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Example of a flow that does not preserve volume measure (autonomous ode)

Consider the autonomous differential equation in $mathcal{U} = mathbb{R} times (0, +infty)$ given by

$$x’ = dfrac{x^2}{1+x^2y^2}, y’ = 0.$$

Justify that the respective flow is complete (i.é, defined for all $t in mathbb{R}$). Show that given $T>0$, there exist limited open sets $X subset mathcal{U}$ such that $X_T = {f^t(x,y); (x,y) in X hspace{0.1cm} text{and} hspace{0.1cm} t in left[0,Tright]}$ has infinite volume measure.

Attempt: Consider the initial condition of the equation as $gamma (0) = (x_0,y_0)$. Since $y’ = 0$, I know that $y = y_0$, with $y_0 > 0$. I was able to show that the flow of this autonomous equation is

$$f^t(x_0,y_0) = left( frac{left(-left(t – frac{1}{x_0} + y_0 x _0 right) overline{+} left[left(t – frac{1}{x_0} + y_0 x_0 right) + 4y_0 right]^{frac{1}{2}} right)}{2y_0}, y_0 right),$$

which is defined for every $t in mathbb{R}$. Hence, the respective flow is complete.

Given $T >0$, my initial idea was to consider $X = (a_1, a_2) times (0,y_0) subset mathcal{U}$, with

$$a_1 = frac{left(-left(t – frac{1}{x_0} + y_0 x _0 right) – left[left(t – frac{1}{x_0} + y_0 x_0 right) + 4y_0 right]^{frac{1}{2}} right)}{2y_0}$$

$$ hspace{0.1cm} text{and} hspace{0.1cm}$$

$$ a_2 = frac{left(-left(t – frac{1}{x_0} + y_0 x _0 right) +left[left(t – frac{1}{x_0} + y_0 x_0 right) + 4y_0 right]^{frac{1}{2}} right)}{2y_0}, $$

which is an open and limited set. But I’m not sure if $X_T$ is a set with infinite volume measure. Can anyone help me conclude or even help me construct a different limited open set?

Any help would be appreciated!

Mathematics Asked by Raquel Magalhães on November 12, 2021

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