System of equations and perturbation methods












0














I would like to characterise how the solution of a nonlinear system of equations change if a perturbation term is added.



Namely, I have the system
begin{array}{lcl} 2y [F(x) - varepsilon frac{1}{y(xy-1)} ] & = & 0 \ y^2 [F^prime (x) -varepsilon frac{y}{xy-1}] & = & 0end{array}



In the unperturbed case $varepsilon = 0$ the solution is handy $$ (x_0,0)$$ where $x_0$ is such that $F^prime (x_0) = 0$.



How to describe the solution for small $varepsilon$??



I would have tried to expand the perturbed terms around the solution of the unperturbed system, but the term $frac{1}{y(xy-1)}$ is not even defined there.



Alternatively, following the perturbation theory one could assume that the perturbed solution can be expressed as
begin{array}{lcl} x & = & x_0 + varepsilon f_1 +varepsilon^2 f_2 + dots \ y = & = & varepsilon g_1 +varepsilon^2 g_2 + dots end{array}



and let me substitute in the first equation of the system.
I get
$$ 2(varepsilon g_11 + varepsilon^2 g_2 + dots ) Big[F(x_0+varepsilon f_1 + dots) - varepsilon frac{1}{(varepsilon g_11 + varepsilon^2 g_2 + dots)((x_0+varepsilon f_1 + dots)(varepsilon g_1 + varepsilon^2 g_2 + dots))}Big]$$ which could develop into



$$2(varepsilon g_1 + varepsilon^2 g_2 + dots )Big[[F(x_0) + F ^{prime prime}(x_0)frac{1}{2}varepsilon^2 f_1^2] - varepsilon frac{1}{x_0varepsilon^2g_1^2 + x_0varepsilon^3g_1g_2 + varepsilon^3 f_1g_1^2 + varepsilon^3x_0 g_1g_2 + dots}Big] $$



Now I need to collect terms that are first order in $epsilon$.



But how to do that systematically?
I am struggling to handle the fraction $$- varepsilon frac{1}{x_0varepsilon^2g_1^2 + x_0varepsilon^3g_1g_2 + varepsilon^3 f_1g_1^2 + varepsilon^3x_0 g_1g_2 + dots}$$



I would be most grateful for any hint.



Thanks and Happiest New 2019



EDIT:
I would like to describe one more attempt of mine.
I thought of replacing the original perturbed system
begin{array}{lcl} 2y [F(x) - varepsilon frac{1}{y(xy-1)} ] & = & 0 \ y^2 [F^prime (x) -varepsilon frac{y}{xy-1}] & = & 0end{array}
with the system



begin{array}{lcl} 2y [F(x) - varepsilon (-frac{1}{y}-x) ] & = & 0 \ y^2 [F^prime (x) -varepsilon (-y)] & = & 0end{array}
using the approximations



$$frac{1}{y(xy-1)} approx -frac{1}{y} -x$$ and $$ frac{y}{xy-1} approx -y$$ first-order valid around $y=0$.
Then I get something tractable, would this be a workaround?



EDIT
Following the comment by User121049, I would like to add, should it be of any interest, that the problem I have is equivalent to finding the stationary point of the function



$$ Z(x,y) = y^2 Big[ F(x) - epsilon [log(frac{1}{y}-x) +1)] Big]$$



the system I originally described is obtained by setting the partial derivatives to zero.






systems-of-equations perturbation-theory







share|cite|improve this question

















This question has an open bounty worth +200
reputation from An aedonist ending in 6 days.


This question has not received enough attention.
















