This won’t sound like fun (or Fay!) to start with, but we’ll get there…

I’ve been playing a bit with calculating Lyapunov exponents using Haskell in the previous two articles. My original motivation for looking at these methods was to try to calculate Lyapunov exponents for partial differential equations. That’s pretty hard though, from both a practical and a theoretical point of view (I’m not even 100% sure I know what a Lyapunov exponent means for an infinite-dimensional system). Of course, for computational purposes, we always compute using a finite discretisation of a PDE, and there’s an interest in seeing how Lyapunov exponents calculated for these discretisations converge (or not) as the discretisations become finer. That’s also kind of hard, but we can get started by thinking about computational methods by using ODE systems that we can set up with a variable number of degrees of freedom.

One system that’s been used by a few people^{1} is a driven ring oscillator, like this:

$$\frac{{d}^{2}y}{d{t}^{2}}=-\alpha ({y}^{2}-1)\stackrel{\u0307}{y}-{\omega}^{2}y$$

$$\frac{{d}^{2}{x}_{1}}{d{t}^{2}}=-{d}_{1}\stackrel{\u0307}{{x}_{1}}-\beta [V\u02b9({x}_{1}-{x}_{N})-V\u02b9({x}_{2}-{x}_{1})]+\sigma y$$

$$\frac{{d}^{2}{x}_{i}}{d{t}^{2}}=-{d}_{i}\stackrel{\u0307}{{x}_{i}}-\beta [V\u02b9({x}_{i}-{x}_{i-1})-V\u02b9({x}_{i+1}-{x}_{i})],\phantom{\rule{2em}{0ex}}i=2,\dots ,N$$

Here $y(t)$ is a nonlinear forcing oscillator and the ${x}_{i}$ ($i=1,\dots ,N$) are the “ring” oscillators, with ${x}_{1}$ being forced by the external oscillator $y$. Each of the ring oscillators has a damping term and a term that depends on a potential based on the separation to the neighbouring oscillators in either direction around the ring.

It’s at least partially clear what’s going to happen in a system like this. The forcing is a fairly standard looking nonlinear oscillator and we can pick parameters to ensure that it has a stable limit cycle and provides a periodic forcing to the ring oscillator. The ring oscillator itself obviously has adjacent elements pushing each other around, but what sort of orbits you get isn’t very clear (or not to me, at least).

So, I thought a little bit of a visualisation might be in order. But what to use for the visualisation? It would be nice to have a little JavaScript toy so that I could embed it in a blog post, but that would mean writing JavaScript! There has to be a better way^{2}. Enter Fay, written by Chris Done! Fay is a compiler for a strict subset of Haskell that targets JavaScript. There are a number of other Haskell-to-JS compilers out there, but Fay is distinguished by being very lightweight and having a very nice approach to calling JavaScript functions from Haskell.

Here, I want to talk about the experience of using Fay to build a little toy to explore the behaviour of the ring oscillator described above.

*Note: this uses the HTML5 canvas element and the JavaScript API for manipulating it. If you’re using some crappy old browser (in particular, Internet Explorer 8 or earlier), you won’t see much…*

Here it is (and here is the code, along with a stand-alone HTML page to embed it in). The grey blob and graph trace show the external forcing oscillator, blue shows the ring oscillator that is attached to the external forcing and red shows the other ring oscillators. The graph traces have a mean displacement subtracted for convenience – if you wait long enough, you can see the mean drift of the oscillators around the ring. Have a play…

N =

α =

ω =

d_{even} =

d_{odd} =

β =

σ =

So, how does this work in Fay? Most pure code in Fay is identical to what you would write in Haskell, apart from a couple of caveats related to the fact that Fay is very new and isn’t intended to be a full implementation of Haskell 2010 anyway^{3}.

Here’s an example. We need to integrate the ODE system presented above. To keep things simple, we’ll use a fixed time-step 4th order Runge-Kutta step. Here’s what that looks like in Haskell, for a function `f`

, an initial state `yn`

and a time step `h`

:

```
rk4 :: ([Double] -> [Double]) -> [Double] -> Double -> [Double]
rk4 f yn h = zipWith5 (\y a b c d -> y+(a+2*b+2*c+d)/6) yn k1 k2 k3 k4
where k1 = map mh $ f yn
k2 = map mh $ f (zipWith (+) yn (map half k1))
k3 = map mh $ f (zipWith (+) yn (map half k2))
k4 = map mh $ f (zipWith (+) yn k3)
mh x = h * x
half x = 0.5 * x
```

And the Fay code? Well, it’s just the same! This is the really sweet thing about using something like Fay (or UHC-js or Haste or any of the other Haskell-to-JS compiler approaches) for developing the interactive part of web pages: you write Haskell code, which you can test in GHCi using all the normal methods you would use for developing any other Haskell code. For web applications, there’s another big advantage, which is that code can be shared between the server and client sides of the application, which eliminates the hassle of maintaining both client and server side representations of data structures.

