations

(5.27) $$ \begin{align} { \mathbf{x} } = R_{t-t^\nu}(\mathbf{y},w),\quad t = t^\nu+e^\tau. \end{align} $$

Consider arbitrary times $t_2> t_1 > t^\nu $ and denote

(5.28) $$ \begin{align} \begin{array}{c} { \mathbf{x} }_i = { \mathbf{x} }(t_i), \quad \mathbf{y}_i = \mathbf{y}(t_i), \quad w_i = w(t_i), \quad \tau_i = \log(t_i-t^\nu),\quad i = 1,2. \end{array} \end{align} $$

Recalling that ${\Phi }^t$ and $Y^\tau $ denote the flows of the systems in equations (1.1), (1.2), and (1.5), one has

(5.29) $$ \begin{align} { \mathbf{x} }_2 = \Phi^{t_2-t_1}({ \mathbf{x} }_1), \quad t_2> t_1 > t^\nu, \end{align} $$

and

(5.30) $$ \begin{align} (\mathbf{y}_2,w_2) = Y^{\tau_2-\tau_1}(\mathbf{y}_1,w_1), \quad \tau_2> \tau_1. \end{align} $$

The first expression in equation (5.27) yields

(5.31) $$ \begin{align} { \mathbf{x} }_1 = R_{t_1-t^\nu}(\mathbf{y}_1,w_1), \quad { \mathbf{x} }_2 = R_{t_2-t^\nu}(\mathbf{y}_2,w_2). \end{align} $$

Equations (5.29)–(5.31) provide the conjugation relation between the flows as

(5.32) $$ \begin{align} {\Phi}^{t_2-t_1} = R_{t_2-t^\nu} \circ Y^{\tau_2-\tau_1} \circ R^{-1}_{t_1-t^\nu}, \end{align} $$

where $(\mathbf {y},w) = R^{-1}_t({ \mathbf {x} })$ is the inverse map.

Let us apply equations (5.28) and (5.32) for the stochastically regularized solution given by equations (2.19) and (2.20). We take

(5.33) $$ \begin{align} t_2 = t, \quad t_1 = t^{\nu,*}_{\mathsf e\mathsf s\mathsf c}, \quad t^\nu = t^{\nu,*}_{\mathsf e\mathsf s\mathsf c}-\nu^{1-\alpha} = t_{\mathsf e\mathsf n\mathsf t}^\nu+(T-1)\nu^{1-\alpha} \end{align} $$

for any given time $t> t_b$ . Notice that $t_2> t_1$ for sufficiently small $\nu> 0$ . Then, we use equation (5.32) to rewrite equation (2.20) in the form of three successive measure pushforwards as

(5.34) $$ \begin{align} \mu^\nu_t({ \mathbf{x} }) = ({\Phi}^{t_2-t_1})_* \mu^\nu_{\mathsf e\mathsf s\mathsf c}({ \mathbf{x} }) = (R_{t_2-t^\nu})_* (Y^{\tau_2-\tau_1})_* (R^{-1}_{t_1-t^\nu})_* \mu^\nu_{\mathsf e\mathsf s\mathsf c}({ \mathbf{x} }). \end{align} $$

For the first pushforward, equation (5.33) yields

(5.35) $$ \begin{align} (R^{-1}_{t_1-t^\nu})_* \mu^\nu_{\mathsf e\mathsf s\mathsf c}({ \mathbf{x} }) = (R^{-1}_{\nu^{1-\alpha}})_* \mu^\nu_{\mathsf e\mathsf s\mathsf c}({ \mathbf{x} }). \end{align} $$

Notice from equation (1.6) that $R^{-1}_{\nu ^{1-\alpha }}({ \mathbf {x} }) = R^{-1}_1({ \mathbf {x} }/\nu )$ . Thus, applying equation (2.19), we reduce equation (5.35) to the form

