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

Examine the stability of the rest point $(0,0)$ of the system $$\dot{x}=10\sin x-29y+3y^3,\qquad\dot{y}=5x-14\sin y+y^2,$$ by the method of first approximation (linearisation).

Step 1 · Verification that $(0,0)$ is a rest point

At $(0,0)$ both right-hand sides vanish.

Step 2 · Taylor expansions near $(0,0)$

Using $\sin x=x-\tfrac{x^3}{6}+\cdots$ and $\sin y=y-\tfrac{y^3}{6}+\cdots$: $$\dot x=10x-29y+\underbrace{\left(-\tfrac{5x^3}{3}+3y^3+O(r^5)\right)}_{=:f_2}=O(r^3),$$ $$\dot y=5x-14y+\underbrace{\left(y^2+\tfrac{7y^3}{3}+O(r^5)\right)}_{=:g_2}=O(r^2).$$

Step 3 · Jacobian matrix at $(0,0)$

$$A=\begin{pmatrix}10&-29\\5&-14\end{pmatrix}.$$

Step 4 · Characteristic equation

$$\det(A-\lambda I)=0\implies\lambda^2+4\lambda+5=0\implies\lambda=-2\pm i.$$

Step 5 · Classification of eigenvalues

Complex, negative real part: $\operatorname{Re}(\lambda_{1,2})=-2<0$. Stable focus of the linearisation.

Step 6 · Verification that nonlinear terms are $o(r)$

$|f_2|=O(r^3)$, $|g_2|=O(r^2)$; both are $o(r)$.

Step 7 · Application of Lyapunov's First Method

Since $\operatorname{Re}(\lambda_{1,2})<0$ and the remainders are $o(r)$, the nonlinear rest point is asymptotically stable.

Step 8 · Routh--Hurwitz verification

For $\lambda^2+4\lambda+5$: $a_1=4>0$, $a_2=5>0$.

Step 9 · Direction of rotation

At $(\varepsilon,0)$: $\dot y=5\varepsilon>0$, so the spiral is counter-clockwise.

Step 10 · Behaviour of trajectories

$r(t)\propto e^{-2t}\to0$, spiralling counter-clockwise with angular frequency $\omega=1$ and period $T=2\pi$.

Step 11 · Stability analysis

The rest point is Lyapunov stable and asymptotically stable.

Step 12 · Conclusion

$$\boxed{ (0,0)\text{ is an \textbf{asymptotically stable focus} of the nonlinear system.} }$$ The eigenvalues are $\lambda_{1,2}=-2\pm i$, with $\operatorname{Re}(\lambda_{1,2})=-2<0$. By Lyapunov's First Method, $f_2=O(r^3)$ and $g_2=O(r^2)$ are $o(r)$ and do not affect the conclusion; the rest point is an asymptotically stable focus with counter-clockwise spiralling trajectories, angular frequency $\omega=1$, period $T=2\pi$, and exponential decay rate $\alpha=2$.

Summary Table

Additional Notes

• The Hartman–Grobman theorem guarantees that, because no eigenvalue of the Jacobian is purely imaginary, the nonlinear phase portrait near the rest point is topologically equivalent to that of its linearisation — so the type found for the linear system (node, saddle, or focus) carries over unchanged to the full nonlinear system.