For the ring oscillator, that’s obviously not an issue, since it all lives in the browser, but the representation of our little ODE system in Haskell is quite succinct and clear.

Of course, we can’t hide forever from the fact that we’re living in the swampy impure world of a web browser, so we need some sort of way to talk back and forth to JavaScript. The approach that Fay takes to this foreign function interface (FFI) problem is really nice. Here’s a definition that allows Haskell code to access the JavaScript `Math.sin`

function:

```
sin :: Double -> Double
sin = ffi "Math.sin(%1)"
```

For impure interactions with JavaScript, you can use the `Fay`

monad to impose sequencing. For instance, getting a DOM element from your page by ID is done like this:

```
getElementById :: String -> Fay Element
getElementById = ffi "document['getElementById'](%1)"
```

This can only be called in a monadic context, so ordering of computation is controlled.

The possibilities of this approach are pretty much endless. For the ring oscillator toy, I needed mutable circular buffers to store the data used to draw the graph view. I did this using a simple Haskell record holding information about the size of the buffer, the current number of entries and the position to insert the next entry, along with a JavaScript array to hold the entries:

```
data Array
instance Foreign Array
data Buffer = Buffer { bufSize :: Int
, bufCurSize :: Int
, bufNext :: Int
, bufArr :: Array }
instance Foreign Buffer
newBuf :: Int -> Fay Buffer
newBuf size = do
arr <- newArray size
return $ Buffer size 0 0 arr
bufAdd :: Buffer -> Double -> Fay Buffer
bufAdd (Buffer sz cursz nxt arr) x = do
let cursz' = if cursz < sz then cursz + 1 else sz
setArrayVal arr nxt x
let nxt' = (nxt + 1) `rem` sz
return $ Buffer sz cursz' nxt' arr
bufVal :: Buffer -> Int -> Fay Double
bufVal (Buffer sz cursz nxt arr) i = do
let idx = (if cursz < sz then i else nxt + i) `rem` sz
arrayVal arr idx >>= return
newArray :: Int -> Fay Array
newArray = ffi "new Array(%1)"
setArrayVal :: Array -> Int -> Double -> Fay ()
setArrayVal = ffi "%1[%2]=%3"
arrayVal :: Array -> Int -> Fay Double
arrayVal = ffi "%1[%2]"
```

This stuff is really easy to use, and just takes all of the pain out of developing interactive web pages.

Event handling is nice and simple too. The JavaScript `addEventListener`

method is made accessible as:

```
addEventListener :: Element -> String -> (Event -> Fay Bool) -> Fay ()
addEventListener = ffi "%1['addEventListener'](%2,%3,false)"
```

so that event handlers are written as functions of type `Event -> Fay Bool`

. Because event handlers are just regular Haskell functions, you can partially apply them, in order to pass in extra context needed during event handling from the computation where the event handler is registered. For example, to handle presses of the “go” button, we set up the event handler:

```
addEventListener go "click" $
doGo timerref (animate c cg pref xref gdataref renderrng) framems
```

and implement it like this:

```
doGo :: Ref (Maybe Int) -> Fay () -> Double -> Event -> Fay Bool
doGo tref anim interval _ = do
oldtimer <- readRef tref
case oldtimer of
Nothing -> do
timer <- setInterval anim interval
writeRef tref (Just timer)
Just _ -> return ()
return False
```

Here, `timerref`

is a mutable reference used to communicate the return value from calls to `setInterval`

between different event handlers, the second `anim`

argument to `doGo`

is an action in the `Fay`

monad that deals with animating the views of the model (and which is produced by the `animate`

function, which takes a bunch of arguments that specify the model context for the animation) and the final argument is the number of milliseconds between animation frames. This form of partial application makes setting up event handlers convenient, and replaces the need for global variables for maintaining state.

