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

Examine the character of the rest point $(0,0)$ of the system $$\dot{x}=x-\tfrac{1}{2}y,\qquad\dot{y}=\tfrac{53}{2}x-y.$$

Step 1 · Coefficient matrix

$$A=\begin{pmatrix}1&-\tfrac{1}{2}\\[4pt]\tfrac{53}{2}&-1\end{pmatrix}.$$

Step 2 · Characteristic equation and eigenvalues

$$\det(A-\lambda I)=0\implies\lambda^2+\tfrac{49}{4}=0 \implies\lambda_{1,2}=\pm\tfrac{7}{2}\,i.$$

Step 3 · Classification of the rest point

The eigenvalues are purely imaginary. Since the system is linear, the rest point $(0,0)$ is a centre exactly. All trajectories are closed curves encircling the origin.

Step 4 · Direction of rotation

Evaluating $\dot y$ at $(1,0)$ gives $\dot y=\tfrac{53}{2}>0$, so the trajectories rotate counter-clockwise.

Step 5 · General real solution

With eigenvector $\mathbf v=(1,2)^\top+i(0,7)^\top$ for $\lambda=-\tfrac72i$: $$x(t)=C_1\cos\tfrac72t+C_2\sin\tfrac72t,$$ $$y(t)=(2C_1-7C_2)\cos\tfrac72t+(2C_2+7C_1)\sin\tfrac72t.$$

Step 6 · First integral

Seeking a quadratic invariant gives $a=53$, $b=-4$, $c=1$, so the family of integral curves is $$53x^2-4xy+y^2=K,\quad K>0,$$ a family of ellipses (positive definite since $16-212=-196<0$ with positive leading coefficient).

Step 7 · Behaviour of trajectories

All trajectories are closed ellipses $53x^2-4xy+y^2=K$, traversed counter-clockwise with period $T=\dfrac{2\pi}{7/2}=\dfrac{4\pi}{7}$. No trajectory approaches or recedes from the origin.

Step 8 · Stability analysis

The rest point is stable in the sense of Lyapunov (for any $\varepsilon>0$, choose $K$ small enough that the ellipse lies within the $\varepsilon$-ball) but not asymptotically stable.

Step 9 · Conclusion

$$\boxed{ (0,0)\text{ is a \textbf{centre} (stable, not asymptotically stable).}\quad \lambda_{1,2}=\pm\tfrac{7}{2}\,i. }$$ All trajectories are closed ellipses $53x^2-4xy+y^2=K$ ($K>0$) traversed counter-clockwise with period $T=\tfrac{4\pi}{7}$. The origin is Lyapunov stable but not asymptotically stable.

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.