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Nonlinear differential equations are those where the unknown function and its derivatives appear to a power other than one or are multiplied together. These types of equations can model complex systems where changes are not proportional. Understanding them is crucial as linear methods don't apply directly. In our system, nonlinear terms like \[ y_1 y_3 \] in \[\dot{y}_2=y_{1}-y_{2}-y_{1}y_{3}\] introduce complexity. They cause interactions between equations, which can create unique, sometimes unpredictable, behavior in solutions. This interaction makes solving these problems analytically challenging, often necessitating numerical methods for solution, which in turn increase the need to consider factors like stiffness.

Examining initial conditions is a critical step in the analysis of differential equations. They set the starting point of the system's evolution. By plugging in the initial conditions, namely\[ y_1(0)=0, y_2(0)=1, y_3(0)=1 \], we determine the starting rates of change for our variables. This gives insights into how the system will behave immediately from the start. As we saw, \[ \dot{y}_{3}(0) = -100 \] is dominant, indicating a significant initial change in \[ y_3 \] compared to other variables. Initial conditions help us anticipate initial behavior, but also highlight potential stiffness if some variables change much faster than others initially.

Numerical stability refers to the behavior of the numerical solution of differential equations as it progresses over time. For stiff systems, certain numerical methods become unstable unless very small step sizes are used. This instability arises when the fast-changing components dominate the solution, causing large errors in the simulation. With our problem, the term \[-100 y_3\] in \[\dot{y}_3\] is a source of fast dynamics, suggesting potential stability issues. Understanding these components is crucial in choosing the appropriate numerical methods, like implicit methods, which often handle stiffness better than explicit methods in these scenarios.

Studying system dynamics involves observing how the system evolves over time. By examining the differential equations,\[\begin{aligned}\dot{y}_{1} &= -0.01y_{1} + 0.01y_{2} \\dot{y}_{2} &= y_{1} - y_{2} - y_{1}y_{3} \\dot{y}_{3} &= y_{1}y_{2} - 100y_{3}\end{aligned}\],we understand how different factors influence the rates of change of each variable. The large negative rate for \[ y_3 \] suggests fast decay, contrasting the slower dynamics in \[ y_1 \] and \[ y_2 \]. Taking into account \[ y_1 y_2 \]'s role in \[\dot{y}_3\], these insights help predict future states and stability. Analyzing dynamics is crucial for understanding not just whether a system is stiff, but how external changes may affect its behavior over time.