  • "How to understand what happens to the solution for small ϵ" -- what do you mean by "what happens"? Also, both perturbed and unperturbed case, any $(x, 0)$ seems to be a solution
    – Peter Franek
    2 days ago












  • I would like to know how the solution changes when $epsilon$ is small, compared to case when $epsilon$ is $0$. I do not understand how $(x,0)$ can be a solution in the perturbed case, as the term $frac{1}{y(xy-1)} $ is not even defined for $y=0$.
    – An aedonist
    2 days ago












  • At the end of the day, I would like to have a closed form solution of the nonlinear system. That looking unfeasible, I would be content in having a solution for "small" $epsilon$.
    – An aedonist
    2 days ago












  • Are you looking for the extrema of $y^2F(x)$ or is the form a coincidence?
    – user121049
    14 hours ago










  • It is not a coincidence,the unperturbed problem comes exactly for the minimisation problem you mentioned
    – An aedonist
    14 hours ago
















0














I would like to characterise how the solution of a nonlinear system of equations change if a perturbation term is added.



Namely, I have the system
begin{array}{lcl} 2y [F(x) - varepsilon frac{1}{y(xy-1)} ] & = & 0 \ y^2 [F^prime (x) -varepsilon frac{y}{xy-1}] & = & 0end{array}



In the unperturbed case $varepsilon = 0$ the solution is handy $$ (x_0,0)$$ where $x_0$ is such that $F^prime (x_0) = 0$.



How to describe the solution for small $varepsilon$??



I would have tried to expand the perturbed terms around the solution of the unperturbed system, but the term $frac{1}{y(xy-1)}$ is not even defined there.



Alternatively, following the perturbation theory one could assume that the perturbed solution can be expressed as
begin{array}{lcl} x & = & x_0 + varepsilon f_1 +varepsilon^2 f_2 + dots \ y = & = & varepsilon g_1 +varepsilon^2 g_2 + dots end{array}



and let me substitute in the first equation of the system.
I get
$$ 2(varepsilon g_11 + varepsilon^2 g_2 + dots ) Big[F(x_0+varepsilon f_1 + dots) - varepsilon frac{1}{(varepsilon g_11 + varepsilon^2 g_2 + dots)((x_0+varepsilon f_1 + dots)(varepsilon g_1 + varepsilon^2 g_2 + dots))}Big]$$ which could develop into



$$2(varepsilon g_1 + varepsilon^2 g_2 + dots )Big[[F(x_0) + F ^{prime prime}(x_0)frac{1}{2}varepsilon^2 f_1^2] - varepsilon frac{1}{x_0varepsilon^2g_1^2 + x_0varepsilon^3g_1g_2 + varepsilon^3 f_1g_1^2 + varepsilon^3x_0 g_1g_2 + dots}Big] $$



Now I need to collect terms that are first order in $epsilon$.



But how to do that systematically?
I am struggling to handle the fraction $$- varepsilon frac{1}{x_0varepsilon^2g_1^2 + x_0varepsilon^3g_1g_2 + varepsilon^3 f_1g_1^2 + varepsilon^3x_0 g_1g_2 + dots}$$



I would be most grateful for any hint.



Thanks and Happiest New 2019



EDIT:
I would like to describe one more attempt of mine.
I thought of replacing the original perturbed system
begin{array}{lcl} 2y [F(x) - varepsilon frac{1}{y(xy-1)} ] & = & 0 \ y^2 [F^prime (x) -varepsilon frac{y}{xy-1}] & = & 0end{array}
with the system



begin{array}{lcl} 2y [F(x) - varepsilon (-frac{1}{y}-x) ] & = & 0 \ y^2 [F^prime (x) -varepsilon (-y)] & = & 0end{array}
using the approximations



$$frac{1}{y(xy-1)} approx -frac{1}{y} -x$$ and $$ frac{y}{xy-1} approx -y$$ first-order valid around $y=0$.
Then I get something tractable, would this be a workaround?



EDIT
Following the comment by User121049, I would like to add, should it be of any interest, that the problem I have is equivalent to finding the stationary point of the function



$$ Z(x,y) = y^2 Big[ F(x) - epsilon [log(frac{1}{y}-x) +1)] Big]$$



the system I originally described is obtained by setting the partial derivatives to zero.






systems-of-equations perturbation-theory







share|cite|improve this question

















This question has an open bounty worth +200
reputation from An aedonist ending in 6 days.


