I believe that the method of solution to your problem is called the method of "dominant balance", and in this case, "singular dominant balance." If you do a web search for that, you should be able to find the information you need.

This method will you give a perturbative solution to as high a degree as you have the propensity to calculate. You can analyze this solution to answer various questions that you implied in your original question, such as what the decay behavior of the solution is, which continuity and smoothness properties it has, etc...

You may also want to search for "catastrophe theory", which catalogs the types of bifurcations that happen in systems such as you have described. This is a one-dimensional bifurcation problem, which are well-studied.

I believe that the method of solution to your problem is called the method of "dominant balance", and in this case, "singular dominant balance." If you do a web search for that, you should be able to find the information you need.

This method will you give a perturbative solution to as high a degree as you have the propensity to calculate. You can analyze this solution to answer various questions that you implied in your original question, such as what the decay behavior of the solution is, which continuity and smoothness properties it has, etc...

If you want to study the solutions of a large class of coefficient functions, not just a specific set, you can leave arbitrary constants in a solution "ansatz" and then develop a parameterized family of solutions. Note that the algebraic expressions involved in finding the simple-looking solutions grow exponentially in the number terms which end up simplifying in the end. Computer algebra is needed to find the simplified form of these solutions, lest you go mad and kill many trees.

You may also want to search for "catastrophe theory", which catalogs the types of bifurcations that happen in systems such as you have described. This is a one-dimensional bifurcation problem, which are well-studied.

I believe that the method of solution to your problem is called the method of "dominant balance", and in this case, "singular dominant balance." If you do a web search for that, you should be able to find the information you need.

You may also want to search for "catastrophe theory", which catalogs the types of bifurcations that happen in systems such as you have described.

I believe that the method of solution to your problem is called the method of "dominant balance", and in this case, "singular dominant balance." If you do a web search for that, you should be able to find the information you need.

This method will you give a perturbative solution to as high a degree as you have the propensity to calculate. You can analyze this solution to answer various questions that you implied in your original question, such as what the decay behavior of the solution is, which continuity and smoothness properties it has, etc...

You may also want to search for "catastrophe theory", which catalogs the types of bifurcations that happen in systems such as you have described. This is a one-dimensional bifurcation problem, which are well-studied.