An attractor for blow-up solutions to a cubic oscillator
(Related to this MathOverflow question).
Consider the nonlinear ODE
$$tag{1}
frac{d^2u}{dt^2}+u=u^3, qquad tinmathbb R,$$
which has the conserved quantity
$$tag{2}
E=frac12 u'^2+frac12 u^2 -frac14u^4.$$
Consider only these solutions that satisfy $u>0$ and $u'>0$.
I am interested in solutions that blow-up at time $T$, in the sense that
$$
lim_{t nearrow T} u(t)=+infty.$$
The following family describes all solutions with $E=0$, parametrized by the blow-up time;
$$
u_{0, T}(t):=frac{sqrt{2}}{sin(T-t)}.$$
The phase portrait suggests that the trajectories of these solutions are an attractor for (1);
and this is reasonable, because (2) yields
$$
u'=sqrt{2E -u^2 + frac12u^4} = frac{u^2}{sqrt2} + O(1), $$
so when $u$ is very big I expect $u$ to be indistinguishable from the unique solution to
$v'=frac{v^2}{sqrt 2} $ that blows up at $T$, that is, $$v_T(t)=frac{sqrt2}{T-t}, $$
and $v_T$ is asymptotically equivalent to $u_{0, T}$ as $tnearrow T$.
I would like to obtain a more precise, quantitative version of this attraction.
Question. Let $u$ be a solution to (1) such that $u(t)to +infty$ as $tnearrow T$. Is it true that $$|u(t)-u_{0, T}(t)|to 0,qquad text{as }tnearrow T?$$ Is it true that $$frac{u(t)}{u_{0,T}(t)}to 1,qquad text{as }tnearrow T?$$
Remark. The equation (1) is a special case of Duffing equation.
differential-equations analysis
asked Nov 24 at 21:02
Giuseppe Negro
17.3k330122
add a comment |
(Related to this MathOverflow question).
Consider the nonlinear ODE
$$tag{1}
frac{d^2u}{dt^2}+u=u^3, qquad tinmathbb R,$$
which has the conserved quantity
$$tag{2}
E=frac12 u'^2+frac12 u^2 -frac14u^4.$$
Consider only these solutions that satisfy $u>0$ and $u'>0$.
I am interested in solutions that blow-up at time $T$, in the sense that
$$
lim_{t nearrow T} u(t)=+infty.$$
The following family describes all solutions with $E=0$, parametrized by the blow-up time;
$$
u_{0, T}(t):=frac{sqrt{2}}{sin(T-t)}.$$
The phase portrait suggests that the trajectories of these solutions are an attractor for (1);
and this is reasonable, because (2) yields
$$
u'=sqrt{2E -u^2 + frac12u^4} = frac{u^2}{sqrt2} + O(1), $$
so when $u$ is very big I expect $u$ to be indistinguishable from the unique solution to
$v'=frac{v^2}{sqrt 2} $ that blows up at $T$, that is, $$v_T(t)=frac{sqrt2}{T-t}, $$
and $v_T$ is asymptotically equivalent to $u_{0, T}$ as $tnearrow T$.
I would like to obtain a more precise, quantitative version of this attraction.
Question. Let $u$ be a solution to (1) such that $u(t)to +infty$ as $tnearrow T$. Is it true that $$|u(t)-u_{0, T}(t)|to 0,qquad text{as }tnearrow T?$$ Is it true that $$frac{u(t)}{u_{0,T}(t)}to 1,qquad text{as }tnearrow T?$$
Remark. The equation (1) is a special case of Duffing equation.
differential-equations analysis
asked Nov 24 at 21:02
Giuseppe Negro
17.3k330122
2
This equation has an exact set of solutions using the Jacobi elliptic function $operatorname{dn}$.
– Jon
Nov 24 at 21:14
@Jon: Thank you, that looks useful. Unfortunately, I don't know a word about these functions. As far as I understand from skimming some references, such as this book (Kolvalcic-Brennan (eds), "The Duffing equation"), the Jacobi elliptic functions describe the periodic solutions, that is, those that do not blow up. Am I wrong?
