To determine: a) The origin [left(0, 0right)] is a critical point of the systems. [frac{dx}{dt}=y + x left(x^{2} + y^{2}right), frac {dx}{dt}= -x + y

Question

nicekikah

Answered question

2020-11-08

To determine:
a) The origin $[\begin{array}{cc} 0 & 0 \end{array}]$ is a critical point of the systems.
$[\frac{d x}{d t} = y + x (x^{2} + y^{2}), \frac{d x}{d t} = - x + y (x^{2} + y^{2})] and [\frac{d x}{d t} = y - x (x^{2} + y^{2}), \frac{d x}{d t} = - x - y (x^{2} + y^{2})]$ . Futhermore, it is a center of the corresponding linear system.
b) The systems $[\frac{d x}{d t} = y + x (x^{2} + y^{2}), \frac{d x}{d t} = - x + y (x^{2} + y^{2})] and [\frac{d x}{d t} = y - y + x (x^{2} + y^{2}), \frac{d x}{d t} = - x - y (x^{2} + y^{2})]$ are almost linear.
c) To prove: $[\frac{d r}{d t} < 0] and [r \to 0 a s t \to \infty],$ hence the critical point for the system $[\frac{d x}{d t} = y - x (x^{2} + y^{2}), \frac{d x}{d t} = - x - y (x^{2} + y^{2})]$ is asymptotically stable and the solution of the initial value problem for $[r w i t h r = r_{0} a t t = 0]$ becomes unbounded as $[t \to \frac{1}{2} r \frac{2}{0}]$ , hence the critical point for the system

Answer & Explanation

falhiblesw

Skilled2020-11-09Added 97 answers

a)
The systems of equations are $[\frac{d x}{d t} = y + x (x^{2} + y^{2}), \frac{d x}{d t} = - x + y (x^{2} + y^{2}) and \frac{d x}{d t} = y - x (x^{2} + y^{2}), \frac{d x}{d t} = - x - y (x^{2} + y^{2})] .$
Formula used: The points, if any, where $[f (x) = 0]$ are called critical pointsof the autonomous system $[x^{p r i m e} = f (x)] .$
Proof: The critical points of the system $[\frac{d x}{d t} = y + x (x^{2} + y^{2}), \frac{d y}{d t} = - x + y (x^{2} + y^{2})]$ are found by solving the equations $[y + x (x^{2} + y^{2}) = 0 and - x + y (x^{2} + y^{2}) = 0] .$
From the equation $[x^{2} + y^{2} = - \frac{y}{x}]$ and from the second equation $[x^{2} + y^{2} = \frac{x}{y}],$
$\Rightarrow - \frac{y}{x} = \frac{x}{y}$
$\Rightarrow x^{2} + y^{2} = 0$
Thus, the only critical point is $[\begin{array}{cc} 0 & 0 \end{array}] .$
Now,
The critical points of the system $[\frac{d x}{d t} = y - x (x^{2} + y^{2}), \frac{d y}{d t} = - x - y (x^{2} + y^{2})]$ are found by solving the equations

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