(5.36) $$ \begin{align} (R^{-1}_{t_1-t^\nu})_* \mu^\nu_{\mathsf e\mathsf s\mathsf c}({ \mathbf{x} }) = (R^{-1}_1)_* \mu^\nu_f({ \mathbf{x} }), \quad \mathrm{d} \mu_f^\nu({ \mathbf{x} }) = f^\nu_{\mathsf e\mathsf s\mathsf c}({ \mathbf{x} })\mathrm{d} { \mathbf{x} }, \end{align} $$

where $\mu ^\nu _f$ denotes the absolutely continuous probability measure with the density $f^\nu _{\mathsf e\mathsf s\mathsf c}$ . Finally, using equations (5.28), (5.33), and (5.36) in equation (5.34) yields

(5.37) $$ \begin{align} \mu^\nu_t({ \mathbf{x} }) = (R_{t-t^\nu})_* (Y^{\tau^\nu})_* (R^{-1}_1)_* \mu^\nu_f({ \mathbf{x} }) \end{align} $$

with

(5.38) $$ \begin{align} \tau^\nu = \tau_2-\tau_1 = \log \frac{t-t_{\mathsf e\mathsf n\mathsf t}^\nu-(T-1)\nu^{1-\alpha}}{\nu^{1-\alpha}}. \end{align} $$

In the inviscid limit, from equations (2.5), (5.33), and (5.38), one has

(5.39) $$ \begin{align} \lim_{\nu \searrow 0} t^\nu = t_b,\quad \lim_{\nu \searrow 0} \tau^\nu = +\infty. \end{align} $$

It remains to take the limit $\nu \searrow 0$ in equation (5.37). The convergence of entry times from equation (2.5) and Proposition 2.2 yields

(5.40) $$ \begin{align} \lim_{\nu \searrow 0} \mathbf{y}_{\mathsf e\mathsf n\mathsf t}^\nu = \mathbf{y}_-, \end{align} $$

where $\mathcal {A}_- = \{\mathbf {y}_-\}$ denotes the fixed-point attractor and $\mathbf {y}_{\mathsf e\mathsf n\mathsf t}^\nu = { \mathbf {x} }_{\mathsf e\mathsf n\mathsf t}^\nu /\nu $ correspond to entry points. Since the map ${\Psi }_{\mathsf R}$ in equation (2.14) is continuous, the limit in equation (5.40) implies

(5.41) $$ \begin{align} f_{\mathsf e\mathsf s\mathsf c}^\nu \xrightarrow{L^1} f_- \quad \textrm{as } \nu \searrow 0, \end{align} $$

where

(5.42) $$ \begin{align} f_{\mathsf e\mathsf s\mathsf c}^\nu = {\Psi}_{\mathsf R} (\mathbf{y}_{\mathsf e\mathsf n\mathsf t}^\nu),\quad f_- = {\Psi}_{\mathsf R} (\mathbf{y}_-). \end{align} $$

Using this limiting function, we rewrite equation (5.37) as

(5.43) $$ \begin{align} \mu^\nu_t({ \mathbf{x} }) = (R_{t-t^\nu})_* [ (Y^{\tau^\nu})_* (R^{-1}_1)_* \mu_-({ \mathbf{x} })+ (Y^{\tau^\nu})_* (R^{-1}_1)_* \Delta\mu_f^\nu({ \mathbf{x} }) ], \end{align} $$

where we introduced the probability measure $\mathrm {d} \mu _-({ \mathbf {x} }) = f_-({ \mathbf {x} })\mathrm {d} { \mathbf {x} }$ and the signed measure for the difference $\Delta \mu _f^\nu ({ \mathbf {x} }) = \mu _f^\nu ({ \mathbf {x} })-\mu _-({ \mathbf {x} })$ . Now we can take the inviscid limit $\nu \searrow 0$ for the expression in square parentheses of equation (5.43), where the times of pushforwards behave as equation (5.39). Since the measure $(R^{-1}_1)_* \mu _-({ \mathbf {x} })$ does not depend on $\nu $ , the first term in square parentheses converges to $\mu ^{\prime }_{\mathsf {phys}}$ by the convergence to equilibrium property. The remaining term vanishes in the limit $\nu \searrow 0$ , because the flow conserves the $L^1$ norm of the density function, and this norm vanishes by the property in equation (5.41). This yields the limit in equation (1.18) with the measure in equation (1.15).

