Chaotic does not mean ultra-random as some people seem to think. Chaotic systems are fundamentally deterministic dynamical systems that have the property of exhibiting wildly different outcomes as a function of the initial conditions. Small errors in estimation/ measurement of initial conditions can result in unusably large differences in predicted results. If the prediction time scale is much smaller than the Lyapunov time, and initial conditions are measured with precision, predicted outcomes can have very good fidelity , and this is how they're tamed analytically.

"Chaos theory concerns deterministic systems whose behavior can, in principle, be predicted. Chaotic systems are predictable for a while and then 'appear' to become random. The amount of time that the behavior of a chaotic system can be effectively predicted depends on three things: how much uncertainty can be tolerated in the forecast, how accurately its current state can be measured, and a time scale depending on the dynamics of the system, called the Lyapunov time. Some examples of Lyapunov times are: chaotic electrical circuits, about 1 millisecond; weather systems, a few days (unproven); the inner solar system, 4 to 5 million years.[19] In chaotic systems, the uncertainty in a forecast increases exponentially with elapsed time. Hence, mathematically, doubling the forecast time more than squares the proportional uncertainty in the forecast. This means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When meaningful predictions cannot be made, the system appears random."

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

On 6/29/2021 3:10 PM, Fred Bloggs wrote:
> Chaotic does not mean ultra-random as some people seem to think. Chaotic systems are fundamentally deterministic dynamical systems that have the property of exhibiting wildly different outcomes as a function of the initial conditions. Small errors in estimation/ measurement of initial conditions can result in unusably large differences in predicted results. If the prediction time scale is much smaller than the Lyapunov time, and initial conditions are measured with precision, predicted outcomes can have very good fidelity , and this is how they're tamed analytically.
>
> "Chaos theory concerns deterministic systems whose behavior can, in principle, be predicted. Chaotic systems are predictable for a while and then 'appear' to become random. The amount of time that the behavior of a chaotic system can be effectively predicted depends on three things: how much uncertainty can be tolerated in the forecast, how accurately its current state can be measured, and a time scale depending on the dynamics of the system, called the Lyapunov time. Some examples of Lyapunov times are: chaotic electrical circuits, about 1 millisecond; weather systems, a few days (unproven); the inner solar system, 4 to 5 million years.[19] In chaotic systems, the uncertainty in a forecast increases exponentially with elapsed time. Hence, mathematically, doubling the forecast time more than squares the proportional uncertainty in the forecast. This means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When meaningful predictions cannot be made, the system appears random."
>
> https://en.wikipedia.org/wiki/Chaos_theory
>
See Strogatz for a not-mathematically-overbearing introduction; the
basics of linear algebra and first-order ODEs on the calculus side
should be good:
<http://www.stevenstrogatz.com/books/nonlinear-dynamics-and-chaos-with-applications-to-physics-biology-chemistry-and-engineering>

Fred Bloggs wrote:
> Chaotic does not mean ultra-random as some people seem to think.
> Chaotic systems are fundamentally deterministic dynamical systems
> that have the property of exhibiting wildly different outcomes as a
> function of the initial conditions. Small errors in estimation/
> measurement of initial conditions can result in unusably large
> differences in predicted results. If the prediction time scale is
> much smaller than the Lyapunov time, and initial conditions are
> measured with precision, predicted outcomes can have very good
> fidelity , and this is how they're tamed analytically.
>
> "Chaos theory concerns deterministic systems whose behavior can, in
> principle, be predicted. Chaotic systems are predictable for a while
> and then 'appear' to become random. The amount of time that the
> behavior of a chaotic system can be effectively predicted depends on
> three things: how much uncertainty can be tolerated in the forecast,
> how accurately its current state can be measured, and a time scale
> depending on the dynamics of the system, called the Lyapunov time.
> Some examples of Lyapunov times are: chaotic electrical circuits,
> about 1 millisecond; weather systems, a few days (unproven); the
> inner solar system, 4 to 5 million years.[19] In chaotic systems, the
> uncertainty in a forecast increases exponentially with elapsed time.
> Hence, mathematically, doubling the forecast time more than squares
> the proportional uncertainty in the forecast. This means, in
> practice, a meaningful prediction cannot be made over an interval of
> more than two or three times the Lyapunov time. When meaningful
> predictions cannot be made, the system appears random."
>
> https://en.wikipedia.org/wiki/Chaos_theory
>

The kicker is that chaotic systems are very powerful noise amplifiers,
and there is a fundamental limit to the precision of the initial conditions.

If you have a frictionless pool table with perfectly elastic collisions,
and break the rack normally, the position uncertainties of the balls
grow exponentially with time. That is, a small error in aiming makes
both balls go off at a slightly incorrect angle. That angular error
grows linearly as the balls roll, causing the next collisions to be
further off in aim. That causes more angular error, which turns into
more aiming error on the next collisions, and so on.

