Consider a dynamical system $\dot{x} = f(x,p)$, where $p \in R^n$ is a bunch of parameters and $f$ is a smooth function, both in $x$ and $p$. Given an initial value problem where $x(0) = x_0$, it can easily be seen that any bounded solution on a finite time-interval is continuous w.r.t. $p$ (proof by Gronwall-estimate).

But what happens when we consider the unstable manifold of a fixed-point? For simplicity, lets consider $f(0,p) = 0 \, \forall p$, and $0$ has an unstable manifold $M^-_p$ of dimension $1$ for all $p$. Given some $p_0$ and a bounded segment of $M^-_{p_0}$ that includes $0$, is there any sort of continuity as we change $p$?

I expect a structurally similar result as in the initial value case, though this case is technically more delicate. Any help or key word that hints in the right direction is appreciated, as I am no expert in this field.

EDIT: An approach that came to my mind is to prove that there exists some kind of linearisation which is valid in a very small ball $B_\delta(0)$ that does NOT depend on $p$. Given this, one can smoothly change the coordinates in this area.

Consider a dynamical system $\dot{x} = f(x,p)$, where $p \in R^n$ is a bunch of parameters and $f$ is a smooth function, both in $x$ and $p$. Given an initial value problem where $x(0) = x_0$, it can easily be seen that any bounded solution on a finite time-interval is continuous w.r.t. $p$ (proof by Gronwall-estimate).

But what happens when we consider the unstable manifold of a fixed-point? For simplicity, lets consider $f(0,p) = 0 \, \forall p$, and $0$ has an unstable manifold $M^-_p$ of dimension $1$ for all $p$. Given some $p_0$ and a bounded segment of $M^-_{p_0}$ that includes $0$, is there any sort of continuity as we change $p$?

I expect a structurally similar result as in the initial value case, though this case is technically more delicate. Any help or key word that hints in the right direction is appreciated, as I am no expert in this field.

EDIT: An approach that came to my mind is to prove that there exists some kind of linearisation which is valid in a ball $B_\delta(0)$ that does NOT depend on $p$. Given this, one can smoothly change the coordinates in this area.

Consider a dynamical system $\dot{x} = f(x,p)$, where $p \in R^n$ is a bunch of parameters and $f$ is a smooth function, both in $x$ and $p$. Given an initial value problem where $x(0) = x_0$, it can easily be seen that any bounded solution on a finite time-interval is continuous w.r.t. $p$ (proof by Gronwall-estimate).

But what happens when we consider the unstable manifold of a fixed-point? For simplicity, lets consider $f(0,p) = 0 \, \forall p$, and $0$ has an unstable manifold $M^-_p$ of dimension $1$ for all $p$. Given some $p_0$ and a bounded segment of $M^-_{p_0}$ that includes $0$, is there any sort of continuity as we change $p$?

I expect a structurally similar result as in the initial value case, though this case is technically more delicate. Any help or key word that hints in the right direction is appreciated  :)

Consider a dynamical system $\dot{x} = f(x,p)$, where $p \in R^n$ is a bunch of parameters and $f$ is a smooth function, both in $x$ and $p$. Given an initial value problem where $x(0) = x_0$, it can easily be seen that any bounded solution on a finite time-interval is continuous w.r.t. $p$ (proof by Gronwall-estimate).

But what happens when we consider the unstable manifold of a fixed-point? For simplicity, lets consider $f(0,p) = 0 \, \forall p$, and $0$ has an unstable manifold $M^-_p$ of dimension $1$ for all $p$. Given some $p_0$ and a bounded segment of $M^-_{p_0}$ that includes $0$, is there any sort of continuity as we change $p$?

I expect a structurally similar result as in the initial value case, though this case is technically more delicate. Any help or key word that hints in the right direction is appreciated, as I am no expert in this field.

EDIT: An approach that came to my mind is to prove that there exists some kind of linearisation which is valid in a very small ball $B_\delta(0)$ that does NOT depend on $p$. Given this, one can smoothly change the coordinates in this area.

Consider a dynamical system $\dot{x} = f(x,p)$, where $p \in R^n$ is a bunch of parameters and $f$ is a smooth function, both in $x$ and $p$. Given an initial value problem where $x(0) = x_0$, it can easily be seen that any bounded solution on a finite time-interval is continuous w.r.t. $p$ (proof by Gronwall-estimate).

But what happens when we consider the unstable manifold of a fixed-point? For simplicity, lets consider $f(0,p) = 0 \, \forall p$, and $0$ has an unstable manifold $M^-_p$ of dimension $1$ for all $p$. Given some $p_0$ and a bounded segment of $M^-_{p_0}$ that includes $0$, is there any sort of continuity as we change $p$?

I expect something as in the initial value case, but google did not provide any answer so far. Any help or key word that hints in the right direction is appreciated :)

Consider a dynamical system $\dot{x} = f(x,p)$, where $p \in R^n$ is a bunch of parameters and $f$ is a smooth function, both in $x$ and $p$. Given an initial value problem where $x(0) = x_0$, it can easily be seen that any bounded solution on a finite time-interval is continuous w.r.t. $p$ (proof by Gronwall-estimate).

But what happens when we consider the unstable manifold of a fixed-point? For simplicity, lets consider $f(0,p) = 0 \, \forall p$, and $0$ has an unstable manifold $M^-_p$ of dimension $1$ for all $p$. Given some $p_0$ and a bounded segment of $M^-_{p_0}$ that includes $0$, is there any sort of continuity as we change $p$?

I expect a structurally similar result as in the initial value case, though this case is technically more delicate. Any help or key word that hints in the right direction is appreciated :)