5.3 Proof of Proposition 4.1

We will formulate the proof for the first part of the proposition, such that it can be extended later for $C^k$ -perturbed systems.

The system in equation (1.5) considered for $(\mathbf {y},w) \in \mathcal {V} \times \mathbb {R}^+$ with $F_{\mathsf r}(\mathbf {y}) \equiv F_0$ takes the form

(5.44) $$ \begin{align} \frac{\mathrm{d} \mathbf{y}}{\mathrm{d} \tau} = w\mathbf{F}_{\mathsf s}(\mathbf{y}),\quad \frac{\mathrm{d} w}{\mathrm{d} \tau} = w+(\alpha-1)F_{0}w^2. \end{align} $$

The second equation in equation (5.44) has the fixed point attractor $w = W_0 := [(1-\alpha )F_{0}]^{-1}> 0$ with the basin $w> 0$ . Recall that $\mathcal {B}(\mathcal {A}_+) \subset \mathcal {V} \subset \mathbb {R}^{d-1}$ . Equation (1.10) with $F_{\mathsf r}(\mathbf {y}) \equiv F_0$ defines the function $G:\mathcal {B}(\mathcal {A}_+) \mapsto \mathbb {R}^+$ as

(5.45) $$ \begin{align} G(\mathbf{y}) \equiv W_0. \end{align} $$

We define the corresponding graph as

(5.46) $$ \begin{align} \mathcal{G}(\mathcal{A}_+) = \{(\mathbf{y},w): \mathbf{y} \in \mathcal{B}(\mathcal{A}_+),\, w = G(\mathbf{y})\}, \end{align} $$

which is the invariant manifold for the system in equation (5.44). The attractor $\mathcal {A}^{\prime }_+ \subset \mathcal {G}(\mathcal {A}_+)$ is given by equation (1.9).

Linearization of the system in equation (5.44) at any point of $\mathcal {G}(\mathcal {A}_+)$ takes the form

(5.47) $$ \begin{align} \frac{d}{\mathrm{d} \tau} \left(\begin{array}{c} \delta{\mathbf{y}}\\[3pt] \delta{w} \end{array}\right) = \left(\begin{array}{cc} W_0\nabla\mathbf{F}_s &\ \ \mathbf{F}_s(\mathbf{y}) \\[5pt] \mathbf{0}& -1 \end{array}\right) \left(\begin{array}{c} \delta\mathbf{y}\\[3pt] \delta{w} \end{array}\right), \end{align} $$

where $(\delta \mathbf {y},\delta w) \in \mathbb {R}^{d-1} \times \mathbb {R}$ is an infinitesimal perturbation in the tangent space. It is straightforward to verify that the system in equation (5.47) has a solution

(5.48) $$ \begin{align} \left(\begin{array}{c} \delta\mathbf{y}\\[3pt] \delta{w} \end{array}\right) = e^{-\tau}\left(\begin{array}{c} \mathbf{F}_s(\mathbf{y}) \\[3pt] -1 \end{array}\right), \end{align} $$

which provides the eigenvalue $-1$ with the corresponding eigenvector. The eigenvector defines the linear space $E^{ss}$ transversal to the graph $\mathcal {G}(\mathcal {A}_+)$ , and it will play the role of strong stable (contracting) direction. Remaining eigenvalues are determined by the Jacobian matrix $W_0\nabla \mathbf {F}_s$ with the corresponding linear invariant space $E^c = \mathbb {R}^{d-1} \times \{0\}$ tangent to the graph $\mathcal {G}(\mathcal {A}_+)$ . Assumptions in equation (4.1) imply that eigenvalues of $W_0\nabla \mathbf {F}_s$ with $W_0 = [(1-\alpha )F_{0}]^{-1}$ have absolute values smaller than unity.

