Assessing stability or instability of a system of equations with complex eigenvaluesHaving this system , we get clearly two complex eigenvalues. If one has to assess the stability of the system at these eigenvalues, we have for the matrix A:abcdthat&nbsp;T2−4Δ≥&nbsp;or&nbsp;≤&nbsp;or&nbsp;=0, where&nbsp;T=a+c, while&nbsp;Δ=ad−bc=DetA. But with complex eigenvalues&nbsp;±2i, T which must be real, becomes complex and&nbsp;δ&nbsp;is always 0, when it would vary from greater than or lesser than zero for real eigenvalues. How do we solve this with complex eigenvalues?

Question

Assessing stability or instability of a system of equations with complex eigenvaluesHaving this system , we get clearly two complex eigenvalues. If one has to assess the stability of the system at these eigenvalues, we have for the matrix A:abcdthat&amp;nbsp;T2−4Δ≥&amp;nbsp;or&amp;nbsp;≤&amp;nbsp;or&amp;nbsp;=0, where&amp;nbsp;T=a+c, while&amp;nbsp;Δ=ad−bc=DetA. But with complex eigenvalues&amp;nbsp;±2i, T which must be real, becomes complex and&amp;nbsp;δ&amp;nbsp;is always 0, when it would vary from greater than or lesser than zero for real eigenvalues. How do we solve this with complex eigenvalues?

Habrmanh6h · Accepted Answer

If the eigenvalues are pure imaginary, then the system has only a center. If the eigenvalues are complex, the non-zero real part (call it α) of the term T2−4Δ defines whether it is a stable or unstable spiral - for α&amp;gt;0, then it is a unstable spiral point, and inversely, α&amp;lt;0 then it is a stable spiral.

Assessing stability or instability of a system of

Answered question

Answer & Explanation

New Questions in Calculus 2