Finding Eigenvalues for Different r Values

[Dustinsfl]

$$
\mathcal{J} = \begin{pmatrix}
-\sigma & \sigma & 0\\
1 & -1 & -\sqrt{b(r - 1)}\\
\sqrt{b(r - 1)} & \sqrt{b(r - 1)} & - b
\end{pmatrix}
$$
From a quick try and error, I was able to find that when $r = 1.3456171$ we will have 3 negative eigenvalues.
But when $r = 1.3456172$, there will be a complex-conjugate pair of eigenvalues.
Is there a mathematically more elegant way to determine this r value?
$b = \frac{8}{3}$ and $\sigma = 10$

 

[Opalg]

$$
\mathcal{J} = \begin{pmatrix}
-\sigma & \sigma & 0\\
1 & -1 & -\sqrt{b(r - 1)}\\
\sqrt{b(r - 1)} & \sqrt{b(r - 1)} & - b
\end{pmatrix}
$$
From a quick try and error, I was able to find that when $r = 1.3456171$ we will have 3 negative eigenvalues.
But when $r = 1.3456172$, there will be a complex-conjugate pair of eigenvalues.
Is there a mathematically more elegant way to determine this r value?
$b = \frac{8}{3}$ and $\sigma = 10$

This is an interesting problem. The eigenvalue equation is $\det(\mathcal{J} - \lambda I) = 0$, which simplifies to $$\lambda^3 + (b+\sigma+1)\lambda^2 + b(r+\sigma)\lambda + 2b\sigma(r-1) = 0.\qquad(1)$$ At the critical value of $r$ (the one where the bifurcation occurs, from three real roots to a complex conjugate pair), the eigenvalue equation will have a repeated root. That root will also be a root of the derived equation. So differentiate the eigenvalue equation to get $$3\lambda^2 +2(b+\sigma+1)\lambda + b(r+\sigma) = 0.\qquad(2)$$ Solve those equations together, to find that $$\lambda = \frac{18b\sigma(r-1) - b(b+\sigma+1)(r+\sigma)}{2(b+\sigma+1)^2 -6b(r+\sigma)}.$$ The next stage is to substitute that value of $\lambda$ into equation (2), to get a cubic equation for $r$. If you have stamina and patience enough to do that, you should find that, for the given values of $b$ and $\sigma$, one of the solutions is $r = 1.345617...$. (Rather you than me, though. (Yawn) )

 

[Dustinsfl]

Solve those equations together, to find that $$\lambda = \frac{18b\sigma(r-1) - b(b+\sigma+1)(r+\sigma)}{2(b+\sigma+1)^2 -6b(r+\sigma)}.$$

How did you get this lambda? I solved for the quadratic and got something different. I also set the quadratic and cubic equal and then solved for lambda but didn't arrive at this lambda either.

 

[Opalg]

How did you get this lambda? I solved for the quadratic and got something different. I also set the quadratic and cubic equal and then solved for lambda but didn't arrive at this lambda either.

Starting from $$\lambda^3 + (b+\sigma+1)\lambda^2 + b(r+\sigma)\lambda + 2b\sigma(r-1) = 0\qquad(1)$$ and $$3\lambda^2 +2(b+\sigma+1)\lambda + b(r+\sigma) = 0,\qquad(2)$$ I multiplied (1) by 3, and (2) by $\lambda$, and subtracted, getting $$(b+\sigma+1)\lambda^2 + b(r+\sigma)\lambda + 6b\sigma(r-1) = 0.\qquad(3)$$ I then multiplied (2) by $(b+\sigma+1)$, and (3) by 3, and again subtracted, to get the equation $$\lambda = \frac{18b\sigma(r-1) - b(b+\sigma+1)(r+\sigma)}{2(b+\sigma+1)^2 -6b(r+\sigma)}.$$

 

[blue_raver22]

There are a few different approaches that can be used to find the eigenvalues for different values of r in this matrix. One method is to use the characteristic polynomial, which is found by taking the determinant of the matrix $\mathcal{J} - \lambda I$, where $\lambda$ is the eigenvalue and $I$ is the identity matrix. This polynomial can then be solved for values of r that result in specific eigenvalues.

Another approach is to use the fact that the trace of a matrix is equal to the sum of its eigenvalues. In this case, the trace of the matrix is $-11 + \sqrt{b(r-1)}$, so if we can find values of r that make the trace equal to the desired eigenvalues, we can determine the corresponding eigenvalues.

Additionally, we can use the fact that the determinant of a matrix is equal to the product of its eigenvalues. By setting the determinant of $\mathcal{J}$ equal to 0, we can find values of r that result in eigenvalues of 0, which can then be used to find the remaining eigenvalues.

Overall, there are multiple ways to find the eigenvalues for different values of r in this matrix, and the most appropriate method may depend on the specific values of r and the desired eigenvalues.