We showed that, at each point of the graph in equation (5.46), there exists a splitting $E^{ss}\oplus E^c$ of the tangent space, which is invariant for the linearized system in equation (5.47) and such that $E^{ss}$ dominates (contracts stronger than) the so-called central directions in $E^c$ . It follows from the stable manifold theorem that each point of $\mathcal {G}(\mathcal {A}_+)$ has a one-dimensional strong stable invariant manifold, which is tangent to $E^{ss}$ ; for background on the invariant manifold theory, see [Reference Shub44, Ch. 6] for discrete systems and [Reference Wiggins47, §4.5] for flows. Such a structure can be described locally by a homeomorphism $\rho : \mathcal {U}\times (-\delta , \delta ) \mapsto \mathcal {U}'$ , where $\mathcal {U}$ and $\mathcal {U}'$ are respectively some trapping neighborhoods of the attractors $\mathcal {A}_+$ and $\mathcal {A}^{\prime }_+$ , and $\delta> 0$ is some (small) number. Here, the fibers $\rho (\mathbf {y},\xi )$ for fixed $\mathbf {y}$ are local $C^1$ -parameterizations of the strong stable manifolds starting on the graph $\rho (\mathbf {y},0) \in \mathcal {G}(\mathcal {A}_+)$ .

Let $Y^\tau $ be the flow of the system in equation (5.44). We denote by $Y^\tau _{\rho }=\rho ^{-1}\circ Y^\tau \circ \rho $ the flow, which is defined in $\mathcal {U}\times (-\delta , \delta )$ and conjugated to $Y^\tau $ . By construction, this new flow $Y^\tau _{\rho }$ has the attractor $\mathcal {A}_\rho = \{(\mathbf {y},0):\ \mathbf {y} \in \mathcal {A}_+\}$ with the physical measure

(5.49) $$ \begin{align} \mathrm{d} \mu_\rho(\mathbf{y},\xi) = \mathrm{d} \mu_{\mathsf{phys}}(\mathbf{y})\, \delta(\xi)\mathrm{d} \xi, \end{align} $$

where $\delta (\xi )$ is the Dirac delta-function and $\mu _{\mathsf {phys}}$ is the physical measure of the attractor $\mathcal {A}_+$ . Straight segments $(\mathbf {y},\xi )$ with fixed $\mathbf {y}$ and $\xi \in (-\delta ,\delta )$ correspond to strong stable manifolds for the new flow $Y^\tau _{\rho }$ . Moreover, since strong stable manifolds have constant eigenvalue $-1$ , $Y^\tau _{\rho }$ has uniform contraction along strong stable manifolds to the plane $\xi = 0$ in a sufficiently small neighborhood $\mathcal {U}\times (-\delta , \delta )$ .

Now, the property of convergence to equilibrium for the flow $Y^\tau $ follows from the same property for $Y^\tau _{\rho }$ , where the latter is established as follows. The condition of convergence to equilibrium in equation (1.13) for the new system becomes

(5.50) $$ \begin{align} \lim_{\tau\rightarrow +\infty}\int \varphi \circ Y^\tau_\rho \, \mathrm{d} \mu(\mathbf{y},\xi) = \int \varphi \,\mathrm{d} \mu_\rho(\mathbf{y},\xi) = \int \varphi(\mathbf{y},0) \,\mathrm{d} \mu_{\mathsf{phys}}(\mathbf{y}), \end{align} $$

where we used equation (5.49) and integrated the Dirac delta-function. It is enough to verify this condition for absolutely continuous probability measures $\mu (\mathbf {y},\xi )$ supported in the trapping neighborhood $\mathcal {U}\times (-\delta , \delta )$ . Using properties of strong stable manifolds for the flow $Y^\tau _\rho $ , the integral in the left-hand side of equation (5.50) can be written as

