Chaos in FF metastability

I've been reading about chaos theory and it occurred to me that
metastability might be a chaotic process.  It seems something as simple
as a damped, driven pendulum (a grandfather clock) can exhibit chaos.
The pendulum can swing stabily if given enough energy initially so that
it is driven and remains above a threshold point.  But if released well
below the threshold it will decay to a static point.  It will exhibit
chaotic behaviour when released near the threshold point, rising and
falling in ampitude and never achieving a stable period, but never
decaying to a static point either.

Does anyone know if a FF driven into metastability meets the criteria
for chaos?  Are there factors that prevent a FF output from being
chaotic even in metastability?

"rickman" <spamgoeshere4@yahoo.com> wrote in message
news:1151941283.587968.307520@v61g2000cwv.googlegroups.com...
> Does anyone know if a FF driven into metastability meets the criteria
> for chaos?  Are there factors that prevent a FF output from being
> chaotic even in metastability?

I would imagine that metastable flip-flops might well be capable of chaotic
behaviour but, like you, I didn't find any published reference for this.

Like a "classical" chaotic system such as a double-pendulum, whether any
chaotic behaviour actually occurs will depend on the physical parameters and
initial conditions of the system. So even if the equations governing the
flip-flop's state permit chaotic behaviour, it might never appear under
"normal" circumstances.

I am not a semiconductor physicist (I'm not *any* kind of physicist) so I
can only speculate! :) Would also be interested to know the answer to this.

    -Ben-

"rickman" <spamgoeshere4@yahoo.com> wrote:

>I've been reading about chaos theory and it occurred to me that
>metastability might be a chaotic process.  It seems something as simple
>as a damped, driven pendulum (a grandfather clock) can exhibit chaos.
>The pendulum can swing stabily if given enough energy initially so that
>it is driven and remains above a threshold point.  But if released well
>below the threshold it will decay to a static point.  It will exhibit
>chaotic behaviour when released near the threshold point, rising and
>falling in ampitude and never achieving a stable period, but never
>decaying to a static point either.
>
>Does anyone know if a FF driven into metastability meets the criteria
>for chaos?  Are there factors that prevent a FF output from being
>chaotic even in metastability?

Thanks for the interesting question.


Metastablility is a whole lot of different things under one name, with
the one thing in common being the output is expected to be digital and
the process for getting there is analog.

Some of the TTL FFs have behavior in metastable cases that might well
be chaotic.  TTL can do all sorts of weird things when metastable,
including oscillating.  To prove that one of those weird things was
both metastable and chaotic could be done if someone could find a
period*3 variation in the oscillation after a metastable event.

Standard IC CMOS FFs on the other hand, I don't think so.  The
internal nodes and output do not oscillate, due to the design, as far
as I understand it.  No oscillation, no chaos.

One could clearly design a FF in any technology that would have
chaotic metastable behavior.  Or a larger circuit with chaotic
behavior that depended on the metastability characteristics of the FF.


This brings up a different sort of questions:

Would there ever be a reason to design a FF to have chaotic metastable
behavior?  I can't think of any, but perhaps I'm missing something.


Are there any useful chaotic circuits that depend on the metastability
characteristics of one or more FFs?

A google search finds this:

http://www.ee.surrey.ac.uk/Personal/D.Jefferies/reliability/reliability.html


--
Phil Hays (Xilinx, but speaking for himself)

Phil Hays wrote:
> "rickman" <spamgoeshere4@yahoo.com> wrote:
> 
> 
>>I've been reading about chaos theory and it occurred to me that
>>metastability might be a chaotic process.  It seems something as simple
>>as a damped, driven pendulum (a grandfather clock) can exhibit chaos.
>>The pendulum can swing stabily if given enough energy initially so that
>>it is driven and remains above a threshold point.  But if released well
>>below the threshold it will decay to a static point.  It will exhibit
>>chaotic behaviour when released near the threshold point, rising and
>>falling in ampitude and never achieving a stable period, but never
>>decaying to a static point either.
>>
>>Does anyone know if a FF driven into metastability meets the criteria
>>for chaos?  Are there factors that prevent a FF output from being
>>chaotic even in metastability?

  Sounds like an ideal target for experimentation :)

ie build a significant number of cells that you make deliberately
metastable, and store the results.
  I'd do this one-at-a-time, to avoid cell coupling effects.
( ie, if you have 512 cells, use 512 edges to get the results )

  Of course, actually hitting the very narrow metastable zone
is going to be non trivial. It would need a 'deliberate seeking'
circuit design.


