Examine the stability of the rest point of the system…

Examine the stability of the rest point of the system $$\dot{x}=\ln(4y+e^{-3x}),\qquad\dot{y}=2y-1+\sqrt{1-6x},$$ by the method of first approximation (linearisation).

Step 1 · Finding the rest point

Testing $(x,y)=(0,0)$: $4(0)+e^0=1$, so $\ln1=0$; and $2(0)-1+\sqrt{1-0}=-1+1=0$. The point $(0,0)$ is a rest point.

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

Using $e^{-3x}=1-3x+\tfrac{9x^2}{2}+O(x^3)$ and $\ln(1+u)=u-\tfrac{u^2}{2}+O(u^3)$ with $u=-3x+4y+\tfrac{9x^2}{2}+\cdots$: $$\dot{x}=-3x+4y+\underbrace{12xy-8y^2+O(r^3)}_{=:f_2}.$$ Using $\sqrt{1-6x}=1-3x-\tfrac{9x^2}{2}+O(x^3)$: $$\dot{y}=-3x+2y+\underbrace{\left(-\tfrac{9x^2}{2}+O(r^3)\right)}_{=:g_2}.$$ Both $f_2,g_2=O(r^2)=o(r)$ as $r\to0$.

Step 3 · Jacobian matrix at $(0,0)$

Direct differentiation confirms $$A=\begin{pmatrix}-3&4\\-3&2\end{pmatrix}.$$

Step 4 · Characteristic equation

$$\det(A-\lambda I)=0\implies\lambda^2+\lambda+6=0 \implies\lambda=\frac{-1\pm\sqrt{-23}}{2}=-\tfrac12\pm\tfrac{\sqrt{23}}{2}i.$$

Step 5 · Classification of eigenvalues

Complex with negative real part: $\operatorname{Re}(\lambda_{1,2})=-\tfrac12<0$. The linearised system has a stable focus at the origin.

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

For $r\leq1$: $|f_2|\leq20r^2$, $|g_2|\leq5r^2$, so $\|F_2\|/r\leq Cr\to0$.

Step 7 · Application of Lyapunov's First Method

Since $\operatorname{Re}(\lambda_{1,2})=-\tfrac12<0$ and $f_2,g_2=O(r^2)=o(r)$, the rest point of the nonlinear system is asymptotically stable.

Step 8 · Routh--Hurwitz verification

For $\lambda^2+\lambda+6$: $a_1=1>0$, $a_2=6>0$ — both satisfied.

Step 9 · Direction of rotation

At $(\varepsilon,0)$: $\dot x\approx-3\varepsilon<0$; at $(0,\varepsilon)$: $\dot y=2\varepsilon>0$. Trajectories move into the third quadrant from the positive $x$-axis: the spiral is clockwise.

Step 10 · Behaviour of trajectories

The distance from the origin decays as $r(t)\propto e^{-t/2}$, spiralling clockwise with angular frequency $\omega=\tfrac{\sqrt{23}}{2}$ and period $T=\tfrac{4\pi}{\sqrt{23}}$.

Step 11 · Stability analysis

Since $\operatorname{Re}(\lambda_{1,2})=-\tfrac12<0$, 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 of the Jacobian are $\lambda_{1,2}=-\tfrac12\pm\tfrac{\sqrt{23}}{2}i$, with $\operatorname{Re}(\lambda_{1,2})=-\tfrac12<0$. By Lyapunov's First Method, the nonlinear remainders $f_2,g_2=O(r^2)$ are $o(r)$ and do not affect the conclusion: the rest point is an asymptotically stable focus with clockwise spiralling trajectories, angular frequency $\omega=\tfrac{\sqrt{23}}{2}$, 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.