CDS - Classification of the Steady States of a 2D Linear System • The Plane Delta-Tau

Remember:

If we have complex conjugate eigenvalues, meaning: ${\lambda }_{1,2}=a\pm ib$, then we will obtain form the general solution in 2D $\left(x(t)=\alpha {\vec{v}}_1{e}^{{\lambda }_{1}t}+\beta {\vec{v}}_2{e}^{{\lambda }_{2}t}\right)$ can be rewritten in a sinusolidal form, using the Euler formula:$e^{a\pm ib}=e^{at}\left(\cos (bt)+i\sin (bt)\right)$Such terms represent growing oscillations for $a=\mathrm{Re}(\lambda )>0$ and decaying oscillations for for $a=\mathrm{Re}(\lambda )<0$, while if the real part is equal to $0$ we will have non-changing oscillation.

We can classify the steady states of a 2D linear systems, using the following notation After calculating the eigenvalues ${\lambda }_1$ and ${\lambda }_2$, we define $\tau$ (the trace matrix of $A$) as:$\tau ={\lambda }_1+{\lambda }_2$And $\Delta$ (the determinant of $A$) as:$\Delta ={\lambda }_1{\lambda }_2$Then we can define the type of steady state and its stability using this graph:

---

• In case we deal with complex eigenvalues.
• However, again, we will not use this.not-sure-about-this
• It is important to understand the difference between positive/negative and real/complex eigenvalues.IMPORTANT