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

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

Step 3 · Classification of the rest point

The eigenvalues are complex with negative real part: $\operatorname{Re}(\lambda_{1,2})=-\tfrac12<0$. 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 counter-clockwise.

Step 5 · Eigenvector and general real solution

For $\lambda_1=-\tfrac12+\tfrac{\sqrt3}{2}i$: eigenvector $\mathbf v^{(1)}=(-1+\sqrt3 i,\,2)^\top=(-1,2)^\top+i(\sqrt3,0)^\top$. With $\alpha=-\tfrac12$, $\beta=\tfrac{\sqrt3}{2}$: $$x(t)=e^{-t/2}\!\left[(-C_1+\sqrt3C_2)\cos\tfrac{\sqrt3}{2}t +(-C_2-\sqrt3C_1)\sin\tfrac{\sqrt3}{2}t\right],$$ $$y(t)=e^{-t/2}\!\left[2C_1\cos\tfrac{\sqrt3}{2}t+2C_2\sin\tfrac{\sqrt3}{2}t\right].$$

Step 6 · Behaviour of trajectories

The distance from the origin satisfies $r(t)\propto e^{-t/2}\to0$ as $t\to+\infty$: all trajectories spiral inward. The angular frequency is $\omega=\tfrac{\sqrt3}{2}$, giving period $T=\tfrac{4\pi}{\sqrt3}$. As $t\to-\infty$, $r(t)\to+\infty$.

Step 7 · Stability analysis

Since $\operatorname{Re}(\lambda_{1,2})=-\tfrac12<0$, the rest point is both stable in the sense of Lyapunov and asymptotically stable: every trajectory converges to the origin as $t\to+\infty$.

Step 8 · Conclusion

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