Computing the Poincare Map of a Dynamical System

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Consider the two-dimensional system $dotr = r(1 − r)$ and $dottheta = 1$ Compute the
Poincare map for the local section $S = (r, θ) $.

I'm not really sure how to proceed. I know what a Poincare map is (definitionally), but I'm not sure how to compute it for this particular system.

differential-equations dynamical-systems

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asked Jul 19 at 22:37

rubikscube09

869617

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up vote
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favorite

1

Consider the two-dimensional system $dotr = r(1 − r)$ and $dottheta = 1$ Compute the
Poincare map for the local section $S = (r, θ) $.

I'm not really sure how to proceed. I know what a Poincare map is (definitionally), but I'm not sure how to compute it for this particular system.

differential-equations dynamical-systems

share|cite|improve this question

asked Jul 19 at 22:37

rubikscube09

869617

add a comment | 

up vote
1
down vote

favorite

1

up vote
1
down vote

favorite

1

1

Consider the two-dimensional system $dotr = r(1 − r)$ and $dottheta = 1$ Compute the
Poincare map for the local section $S = (r, θ) $.

I'm not really sure how to proceed. I know what a Poincare map is (definitionally), but I'm not sure how to compute it for this particular system.

differential-equations dynamical-systems

share|cite|improve this question

asked Jul 19 at 22:37

rubikscube09

869617

Consider the two-dimensional system $dotr = r(1 − r)$ and $dottheta = 1$ Compute the
Poincare map for the local section $S = (r, θ) $.

I'm not really sure how to proceed. I know what a Poincare map is (definitionally), but I'm not sure how to compute it for this particular system.

differential-equations dynamical-systems

share|cite|improve this question

asked Jul 19 at 22:37

rubikscube09

869617

share|cite|improve this question

share|cite|improve this question

asked Jul 19 at 22:37

rubikscube09

869617

asked Jul 19 at 22:37

rubikscube09

869617

asked Jul 19 at 22:37

rubikscube09

869617

869617

add a comment | 

add a comment | 

2 Answers
2

active

oldest

votes

up vote
2
down vote



accepted

The Poincaré map $f(r_0)$ is the radius of the first intersection with $S$ of the solution starting at $(0,r_0)$ :

Note that this intersection is at $theta=t=2pi$.

The solution of the initial value problem
$$
dot r= r(1-r),quad r(0)=r_0
$$
is
equal to
$$r(t)= frac11-fracr_0 - 1r_0 e^-t ,$$
thus, the radius of the next intersection is
$$
r(2pi)= frac11-fracr_0 - 1r_0 e^-2pi .
$$
Finally, the Poincaré map is
$$
f(r_0)= frac11-fracr_0 - 1r_0 e^-2pi .
$$

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answered Jul 20 at 7:01

AVK

1,7201415

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up vote
2
down vote

Hints:

• This is in polar coordinates, because it would not make sense otherwise.


• The Poincaré map gives the relation between one intersection of $S$ and the next.


• $dotr=r(1-r)$ can be solved analytically.


• The two differential equations are uncoupled, thus allowing you to easily determine the time points of intersections.

share|cite|improve this answer

answered Jul 20 at 7:02

Wrzlprmft

2,71111132

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2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
2
down vote



accepted

The Poincaré map $f(r_0)$ is the radius of the first intersection with $S$ of the solution starting at $(0,r_0)$ :

Note that this intersection is at $theta=t=2pi$.

The solution of the initial value problem
$$
dot r= r(1-r),quad r(0)=r_0
$$
is
equal to
$$r(t)= frac11-fracr_0 - 1r_0 e^-t ,$$
thus, the radius of the next intersection is
$$
r(2pi)= frac11-fracr_0 - 1r_0 e^-2pi .
$$
Finally, the Poincaré map is
$$
f(r_0)= frac11-fracr_0 - 1r_0 e^-2pi .
$$

share|cite|improve this answer

answered Jul 20 at 7:01

AVK

1,7201415

add a comment | 

up vote
2
down vote



accepted

The Poincaré map $f(r_0)$ is the radius of the first intersection with $S$ of the solution starting at $(0,r_0)$ :

Note that this intersection is at $theta=t=2pi$.