This question has not received enough attention.
















  • "How to understand what happens to the solution for small ϵ" -- what do you mean by "what happens"? Also, both perturbed and unperturbed case, any $(x, 0)$ seems to be a solution
    – Peter Franek
    2 days ago












  • I would like to know how the solution changes when $epsilon$ is small, compared to case when $epsilon$ is $0$. I do not understand how $(x,0)$ can be a solution in the perturbed case, as the term $frac{1}{y(xy-1)} $ is not even defined for $y=0$.
    – An aedonist
    2 days ago












  • At the end of the day, I would like to have a closed form solution of the nonlinear system. That looking unfeasible, I would be content in having a solution for "small" $epsilon$.
    – An aedonist
    2 days ago












  • Are you looking for the extrema of $y^2F(x)$ or is the form a coincidence?
    – user121049
    14 hours ago










  • It is not a coincidence,the unperturbed problem comes exactly for the minimisation problem you mentioned
    – An aedonist
    14 hours ago














0












0








0







I would like to characterise how the solution of a nonlinear system of equations change if a perturbation term is added.



Namely, I have the system
begin{array}{lcl} 2y [F(x) - varepsilon frac{1}{y(xy-1)} ] & = & 0 \ y^2 [F^prime (x) -varepsilon frac{y}{xy-1}] & = & 0end{array}



In the unperturbed case $varepsilon = 0$ the solution is handy $$ (x_0,0)$$ where $x_0$ is such that $F^prime (x_0) = 0$.



How to describe the solution for small $varepsilon$??



I would have tried to expand the perturbed terms around the solution of the unperturbed system, but the term $frac{1}{y(xy-1)}$ is not even defined there.



Alternatively, following the perturbation theory one could assume that the perturbed solution can be expressed as
begin{array}{lcl} x & = & x_0 + varepsilon f_1 +varepsilon^2 f_2 + dots \ y = & = & varepsilon g_1 +varepsilon^2 g_2 + dots end{array}



and let me substitute in the first equation of the system.
I get
$$ 2(varepsilon g_11 + varepsilon^2 g_2 + dots ) Big[F(x_0+varepsilon f_1 + dots) - varepsilon frac{1}{(varepsilon g_11 + varepsilon^2 g_2 + dots)((x_0+varepsilon f_1 + dots)(varepsilon g_1 + varepsilon^2 g_2 + dots))}Big]$$ which could develop into



$$2(varepsilon g_1 + varepsilon^2 g_2 + dots )Big[[F(x_0) + F ^{prime prime}(x_0)frac{1}{2}varepsilon^2 f_1^2] - varepsilon frac{1}{x_0varepsilon^2g_1^2 + x_0varepsilon^3g_1g_2 + varepsilon^3 f_1g_1^2 + varepsilon^3x_0 g_1g_2 + dots}Big] $$



Now I need to collect terms that are first order in $epsilon$.



But how to do that systematically?
I am struggling to handle the fraction $$- varepsilon frac{1}{x_0varepsilon^2g_1^2 + x_0varepsilon^3g_1g_2 + varepsilon^3 f_1g_1^2 + varepsilon^3x_0 g_1g_2 + dots}$$



I would be most grateful for any hint.



Thanks and Happiest New 2019



EDIT:
I would like to describe one more attempt of mine.
I thought of replacing the original perturbed system
begin{array}{lcl} 2y [F(x) - varepsilon frac{1}{y(xy-1)} ] & = & 0 \ y^2 [F^prime (x) -varepsilon frac{y}{xy-1}] & = & 0end{array}
with the system



begin{array}{lcl} 2y [F(x) - varepsilon (-frac{1}{y}-x) ] & = & 0 \ y^2 [F^prime (x) -varepsilon (-y)] & = & 0end{array}
using the approximations



$$frac{1}{y(xy-1)} approx -frac{1}{y} -x$$ and $$ frac{y}{xy-1} approx -y$$ first-order valid around $y=0$.
Then I get something tractable, would this be a workaround?