(5.51) $$ \begin{align} \int \varphi \circ Y^\tau_\rho \, \mathrm{d} \mu(\mathbf{y},\xi) = \int {\varphi ( Y^\tau_\rho(\mathbf{y},0) ) \, \mathrm{d} \mu(\mathbf{y},\xi)} +\int \varphi_1 \circ Y^\tau_\rho \, \mathrm{d} \mu(\mathbf{y},\xi), \end{align} $$

where we introduced the function $\varphi _1(\mathbf {y},\xi ) = \varphi (\mathbf {y},\xi )-\varphi (\mathbf {y},0)$ . Since the flow $Y^\tau _\rho $ has the property of uniform contraction to the plane $\xi = 0$ , where $\varphi _1 = 0$ , the last integral in equation (5.51) vanishes in the limit $\tau \to +\infty $ . For the first integral on the right-hand side of equation (5.51), we write

(5.52) $$ \begin{align} \int {\varphi ( Y^\tau_\rho(\mathbf{y},0) ) \, \mathrm{d} \mu(\mathbf{y},\xi)} = \int {\varphi ( Y^\tau_\rho(\mathbf{y},0) ) \, \mathrm{d} \mu_{\mathsf{int}}(\mathbf{y})}, \end{align} $$

where $\mu _{\mathsf {int}}(\mathbf {y})$ is obtained from the measure $\mu (\mathbf {y},\xi )$ by integration with respect to $\xi $ . The last integral in equation (5.52) corresponds to the flow $Y^\tau _\rho $ restricted to the invariant plane $\xi = 0$ , and it is conjugate to the original flow $Y^\tau $ restricted to the graph in equation (5.46) with the constant function in equation (5.45). The latter becomes the flow $X^s$ of the system in equation (1.7) after the scaling of time with the constant factor $W_0$ . Therefore, we reduced equation (5.50) to the analogous condition of convergence to equilibrium for the system in equation (1.7), which holds by our assumptions. This proves the first part of the proposition.

For the proof of the $C^k$ -robust convergence to equilibrium, we will need the following lemma.

Lemma 5.3. Consider an attractor $\mathcal {A}_+$ of the system in equation (1.7) with $C^k$ -functions $\mathbf {F}_{\mathsf s}: \mathcal {V}\mapsto \mathbb {R}^{d-1} $ and $F_{\mathsf r}: \mathcal {V} \mapsto \mathbb {R}$ satisfying the conditions

(5.53) $$ \begin{align} \| \nabla \mathbf{F}_{\mathsf s}\| < M,\quad F_{\mathsf r}(\mathbf{y})> m > 0 \end{align} $$

for any $\mathbf {y} \in \mathcal {B}(\mathcal {A}_+)$ and positive constants m and M such that $M < (1-\alpha )m/k$ . Then, we have the following.

1. (i) Equation (1.10) defines the $C^k$ -differentiable function $G:\mathcal {B}(\mathcal {A}_+) \mapsto \mathbb {R}^+$ .

2. (ii) Let $\mathbf {y}(\tau )$ be the solution of the equation

   (5.54) $$ \begin{align} \frac{\mathrm{d} \mathbf{y}}{\mathrm{d} \tau}= G(\mathbf{y})\mathbf{F}_{\mathsf s}(\mathbf{y}) \end{align} $$
   for arbitrary initial condition $\mathbf {y}_0 \in \mathcal {B}(\mathcal {A}_+)$ . Then $w(\tau ) = G(\mathbf {y}(\tau ))$ satisfies equation (1.5).

3. (iii) Sufficiently small $C^k$ -perturbations of $\mathbf {F}_{\mathsf s}$ and $F_{\mathsf r}$ yield small $C^k$ -perturbations of G.

Proof. The above lemma is related to the general statements of the invariant manifold theory as stated in [Reference Hirsch, Pugh and Shub28] for discrete systems and in [Reference Wiggins47] for flows. Below, for completeness, we present a direct proof for arbitrary functions $F_{\mathsf r}$ satisfying equation (5.53). Let us first consider the case $k = 1$ .