The solution of the initial value problem
$$
dot r= r(1-r),quad r(0)=r_0
$$
is
equal to
$$r(t)= frac11-fracr_0 - 1r_0 e^-t ,$$
thus, the radius of the next intersection is
$$
r(2pi)= frac11-fracr_0 - 1r_0 e^-2pi .
$$
Finally, the Poincaré map is
$$
f(r_0)= frac11-fracr_0 - 1r_0 e^-2pi .
$$

share|cite|improve this answer

answered Jul 20 at 7:01

AVK

1,7201415

add a comment | 

up vote
2
down vote



accepted

up vote
2
down vote



accepted

The Poincaré map $f(r_0)$ is the radius of the first intersection with $S$ of the solution starting at $(0,r_0)$ :

Note that this intersection is at $theta=t=2pi$.

The solution of the initial value problem
$$
dot r= r(1-r),quad r(0)=r_0
$$
is
equal to
$$r(t)= frac11-fracr_0 - 1r_0 e^-t ,$$
thus, the radius of the next intersection is
$$
r(2pi)= frac11-fracr_0 - 1r_0 e^-2pi .
$$
Finally, the Poincaré map is
$$
f(r_0)= frac11-fracr_0 - 1r_0 e^-2pi .
$$

share|cite|improve this answer

answered Jul 20 at 7:01

AVK

1,7201415

The Poincaré map $f(r_0)$ is the radius of the first intersection with $S$ of the solution starting at $(0,r_0)$ :

Note that this intersection is at $theta=t=2pi$.

The solution of the initial value problem
$$
dot r= r(1-r),quad r(0)=r_0
$$
is
equal to
$$r(t)= frac11-fracr_0 - 1r_0 e^-t ,$$
thus, the radius of the next intersection is
$$
r(2pi)= frac11-fracr_0 - 1r_0 e^-2pi .
$$
Finally, the Poincaré map is
$$
f(r_0)= frac11-fracr_0 - 1r_0 e^-2pi .
$$

share|cite|improve this answer

answered Jul 20 at 7:01

AVK

1,7201415

share|cite|improve this answer

share|cite|improve this answer

answered Jul 20 at 7:01

AVK

1,7201415

answered Jul 20 at 7:01

AVK

1,7201415

answered Jul 20 at 7:01

AVK

1,7201415

1,7201415

add a comment | 

add a comment | 

up vote
2
down vote

Hints:

• This is in polar coordinates, because it would not make sense otherwise.


• The Poincaré map gives the relation between one intersection of $S$ and the next.


• $dotr=r(1-r)$ can be solved analytically.


• The two differential equations are uncoupled, thus allowing you to easily determine the time points of intersections.

share|cite|improve this answer

answered Jul 20 at 7:02

Wrzlprmft

2,71111132

add a comment | 

up vote
2
down vote

Hints:

• This is in polar coordinates, because it would not make sense otherwise.


• The Poincaré map gives the relation between one intersection of $S$ and the next.


• $dotr=r(1-r)$ can be solved analytically.


• The two differential equations are uncoupled, thus allowing you to easily determine the time points of intersections.

share|cite|improve this answer

answered Jul 20 at 7:02

Wrzlprmft

2,71111132

add a comment | 

up vote
2
down vote

up vote
2
down vote

Hints:

• This is in polar coordinates, because it would not make sense otherwise.


• The Poincaré map gives the relation between one intersection of $S$ and the next.


• $dotr=r(1-r)$ can be solved analytically.


• The two differential equations are uncoupled, thus allowing you to easily determine the time points of intersections.

share|cite|improve this answer

answered Jul 20 at 7:02

Wrzlprmft

2,71111132

Hints:

• This is in polar coordinates, because it would not make sense otherwise.


• The Poincaré map gives the relation between one intersection of $S$ and the next.


• $dotr=r(1-r)$ can be solved analytically.


• The two differential equations are uncoupled, thus allowing you to easily determine the time points of intersections.

share|cite|improve this answer

answered Jul 20 at 7:02

Wrzlprmft

2,71111132

share|cite|improve this answer

share|cite|improve this answer

answered Jul 20 at 7:02

Wrzlprmft

2,71111132

answered Jul 20 at 7:02

Wrzlprmft

2,71111132

answered Jul 20 at 7:02

Wrzlprmft

2,71111132

2,71111132

add a comment | 

add a comment | 

 

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