differential equations - What happens to the solutions of a fourth-order boundary-value problem as you turn off the fourth-order coefficient?

Background

Lagrangian mechanics on $mathbb R^n$ is usually defined by picking a Lagrangian function $L: {rm T}mathbb R^n to mathbb R$, where ${rm T}mathbb R^n = mathbb R^{2n}$ is the tangent bundle of the configuration space $mathbb R^n$. Such a function determines the Euler-Lagrange equations:
$$frac{d}{dt}left[ frac{partial L}{partial v^i}bigl( dotgamma(t), gamma(t)bigr) right] - frac{partial L}{partial q^i} bigl( dotgamma(t), gamma(t)bigr) = 0$$
Here $(v^i,q^i)$ for $i=1,dots,n$ are the standard coordinates on ${rm T}mathbb R^n$, $gamma: [0,T] to mathbb R^n$ is a smooth function, and $dotgamma^i(t) = frac{dgamma^i}{dt}$. Suppose that the matrix $frac{partial^2 L}{partial v^ipartial v^j}(v,q)$ is invertible for any $(v,q) in {rm T}mathbb R^n$. Then the Euler-Lagrange equations are a nondegenerate second-order differential equation on $mathbb R^n$. I am interested in the boundary-value problem for $L$. Namely, fix $T > 0$ and $q_1,q_2 in mathbb R^n$; the BVP asks to find the set $C(q_1,q_2,T)$ of all paths $gamma: [0,T] to mathbb R^n$ with $gamma(0) = q_1$ and $gamma(t) = q_2$. Generically, this is a discrete set.

My question

Suppose that if instead of the Euler-Lagrange equations above, I pick some small parameter $epsilon$ and consider the differential equation
$$frac{d}{dt}left[ frac{partial L}{partial v^i}bigl( dotgamma(t), gamma(t)bigr) right] - frac{partial L}{partial q^i} bigl( dotgamma(t), gamma(t)bigr) = epsilongamma^{(4)}(t)^i$$
where $gamma^{(4)}(t)^i$ is the $i$th component of the fourth derivative of $gamma$ with respect to $t$ (the "jounce", a word I just learned from Wikipedia). For $epsilon neq 0$, the EL equations are a nondegenerate fourth-order differential equation, and so generically solutions to the boundary-value-problem above form a two-dimensional family. To restrict to a discrete set, we should fix more boundary values. Pick $(v_1,q_1), (v_2,q_2) in {rm T}mathbb R^n$ and $T > 0$, and define $C_epsilon(v_1,q_1,v_2,q_2,T)$ to be the set of solutions $gamma$ to the $epsilon$-dependent EL equations with $(dotgamma(0),gamma(0)) = (v_1,_1)$ and $(dotgamma(T),gamma(T)) = (v_2,q_2)$.

My question is: as $epsilon to 0$, in what sense do we have $C_epsilon(v_1,q_1,v_2,q_2,T) to C(q_1,q_2,T)$?

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