> Thanks for the interesting question.
> 
> 
> Metastablility is a whole lot of different things under one name, with
> the one thing in common being the output is expected to be digital and
> the process for getting there is analog.
> 
> Some of the TTL FFs have behavior in metastable cases that might well
> be chaotic.  TTL can do all sorts of weird things when metastable,
> including oscillating.  To prove that one of those weird things was
> both metastable and chaotic could be done if someone could find a
> period*3 variation in the oscillation after a metastable event.
> 
> Standard IC CMOS FFs on the other hand, I don't think so.  The
> internal nodes and output do not oscillate, due to the design, as far
> as I understand it.  No oscillation, no chaos.

Yes and no. They are regenerative, and they are also analog,
and there has to be threshold noise in there as well...
So there is chaos in the settling time/final value.

> 
> One could clearly design a FF in any technology that would have
> chaotic metastable behavior.  Or a larger circuit with chaotic
> behavior that depended on the metastability characteristics of the FF.
> 
> 
> This brings up a different sort of questions:
> 
> Would there ever be a reason to design a FF to have chaotic metastable
> behavior?  I can't think of any, but perhaps I'm missing something.

If you mean never settle, that would be very hard.

But there is a wide area of usage for seeding random number generators.
Some devices use local oscillators for this, but they are power hungry,
and less than ideally random.

-jg

Jim Granville wrote:
> Phil Hays wrote:
> > "rickman" <spamgoeshere4@yahoo.com> wrote:
> >
> >
> >>I've been reading about chaos theory and it occurred to me that
> >>metastability might be a chaotic process.  It seems something as simple
> >>as a damped, driven pendulum (a grandfather clock) can exhibit chaos.
> >>The pendulum can swing stabily if given enough energy initially so that
> >>it is driven and remains above a threshold point.  But if released well
> >>below the threshold it will decay to a static point.  It will exhibit
> >>chaotic behaviour when released near the threshold point, rising and
> >>falling in ampitude and never achieving a stable period, but never
> >>decaying to a static point either.
> >>
> >>Does anyone know if a FF driven into metastability meets the criteria
> >>for chaos?  Are there factors that prevent a FF output from being
> >>chaotic even in metastability?
>
>   Sounds like an ideal target for experimentation :)
>
> ie build a significant number of cells that you make deliberately
> metastable, and store the results.
>   I'd do this one-at-a-time, to avoid cell coupling effects.
> ( ie, if you have 512 cells, use 512 edges to get the results )
>
>   Of course, actually hitting the very narrow metastable zone
> is going to be non trivial. It would need a 'deliberate seeking'
> circuit design.
>
>
> > Thanks for the interesting question.
> >
> >
> > Metastablility is a whole lot of different things under one name, with
> > the one thing in common being the output is expected to be digital and
> > the process for getting there is analog.
> >
> > Some of the TTL FFs have behavior in metastable cases that might well
> > be chaotic.  TTL can do all sorts of weird things when metastable,
> > including oscillating.  To prove that one of those weird things was
> > both metastable and chaotic could be done if someone could find a
> > period*3 variation in the oscillation after a metastable event.
> >
> > Standard IC CMOS FFs on the other hand, I don't think so.  The
> > internal nodes and output do not oscillate, due to the design, as far
> > as I understand it.  No oscillation, no chaos.
>
> Yes and no. They are regenerative, and they are also analog,
> and there has to be threshold noise in there as well...
> So there is chaos in the settling time/final value.
>
> >
> > One could clearly design a FF in any technology that would have
> > chaotic metastable behavior.  Or a larger circuit with chaotic
> > behavior that depended on the metastability characteristics of the FF.
> >
> >
> > This brings up a different sort of questions:
> >
> > Would there ever be a reason to design a FF to have chaotic metastable
> > behavior?  I can't think of any, but perhaps I'm missing something.
>
> If you mean never settle, that would be very hard.
>
> But there is a wide area of usage for seeding random number generators.
> Some devices use local oscillators for this, but they are power hungry,
> and less than ideally random.
>
> -jg

When I was so much younger, an elder colleague now retired explained a
little experiment he used to do for the new lads that weren't quite
sure about metastability, must have been early 70s.

Using classic 4000 series RCA cmos devices, he would set up a flop to
go meta stable and keep it there by watching it come out. It needed a
servo loop that would put a voltage onto Din with a D/A converter and
restore the flop back to the middle state if it came out adjusting the
servo to maximize the period. Can't recall how long though. You could
probably repeat that today if you could find real 18v style cosmac
parts.

