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

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

Step 1 · Coefficient matrix

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

Step 2 · Characteristic equation and eigenvalues

$$\det(A-\lambda I)=0\implies(1-\lambda)^2+1=0\implies\lambda_{1,2}=1\pm i.$$

Step 3 · Classification of the rest point

The eigenvalues are complex with positive real part: $\operatorname{Re}(\lambda_{1,2})=1>0$. The rest point $(0,0)$ is an unstable focus: all trajectories spiral away from 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 · Polar form and exact spiral

Introducing polar coordinates $x=r\cos\theta$, $y=r\sin\theta$ and differentiating $r^2=x^2+y^2$ and $\theta=\arctan(y/x)$ along the flow gives the decoupled system $$\dot r=r,\qquad\dot\theta=-1.$$ Solving exactly: $r(t)=r_0e^{t}$, $\theta(t)=\theta_0-t$, so eliminating $t$ gives the family of logarithmic spirals $$r=C\,e^{-\theta},\quad C>0.$$

Step 6 · Behaviour of trajectories

From $\dot r=r>0$: as $t\to+\infty$, $r\to+\infty$ (outward spiral); as $t\to-\infty$, $r\to0$ (trajectories approach the origin while completing infinitely many clockwise revolutions). The angular speed is constant, $|\dot\theta|=1$, giving period $T=2\pi$ and exponential radial growth rate $\alpha=1$.

Step 7 · Stability analysis

Since $\operatorname{Re}(\lambda_{1,2})=1>0$, the rest point is unstable: it is not Lyapunov stable.

Step 8 · Conclusion

$$\boxed{ (0,0)\text{ is an \textbf{unstable focus}.}\quad \lambda_{1,2}=1\pm i,\quad\operatorname{Re}(\lambda)=1>0. }$$ In polar coordinates the system reduces to $\dot{r}=r$, $\dot{\theta}=-1$, giving the exact family of logarithmic spirals $r=Ce^{-\theta}$ ($C>0$). All trajectories spiral clockwise outward with angular frequency $\omega=1$ and exponential growth rate $\alpha=1$, confirming instability.

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.