Let $\dot{\boldsymbol{x}}=\boldsymbol{F}(\boldsymbol{x};p)$ be an autonomous dynamical system defined on $[a,b]^n$ ($-\infty<a<B<\infty)$; $p\in\mathbb{R}$ is some fixed parameter. Suppose that $\boldsymbol{s}$ is an asymptotically stable steady state for any choice of $p$; given $p$, let $B_p\subseteq [a,b]^n$ denote its basin of attraction.

My question relates to whether I can use a result from Alberto and Chiang (2015, p. 198) to characterize how the Lebesgue measure of $B_p$ varies with $p$. This result is stated below, but a definition is needed first.

Definition: $V:\mathbb{R}^n\to\mathbb{R}$ is a local energy function (around $\boldsymbol{s}$) if $\exists W\subseteq \mathbb{R}^n$ s.t.
$\quad 1.$ $W$ is open and contains $\boldsymbol{s}$
$\quad 2.$ $\frac{d}{dt} V (x(t)) \leq 0 \ \, \forall x(t)\in W$
$\quad 3.$ $\left\{t\geq 0: \frac{d}{dt}V(x(t))=0\right\}$ has zero Lebesgue measure $\forall x(t)\in W$ s.t. $\underbrace{F(x(t);p)\neq 0}_{\text{i.e. not a steady state.}}$
Result: (Alberto and Chiang, 2015, p. 198): Suppose there exists a $V$ and $W$ that satisfy the above definition. For each $d\in\mathbb{R}$, let $S_d$ be the connected component of $\{\boldsymbol{x}\in[a,b]^n:V(\boldsymbol{x})\leq d\}$. Then $S^*:=\max_d\{S_d:S_d\subseteq W\}$ is "the largest estimate of [$B_p$] one can obtain via ... $V(\cdot)$."

Finally, my question: Suppose that I can choose a $V$ and $W$ that satisfy the above definition for all $p$ in an $\varepsilon$-ball centered at some $p_0\in\mathbb{R}$. Say I then apply the above method for an arbitrarily chosen $p\in (p_0-\varepsilon,p_0+\varepsilon)$; denote the resultant set $S^*_p$. Letting $\mu_{_{\mathcal{L}}}$ denote the Lebesgue measure, suppose that I can take the derivative $\frac{\partial}{\partial p} \mu_{_{\mathcal{L}}} (S^*_p)$, and find that it is, say, positive at $p=p_0$. Does this imply that $\mu_{_{\mathcal{L}}}(B_p)$ is increasing in $p$ at $p=p_0$?

Since $S_p^*$ is a subset of $B_p$, one would naturally worry that increasing $p$ could cause $S_p^*$ to increase/decrease and for $B_p$ to do the opposite. Can this indeed happen here? Or is $S_p^*$ constructed in a way that precludes this?

Any help on this would be very much appreciated. Thank you very much in advance to anyone who takes the time to consider my question.

Let $\dot{\boldsymbol{x}}=\boldsymbol{F}(\boldsymbol{x};p)$ be an autonomous dynamical system defined on $[a,b]^n$ ($-\infty<a<B<\infty)$; $p\in\mathbb{R}$ is some fixed parameter. Suppose that $\boldsymbol{s}$ is an asymptotically stable steady state for any choice of $p$; given $p$, let $B_p\subseteq [a,b]^n$ denote its basin of attraction.

My question relates to whether I can use a result from Alberto and Chiang (2015, p. 198) to characterize how the Lebesgue measure of $B_p$ varies with $p$. This result is stated below, but a definition is needed first.

Definition: $V:\mathbb{R}^n\to\mathbb{R}$ is a local energy function (around $\boldsymbol{s}$) if $\exists W\subseteq \mathbb{R}^n$ s.t.
$\quad 1.$ $W$ is open and contains $\boldsymbol{s}$
$\quad 2.$ $\frac{d}{dt} V (x(t)) \leq 0 \ \, \forall x(t)\in W$
$\quad 3.$ $\left\{t\geq 0: \frac{d}{dt}V(x(t))=0\right\}$ has zero Lebesgue measure $\forall x(t)\in W$ s.t. $\underbrace{F(x(t);p)\neq 0}_{\text{i.e. not a steady state.}}$
Result: (Alberto and Chiang, 2015, p. 198): Suppose there exists a $V$ and $W$ that satisfy the above definition. For each $d\in\mathbb{R}$, let $S_d$ be the connected component of $\{\boldsymbol{x}\in[a,b]^n:V(\boldsymbol{x})\leq d\}$. Then $S^*:=\max_d\{S_d:S_d\subseteq W\}$ is "the largest estimate of [$B_p$] one can obtain via ... $V(\cdot)$."

Finally, my question: Suppose that I can choose a $V$ and $W$ that satisfy the above definition for all $p$ in an $\varepsilon$-ball centered at some $p_0\in\mathbb{R}$. Say I then apply the above method for an arbitrarily chosen $p\in (p_0-\varepsilon,p_0+\varepsilon)$; denote the resultant set $S^*_p$. Letting $\mu_{_{\mathcal{L}}}$ denote the Lebesgue measure, suppose that I can take the derivative $\frac{\partial}{\partial p} \mu_{_{\mathcal{L}}} (S^*_p)$, and find that it is, say, positive at $p=p_0$. Does this imply that $\mu_{_{\mathcal{L}}}(B_p)$ is increasing in $p$ at $p=p_0$?

Since $S_p^*$ is a subset of $B_p$, one would naturally worry that increasing $p$ could cause $S_p^*$ to increase/decrease and for $B_p$ to do the opposite. Can this indeed happen here? Or is $S_p^*$ constructed in a way that precludes this?

Any help on this would be very much appreciated. Thank you very much in advance to anyone who takes the time to consider my question.

$^*\!$ Note: I realized that "comparative statics" is not standard math terminology. TL;DR, economists basically mean "$\frac{\partial}{\partial\ \text{parameter}} \text{(thing in model)} $," but if you would like to know more here's the Wikipedia page! 😄