– Giuseppe Negro
Nov 25 at 17:28
1
Yes, you are right. These are periodical solutions. A prototypical one is given by the solution of the pendulum equation. The $operatorname{dn}$ never becomes zero.
– Jon
Nov 25 at 23:06
This question is related to the present one.
– Giuseppe Negro
Dec 4 at 14:09
add a comment |
(Related to this MathOverflow question).
Consider the nonlinear ODE
$$tag{1}
frac{d^2u}{dt^2}+u=u^3, qquad tinmathbb R,$$
which has the conserved quantity
$$tag{2}
E=frac12 u'^2+frac12 u^2 -frac14u^4.$$
Consider only these solutions that satisfy $u>0$ and $u'>0$.
I am interested in solutions that blow-up at time $T$, in the sense that
$$
lim_{t nearrow T} u(t)=+infty.$$
The following family describes all solutions with $E=0$, parametrized by the blow-up time;
$$
u_{0, T}(t):=frac{sqrt{2}}{sin(T-t)}.$$
The phase portrait suggests that the trajectories of these solutions are an attractor for (1);
and this is reasonable, because (2) yields
$$
u'=sqrt{2E -u^2 + frac12u^4} = frac{u^2}{sqrt2} + O(1), $$
so when $u$ is very big I expect $u$ to be indistinguishable from the unique solution to
$v'=frac{v^2}{sqrt 2} $ that blows up at $T$, that is, $$v_T(t)=frac{sqrt2}{T-t}, $$
and $v_T$ is asymptotically equivalent to $u_{0, T}$ as $tnearrow T$.
I would like to obtain a more precise, quantitative version of this attraction.
Question. Let $u$ be a solution to (1) such that $u(t)to +infty$ as $tnearrow T$. Is it true that $$|u(t)-u_{0, T}(t)|to 0,qquad text{as }tnearrow T?$$ Is it true that $$frac{u(t)}{u_{0,T}(t)}to 1,qquad text{as }tnearrow T?$$
Remark. The equation (1) is a special case of Duffing equation.
differential-equations analysis
asked Nov 24 at 21:02
Giuseppe Negro
17.3k330122
(Related to this MathOverflow question).
Consider the nonlinear ODE
$$tag{1}
frac{d^2u}{dt^2}+u=u^3, qquad tinmathbb R,$$
which has the conserved quantity
$$tag{2}
E=frac12 u'^2+frac12 u^2 -frac14u^4.$$
Consider only these solutions that satisfy $u>0$ and $u'>0$.
I am interested in solutions that blow-up at time $T$, in the sense that
$$
lim_{t nearrow T} u(t)=+infty.$$
The following family describes all solutions with $E=0$, parametrized by the blow-up time;
$$
u_{0, T}(t):=frac{sqrt{2}}{sin(T-t)}.$$
The phase portrait suggests that the trajectories of these solutions are an attractor for (1);
and this is reasonable, because (2) yields
$$
u'=sqrt{2E -u^2 + frac12u^4} = frac{u^2}{sqrt2} + O(1), $$
so when $u$ is very big I expect $u$ to be indistinguishable from the unique solution to
$v'=frac{v^2}{sqrt 2} $ that blows up at $T$, that is, $$v_T(t)=frac{sqrt2}{T-t}, $$
and $v_T$ is asymptotically equivalent to $u_{0, T}$ as $tnearrow T$.
I would like to obtain a more precise, quantitative version of this attraction.
Question. Let $u$ be a solution to (1) such that $u(t)to +infty$ as $tnearrow T$. Is it true that $$|u(t)-u_{0, T}(t)|to 0,qquad text{as }tnearrow T?$$ Is it true that $$frac{u(t)}{u_{0,T}(t)}to 1,qquad text{as }tnearrow T?$$
Remark. The equation (1) is a special case of Duffing equation.
differential-equations analysis
differential-equations analysis
asked Nov 24 at 21:02
Giuseppe Negro
17.3k330122
asked Nov 24 at 21:02
Giuseppe Negro
17.3k330122
edited Dec 10 at 21:32
asked Nov 24 at 21:02
Giuseppe Negro
17.3k330122
asked Nov 24 at 21:02
Giuseppe Negro
17.3k330122
asked Nov 24 at 21:02
Giuseppe Negro
17.3k330122
17.3k330122
2
This equation has an exact set of solutions using the Jacobi elliptic function $operatorname{dn}$.
