Discrete dynamical systems - Invertible maps

[Joppy]

Hi (Sleepy),

I suspect this is trivial, but I couldn't find any info onlin.

Consider the folowing map: $\phi_{n+1} = f(\phi_n ; \Theta, a) = (\phi_n + \Theta + a \sin \phi_n) \mod 2\pi$.

I need to check if is invertible: $\phi_n = f^{-1} (\phi_{n+1}; \Theta, a)$ when a = 1/2 or 3/2.

[DESMOS=-0.626372053161002,7.429191250095138,-0.30385709978601394,7.751706203470127]y=\operatorname{mod}\left(x\ +\ b+\frac{3}{2}\sin\left(x\right),2\pi\right);b=0;y=\operatorname{mod}\left(x\ +\ b+\frac{1}{2}\sin\left(x\right),2\pi\right);[/DESMOS]

I figure show that the map is injective, and then find it's inverse as we would with any other function (from the graph we see that for $a = 1/2$, the map is injective).

Which would just showing: $\phi_n + a \sin (\phi_n) = \phi_{n+1} + a \sin (\phi_{n+1}) \implies \phi_n = \phi_{n+1}$? Is there some neat substitution which will help out this step?

 

[I like Serena]

Hey Joppy!

I see we have a $\phi_n$ that gets summed with $\sin\phi_n$.
There's no simple inverse for something like that.

To find if $f$ is invertible, we can check if it's strictly monotone (and possibly non-strictly monotone in countable many points).
That means taking the derivative with respect to $\phi_n$ and see for which values of $a$ that is sufficiently monotone (either always positive or always negative) ...

Oh, and I'm guessing we restrict ourself to $\phi_n \in [0,2\pi)$.
Otherwise it's definitely not invertible. (Nerd)

 

[Joppy]

To find if $f$ is invertible, we can check if it's strictly monotone (and possibly non-strictly monotone in countable many points).
That means taking the derivative with respect to $\phi_n$ and see for which values of $a$ that is sufficiently monotone (either always positive or always negative) ...

Oh yeah... (Wasntme).

Thanks! Will do that and post back if any troubles. Probbaly I don't know how to find a bound for a (i'd like to find a range of a s.t. the map is invertible, or something like this).

 

[I like Serena]

Just realized that the $\bmod{2\pi}$ has a nasty impact.
For starters, it means that $\phi_n$ has a domain of $[0,2\pi)$ due to its recursive nature.
And if the function value gets above $2\pi$ (or below $0$), anywhere, it gets truncated, probably causing the function to become non-invertible.
So I think we have to consider cases where this can happen.

 

[S.G. Janssens]

Joppy, were you given the name of this map? It is quite well-known and if you have some difficulties studying it, as alluded to already a bit in post #4, then that is not surprising.

 

[Joppy]

Joppy, were you given the name of this map? It is quite well-known and if you have some difficulties studying it, as alluded to already a bit in post #4, then that is not surprising.

I was not. I thought it might have been a discrete equivalent of the forced oscillator or something like that, but it lacks a 'second difference' term. I'd love to know what it's called :)! Thanks.

Btw, I'm looking into the invertibility as it (along with the dimension of the system) will tell us whether or not chaos is possible. I also need to check if the system is conservative ($\mathbf{J(x)} = |\det(\partial f(\phi_n)/\partial \phi_n)| = 1$) or dissipative. Haven't had time yet, back soon :).

Just realized that the $\bmod{2\pi}$ has a nasty impact.
For starters, it means that $\phi_n$ has a domain of $[0,2\pi)$ due to its recursive nature.
And if the function value gets above $2\pi$ (or below $0$), anywhere, it gets truncated, probably causing the function to become non-invertible.
So I think we have to consider cases where this can happen.

