Many practical ordinary differential equations show a puzzling phenomena when attempting to approximate them: stiffness, which is both well understood and completely undefined in the current literature. It is known that when a problem is stiff, more simple and conventional numerical methods of approximation will have huge computational demands in order to approximate the problem. This comes from these simpler, explicit methods having finite regions of absolute stability. Methods with infinite regions of absolute stability, on the other hand, are much more efficient at approximating
these stiff problems. Although less efficient for problems that are not stiff, they are greatly useful for approximating problems when they are known to be stiff. But how do we know if the stiff in the first place? We cannot apply the definition of a stiff problem to check; it has no agreed upon definition. This thesis instead chose to analyze for stiffness based on the characteristics that stiff problems tend to exhibit. We used and implemented into MATLAB three common approaches: the step ratio approach, the eigenvalue ratio approach, and the stiffness indicator approach. We then applied these approaches on multiple initial value problems to analyze for stiffness, including linear and non-linear systems, as well as two chaos systems. Our analysis revealed that some of the approaches implemented were better fitted for certain types of problems than others, and that the results of the analysis also showed multiple systems’ stiffness changing in correlation with one or multiple of its state variables. The two chaos systems also saw oscillatory behavior within the stiffness of their solutions, changing in frequency as function parameters were altered.