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Method of averaging : approximation of solution in phase space over time $frac1varepsilon $. What does it mean?

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1

$begingroup$

I'm reading the wikipedia page concerning method averaging. The system has the following form $$dot x=varepsilon f(x,t,varepsilon ),quad 0<varepsilon ll 1,tag1$$
of a phase space variable $dot x$.

Q1) What mean a phase space variable $dot x$ ?

The fast oscillation is given by $f$ versus slow drift of $dot x$.

Q2) I don't really understand what they mean by fast oscillation VS slow drift of $dot x$. Any idea ?

Then suggest to solve $$dot y=varepsilon frac1Tint_0^T f(x,t,0)dt=:varepsilon bar f(y),$$
and they say : $y$ approximates the solution curves of $dot x$ inside a connected and compact region of the phase space and over time of $frac1varepsilon $.

Q3) Could someone tell me what it mean ? What is the solution curve of $dot x$ ? Do they mean the solution of the equation $(1)$ ?

Q4) After, I don't get what they mean by : inside a connected and compact region of the phase space and over time of $frac1varepsilon $.

real-analysis ordinary-differential-equations intuition

share|cite|improve this question

edited 1 hour ago

Alex Ravsky

42.6k32383

asked Mar 14 at 15:37

PierrePierre

12811

$endgroup$

This question has an open bounty worth +50
reputation from Pierre ending ending at 2019-03-23 16:18:11Z">in 2 days.

This question has not received enough attention.

add a comment | 

1

$begingroup$

I'm reading the wikipedia page concerning method averaging. The system has the following form $$dot x=varepsilon f(x,t,varepsilon ),quad 0<varepsilon ll 1,tag1$$
of a phase space variable $dot x$.

Q1) What mean a phase space variable $dot x$ ?

The fast oscillation is given by $f$ versus slow drift of $dot x$.

Q2) I don't really understand what they mean by fast oscillation VS slow drift of $dot x$. Any idea ?

Then suggest to solve $$dot y=varepsilon frac1Tint_0^T f(x,t,0)dt=:varepsilon bar f(y),$$
and they say : $y$ approximates the solution curves of $dot x$ inside a connected and compact region of the phase space and over time of $frac1varepsilon $.

Q3) Could someone tell me what it mean ? What is the solution curve of $dot x$ ? Do they mean the solution of the equation $(1)$ ?

Q4) After, I don't get what they mean by : inside a connected and compact region of the phase space and over time of $frac1varepsilon $.

real-analysis ordinary-differential-equations intuition

share|cite|improve this question

edited 1 hour ago

Alex Ravsky

42.6k32383

asked Mar 14 at 15:37

PierrePierre

12811

$endgroup$

This question has an open bounty worth +50
reputation from Pierre ending ending at 2019-03-23 16:18:11Z">in 2 days.

This question has not received enough attention.

add a comment | 

$begingroup$

I'm reading the wikipedia page concerning method averaging. The system has the following form $$dot x=varepsilon f(x,t,varepsilon ),quad 0<varepsilon ll 1,tag1$$
of a phase space variable $dot x$.

Q1) What mean a phase space variable $dot x$ ?

The fast oscillation is given by $f$ versus slow drift of $dot x$.

Q2) I don't really understand what they mean by fast oscillation VS slow drift of $dot x$. Any idea ?

Then suggest to solve $$dot y=varepsilon frac1Tint_0^T f(x,t,0)dt=:varepsilon bar f(y),$$
and they say : $y$ approximates the solution curves of $dot x$ inside a connected and compact region of the phase space and over time of $frac1varepsilon $.

Q3) Could someone tell me what it mean ? What is the solution curve of $dot x$ ? Do they mean the solution of the equation $(1)$ ?

Q4) After, I don't get what they mean by : inside a connected and compact region of the phase space and over time of $frac1varepsilon $.

real-analysis ordinary-differential-equations intuition

share|cite|improve this question

edited 1 hour ago

Alex Ravsky

42.6k32383

asked Mar 14 at 15:37

PierrePierre

12811

$endgroup$

share|cite|improve this question

edited 1 hour ago

Alex Ravsky

42.6k32383

asked Mar 14 at 15:37

PierrePierre

12811

share|cite|improve this question

edited 1 hour ago

Alex Ravsky

42.6k32383

asked Mar 14 at 15:37

PierrePierre

12811

edited 1 hour ago

Alex Ravsky

42.6k32383

edited 1 hour ago

Alex Ravsky

42.6k32383

asked Mar 14 at 15:37

PierrePierre

12811

asked Mar 14 at 15:37

PierrePierre

12811