John Jakson

I have spent some time in analyzing and measuring metastability. Maybe
I can point out some aspects:
I usually explain metastability by flicking a sharp pen so that it
either tips forward or backwards on the table.
There is a very, very small chance of giving it just so much energy
that it ends up standing on its tip with zero end-velocity. How long it
stays there depends on the mass of the pen, the polar momentum,
gravity, and thermal noise.
The electrical equivalent is the cross-coupled latch. It might end up
in the metastable balanced state, and the duration of the stay depends
on the gain-bandwidth product of the latch feedback, plus the system
noise. A CMOS latch is very simple, and has much higher gain x
bandwidth than the old TTL circuits, which could behave quite badly
(oscillating,...)
I have measured modern (well, kind of modern: Virtex-2Pro) latches, and
I found that they very, very rarely stay in the metastable state longer
than 3 ns. To get them to do this requires a very precise timing
relationship between the data and the clock input. The capture window
is substatially less than a femtosecond, which is a millionth of a
nanosecond. I have not found a way to do this in a deterministic way,
but it is quite easy to do it statistically, just use lots of
asynchronous MHz on clock and data, and capture the rare occurance of
en extra delay. See the Xilinx app note XAPP094. These measurements
nicely quantify the metastability risk.
Now back to Chaos...
Peter Alfke

Peter Alfke wrote:
> I have spent some time in analyzing and measuring metastability. Maybe
> I can point out some aspects:
> I usually explain metastability by flicking a sharp pen so that it
> either tips forward or backwards on the table.
> There is a very, very small chance of giving it just so much energy
> that it ends up standing on its tip with zero end-velocity. How long it
> stays there depends on the mass of the pen, the polar momentum,
> gravity, and thermal noise.

You may use this as an analogy, but it is not mathematically
equivalent.  The pencil model is not capable of any form of oscillation
as it is too simple.  But in my original post I explain that something
as simple as a damped, driven pendulum *is* capable of chaotic
operation.  That is why I ask if the "cross-coupled latch" has a
similar type of complexity as the pendulum and is capable of chaotic
operation.

> The electrical equivalent is the cross-coupled latch. It might end up
> in the metastable balanced state, and the duration of the stay depends
> on the gain-bandwidth product of the latch feedback, plus the system
> noise. A CMOS latch is very simple, and has much higher gain x
> bandwidth than the old TTL circuits, which could behave quite badly
> (oscillating,...)

Two mistakes perhaps?  The latch is not equivalent to the pencil and I
believe the noise is not a factor in resolving the metastability.  This
was discussed here once ad nauseum and I never read an explanation that
was other than the seat of the pants on why noise would actually help
to resolve metastability.  Please correct me if I missed something and
am wrong about this.


> I have measured modern (well, kind of modern: Virtex-2Pro) latches, and
> I found that they very, very rarely stay in the metastable state longer
> than 3 ns. To get them to do this requires a very precise timing
> relationship between the data and the clock input. The capture window
> is substatially less than a femtosecond, which is a millionth of a
> nanosecond. I have not found a way to do this in a deterministic way,
> but it is quite easy to do it statistically, just use lots of
> asynchronous MHz on clock and data, and capture the rare occurance of
> en extra delay. See the Xilinx app note XAPP094. These measurements
> nicely quantify the metastability risk.
> Now back to Chaos...

Yes, chaos!  The book I read did not give any real math details on the
requirements for producing chaos, but there were several very simple
examples.  They talk about some very simple examples with three
"attractors" where any point around the borderline between them is
always adjacent to points which will take the system to any of the
three states.  But with only two attractors, a FF may be too simple a
system to show chaos.  Good thing we aren't combining metastability
with multivalued logic. ;^)

Jim Granville <no.spam@designtools.co.nz> wrote:

>Phil Hays wrote:
>> Metastablility is a whole lot of different things under one name, with
>> the one thing in common being the output is expected to be digital and
>> the process for getting there is analog.
>> 
>> Some of the TTL FFs have behavior in metastable cases that might well
>> be chaotic.  TTL can do all sorts of weird things when metastable,
>> including oscillating.  To prove that one of those weird things was
>> both metastable and chaotic could be done if someone could find a
>> period*3 variation in the oscillation after a metastable event.
>> 
>> Standard IC CMOS FFs on the other hand, I don't think so.  The
>> internal nodes and output do not oscillate, due to the design, as far
>> as I understand it.  No oscillation, no chaos.
>
>Yes and no. They are regenerative, and they are also analog,
>and there has to be threshold noise in there as well...
>So there is chaos in the settling time/final value.

