Dynamical Systems: Stability of Fixed Points

Watch
Announcements
#1

Just to make sure, am I able to knock parts (a) and (b) down with one function; i.e. , where the fixed point is .

This point is both an attractor and repeller since starting with will always take you to the fixed point no matter how far off you start, while with you will always repel away from the fixed point no matter how close or far you start as well.

Diagram:

0
2 years ago
#2
I would have thought not; don't you require an attractor to have a basin of attraction that contains an open neighbourhood?
0
2 years ago
#3
(Original post by Gregorius)
I would have thought not; don't you require an attractor to have a basin of attraction that contains an open neighbourhood?
Would be my gut reaction as well, you'd need it to attract / diverge on both sides. When the function has a unity gradient, its generally classified as neither.
0
#4
(Original post by Gregorius)
I would have thought not; don't you require an attractor to have a basin of attraction that contains an open neighbourhood?
Hm, the definition I was given makes no mention of that.

Def (attractor): A fixed point is called an attractor if it attracts orbit with initial close to . In other words, such that .

Spoiler:
Show

0
2 years ago
#5
(Original post by RDKGames)
Hm, the definition I was given makes no mention of that.

Def (attractor): A fixed point is called an attractor if it attracts orbit with initial close to . In other words, such that .

Spoiler:
Show

It would only attract on one side though (as you point out in the OP), not all initial points in a ball around the "attractor".
0
#6
(Original post by mqb2766)
It would only attract on one side though (as you point out in the OP), not all initial points in a ball around the "attractor".
So you're saying that the given definition is incorrect?
0
2 years ago
#7
(Original post by RDKGames)
So you're saying that the given definition is incorrect?
The definition sounds fine to me. It would have to attract on both sides to satisfy the defn?
If all the curves have to have a gradient of 1 at the fixed point, your example sounds like c)
Can you not create two antisymmetric curves (about 1) by cutting your example into two, where one will attract and the other diverge?
0
#8
(Original post by mqb2766)
The definition sounds fine to me. It would have to attract on both sides to satisfy the defn?
But I don't see where in the definition it requires that it must attract on both sides of the fixed point.
It just says 'close to ' but then being as close as we want from only one side is sufficient to satisfy this.

If all the curves have to have a gradient of 1 at the fixed point, your example sounds like c)
Can you not create two antisymmetric curves (about 1) by cutting your example into two, where one will attract and the other diverge?
But ok, in the case that we need a neighbourhood around to attract then indeed I can just split the current curve so that:

a) [attractor] can have

and looks like
Spoiler:
Show

b) [repeller] can have

and looks like
Spoiler:
Show

and so c) is obviously fit by my original example in OP.
0
2 years ago
#9
(Original post by RDKGames)
But I don't see where in the definition it requires that it must attract on both sides of the fixed point.
It just says 'close to ' but then being as close as we want from only one side is sufficient to satisfy this.

But ok, in the case that we need a neighbourhood around to attract then indeed I can just split the current curve so that:

a) [attractor] can have

and looks like
Spoiler:
Show

b) [repeller] can have

and looks like
Spoiler:
Show

and so c) is obviously fit by my original example in OP.
Looks good to me.

A fixed point x^{*} is called an attractor if it attracts orbit with initial x_0 close to x^{*}. In other words, such that

You can pick an x_0 close to x* and make it diverge. The sign of is irrelevant in the definition, but in your example it is relevant?
1
2 years ago
#10
(Original post by RDKGames)
But I don't see where in the definition it requires that it must attract on both sides of the fixed point.

Note carefully the modulus signs.
1
#11
(Original post by mqb2766)
Looks good to me.

A fixed point x^{*} is called an attractor if it attracts orbit with initial x_0 close to x^{*}. In other words, such that

You can pick an x_0 close to x* and make it diverge. The sign of is irrelevant in the definition, but in your example it is relevant?
(Original post by Gregorius)

Note carefully the modulus signs.
Ok thanks, I was overlooking and misunderstanding that a bit there for a moment.
2
X

new posts
Back
to top
Latest
My Feed

Oops, nobody has postedin the last few hours.

Why not re-start the conversation?

see more

See more of what you like onThe Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

Poll

Join the discussion

How would you feel if uni students needed to be double vaccinated to start in Autumn?

I'd feel reassured about my own health (33)
14.8%
I'd feel reassured my learning may be less disrupted by isolations/lockdowns (68)
30.49%
I'd feel less anxious about being around large groups (27)
12.11%
I don't mind if others are vaccinated or not (19)
8.52%
I'm concerned it may disadvantage some students (13)
5.83%
I think it's an unfair expectation (60)
26.91%
Something else (tell us in the thread) (3)
1.35%