Limit sets of a gradient field
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I am trying to solve this question on J. Sotomayor's book on ODEs.
Define $X=nabla f$, $f$ being defined in an open subset $Delta subset mathbb R^n$. Prove that $X$ has no periodic orbits. And, if $X$ have only isolated singular points, show that is $pin Delta$ then the limit set of $p$ is empty or is a singular point.
About the first statement: if $gamma$ is a (non-constant) periodic orbit, then, for some $T>0$, $gamma(0)=gamma(T)$. Therefore:
$$0=f(gamma(T))-f(gamma(0))=int_0^Tnabla f(gamma(t)) cdotgamma'(t)dt=int_0^Tnabla f(gamma(t)) cdot nabla f(gamma(t)) dt =$$
$$=int_0^Tvert{nabla f(gamma(t))vert^2dt>0 }$$
and this is an absurd.
But I am having some troubles in the second part. I have some ideas.
If the orbit $gamma_p$ passing through $p$ is not periodic then it is constant or it is injective. If $y_p$ is constant, $p$ is a singular point and $omega(p)=p$.
The trouble is when $gamma_p$ is one-to-one. What I've been trying to do is to prove that in this case $q in omega(p)$ only if
$$lim_{t to infty} gamma_p(t)=q$$
and then using the fact that
$$f(q)-f(gamma(0))=int_0^infty vertnabla f(gamma(t))vert^2dt$$
But the integral on the right side converges only if
$$lim_{t to infty} nabla f(gamma_p(t))=nabla f(q)=0$$
and therefore $q$ is a singular point.
Is this correct? If it is, any hints of how to complete the missing step? It seems pretty intuitive to me, but I can't formalize it.
ordinary-differential-equations vector-analysis
asked Dec 1 '18 at 0:09
Célio AugustoCélio Augusto
142
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add a comment |
$begingroup$
I am trying to solve this question on J. Sotomayor's book on ODEs.
Define $X=nabla f$, $f$ being defined in an open subset $Delta subset mathbb R^n$. Prove that $X$ has no periodic orbits. And, if $X$ have only isolated singular points, show that is $pin Delta$ then the limit set of $p$ is empty or is a singular point.
About the first statement: if $gamma$ is a (non-constant) periodic orbit, then, for some $T>0$, $gamma(0)=gamma(T)$. Therefore:
$$0=f(gamma(T))-f(gamma(0))=int_0^Tnabla f(gamma(t)) cdotgamma'(t)dt=int_0^Tnabla f(gamma(t)) cdot nabla f(gamma(t)) dt =$$
$$=int_0^Tvert{nabla f(gamma(t))vert^2dt>0 }$$
and this is an absurd.
But I am having some troubles in the second part. I have some ideas.
If the orbit $gamma_p$ passing through $p$ is not periodic then it is constant or it is injective. If $y_p$ is constant, $p$ is a singular point and $omega(p)=p$.
The trouble is when $gamma_p$ is one-to-one. What I've been trying to do is to prove that in this case $q in omega(p)$ only if
$$lim_{t to infty} gamma_p(t)=q$$
and then using the fact that
$$f(q)-f(gamma(0))=int_0^infty vertnabla f(gamma(t))vert^2dt$$
But the integral on the right side converges only if
$$lim_{t to infty} nabla f(gamma_p(t))=nabla f(q)=0$$
and therefore $q$ is a singular point.
Is this correct? If it is, any hints of how to complete the missing step? It seems pretty intuitive to me, but I can't formalize it.
ordinary-differential-equations vector-analysis
asked Dec 1 '18 at 0:09
Célio AugustoCélio Augusto
142
$endgroup$
$begingroup$
I managed to proof that f is constant in the limit sets, I wonder if that hepls
$endgroup$
– Célio Augusto
Dec 2 '18 at 20:13
add a comment |
$begingroup$
I am trying to solve this question on J. Sotomayor's book on ODEs.
Define $X=nabla f$, $f$ being defined in an open subset $Delta subset mathbb R^n$. Prove that $X$ has no periodic orbits. And, if $X$ have only isolated singular points, show that is $pin Delta$ then the limit set of $p$ is empty or is a singular point.