(i) Changing signs of the integration variables $s_1$ and $s_2$ in equation (1.10) yields

(5.55) $$ \begin{align} G(\mathbf{y}) = \lim_{s \to -\infty} G_s(\mathbf{y}),\quad G_s(\mathbf{y}) = \int_s^0 \exp\bigg[(\alpha-1)\int_{s_1}^0 F_{\mathsf r}\circ X^{s_2}(\mathbf{y})\, \mathrm{d} s_2\bigg]\,\mathrm{d} s_1, \end{align} $$

where we introduced the function $G_s: \mathcal {V} \mapsto \mathbb {R}^+$ . By construction, $G_s$ is a $C^1$ -function for any s. The second condition in equation (5.53) implies the uniform convergence of the limit in equation (5.55) for $\mathbf {y} \in \mathcal {B}(\mathcal {A}_+)$ . Hence, the limiting function G is continuous in $\mathcal {B}(\mathcal {A}_+)$ .

Computing the Jacobian matrix $\nabla G_s$ in equation (5.55) at a given point $\mathbf {y}$ yields

(5.56) $$ \begin{align} \nabla G_s = (\alpha-1) \int_{s}^0 \bigg(\int_{s_1}^0 \nabla ( F_{\mathsf r}\circ X^{s_2})\, \mathrm{d} s_2\bigg) \exp\bigg[(\alpha-1)\int_{s_1}^0 F_{\mathsf r}\circ X^{s_2}(\mathbf{y})\,\mathrm{d} s_2\bigg]\,\mathrm{d} s_1, \end{align} $$

where

(5.57) $$ \begin{align} \nabla ( F_{\mathsf r}\circ X^{s_2}) = (\nabla F_{\mathsf r})_{X^{s_2}(\mathbf{y})} \nabla X^{s_2}, \end{align} $$

and $(\nabla F_{\mathsf r})_{X^{s_2}(\mathbf {y})}$ denotes the gradient vector $\nabla F_{\mathsf r}$ computed at $X^{s_2}(\mathbf {y})$ .

Since $X^s$ is the flow of the system in equation (1.7), by the classical theory of ordinary differential equations, the Jacobian matrix $\nabla X^s$ satisfies the linear Cauchy problem

(5.58) $$ \begin{align} \frac{\mathrm{d} }{\mathrm{d} s} \,\nabla X^s = (\nabla \mathbf{F}_{\mathsf s})_{X^{s}(\mathbf{y})} \nabla X^s, \quad \nabla X^0 = \mathbf{I}, \end{align} $$

where $\mathbf {I}$ is the identity matrix and $(\nabla \mathbf {F}_{\mathsf s})_{X^{s}(\mathbf {y})}$ is the Jacobian matrix $\nabla \mathbf {F}_{\mathsf s}$ at $X^{s}(\mathbf {y})$ . Using equation (5.58) for negative s, the first bound of equation (5.53), and Grönwall’s inequality, we estimate

(5.59) $$ \begin{align} \| \nabla X^{s} \| \leq e^{-M s},\quad s \le 0. \end{align} $$

Let $M_{\mathsf r} = \max _{\mathbf {y} \in \mathcal {V}} \|\nabla F_{\mathsf r}\| \ge 0$ . Using equations (5.56), (5.57), and (5.59) with the bounds in equation (5.53) and recalling that $\alpha < 1$ , $s_1 \le 0$ and $s_2 \le 0$ , we obtain

(5.60) $$ \begin{align} \|\nabla G_s-\nabla G_{s'} \| \leq (1-\alpha) \int_{s}^{s'} \bigg(\int_{s_1}^0 M_{\mathsf r} e^{-M s_2}\, \mathrm{d} s_2\bigg) e^{(1-\alpha)m s_1}\,\mathrm{d} s_1 \end{align} $$

for any $s < s' < 0$ . Integrating the right-hand side of equation (5.60) and taking into account that

(5.61) $$ \begin{align} (1-\alpha)m> (1-\alpha)m-M > 0 \end{align} $$

by the conditions of the lemma, one can show the Cauchy convergence of the gradients $\nabla G_s$ in the limit $s \to -\infty $ . Since the bound in equation (5.60) does not depend on $\mathbf {y}$ , the convergence is uniform in $\mathcal {B}(\mathcal {A}_+)$ . This proves the continuity of the limiting gradient $\nabla G$ in equation (5.55).