Take a look at the `render`

and `renderGraph`

functions in `RingOscillator.hs`

. The Fay FFI makes accessing the JavaScript HTML5 canvas API a real breeze. Combined with being able to express computational elements of the render functions in Haskell, this is a real win. Here’s an example from `renderGraph`

, where we need to calculate the positions of the time ticks on the scrolling animated graph and render them:

```
let ticks = takeWhile (\t -> t <= floor (ts + gwwtime)) [floor ts + 1..]
let tickxs = map (\t -> (fromIntegral t - ts) * pxpert) ticks
forM_ (zip ticks tickxs) $ \(t,x) -> do
moveTo cg (x,gwh/2-ticklen/2)
lineTo cg (x,gwh/2+ticklen/2)
let txt = show t
txtw <- measureText cg txt
fillText cg txt (x-txtw/2,gwh/2+2*ticklen) Nothing
```

Here, `ts`

is the current earliest time displayed on the graph, `gwwtime`

is the total width of the graph in time units, `pxpert`

is the number of pixels per time unit, `cg`

is the graphics context used to draw into the graph canvas and `gwh`

is the height of the graph canvas.

It’s Haskell in the browser. It’s great. In more detail:

The FFI is a really good idea. While it’s perhaps not quite as type-safe as the GHC FFI, it is

*really*easy to use. So much of the pain of programming in heterogeneous environments comes from boundary issues that it’s very cool to have it feel so effortless here.It works. It’s an early stage project, so there are bugs, but they’re not hard to track down – the JavaScript code produced by the compiler is quite readable, and it’s easy to write test cases that you can run in a terminal with Node.js. I’ve had a go at fixing a couple of very minor issues in Fay and I was pleasantly surprised at how painless the workflow was.

There is Yesod integration. Again, very new, but it’s starting to look quite feasible to write web apps end-to-end in Haskell. Minimising the client/server divide is potentially quite a big deal, and anything that reduces the number of languages needed to write web apps has to be a good thing (especially if the language we eliminate is JavaScript…).

The JavaScript code that Fay produces isn’t the fastest in the world. But honestly, who cares? I view this as more a proof of concept than anything else. There are various ideas floating around for how to make better backends

^{4}but, for the meantime, Fay is here and it works.

Oh, and did I learn anything about the ring oscillator? Well, a little bit. There’s this canonical drift of the oscillators around the ring, which is probably associated with a zero Lyapunov exponent somewhere. There are interesting “locked” modes that seem to be related to the symmetry of the system (perhaps because the inter-oscillator interaction potential is symmetric?). There are these weird things called “chimera” states that appear in some nonlinear coupled oscillators^{5}, where some of the degrees of freedom become synchronously locked while the rest continue to evolve chaotically – I don’t know if this is a very simple example of that kind of phenomenon.

I’ve not yet convinced myself that it will be possible (or easy anyway) to compute the Lyapunov exponents of this sort of system using the approach of Rangarajan et al. that I used for the smaller systems in the earlier articles. Although in theory, it’s possible to compute a subset of the Lyapunov exponents using a smaller system than the $n(n+3)/2$ degree of freedom system needed for the whole spectrum, it’s not clear how to do that in an efficient way. More thinking required, or perhaps I’ll have to try a cleverer method…

I saw it in T. J. Bridges & S. Reich (2001). Computing Lyapunov exponents on a Stiefel manifold.

*Physica D***156**(3-4), 219-238.↩I also had a bit of a play with Elm before settling on Fay. Elm is neat and I really like the idea of using functional reactive programming, but as a language it’s just different enough from Haskell that I was finding it hard to write things quickly and just similar enough to Haskell that I felt like I ought to be able to write things quickly.↩

In particular, Fay doesn’t have type classes yet, which makes some things tricky to express. Even though there is no

`Monad`

type class though, it does have*a*monad, called`Fay`

, which is used for sequencing operations that have to deal with potentially impure effects in the JavaScript ecosystem. That means that Haskell monad utility functions (`mapM`

, etc.) can be carried across to Fay with only minor changes.↩Some of these ideas sound really quite cheeky: one approach that’s been suggested is to use the front end of GHC to produce a type-annotated AST, then to strip out all the GHC-specific bits, replacing them with Fay-specific things. You could then feed the resulting modified AST back into GHC, let it do all its optimisation magic and take either the resulting Core or STG assembler output and compile

*that*to JavaScript. That’s kind of a scary proposition, since it would involve wallowing in the murky depths of the GHC API, but it would mean that a). you could compile all of GHC’s fancy type system extensions and b). you would get the benefit of all of GHC’s optimisation passes (including things like list fusion).↩D. M. Abrams & S. H. Strogatz (2006). Chimera states in a ring of nonlocally coupled oscillators.

*Int. J. Bifurc. Chaos***16**(1), 21-37.↩

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