91Ó°ÊÓ

Consider the non-linear system: $$ \begin{aligned} &(\mathrm{dx} / \mathrm{dt})=\sin \mathrm{x}-4 \mathrm{y} \\ &(\mathrm{dy} / \mathrm{dt})=\sin 2 \mathrm{x}-5 \mathrm{y} \end{aligned} $$ Find the critical point and discuss its nature.

Find all the real critical points of the nonlinear system: $$ \begin{aligned} &(\mathrm{dx} / \mathrm{dt})=8 \mathrm{x}-\mathrm{y}^{2} \\ &(\mathrm{dy} / \mathrm{dt})=-6 \mathrm{y}+6 \mathrm{x}^{2} \end{aligned} $$ and determine the type and stability of each of these critical points.

The two non-linear systems $$ \begin{aligned} &(d x / d t)=-y-x^{2} \\ &(d y / d t)=x \\ &(d x / d t)=-y-x^{3} \\ &(d y / d t)=x \end{aligned} $$ have the critical point \((0,0)\). Discuss the nature and stability of the critical point.

The two non-linear systems: $$ \begin{aligned} &(\mathrm{dx} / \mathrm{dt})=2 \mathrm{xy} ; \\ &(\mathrm{dy} / \mathrm{dt})=3 \mathrm{y}^{2}-\mathrm{x}^{2} \\ &(\mathrm{dx} / \mathrm{dt})=\mathrm{x}^{2} \\ &(\mathrm{dy} / \mathrm{dt})=2 \mathrm{y}^{2}-\mathrm{xy} . \end{aligned} $$ have the critical point \((0,0)\). Discuss the nature and stability of the critical point.

Classify the following Liapunov functions: $$ \begin{aligned} &E(x, y)=x^{2}+y^{2} \\ &E(x, y)=x^{2} \\ &E(x, y)=-x^{2}-y^{2} \\ &E(x, y)=-y^{2} \end{aligned} $$

Classify the critical point of the system: $$ \begin{aligned} &(\mathrm{dx} / \mathrm{dt})=-\mathrm{x}+\mathrm{y}^{2} \\ &(\mathrm{dy} / \mathrm{dt})=-\mathrm{y}+\mathrm{x}^{2} \end{aligned} $$ using Liapunov's function.

Consider the non-linear conservative system $$ \left[\left(d^{2} x\right) /\left(d t^{2}\right)\right]=4 x^{3}-4 x $$ Find the critical points of the system.

Solve the initial-value problem for a non-linear spring, $$ \begin{array}{rl} x+\mathrm{kx}+\mathrm{k}^{\prime} \mathrm{x}^{3}=0 & \mathrm{x}(0)=\mathrm{c} \\\ & \mathrm{x}(0)=0, \end{array} $$ using the perturbation series method.

Solve Van der Pol's equation: $$ x-\varepsilon\left(1-x^{2}\right) x+a x=0 $$ using the perturbation series method a is to be chosen to make the solution periodic. Let \(\mathrm{x}(0)=0, \mathrm{x}(0)=\mathrm{u}_{0}\)