– Jon
Nov 24 at 21:14
@Jon: Thank you, that looks useful. Unfortunately, I don't know a word about these functions. As far as I understand from skimming some references, such as this book (Kolvalcic-Brennan (eds), "The Duffing equation"), the Jacobi elliptic functions describe the periodic solutions, that is, those that do not blow up. Am I wrong?
– Giuseppe Negro
Nov 25 at 17:28
1
Yes, you are right. These are periodical solutions. A prototypical one is given by the solution of the pendulum equation. The $operatorname{dn}$ never becomes zero.
– Jon
Nov 25 at 23:06
This question is related to the present one.
– Giuseppe Negro
Dec 4 at 14:09
add a comment |
2
This equation has an exact set of solutions using the Jacobi elliptic function $operatorname{dn}$.
– Jon
Nov 24 at 21:14
@Jon: Thank you, that looks useful. Unfortunately, I don't know a word about these functions. As far as I understand from skimming some references, such as this book (Kolvalcic-Brennan (eds), "The Duffing equation"), the Jacobi elliptic functions describe the periodic solutions, that is, those that do not blow up. Am I wrong?
– Giuseppe Negro
Nov 25 at 17:28
1
Yes, you are right. These are periodical solutions. A prototypical one is given by the solution of the pendulum equation. The $operatorname{dn}$ never becomes zero.
– Jon
Nov 25 at 23:06
This question is related to the present one.
– Giuseppe Negro
Dec 4 at 14:09
2
2
This equation has an exact set of solutions using the Jacobi elliptic function $operatorname{dn}$.
– Jon
Nov 24 at 21:14
This equation has an exact set of solutions using the Jacobi elliptic function $operatorname{dn}$.
– Jon
Nov 24 at 21:14
@Jon: Thank you, that looks useful. Unfortunately, I don't know a word about these functions. As far as I understand from skimming some references, such as this book (Kolvalcic-Brennan (eds), "The Duffing equation"), the Jacobi elliptic functions describe the periodic solutions, that is, those that do not blow up. Am I wrong?
– Giuseppe Negro
Nov 25 at 17:28
@Jon: Thank you, that looks useful. Unfortunately, I don't know a word about these functions. As far as I understand from skimming some references, such as this book (Kolvalcic-Brennan (eds), "The Duffing equation"), the Jacobi elliptic functions describe the periodic solutions, that is, those that do not blow up. Am I wrong?
– Giuseppe Negro
Nov 25 at 17:28
1
1
Yes, you are right. These are periodical solutions. A prototypical one is given by the solution of the pendulum equation. The $operatorname{dn}$ never becomes zero.
– Jon
Nov 25 at 23:06
Yes, you are right. These are periodical solutions. A prototypical one is given by the solution of the pendulum equation. The $operatorname{dn}$ never becomes zero.
– Jon
Nov 25 at 23:06
This question is related to the present one.
– Giuseppe Negro
Dec 4 at 14:09
This question is related to the present one.
– Giuseppe Negro
Dec 4 at 14:09
add a comment |
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2
This equation has an exact set of solutions using the Jacobi elliptic function $operatorname{dn}$.
– Jon
Nov 24 at 21:14
@Jon: Thank you, that looks useful. Unfortunately, I don't know a word about these functions. As far as I understand from skimming some references, such as this book (Kolvalcic-Brennan (eds), "The Duffing equation"), the Jacobi elliptic functions describe the periodic solutions, that is, those that do not blow up. Am I wrong?
– Giuseppe Negro
Nov 25 at 17:28
1
Yes, you are right. These are periodical solutions. A prototypical one is given by the solution of the pendulum equation. The $operatorname{dn}$ never becomes zero.
– Jon
Nov 25 at 23:06
This question is related to the present one.
– Giuseppe Negro
Dec 4 at 14:09