I don't have it handy, but I recall reading a calculation that showed
that after 30 seconds of this, the amplified Heisenberg uncertainty
became larger than the diameter of a ball, meaning that you could no
longer--even in principle--predict which balls would collide next.

Information is appearing in the universe at a very high rate all the time.

Like many chaotic systems, the motions of these ideal pool balls have
constraints, such as that the total energy of the system is constant.
The balls will bounce around and bounce around forever, but none of them
can ever jump off the table and hit the ceiling. As in epidemics,
exponential growth just applies to the early stages. ;)

Cheers

Phil Hobbs

--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510

http://electrooptical.net
http://hobbs-eo.com

On Tuesday, June 29, 2021 at 4:01:05 PM UTC-4, Phil Hobbs wrote:
> Fred Bloggs wrote:
> > Chaotic does not mean ultra-random as some people seem to think.
> > Chaotic systems are fundamentally deterministic dynamical systems
> > that have the property of exhibiting wildly different outcomes as a
> > function of the initial conditions. Small errors in estimation/
> > measurement of initial conditions can result in unusably large
> > differences in predicted results. If the prediction time scale is
> > much smaller than the Lyapunov time, and initial conditions are
> > measured with precision, predicted outcomes can have very good
> > fidelity , and this is how they're tamed analytically.
> >
> > "Chaos theory concerns deterministic systems whose behavior can, in
> > principle, be predicted. Chaotic systems are predictable for a while
> > and then 'appear' to become random. The amount of time that the
> > behavior of a chaotic system can be effectively predicted depends on
> > three things: how much uncertainty can be tolerated in the forecast,
> > how accurately its current state can be measured, and a time scale
> > depending on the dynamics of the system, called the Lyapunov time.
> > Some examples of Lyapunov times are: chaotic electrical circuits,
> > about 1 millisecond; weather systems, a few days (unproven); the
> > inner solar system, 4 to 5 million years.[19] In chaotic systems, the
> > uncertainty in a forecast increases exponentially with elapsed time.
> > Hence, mathematically, doubling the forecast time more than squares
> > the proportional uncertainty in the forecast. This means, in
> > practice, a meaningful prediction cannot be made over an interval of
> > more than two or three times the Lyapunov time. When meaningful
> > predictions cannot be made, the system appears random."
> >
> > https://en.wikipedia.org/wiki/Chaos_theory
> >
> The kicker is that chaotic systems are very powerful noise amplifiers,
> and there is a fundamental limit to the precision of the initial conditions.
>
> If you have a frictionless pool table with perfectly elastic collisions,
> and break the rack normally, the position uncertainties of the balls
> grow exponentially with time. That is, a small error in aiming makes
> both balls go off at a slightly incorrect angle. That angular error
> grows linearly as the balls roll, causing the next collisions to be
> further off in aim. That causes more angular error, which turns into
> more aiming error on the next collisions, and so on.
>
> I don't have it handy, but I recall reading a calculation that showed
> that after 30 seconds of this, the amplified Heisenberg uncertainty
> became larger than the diameter of a ball, meaning that you could no
> longer--even in principle--predict which balls would collide next.
>
> Information is appearing in the universe at a very high rate all the time..
>
> Like many chaotic systems, the motions of these ideal pool balls have
> constraints, such as that the total energy of the system is constant.
> The balls will bounce around and bounce around forever, but none of them
> can ever jump off the table and hit the ceiling. As in epidemics,
> exponential growth just applies to the early stages. ;)

I'm mainly concerned with short term (3 days) weather prediction. The European model streams real time weather data into the simulation continuously, which I assume means the initial conditions are constantly being reset. There use tens if not hundreds of thousands of real time sensor inputs, terrestrial and satellite. If a butterfly flaps its wings in the Himalayas, they know about it. The price they pay is 24 hour turnaround, but they are consistently dead on accurate. It's a gold standard of weather prediction right now.

>
> Cheers
>
> Phil Hobbs
>
> --
> Dr Philip C D Hobbs
> Principal Consultant
> ElectroOptical Innovations LLC / Hobbs ElectroOptics
> Optics, Electro-optics, Photonics, Analog Electronics
> Briarcliff Manor NY 10510
>
> http://electrooptical.net
> http://hobbs-eo.com

