Wrote most of Kessler Syndrome Section

CurrentWriting/sections/02_KesslerSyndrome.tex

@@ -36,19 +36,45 @@ It is easily shown that this criteria is sufficient to guarantee Rao and Rondina
 It has three primary benefits:
 \begin{itemize}
     \item  % Can be solved for algebraically or numerically for a given, bounded state space.
-        The kessler region can be numerically described within bounded state spaces.
+        The $\epsilon$-kessler region can be numerically described within bounded state spaces.
     \item % This is what you would actually compute.
-        In a Computational General Equilibrium Model, as most models of any complexity will be,
+        In a computational model, as most models of any complexity will be,
         you cannot check for divergence numerically.
         The condition given is a basic guarantee of the divergent behavior that is
         required for Kessler Syndrome and acknowledges computational limitations.
     \item Finally, a slow divergence is no divergence in the grand scheme of things.
         It is possible to have a mathematically divergent function, but one that is so slow,
         there is no noticable degree of debris growth before Sol enters a red giant phase.
-        In this specification, it is possible to choose $\epsilon$ such that the divergent behavior 
+        In this specification, it is possible to choose $\epsilon$ such that the divergent behaviors
-        has an impact on a meaningful timescale.
+        identified have an impact on a meaningful timescale.
-%    \item % Stochastic versions could might be describable as martigales.
 \end{itemize}
 
 
+% Issue with this approach: What about cyclical behaviors?
+% Autocatalysis leads to high debris leads to reduced launches 
+% which leads to debris decay leads to increased launches leads to Autocatalysis
+There is at least one issue with this definition of $\epsilon$-kessler regions.
+Let's define a ``proto-kesslerian'' region as the stock and debris levels such that:
+\begin{align}
+    \kappa = \left\{ \{s^j_t\}, D_t : 
+            D_{t+1} - D_{t} \geq \epsilon > 0 \right\}
+\end{align}
+It may be, under certain situations, the case that optimal launch rates cycle along with
+debris and stock levels, leading to a cycle in and out of the proto-kesslerian regions.
+This is an issue because, assumning a stable cycle, Rao's definition
+of the kessler region would capture this behavior, but the $\epsilon$-kessler definition 
+would not.
+I believe, but have not verified, that some choices of $\epsilon$, although permitting cycles,
+would relegate them to levels with minimal economic impact.
+%Maybe can be studies by phase or flow diagrams?
+
+
+%Area of research: What makes a good \epsilon?
+This leads to the important question of ``What makes a good value of $\epsilon$?''
+One method, in the spirit of \cite{Adilov2018}, is to choose a change in debris, $D_{t+1} - D_t$, such that
+the loss of satellites in periods $t+1$ to $t+k$ is increased by or to a certain percentage, say 50\%.
+I've put very little thought into addressing this general question so far,
+and need to analyze the implications of different choice rules.
+
+
 \end{document}