Consider a continuous time nonlinear dynamical system
$$\dot X = f(X, t), \quad X = (x_1, \dotsc, x_N),\tag{1}$$
and a particular solution $X(t)$ with initial value $X(0) = X_0$. Moreover, suppose that, for every $r > 0$, I can obtain a solution $Y_r(t)$ to the related system
$$\dot Y = f(Y, t) + \frac{1}{r} g(Y,t),\tag{2}$$
with initial value
$$Y_r(0) = X_0 + O(r^{-1}).\tag{3}$$
One would naively expect that $Y_r(t) \to X(t)$ pointwise as $r \to \infty$, but is this necessarily the case? I suspect not, but I cannot think of a counterexample. Are there some conditions on $f$ and $g$ (continuity, differentiability, …) that would guarantee such convergence of the solutions?
I would also appreciate helpful reference suggestions and/or search terms.
Best Answer
In case of Lipschitz continous $f+\tfrac{g}{r}$ and continous $g$, I was able to show the desired result. If we consider the difference between $X$ and $Y_r$ we obtain $$ \tfrac{\partial}{\partial t}(X-Y_r)=f(X,t)-f(Y_r,t)-\tfrac{1}{r}g(Y_r,t)\\ \Rightarrow \tfrac{\partial}{\partial t}\tfrac{1}{2}|X-Y_r|^2=\left(f(X,t)+\tfrac{1}{r}g(X,t)-f(Y_r,t)-\tfrac{1}{r}g(Y_r,t)\right)\cdot(X-Y_r)-\tfrac{1}{r}g(X,t)\cdot(X-Y_r)\\ \Rightarrow \tfrac{\partial}{\partial t}|X-Y_r|^2\leq 2 L|X-Y_r|^2+\tfrac{2}{r}|g(X,t)|\cdot|X-Y_r|\\ u(t)-(O(\tfrac{1}{r}))^2\leq \int_0^t2 Lu(s)ds+\tfrac{2}{r}\int_0^t|g(X,s)|\sqrt{u(s)}ds $$ where $u(t)=|X(s)-Y(s)|^2$. A nonlinear version of Gronwall's Lemma e.g. here, Theorem 2.4 yields $$ u(t)\leq e^{2Lt}\left(O(\tfrac{1}{r})+\tfrac{1}{r}\int_0^t|g(X(s),s)|e^{2Ls}ds\right)^2 $$
this proves the desired convergence.
Remark: The lipschitz continuity of $f+\tfrac{g}{r}$ can be replaced by $f$ lipschitz and $|g(y,t)|\leq C(t)$ uniformly bounded in $y$.