Noise and such isn't chaos, in the mathamatical sense of the term.

http://en.wikipedia.org/wiki/Chaos_theory


--
Phil Hays (Xilinx, but speaking for himself)

I am so happy to have removed the mystery from metastability. I do not
want to get chaos back in.
It is the simplicity of the CMOS latch structure that causes
metastability to be so well-behaved and incapable of any oscillation.
Nobody has ever reported oscillation in CMOS latches (but well in TTL
structures that are more complex). Hurray for simplicity...
I think the pen analogy is valid...
Peter Alfke
=====================
rickman wrote:
> Peter Alfke wrote:
> > I have spent some time in analyzing and measuring metastability. Maybe
> > I can point out some aspects:
> > I usually explain metastability by flicking a sharp pen so that it
> > either tips forward or backwards on the table.
> > There is a very, very small chance of giving it just so much energy
> > that it ends up standing on its tip with zero end-velocity. How long it
> > stays there depends on the mass of the pen, the polar momentum,
> > gravity, and thermal noise.
>
> You may use this as an analogy, but it is not mathematically
> equivalent.  The pencil model is not capable of any form of oscillation
> as it is too simple.  But in my original post I explain that something
> as simple as a damped, driven pendulum *is* capable of chaotic
> operation.  That is why I ask if the "cross-coupled latch" has a
> similar type of complexity as the pendulum and is capable of chaotic
> operation.
>
> > The electrical equivalent is the cross-coupled latch. It might end up
> > in the metastable balanced state, and the duration of the stay depends
> > on the gain-bandwidth product of the latch feedback, plus the system
> > noise. A CMOS latch is very simple, and has much higher gain x
> > bandwidth than the old TTL circuits, which could behave quite badly
> > (oscillating,...)
>
> Two mistakes perhaps?  The latch is not equivalent to the pencil and I
> believe the noise is not a factor in resolving the metastability.  This
> was discussed here once ad nauseum and I never read an explanation that
> was other than the seat of the pants on why noise would actually help
> to resolve metastability.  Please correct me if I missed something and
> am wrong about this.
>
>
> > I have measured modern (well, kind of modern: Virtex-2Pro) latches, and
> > I found that they very, very rarely stay in the metastable state longer
> > than 3 ns. To get them to do this requires a very precise timing
> > relationship between the data and the clock input. The capture window
> > is substatially less than a femtosecond, which is a millionth of a
> > nanosecond. I have not found a way to do this in a deterministic way,
> > but it is quite easy to do it statistically, just use lots of
> > asynchronous MHz on clock and data, and capture the rare occurance of
> > en extra delay. See the Xilinx app note XAPP094. These measurements
> > nicely quantify the metastability risk.
> > Now back to Chaos...
>
> Yes, chaos!  The book I read did not give any real math details on the
> requirements for producing chaos, but there were several very simple
> examples.  They talk about some very simple examples with three
> "attractors" where any point around the borderline between them is
> always adjacent to points which will take the system to any of the
> three states.  But with only two attractors, a FF may be too simple a
> system to show chaos.  Good thing we aren't combining metastability
> with multivalued logic. ;^)

"Peter Alfke" <alfke@sbcglobal.net> wrote in message 
news:1151994135.380653.322220@h44g2000cwa.googlegroups.com...
>I am so happy to have removed the mystery from metastability. I do not
> want to get chaos back in.
> It is the simplicity of the CMOS latch structure that causes
> metastability to be so well-behaved and incapable of any oscillation.
> Nobody has ever reported oscillation in CMOS latches (but well in TTL
> structures that are more complex). Hurray for simplicity...
> I think the pen analogy is valid...
> Peter Alfke
> =====================
Hi Peter,
I agree with you, the pen example is rather good. Over the years, people 
like Xilinx have been making the tip of the pen sharper and sharper.
On to chaos: I don't think a system needs oscillation of each component 
comprising the system in order to be chaotic. The resultant system changing 
state is the chaotic thing, rather than the FF oscillating. For example :-
http://en.wikipedia.org/wiki/Logistic_map
In this scenario, each individual doesn't oscillate, they just live or die, 
but the system is chaotic.
Also, what matters is the sensitivity to initial conditions. Again see the 
previous example.
The state of the system has to be dense, but that's possible to build with a 
logic circuit. e.g. all those computer simulations of Lorenz systems. If you 
had a lot of pens, and the flick of each one depended on the outcome of the 
previous pen's flick, I think that could be chaotic, (And not only in the 
mathematical sense if you use fountain pens!) if the flick impulse each time 
is very close to the 'metastable' impulse.
In conclusion, I think an individual CMOS FF given a single metastable clock 
event doesn't comprise a chaotic system, but you can make a chaotic system 
with several of them, no trouble.
Cheers, Syms. 



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