Is there a theory of being stable around a path? I've been trying to learn a bit about dynamical systems and stability. Many definitions center around the stability at an equilibrium point $x^*$ of the system $x' = f(x)$. Intuitively this makes sense to me since if $x(0) = x^*$ then $x(t) = x^*$ for every $t \geq 0$, and if $x^*$ is asymptotically stable then all paths in a neighborhood around $x^*%$ converge to it in due time. In addition, this has the nice benefit that we may use Lyapunov's work to show stability.
Can this definition be extended to the case where the "attracting" point depends in time (so that $x^*(t)$ is now an arbitrary solution) ? For example, in the below figure let the red line depicts $x^*(t)$, the attracting point at time $t$. Then the blue line shows $x(t)$ approaching $x^*(t)$ at every time (in the sense that the gap between the red and blue never grows).

To be a bit more formal, I want $x(t) = x^*(t)$ for all $t \geq 0$ if $x(0) = x^*(0)$, and if instead we have $\left | x(0) - x^*(0) \right | < \delta$ then for all large enough time values $t \geq t_0$ we have $\left | x(t) - x^*(t) \right | < \epsilon$.
Does such a concept exist in literature? Can this be made into the well-studied case $x^* = 0$ via some appropriate transformation? Can a Lyapunov-type result (possibly time-dependent) exist for such a situation?
 A: As @Aphyd points out, this could be done by making a coordinate transform so that a point moving along the trajectory x(t) becomes stationary, while other points (such as x*(t)) moves with the same velocity relative to x(t). Interestingly some systems remain topologically unchanged under such transformation, such as $x'=x, y'=y$. Most of the time a system's stability or topological structure is completely determined by its nonwandering set, which is just a generalized form of stationary points.
A: One can use Lyapunov exponents and the classical smooth ergodic theory to address this matter (with additional substantial quantitative information) (see e.g. Computing Lyapunov Exponents , Lyapunov exponent for 2D map?), at fixed points they reduce to the logarithms of eigenvalues of the linearization of the flow (see Lyapunov exponent of a stable p-cycle.).
More generally, two classical theorems in smooth ergodic theory are relevant: theorems of Oseledets and Pesin. Here is a rough description (see https://encyclopediaofmath.org/wiki/Multiplicative_ergodic_theorem, https://encyclopediaofmath.org/wiki/Pesin_theory for further details and references; the connection to the stability theory of ODE's is, assuming the vector field is twice continuously differentiable, by way of the equations of first variation, see https://encyclopediaofmath.org/wiki/Variational_equations).
Let us assume that we have a bounded set invariant under the flow. Then Oseledets' Ergodic Theorem says that for almost all (with respect to any flow-invariant Borel probability measure on the invariant set) initial conditions $x$, there are real numbers $\chi_x^1, \chi_x^2,...,\chi_x^n$ (possibly with multiplicity, here $n$ is the dimension of the configuration space), called the Lyapunov exponents of the flow at $x$, that dictate the exponential rate at which nearby solutions separate from the one uniquely defined by $x$. As mentioned above, if $x$ happens to be a fixed point, then $\chi_x^i=\ln|\lambda_x^i|$, where $\lambda_x^i$ is the $i$th eigenvalue (counted possibly with multiplicity) of the derivative of the time-$1$ map of the flow at $x$. If $x$ is not fixed, there will be at least one zero Lyapunov exponent, coming from the orbit direction (see Lyapunov exponents in bounded systems or Zero Lyapunov exponent for chaotic systems ; this is essentially why I assumed the existence of a bounded invariant set; one could alternatively require the linearization to be log-integrable).
In particular if the first $s$ Lyapunov exponents at $x$ are negative and the remainder are nonnegative, then there are exactly $s$ directions along which the solution defined by $x$ is asymptotically (exponentially) stable. Then Pesin's Stable Manifold Theorem says that there is an embedded open ball $\mathcal{S}_{x,loc}$.in the configuration space, called the local stable manifold of the flow at $x$, centered at $x$ of dimension exactly $s$ such that any solution with initial condition in $\mathcal{S}_{x,loc}$ will converge exponentially fast to the solution defined by $x$ as time goes to infinity. Further, the embedding defining the local stable manifold is twice continuously differentiable.
