Is the System Stiff? Analyzing Eigenvalues of A

[evinda]

Hello! (Wave)

Let the problem of initial values

$\left\{\begin{matrix}
y_1'(t)=-0.5y_1(t)+0.501 y_2(t), t \in [0,10^3]\\
y_2'(t)=0.501y_1(t)-0.5y_2(t) \\
y_1(0)=1.1 \\
y_2(0)-0.9
\end{matrix}\right. \\ $

The question is if the system is stiff.$$A=\bigl(\begin{smallmatrix}
-0.5 & 0.501\\
0.501 & -0.5
\end{smallmatrix}\bigr)$$

$$y'=Ay$$

where $y'=\binom{y_1'}{y_2'}, \ y=\binom{y_1}{y_2}$.

The eigenvalues are $\lambda_1=-1.001, \lambda_2=0.001$.According the assistant of the professor, since it holds that $|\lambda_1|>> |\lambda_2|$ we deduce that the system is stiff.
But I found the following in my book:>Stiff systems are systems of the form $x'=Ax$ for ehich there are eigenvalues $\lambda_{\mu}$ and $\lambda_{\nu}$ of $A$, with negative real part, such that $|Re{\lambda_{\mu}}|>>|Re{\lambda_{\nu}}|$.So since we found only one negative and one positive eigenvalue, don't we conclude that the system isn't stiff? Or am I wrong?

 

[jvicens]

Hello there! As a fellow scientist, let me offer my perspective on this problem.

Based on the definition of stiff systems that you found in your book, it seems like the system in question may not be considered stiff. However, it is important to note that the concept of stiffness is not always well-defined and can vary depending on the context and the problem at hand.

In this case, it may be helpful to also consider the values of the eigenvalues in addition to their signs. Even though there is only one negative eigenvalue, the magnitude of that eigenvalue is much larger than the positive eigenvalue. This indicates that the system may exhibit behavior similar to that of a stiff system, even if it may not technically meet the strict definition.

Ultimately, whether or not the system is considered stiff may depend on the specific criteria and definitions used by the professor or the assistant. It may be helpful to discuss this further with them to gain a better understanding of their reasoning and perspective on the problem.

I hope this helps and good luck with your analysis!