Is there a formal name for a mathematical model that has an overly sensitive parameter

Question

I'm a software engineer with a bit of mathematics expertise working with an NGO that has developed mathematical models for ecosystem services. It turns out that one of these models is overly sensitive to a parameter. Specifically the output of the model is exponential with respect to that parameter and linear with respect to the others.

Is there a formal name for this concept?

Answer 1

In the study of dynamical systems there is the concept of 'sensitive dependence on initial conditions' (abbreviated SDIC) which says that two initially nearby trajectories will diverge exponentially quickly. This is known as the 'butterfly effect' in popular terminology.

More mathematically, take two trajectories $x_1(t)$, $x_2(t)$ and define the difference between them by $\delta x(t) = |x_1(t)-x_2(t)|$. We say that a system exhibits SDIC if for two initially nearby trajectories (in the sense that $\delta x(0)$ is small) we have

$$\delta x(t) \sim e^{\lambda t}\delta x(0)$$

for some $\lambda>0$, i.e. there is an exponentially fast divergence.

Your case is slightly more complicated because the initial sensitivity is to a parameter, rather than to the initial conditions. However, a dynamical system with state $x$ and a parameter vector $c$

$$\dot{x}=f(x,c)$$

can always be reimagined as a dynamical system in a larger state space by defining $y=(x,c)$ and setting

$$\dot{y} = (f(x,c), 0)$$

in which case you can apply the definitionto conclude that a system with exponential sensitivity to one of the parameters exhibits SDIC.

Answer 2

Complementary answer to Chris Taylor's answer is the word bifurcation. It is usually used for systems where some parameters of the system is changed and the trajectories of the system vary quite drastically qualitatively.

Answer 3

one of these models is overly sensitive to a parameter. Specifically the output of the model is exponential with respect to that parameter and linear with respect to the others.

The term often used to describe this is "stiffness" (of a system of differential equations or other dynamical model). It means a model where some parameters have a different scale of influence than others. A common situation is where there are fast and slow variables that operate on different time scales, and the model shows much sharper sensitivity to the rate parameters for one type of variable than the other.