EDIT
Following the comment by User121049, I would like to add, should it be of any interest, that the problem I have is equivalent to finding the stationary point of the function



$$ Z(x,y) = y^2 Big[ F(x) - epsilon [log(frac{1}{y}-x) +1)] Big]$$



the system I originally described is obtained by setting the partial derivatives to zero.






systems-of-equations perturbation-theory







share|cite|improve this question















I would like to characterise how the solution of a nonlinear system of equations change if a perturbation term is added.



Namely, I have the system
begin{array}{lcl} 2y [F(x) - varepsilon frac{1}{y(xy-1)} ] & = & 0 \ y^2 [F^prime (x) -varepsilon frac{y}{xy-1}] & = & 0end{array}



In the unperturbed case $varepsilon = 0$ the solution is handy $$ (x_0,0)$$ where $x_0$ is such that $F^prime (x_0) = 0$.



How to describe the solution for small $varepsilon$??



I would have tried to expand the perturbed terms around the solution of the unperturbed system, but the term $frac{1}{y(xy-1)}$ is not even defined there.



Alternatively, following the perturbation theory one could assume that the perturbed solution can be expressed as
begin{array}{lcl} x & = & x_0 + varepsilon f_1 +varepsilon^2 f_2 + dots \ y = & = & varepsilon g_1 +varepsilon^2 g_2 + dots end{array}



and let me substitute in the first equation of the system.
I get
$$ 2(varepsilon g_11 + varepsilon^2 g_2 + dots ) Big[F(x_0+varepsilon f_1 + dots) - varepsilon frac{1}{(varepsilon g_11 + varepsilon^2 g_2 + dots)((x_0+varepsilon f_1 + dots)(varepsilon g_1 + varepsilon^2 g_2 + dots))}Big]$$ which could develop into



$$2(varepsilon g_1 + varepsilon^2 g_2 + dots )Big[[F(x_0) + F ^{prime prime}(x_0)frac{1}{2}varepsilon^2 f_1^2] - varepsilon frac{1}{x_0varepsilon^2g_1^2 + x_0varepsilon^3g_1g_2 + varepsilon^3 f_1g_1^2 + varepsilon^3x_0 g_1g_2 + dots}Big] $$



Now I need to collect terms that are first order in $epsilon$.



But how to do that systematically?
I am struggling to handle the fraction $$- varepsilon frac{1}{x_0varepsilon^2g_1^2 + x_0varepsilon^3g_1g_2 + varepsilon^3 f_1g_1^2 + varepsilon^3x_0 g_1g_2 + dots}$$



I would be most grateful for any hint.



Thanks and Happiest New 2019



EDIT:
I would like to describe one more attempt of mine.
I thought of replacing the original perturbed system
begin{array}{lcl} 2y [F(x) - varepsilon frac{1}{y(xy-1)} ] & = & 0 \ y^2 [F^prime (x) -varepsilon frac{y}{xy-1}] & = & 0end{array}
with the system



begin{array}{lcl} 2y [F(x) - varepsilon (-frac{1}{y}-x) ] & = & 0 \ y^2 [F^prime (x) -varepsilon (-y)] & = & 0end{array}
using the approximations



$$frac{1}{y(xy-1)} approx -frac{1}{y} -x$$ and $$ frac{y}{xy-1} approx -y$$ first-order valid around $y=0$.
Then I get something tractable, would this be a workaround?



EDIT
Following the comment by User121049, I would like to add, should it be of any interest, that the problem I have is equivalent to finding the stationary point of the function



$$ Z(x,y) = y^2 Big[ F(x) - epsilon [log(frac{1}{y}-x) +1)] Big]$$



the system I originally described is obtained by setting the partial derivatives to zero.






systems-of-equations perturbation-theory




systems-of-equations perturbation-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 14 hours ago

























asked 2 days ago









An aedonist

2,154618




2,154618






This question has an open bounty worth +200
reputation from An aedonist ending in 6 days.