I've always been curious about this, but never asked since I'm not sure the question makes sense. I'll try: What impact, exactly does 'modulo-ing' the output of a dynamical system have? It's a very common thing to do and allows us to study a system over some finite interval as opposed to the whole real line for example.
But doesn't it also mean that we are gluing the end points (or boundaries) together and are (in a sense) looking at a projection from a higher dimensional phase space on a lower dimensional plane? Or am I talking nonsense (Wondering)(Speechless) (Bigsmile) (Note: I'm not talking about Poincare sections)

 

[S.G. Janssens]

I was not. I thought it might have been a discrete equivalent of the forced oscillator or something like that, but it lacks a 'second difference' term. I'd love to know what it's called :)! Thanks.

Your map from post #1 is equivalent to what is called the "circle map" (or "standard circle map") in the literature. I am saying "equivalent" because the circle map is often written in the slightly different form
\[
\psi_{n+1} = \psi_n + \theta - a\sin{\psi_n} \pmod{2\pi},
\]
so with a minus-sign in front of the non-negative parameter $a$. You can switch between both forms using $\psi_n = \phi_n + \pi$ - that is merely a phase shift - and the fact that $\sin{\phi_n} = \sin(\psi_n - \pi) = - \sin{\psi_n}$.

The circle map is related to Chirikov's standard map, which is two-dimensional. Indeed, the latter map arises as a Poincaré section of a forced (or rather: "kicked") oscillator. Both examples are rather famous and you can find a lot of information about them.

I've always been curious about this, but never asked since I'm not sure the question makes sense. I'll try: What impact, exactly does 'modulo-ing' the output of a dynamical system have? It's a very common thing to do and allows us to study a system over some finite interval as opposed to the whole real line for example.
But doesn't it also mean that we are gluing the end points (or boundaries) together and are (in a sense) looking at a projection from a higher dimensional phase space on a lower dimensional plane? Or am I talking nonsense (Wondering)(Speechless) (Bigsmile) (Note: I'm not talking about Poincare sections)

You should not hesitate to ask such questions, because I think they are good questions. Working modulo $2\pi$ means that we view the map as a self-map of the circle: Angles that differ by a multiple of $2\pi$ are regarded as one and the same state of the dynamical system generated by the map.

As you suggest, we may alternatively study the map as a self-map on $\mathbb{R}$. Strictly speaking, we are then studying another map, which is called the lift (to $\mathbb{R}$) of the original map that was defined on the circle.

 

[Joppy]

Your map from post #1 is equivalent to what is called the "circle map" (or "standard circle map") in the literature. I am saying "equivalent" because the circle map is often written in the slightly different form
\[
\psi_{n+1} = \psi_n + \theta - a\sin{\psi_n} \pmod{2\pi},
\]
so with a minus-sign in front of the non-negative parameter $a$. You can switch between both forms using $\psi_n = \phi_n + \pi$ - that is merely a phase shift - and the fact that $\sin{\phi_n} = \sin(\psi_n - \pi) = - \sin{\psi_n}$.

The circle map is related to Chirikov's standard map, which is two-dimensional. Indeed, the latter map arises as a Poincaré section of a forced (or rather: "kicked") oscillator. Both examples are rather famous and you can find a lot of information about them.

Thanks! Yes I have heard of the circle map actually (I might have even studied it at one point..), but didn't recognize it.

You should not hesitate to ask such questions, because I think they are good questions.

I'm typically extremely stubborn when it comes to asking questions, since the reward of understanding is much more if you do it yourself. But sometimes, especially for elementary things, this can be very inefficient :).

Strictly speaking, we are then studying another map, which is called the lift (to $\mathbb{R}$) of the original map that was defined on the circle.

But the dynamics, or characteristic behaviour of the 'lift map' is the same as our original one right? The only difference being the numerical values of the states (which are instead stretched out across $\mathbb{R}$). However as you suggest, this would no longer be a self-map, and thus not a dynamical system?

 

[Joppy]

However as you suggest, this would no longer be a self-map, and thus not a dynamical system?

Wait, I didn't think that through. The lift map could very well be some function on $\mathbb{R}$ mapped to itself: $f: \mathbb{R} \to \mathbb{R}$, which is still a self map.