Examine the character of the rest point (0,0) of the system…

Examine the character of the rest point $(0,0)$ of the system $$\dot{x}=-2x+3y,\qquad\dot{y}=-x-y.$$

Step 1 · Coefficient matrix

$$A=\begin{pmatrix}-2&3\\-1&-1\end{pmatrix}.$$

Step 2 · Characteristic equation and eigenvalues

$$\det(A-\lambda I)=0\implies\lambda^2+3\lambda+5=0.$$ Discriminant $\Delta=-11<0$, so $$\lambda_{1,2}=-\frac{3}{2}\pm\frac{\sqrt{11}}{2}\,i.$$

Step 3 · Classification of the rest point

The eigenvalues are complex with negative real part: $\operatorname{Re}(\lambda_{1,2})=-\tfrac32<0$. The rest point $(0,0)$ is a stable focus: all trajectories spiral toward the origin as $t\to+\infty$.

Step 4 · Direction of rotation

Evaluating $\dot y$ at $(1,0)$ gives $\dot y=-1<0$, so the trajectories rotate clockwise.

Step 5 · General real solution

With eigenvector $\mathbf v=(1,2)^\top+i(\sqrt{11},0)^\top$ for $\lambda=-\tfrac32-\tfrac{\sqrt{11}}{2}i$, $\alpha=-\tfrac32$, $\beta=\tfrac{\sqrt{11}}{2}$: $$x(t)=e^{-3t/2}\!\left[(C_1-\sqrt{11}C_2)\cos\tfrac{\sqrt{11}}{2}t +(C_2+\sqrt{11}C_1)\sin\tfrac{\sqrt{11}}{2}t\right],$$ $$y(t)=e^{-3t/2}\!\left[2C_1\cos\tfrac{\sqrt{11}}{2}t+2C_2\sin\tfrac{\sqrt{11}}{2}t\right].$$

Step 6 · Behaviour of trajectories

The distance from the origin satisfies $r(t)\propto e^{-3t/2}\to0$ as $t\to+\infty$, confirming inward spiralling with angular frequency $\omega=\tfrac{\sqrt{11}}{2}$ and period $T=\tfrac{4\pi}{\sqrt{11}}$. As $t\to-\infty$, $r(t)\to+\infty$.

Step 7 · Stability analysis

Since $\operatorname{Re}(\lambda_{1,2})=-\tfrac32<0$, the rest point is stable and asymptotically stable.

Step 8 · Conclusion

$$\boxed{ (0,0)\text{ is a \textbf{stable focus} (asymptotically stable).}\quad \lambda_{1,2}=-\tfrac{3}{2}\pm\tfrac{\sqrt{11}}{2}\,i. }$$ All trajectories spiral clockwise toward the origin with angular frequency $\omega=\tfrac{\sqrt{11}}{2}$ and exponential decay rate $\alpha=\tfrac{3}{2}$, confirming asymptotic stability.

Summary Table

Additional Notes

• The classification follows from the standard theory of planar linear autonomous systems, based on the sign and nature of the eigenvalues of the coefficient matrix.