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

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?

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Chaotic? –  user4143 Nov 15 '11 at 7:18
Ill-conditioned? Although this increasingly is applied to matrices. –  daniel Nov 15 '11 at 10:44
@daniel: "ill-conditioned" is a more general term, and not just applicable to linear algebra. :) –  Ｊ. Ｍ. Nov 15 '11 at 16:00

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.

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Thank you. This jogs a memory in a numerical analysis course oh so many years ago. –  Rich Nov 15 '11 at 21:41

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.

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Yes, it's important to point this out too - a drastic change could be the result of a bifurcation rather than SDIC. I did my PhD in bifurcation theory, I really should have remembered to put this in! –  Chris Taylor Nov 15 '11 at 11:06

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.

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