# Dynamical Systems: Stability of Fixed Points

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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:

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#2

I would have thought not; don't you require an attractor to have a basin of attraction that contains an open neighbourhood?

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#3

(Original post by

I would have thought not; don't you require an attractor to have a basin of attraction that contains an open neighbourhood?

**Gregorius**)I would have thought not; don't you require an attractor to have a basin of attraction that contains an open neighbourhood?

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**Gregorius**)

I would have thought not; don't you require an attractor to have a basin of attraction that contains an open neighbourhood?

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

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#5

(Original post by

Hm, the definition I was given makes no mention of that.

**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 .
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(Original post by

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".

**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".

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#7

(Original post by

So you're saying that the given definition is incorrect?

**RDKGames**)So you're saying that the given definition is incorrect?

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?

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(Original post by

The definition sounds fine to me. It would have to attract on both sides to satisfy the defn?

**mqb2766**)The definition sounds fine to me. It would have to attract on both sides to satisfy the defn?

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?

Can you not create two antisymmetric curves (about 1) by cutting your example into two, where one will attract and the other diverge?

a) [attractor] can have

and looks like

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b) [repeller] can have

and looks like

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and so c) is obviously fit by my original example in OP.

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#9

(Original post by

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

b) [repeller] can have

and looks like

and so c) is obviously fit by my original example in OP.

**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.

Edit: For c) in your original example (didn't read your reply properly)

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?

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#10

(Original post by

But I don't see where in the definition it requires that it must attract on both sides of the fixed point.

**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.

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(Original post by

Looks good to me.

Edit: For c) in your original example (didn't read your reply properly)

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?

**mqb2766**)Looks good to me.

Edit: For c) in your original example (didn't read your reply properly)

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?

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