(ii) Consider the pair of functions $\mathbf {y}(\tau )$ and $w(\tau ) = G(\mathbf {y}(\tau ))$ , where $\mathbf {y}(\tau )$ satisfies equation (5.54). Obviously, these functions satisfy the first equation of equation (1.5). The second equation in equation (1.5) can be transformed to the form in equation (5.3) with the time change in equation (5.1). Then, this equation is verified as in Lemma 5.2, taking into account that the integrals converge uniformly for all $\mathbf {y} \in \mathcal {B}(\mathcal {A}_+)$ .

(iii) Using the uniform bound in equation (5.60), one proves that the convergence of integrals in equation (5.56) as $s \to -\infty $ is uniform not only with respect to $\mathbf {y}$ , but also with respect to sufficiently small $C^1$ -perturbations of the functions $\mathbf {F}_{\mathsf s}(\mathbf {y})$ and $F_{\mathsf r}(\mathbf {y})$ . This implies that such perturbations lead to $C^1$ -perturbations of $G(\mathbf {y})$ .

This proof extends to the $C^k$ case for $k> 1$ by computing high-order derivatives of G in the way similar to equations (5.55) and (5.56). Generalizing equation (5.59), one can show that kth-order derivatives of $X^s(\mathbf {y})$ are bounded by $c\exp (-kMs)$ for $s \le 0$ and some coefficient $c> 0$ . We leave details of this rather straightforward derivation to the interested reader.

Consider now a perturbed system in equation (1.5) with $\tilde {\mathbf {F}}_s$ close to $\mathbf {F}_{\mathsf s}$ and $\tilde {F}_r$ close to $F_{\mathsf r}$ in the $C^k$ -metric; here and below, the tildes denote properties of the perturbed system. Conditions of Definition 3 ensure that the perturbed system in equation (1.7) has an attractor $\tilde {\mathcal {A}}_+$ with the physical measure and the convergence to equilibrium property. In turn, the perturbed system in equation (1.5) has the attractor $\tilde {\mathcal {A}}^{\prime }_+$ given by the graph $w = \tilde {G}(\mathbf {y})$ of $\mathbf {y} \in \tilde {\mathcal {A}}_+$ ; see Proposition 1.1. Conditions in equation (4.2) remain valid if the perturbation is sufficiently small. Hence, one can choose m and M satisfying conditions of Lemma 5.3, establishing that the function $\tilde {G}(\mathbf {y})$ is $C^k$ -close to the constant from equation (5.45), and also the graph $w = \tilde {G}(\mathbf {y})$ with $\mathbf {y} \in \mathcal {B}(\tilde {\mathcal {A}}_+)$ is invariant under the flow of the perturbed system in equation (1.5).

Restriction of equation (1.5) to the invariant hyper-surface $w = \tilde {G}(\mathbf {y})$ yields

(5.62) $$ \begin{align} \frac{\mathrm{d} \mathbf{y}}{\mathrm{d} \tau}= \tilde{G}(\mathbf{y})\tilde{\mathbf{F}}_s(\mathbf{y}). \end{align} $$

This system is $C^k$ -close to $\mathrm {d} \mathbf {y}/\mathrm {d} \tau = W_0 \mathbf {F}_{\mathsf s}(\mathbf {y})$ , where the latter is equivalent to $\mathrm {d} \mathbf {y}/\mathrm {d} s = \mathbf {F}_{\mathsf s}(\mathbf {y})$ up to the constant time scaling. Since the attractor $\mathcal {A}_+$ of the unperturbed system in equation (1.7) is assumed to have a physical measure with the $C^k$ -robust convergence to equilibrium, the attractor $\tilde {\mathcal {A}}_+$ of the perturbed system in equation (5.62) has a physical measure with the property of convergence to equilibrium, provided that the perturbation is sufficiently small. For concluding the proof, one should notice that all arguments in the first part of the proof (based on the invariant manifold theory) remain valid for small $C^k$ perturbations of the system and of the graph in equation (5.46).