Singular perturbation theory in non-standard form The Next CEO of Stack OverflowPerturbation Analysis for Linear Linearly-Perturbed ODEsIVP Perturbation With Small Non-Linear Termnon-homogeneous constant co-efficient 2nd order linear differential equationFirst-term approximation for singular perturbation of ODE (with two turning points)Boundary Layer, leading order, Pertubation Theory, Differential EquationsMultiscale analysis with non-integer exponentsFormal proof of Lyapunov stabilityDetermine perturbation around saddles for 2D systemPerturbation evolution of a differential equationSystem of equations and perturbation methods
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Singular perturbation theory in non-standard form
The Next CEO of Stack OverflowPerturbation Analysis for Linear Linearly-Perturbed ODEsIVP Perturbation With Small Non-Linear Termnon-homogeneous constant co-efficient 2nd order linear differential equationFirst-term approximation for singular perturbation of ODE (with two turning points)Boundary Layer, leading order, Pertubation Theory, Differential EquationsMultiscale analysis with non-integer exponentsFormal proof of Lyapunov stabilityDetermine perturbation around saddles for 2D systemPerturbation evolution of a differential equationSystem of equations and perturbation methods
$begingroup$
Singular perturbation theory in ODE's is a well treated and highly studied subject. Most of the references I can find take the form,
beginalign
dotx &=fleft( x,z,varepsilon right) \
varepsilon dotz&=gleft( x,z,varepsilon right) \
endalign
where there is an explicit separation of the state variables into slow mode $x in mathbbR^m$ and fast mode $z in mathbbR^n$. I was wondering if anyone can point me to a complete treatment of the more general case:
$$ varepsilon dotx=fleft( x,varepsilon right) $$
where there is no explicit separation or- $$ varepsilon dotx=fleft( x,varepsilon right)+gleft(x right) $$
I believe in the second case a reduction to the standard form can be made by some coordinate change; but how would one go about constructing such a coordinate change? Can such a coordinate change be found in the first case?
A treatment of the linear case:
$$ varepsilon dotx=left( A+varepsilon Bleft( varepsilon right) right)x $$
can be found here but I can't seem to find any material on the most general non-linear case.
ordinary-differential-equations reference-request stability-in-odes perturbation-theory
asked Mar 27 at 18:14
ITAITA
1,053621
$endgroup$
add a comment |
$begingroup$
Singular perturbation theory in ODE's is a well treated and highly studied subject. Most of the references I can find take the form,
beginalign
dotx &=fleft( x,z,varepsilon right) \
varepsilon dotz&=gleft( x,z,varepsilon right) \
endalign
where there is an explicit separation of the state variables into slow mode $x in mathbbR^m$ and fast mode $z in mathbbR^n$. I was wondering if anyone can point me to a complete treatment of the more general case:
$$ varepsilon dotx=fleft( x,varepsilon right) $$
where there is no explicit separation or- $$ varepsilon dotx=fleft( x,varepsilon right)+gleft(x right) $$
I believe in the second case a reduction to the standard form can be made by some coordinate change; but how would one go about constructing such a coordinate change? Can such a coordinate change be found in the first case?
A treatment of the linear case:
$$ varepsilon dotx=left( A+varepsilon Bleft( varepsilon right) right)x $$
can be found here but I can't seem to find any material on the most general non-linear case.
ordinary-differential-equations reference-request stability-in-odes perturbation-theory
asked Mar 27 at 18:14
ITAITA
1,053621
$endgroup$
add a comment |
$begingroup$
Singular perturbation theory in ODE's is a well treated and highly studied subject. Most of the references I can find take the form,
beginalign
dotx &=fleft( x,z,varepsilon right) \
varepsilon dotz&=gleft( x,z,varepsilon right) \
endalign
where there is an explicit separation of the state variables into slow mode $x in mathbbR^m$ and fast mode $z in mathbbR^n$. I was wondering if anyone can point me to a complete treatment of the more general case:
$$ varepsilon dotx=fleft( x,varepsilon right) $$
where there is no explicit separation or- $$ varepsilon dotx=fleft( x,varepsilon right)+gleft(x right) $$
I believe in the second case a reduction to the standard form can be made by some coordinate change; but how would one go about constructing such a coordinate change? Can such a coordinate change be found in the first case?
A treatment of the linear case:
$$ varepsilon dotx=left( A+varepsilon Bleft( varepsilon right) right)x $$
can be found here but I can't seem to find any material on the most general non-linear case.
ordinary-differential-equations reference-request stability-in-odes perturbation-theory
asked Mar 27 at 18:14
ITAITA
1,053621
$endgroup$
Singular perturbation theory in ODE's is a well treated and highly studied subject. Most of the references I can find take the form,
beginalign
dotx &=fleft( x,z,varepsilon right) \
varepsilon dotz&=gleft( x,z,varepsilon right) \
endalign
where there is an explicit separation of the state variables into slow mode $x in mathbbR^m$ and fast mode $z in mathbbR^n$. I was wondering if anyone can point me to a complete treatment of the more general case:
$$ varepsilon dotx=fleft( x,varepsilon right) $$
where there is no explicit separation or- $$ varepsilon dotx=fleft( x,varepsilon right)+gleft(x right) $$
I believe in the second case a reduction to the standard form can be made by some coordinate change; but how would one go about constructing such a coordinate change? Can such a coordinate change be found in the first case?
