# 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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