Examine the stability of the rest point of the system…

Examine the stability of the rest point of the system $$\dot{x}=2x+8\sin y,\qquad\dot{y}=2-e^x-3y-\cos y,$$ by the method of first approximation (linearisation).

Step 1 · Finding the rest point

Testing $(0,0)$: $2(0)+8\sin0=0$; $2-e^0-3(0)-\cos0=2-1-0-1=0$. The point $(0,0)$ is a rest point.

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

Using $\sin y=y-\tfrac{y^3}{6}+O(y^5)$: $$\dot x=2x+8y+\underbrace{\left(-\tfrac{4y^3}{3}+O(r^5)\right)}_{=:f_2}=O(r^3).$$ Using $e^x=1+x+\tfrac{x^2}{2}+\cdots$, $\cos y=1-\tfrac{y^2}{2}+\cdots$: $$\dot y=-x-3y+\underbrace{\left(-\tfrac{x^2}{2}+\tfrac{y^2}{2}+O(r^3)\right)}_{=:g_2}=O(r^2).$$

Step 3 · Jacobian matrix at $(0,0)$

$$A=\begin{pmatrix}2&8\\-1&-3\end{pmatrix}.$$

Step 4 · Characteristic equation

$$\det(A-\lambda I)=0\implies\lambda^2+\lambda+2=0 \implies\lambda=-\tfrac12\pm\tfrac{\sqrt7}{2}i.$$

Step 5 · Classification of eigenvalues

Complex with negative real part: $\operatorname{Re}(\lambda_{1,2})=-\tfrac12<0$. 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 $\|F_2\|/r\to0$.

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 $\lambda^2+\lambda+2$: $a_1=1>0$, $a_2=2>0$.

Step 9 · Direction of rotation

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

Step 10 · Behaviour of trajectories

$r(t)\propto e^{-t/2}\to0$, spiralling clockwise with angular frequency $\omega=\tfrac{\sqrt7}{2}$ and period $T=\tfrac{4\pi\sqrt7}{7}$.

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 $\lambda_{1,2}=-\tfrac12\pm\tfrac{\sqrt7}{2}i$, with $\operatorname{Re}(\lambda_{1,2})=-\tfrac12<0$. 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 clockwise spiralling trajectories, angular frequency $\omega=\tfrac{\sqrt7}{2}$, and exponential decay rate $\alpha=\tfrac12$.

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.