About the first statement: if $gamma$ is a (non-constant) periodic orbit, then, for some $T>0$, $gamma(0)=gamma(T)$. Therefore:
$$0=f(gamma(T))-f(gamma(0))=int_0^Tnabla f(gamma(t)) cdotgamma'(t)dt=int_0^Tnabla f(gamma(t)) cdot nabla f(gamma(t)) dt =$$
$$=int_0^Tvert{nabla f(gamma(t))vert^2dt>0 }$$
and this is an absurd.
But I am having some troubles in the second part. I have some ideas.
If the orbit $gamma_p$ passing through $p$ is not periodic then it is constant or it is injective. If $y_p$ is constant, $p$ is a singular point and $omega(p)=p$.
The trouble is when $gamma_p$ is one-to-one. What I've been trying to do is to prove that in this case $q in omega(p)$ only if
$$lim_{t to infty} gamma_p(t)=q$$
and then using the fact that
$$f(q)-f(gamma(0))=int_0^infty vertnabla f(gamma(t))vert^2dt$$
But the integral on the right side converges only if
$$lim_{t to infty} nabla f(gamma_p(t))=nabla f(q)=0$$
and therefore $q$ is a singular point.
Is this correct? If it is, any hints of how to complete the missing step? It seems pretty intuitive to me, but I can't formalize it.
ordinary-differential-equations vector-analysis
asked Dec 1 '18 at 0:09
Célio AugustoCélio Augusto
142
$endgroup$
I am trying to solve this question on J. Sotomayor's book on ODEs.
Define $X=nabla f$, $f$ being defined in an open subset $Delta subset mathbb R^n$. Prove that $X$ has no periodic orbits. And, if $X$ have only isolated singular points, show that is $pin Delta$ then the limit set of $p$ is empty or is a singular point.
About the first statement: if $gamma$ is a (non-constant) periodic orbit, then, for some $T>0$, $gamma(0)=gamma(T)$. Therefore:
$$0=f(gamma(T))-f(gamma(0))=int_0^Tnabla f(gamma(t)) cdotgamma'(t)dt=int_0^Tnabla f(gamma(t)) cdot nabla f(gamma(t)) dt =$$
$$=int_0^Tvert{nabla f(gamma(t))vert^2dt>0 }$$
and this is an absurd.
But I am having some troubles in the second part. I have some ideas.
If the orbit $gamma_p$ passing through $p$ is not periodic then it is constant or it is injective. If $y_p$ is constant, $p$ is a singular point and $omega(p)=p$.
The trouble is when $gamma_p$ is one-to-one. What I've been trying to do is to prove that in this case $q in omega(p)$ only if
$$lim_{t to infty} gamma_p(t)=q$$
and then using the fact that
$$f(q)-f(gamma(0))=int_0^infty vertnabla f(gamma(t))vert^2dt$$
But the integral on the right side converges only if
$$lim_{t to infty} nabla f(gamma_p(t))=nabla f(q)=0$$
and therefore $q$ is a singular point.
Is this correct? If it is, any hints of how to complete the missing step? It seems pretty intuitive to me, but I can't formalize it.
ordinary-differential-equations vector-analysis
ordinary-differential-equations vector-analysis
asked Dec 1 '18 at 0:09
Célio AugustoCélio Augusto
142
asked Dec 1 '18 at 0:09
Célio AugustoCélio Augusto
142
edited Dec 1 '18 at 18:13
Célio Augusto
asked Dec 1 '18 at 0:09
Célio AugustoCélio Augusto
142
asked Dec 1 '18 at 0:09
Célio AugustoCélio Augusto
142
asked Dec 1 '18 at 0:09
Célio AugustoCélio Augusto
142
142
$begingroup$
I managed to proof that f is constant in the limit sets, I wonder if that hepls
$endgroup$
– Célio Augusto
Dec 2 '18 at 20:13
add a comment |
$begingroup$
I managed to proof that f is constant in the limit sets, I wonder if that hepls
$endgroup$
– Célio Augusto
Dec 2 '18 at 20:13
$begingroup$
I managed to proof that f is constant in the limit sets, I wonder if that hepls
$endgroup$
– Célio Augusto
Dec 2 '18 at 20:13
$begingroup$
I managed to proof that f is constant in the limit sets, I wonder if that hepls
$endgroup$
– Célio Augusto
Dec 2 '18 at 20:13
add a comment |
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$begingroup$
I managed to proof that f is constant in the limit sets, I wonder if that hepls
$endgroup$
– Célio Augusto
Dec 2 '18 at 20:13