This question has not received enough attention.








This question has an open bounty worth +200
reputation from An aedonist ending in 6 days.


This question has not received enough attention.














  • "How to understand what happens to the solution for small ϵ" -- what do you mean by "what happens"? Also, both perturbed and unperturbed case, any $(x, 0)$ seems to be a solution
    – Peter Franek
    2 days ago












  • I would like to know how the solution changes when $epsilon$ is small, compared to case when $epsilon$ is $0$. I do not understand how $(x,0)$ can be a solution in the perturbed case, as the term $frac{1}{y(xy-1)} $ is not even defined for $y=0$.
    – An aedonist
    2 days ago












  • At the end of the day, I would like to have a closed form solution of the nonlinear system. That looking unfeasible, I would be content in having a solution for "small" $epsilon$.
    – An aedonist
    2 days ago












  • Are you looking for the extrema of $y^2F(x)$ or is the form a coincidence?
    – user121049
    14 hours ago










  • It is not a coincidence,the unperturbed problem comes exactly for the minimisation problem you mentioned
    – An aedonist
    14 hours ago


















  • "How to understand what happens to the solution for small ϵ" -- what do you mean by "what happens"? Also, both perturbed and unperturbed case, any $(x, 0)$ seems to be a solution
    – Peter Franek
    2 days ago












  • I would like to know how the solution changes when $epsilon$ is small, compared to case when $epsilon$ is $0$. I do not understand how $(x,0)$ can be a solution in the perturbed case, as the term $frac{1}{y(xy-1)} $ is not even defined for $y=0$.
    – An aedonist
    2 days ago












  • At the end of the day, I would like to have a closed form solution of the nonlinear system. That looking unfeasible, I would be content in having a solution for "small" $epsilon$.
    – An aedonist
    2 days ago












  • Are you looking for the extrema of $y^2F(x)$ or is the form a coincidence?
    – user121049
    14 hours ago










  • It is not a coincidence,the unperturbed problem comes exactly for the minimisation problem you mentioned
    – An aedonist
    14 hours ago
















"How to understand what happens to the solution for small ϵ" -- what do you mean by "what happens"? Also, both perturbed and unperturbed case, any $(x, 0)$ seems to be a solution
– Peter Franek
2 days ago






"How to understand what happens to the solution for small ϵ" -- what do you mean by "what happens"? Also, both perturbed and unperturbed case, any $(x, 0)$ seems to be a solution
– Peter Franek
2 days ago














I would like to know how the solution changes when $epsilon$ is small, compared to case when $epsilon$ is $0$. I do not understand how $(x,0)$ can be a solution in the perturbed case, as the term $frac{1}{y(xy-1)} $ is not even defined for $y=0$.
– An aedonist
2 days ago






I would like to know how the solution changes when $epsilon$ is small, compared to case when $epsilon$ is $0$. I do not understand how $(x,0)$ can be a solution in the perturbed case, as the term $frac{1}{y(xy-1)} $ is not even defined for $y=0$.
– An aedonist
2 days ago














At the end of the day, I would like to have a closed form solution of the nonlinear system. That looking unfeasible, I would be content in having a solution for "small" $epsilon$.
– An aedonist
2 days ago






At the end of the day, I would like to have a closed form solution of the nonlinear system. That looking unfeasible, I would be content in having a solution for "small" $epsilon$.
– An aedonist
2 days ago














Are you looking for the extrema of $y^2F(x)$ or is the form a coincidence?
– user121049
14 hours ago




Are you looking for the extrema of $y^2F(x)$ or is the form a coincidence?
– user121049
14 hours ago












It is not a coincidence,the unperturbed problem comes exactly for the minimisation problem you mentioned
– An aedonist
14 hours ago




It is not a coincidence,the unperturbed problem comes exactly for the minimisation problem you mentioned
– An aedonist
14 hours ago










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