A treatment of the linear case:
$$ varepsilon dotx=left( A+varepsilon Bleft( varepsilon right) right)x $$
can be found here but I can't seem to find any material on the most general non-linear case.
ordinary-differential-equations reference-request stability-in-odes perturbation-theory
ordinary-differential-equations reference-request stability-in-odes perturbation-theory
asked Mar 27 at 18:14
ITAITA
1,053621
asked Mar 27 at 18:14
ITAITA
1,053621
edited yesterday
ITA
asked Mar 27 at 18:14
ITAITA
1,053621
asked Mar 27 at 18:14
ITAITA
1,053621
asked Mar 27 at 18:14
ITAITA
1,053621
1,053621
add a comment |
add a comment |
1 Answer
1
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votes
$begingroup$
A general treatment does not exist at this moment; however, I would advise to keep a close eye on this upcoming title. In the mean time, geometric singular perturbation theory analysis of specific systems in non-standard form can be found here, here, here or here, to name just a few.
answered 2 days ago
Frits VeermanFrits Veerman
7,0562921
$endgroup$
$begingroup$
Thats unfortunate. I did find this article which considers a special form of the second case in the question.
$endgroup$
– ITA
2 days ago
$begingroup$
You'll just have to be patient, I guess. Feel free to accept the answer, btw!
$endgroup$
– Frits Veerman
yesterday
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
A general treatment does not exist at this moment; however, I would advise to keep a close eye on this upcoming title. In the mean time, geometric singular perturbation theory analysis of specific systems in non-standard form can be found here, here, here or here, to name just a few.
answered 2 days ago
Frits VeermanFrits Veerman
7,0562921
$endgroup$
$begingroup$
Thats unfortunate. I did find this article which considers a special form of the second case in the question.
$endgroup$
– ITA
2 days ago
$begingroup$
You'll just have to be patient, I guess. Feel free to accept the answer, btw!
$endgroup$
– Frits Veerman
yesterday
add a comment |
$begingroup$
A general treatment does not exist at this moment; however, I would advise to keep a close eye on this upcoming title. In the mean time, geometric singular perturbation theory analysis of specific systems in non-standard form can be found here, here, here or here, to name just a few.
answered 2 days ago
Frits VeermanFrits Veerman
7,0562921
$endgroup$
$begingroup$
Thats unfortunate. I did find this article which considers a special form of the second case in the question.
$endgroup$
– ITA
2 days ago
$begingroup$
You'll just have to be patient, I guess. Feel free to accept the answer, btw!
$endgroup$
– Frits Veerman
yesterday
add a comment |
$begingroup$
A general treatment does not exist at this moment; however, I would advise to keep a close eye on this upcoming title. In the mean time, geometric singular perturbation theory analysis of specific systems in non-standard form can be found here, here, here or here, to name just a few.
answered 2 days ago
Frits VeermanFrits Veerman
7,0562921
$endgroup$
A general treatment does not exist at this moment; however, I would advise to keep a close eye on this upcoming title. In the mean time, geometric singular perturbation theory analysis of specific systems in non-standard form can be found here, here, here or here, to name just a few.
answered 2 days ago
Frits VeermanFrits Veerman
7,0562921
answered 2 days ago
Frits VeermanFrits Veerman
7,0562921
answered 2 days ago
Frits VeermanFrits Veerman
7,0562921
answered 2 days ago
Frits VeermanFrits Veerman
7,0562921
7,0562921
$begingroup$
Thats unfortunate. I did find this article which considers a special form of the second case in the question.
$endgroup$
– ITA
2 days ago
$begingroup$
You'll just have to be patient, I guess. Feel free to accept the answer, btw!
$endgroup$
– Frits Veerman
yesterday
add a comment |
$begingroup$
Thats unfortunate. I did find this article which considers a special form of the second case in the question.
$endgroup$
– ITA
2 days ago
$begingroup$
You'll just have to be patient, I guess. Feel free to accept the answer, btw!
$endgroup$
– Frits Veerman
yesterday
$begingroup$
Thats unfortunate. I did find this article which considers a special form of the second case in the question.
$endgroup$
– ITA
2 days ago
$begingroup$
Thats unfortunate. I did find this article which considers a special form of the second case in the question.
$endgroup$
– ITA
2 days ago
$begingroup$
You'll just have to be patient, I guess. Feel free to accept the answer, btw!
$endgroup$
– Frits Veerman
yesterday
$begingroup$
You'll just have to be patient, I guess. Feel free to accept the answer, btw!
$endgroup$
– Frits Veerman
yesterday
add a comment |
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