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

Examine the stability of the rest point $(0,0)$ of the system $$\dot{x}=-3x+4y+\sin^3 x-y^2,\qquad \dot{y}=-2x+\sin y+e^y x^2,$$ by the method of first approximation (linearisation).

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

At $(0,0)$: $-3(0)+4(0)+\sin^30-0^2=0$ and $-2(0)+\sin0+e^0\cdot0^2=0$.

Step 2 · Taylor expansions near the origin

$\sin^3x=O(r^3)$, $y^2=O(r^2)$, $\sin y=y+O(r^3)$, $e^yx^2=x^2+O(r^3)=O(r^2)$. So $$\dot x=-3x+4y+\underbrace{\sin^3x-y^2}_{=:f_2},\qquad \dot y=-2x+y+\underbrace{(\sin y-y)+e^yx^2}_{=:g_2},$$ with $f_2,g_2=O(r^2)=o(r)$.

Step 3 · Jacobian matrix at $(0,0)$

Direct differentiation gives $$A=\begin{pmatrix}-3&4\\-2&1\end{pmatrix}.$$

Step 4 · Characteristic equation

$$\det(A-\lambda I)=0\implies\lambda^2-(-3+1)\lambda+\bigl((-3)(1)-(-2)(4)\bigr)=0 \implies\lambda^2+2\lambda+5=0.$$ Discriminant $\Delta=4-20=-16$, so $\lambda=\dfrac{-2\pm4i}{2}=-1\pm2i$.

Step 5 · Classification of eigenvalues

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

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

For $r\leq1$: $|f_2|\leq2r^2$ and $|g_2|\leq(e+1)r^2$, so $\|F_2\|/r\leq Cr\to0$.

Step 7 · Application of the linearisation theorem

Since $\operatorname{Re}(\lambda_{1,2})=-1<0$ and $f_2,g_2=O(r^2)=o(r)$, Lyapunov's First Method gives asymptotic stability of the nonlinear rest point.

Step 8 · Routh--Hurwitz verification

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

Step 9 · Direction of rotation

Evaluating $\dot y$ at $(1,0)$: $\dot y=-2+0+1=-1<0$, so the spiral is clockwise.

Step 10 · Behaviour of trajectories of the nonlinear system

The linearised flow spirals toward the origin with decay rate $e^{-t}$ and angular frequency $\omega=2$ (period $T=\pi$). By Lyapunov's First Method, the nonlinear system inherits this behaviour: every trajectory starting sufficiently close to $(0,0)$ spirals clockwise toward the origin.

Step 11 · Stability analysis

Since $\operatorname{Re}(\lambda_{1,2})=-1<0$, 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 at the origin are $\lambda_{1,2}=-1\pm2i$, with $\operatorname{Re}(\lambda_{1,2})=-1<0$. By Lyapunov's First Method, the nonlinear terms $\sin^3x-y^2=O(r^2)$ and $(\sin y-y)+e^yx^2=O(r^2)$ are $o(r)$ and do not affect the stability conclusion. The rest point is an asymptotically stable focus with clockwise spiralling trajectories, angular frequency $\omega=2$, and exponential decay rate $\alpha=1$.

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.