Mixing
Hartman–Grobman theorem for a two-dimensional system
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I want to find a diffeomorphism $G$ defined on some neighborhood of the origin with $G(0)=0$ such that $G$ transforms the linear system $dotx=x$ to the nonlinear system $doty=y-y^2$. (These are one-dimensional systems).
Any comments or responses are greatly appreciated!
differential-equations differential-geometry dynamical-systems smooth-manifolds
asked Jul 16 at 20:56
Arthur
19812
add a comment |Â
up vote
1
down vote
favorite
I want to find a diffeomorphism $G$ defined on some neighborhood of the origin with $G(0)=0$ such that $G$ transforms the linear system $dotx=x$ to the nonlinear system $doty=y-y^2$. (These are one-dimensional systems).
Any comments or responses are greatly appreciated!
differential-equations differential-geometry dynamical-systems smooth-manifolds
asked Jul 16 at 20:56
Arthur
19812
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I want to find a diffeomorphism $G$ defined on some neighborhood of the origin with $G(0)=0$ such that $G$ transforms the linear system $dotx=x$ to the nonlinear system $doty=y-y^2$. (These are one-dimensional systems).
Any comments or responses are greatly appreciated!
differential-equations differential-geometry dynamical-systems smooth-manifolds
asked Jul 16 at 20:56
Arthur
19812
I want to find a diffeomorphism $G$ defined on some neighborhood of the origin with $G(0)=0$ such that $G$ transforms the linear system $dotx=x$ to the nonlinear system $doty=y-y^2$. (These are one-dimensional systems).
Any comments or responses are greatly appreciated!
differential-equations differential-geometry dynamical-systems smooth-manifolds
asked Jul 16 at 20:56
Arthur
19812
asked Jul 16 at 20:56
Arthur
19812
asked Jul 16 at 20:56
Arthur
19812
asked Jul 16 at 20:56
Arthur
19812
19812
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
3
down vote
accepted
The diffeomorphism is
$$
y=fracxx-1.
$$
Indeed,
$$
dot y=fracdot x(x-1)-xdot x(x-1)^2= fracx(x-1)-x^2(x-1)^2=
fracxx-1-fracx^2(x-1)^2=y-y^2.
$$
This result can be observed by a comparison between the general solutions
$
x= Ce^t
$
and
$
y=fracCe^tCe^t-1
$.
answered Jul 17 at 1:47
AVK
1,7201415
Thanks! I think the domain of this diffeomorphism is $(-infty, 1)$.
– Arthur
Jul 17 at 3:42
@Arthur Yes, $(-infty,1)to (-infty,1)$
– AVK
Jul 17 at 7:28
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
The diffeomorphism is
$$
y=fracxx-1.
$$
Indeed,
$$
dot y=fracdot x(x-1)-xdot x(x-1)^2= fracx(x-1)-x^2(x-1)^2=
fracxx-1-fracx^2(x-1)^2=y-y^2.
$$
This result can be observed by a comparison between the general solutions
$
x= Ce^t
$
and
$
y=fracCe^tCe^t-1
$.
answered Jul 17 at 1:47
AVK
1,7201415
Thanks! I think the domain of this diffeomorphism is $(-infty, 1)$.
– Arthur
Jul 17 at 3:42
@Arthur Yes, $(-infty,1)to (-infty,1)$
– AVK
Jul 17 at 7:28
add a comment |Â
up vote
3
down vote
accepted
The diffeomorphism is
$$
y=fracxx-1.
$$
Indeed,
$$
dot y=fracdot x(x-1)-xdot x(x-1)^2= fracx(x-1)-x^2(x-1)^2=
fracxx-1-fracx^2(x-1)^2=y-y^2.
$$
This result can be observed by a comparison between the general solutions
$
x= Ce^t
$
and
$
y=fracCe^tCe^t-1
$.
answered Jul 17 at 1:47
AVK
1,7201415
Thanks! I think the domain of this diffeomorphism is $(-infty, 1)$.
– Arthur
Jul 17 at 3:42
@Arthur Yes, $(-infty,1)to (-infty,1)$
– AVK
Jul 17 at 7:28
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
The diffeomorphism is
$$
y=fracxx-1.
$$
Indeed,
$$
dot y=fracdot x(x-1)-xdot x(x-1)^2= fracx(x-1)-x^2(x-1)^2=
fracxx-1-fracx^2(x-1)^2=y-y^2.
$$
This result can be observed by a comparison between the general solutions
$
x= Ce^t
$
and
$
y=fracCe^tCe^t-1
$.
answered Jul 17 at 1:47
AVK
1,7201415
The diffeomorphism is
$$
y=fracxx-1.
$$
Indeed,
$$
dot y=fracdot x(x-1)-xdot x(x-1)^2= fracx(x-1)-x^2(x-1)^2=
fracxx-1-fracx^2(x-1)^2=y-y^2.
$$
This result can be observed by a comparison between the general solutions
$
x= Ce^t
$
and
$
y=fracCe^tCe^t-1
$.
answered Jul 17 at 1:47
AVK
1,7201415
answered Jul 17 at 1:47
AVK
1,7201415
answered Jul 17 at 1:47
AVK
1,7201415
answered Jul 17 at 1:47
AVK
1,7201415
1,7201415
Thanks! I think the domain of this diffeomorphism is $(-infty, 1)$.
– Arthur
Jul 17 at 3:42
@Arthur Yes, $(-infty,1)to (-infty,1)$
– AVK
Jul 17 at 7:28
add a comment |Â
Thanks! I think the domain of this diffeomorphism is $(-infty, 1)$.
– Arthur
Jul 17 at 3:42
@Arthur Yes, $(-infty,1)to (-infty,1)$
– AVK
Jul 17 at 7:28
Thanks! I think the domain of this diffeomorphism is $(-infty, 1)$.
– Arthur
Jul 17 at 3:42
Thanks! I think the domain of this diffeomorphism is $(-infty, 1)$.
– Arthur
Jul 17 at 3:42
@Arthur Yes, $(-infty,1)to (-infty,1)$
– AVK
Jul 17 at 7:28
@Arthur Yes, $(-infty,1)to (-infty,1)$
– AVK
Jul 17 at 7:28
add a comment |Â
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