On Tuesday, June 29, 2021 at 3:49:02 PM UTC-4, bitrex wrote:
> On 6/29/2021 3:10 PM, Fred Bloggs wrote:
> > Chaotic does not mean ultra-random as some people seem to think. Chaotic systems are fundamentally deterministic dynamical systems that have the property of exhibiting wildly different outcomes as a function of the initial conditions. Small errors in estimation/ measurement of initial conditions can result in unusably large differences in predicted results. If the prediction time scale is much smaller than the Lyapunov time, and initial conditions are measured with precision, predicted outcomes can have very good fidelity , and this is how they're tamed analytically.
> >
> > "Chaos theory concerns deterministic systems whose behavior can, in principle, be predicted. Chaotic systems are predictable for a while and then 'appear' to become random. The amount of time that the behavior of a chaotic system can be effectively predicted depends on three things: how much uncertainty can be tolerated in the forecast, how accurately its current state can be measured, and a time scale depending on the dynamics of the system, called the Lyapunov time. Some examples of Lyapunov times are: chaotic electrical circuits, about 1 millisecond; weather systems, a few days (unproven); the inner solar system, 4 to 5 million years.[19] In chaotic systems, the uncertainty in a forecast increases exponentially with elapsed time. Hence, mathematically, doubling the forecast time more than squares the proportional uncertainty in the forecast. This means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When meaningful predictions cannot be made, the system appears random."
> >
> > https://en.wikipedia.org/wiki/Chaos_theory
> >
> See Strogatz for a not-mathematically-overbearing introduction; the
> basics of linear algebra and first-order ODEs on the calculus side
> should be good:
>
> <http://www.stevenstrogatz.com/books/nonlinear-dynamics-and-chaos-with-applications-to-physics-biology-chemistry-and-engineering>

That looks good, will take a look at it. All I know about the subject is from a non-linear control text from 15 years ago. Non-linear control dwells on chaotic systems quite a bit too.

Fred Bloggs wrote:
> On Tuesday, June 29, 2021 at 4:01:05 PM UTC-4, Phil Hobbs wrote:
>> Fred Bloggs wrote:
>>> Chaotic does not mean ultra-random as some people seem to think.
>>> Chaotic systems are fundamentally deterministic dynamical
>>> systems that have the property of exhibiting wildly different
>>> outcomes as a function of the initial conditions. Small errors in
>>> estimation/ measurement of initial conditions can result in
>>> unusably large differences in predicted results. If the
>>> prediction time scale is much smaller than the Lyapunov time, and
>>> initial conditions are measured with precision, predicted
>>> outcomes can have very good fidelity , and this is how they're
>>> tamed analytically.
>>>
>>> "Chaos theory concerns deterministic systems whose behavior can,
>>> in principle, be predicted. Chaotic systems are predictable for a
>>> while and then 'appear' to become random. The amount of time that
>>> the behavior of a chaotic system can be effectively predicted
>>> depends on three things: how much uncertainty can be tolerated in
>>> the forecast, how accurately its current state can be measured,
>>> and a time scale depending on the dynamics of the system, called
>>> the Lyapunov time. Some examples of Lyapunov times are: chaotic
>>> electrical circuits, about 1 millisecond; weather systems, a few
>>> days (unproven); the inner solar system, 4 to 5 million
>>> years.[19] In chaotic systems, the uncertainty in a forecast
>>> increases exponentially with elapsed time. Hence, mathematically,
>>> doubling the forecast time more than squares the proportional
>>> uncertainty in the forecast. This means, in practice, a
>>> meaningful prediction cannot be made over an interval of more
>>> than two or three times the Lyapunov time. When meaningful
>>> predictions cannot be made, the system appears random."
>>>
>>> https://en.wikipedia.org/wiki/Chaos_theory
>>>
>> The kicker is that chaotic systems are very powerful noise
>> amplifiers, and there is a fundamental limit to the precision of
>> the initial conditions.
>>
>> If you have a frictionless pool table with perfectly elastic
>> collisions, and break the rack normally, the position uncertainties
>> of the balls grow exponentially with time. That is, a small error
>> in aiming makes both balls go off at a slightly incorrect angle.
>> That angular error grows linearly as the balls roll, causing the
>> next collisions to be further off in aim. That causes more angular
>> error, which turns into more aiming error on the next collisions,
>> and so on.
>>
>> I don't have it handy, but I recall reading a calculation that
>> showed that after 30 seconds of this, the amplified Heisenberg
>> uncertainty became larger than the diameter of a ball, meaning that
>> you could no longer--even in principle--predict which balls would
>> collide next.
>>
>> Information is appearing in the universe at a very high rate all
>> the time.
>>
>> Like many chaotic systems, the motions of these ideal pool balls
>> have constraints, such as that the total energy of the system is
>> constant. The balls will bounce around and bounce around forever,
>> but none of them can ever jump off the table and hit the ceiling.
>> As in epidemics, exponential growth just applies to the early
>> stages. ;)
>
> I'm mainly concerned with short term (3 days) weather prediction. The
> European model streams real time weather data into the simulation
> continuously, which I assume means the initial conditions are
> constantly being reset. There use tens if not hundreds of thousands
> of real time sensor inputs, terrestrial and satellite. If a butterfly
> flaps its wings in the Himalayas, they know about it. The price they
> pay is 24 hour turnaround, but they are consistently dead on
> accurate. It's a gold standard of weather prediction right now.

Okay, but your thread title is completely misleading--if a system is
deterministic, it doesn't stop after three days.

Cheers

Phil Hobbs

--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510

http://electrooptical.net
http://hobbs-eo.com

On Tuesday, June 29, 2021 at 5:55:22 PM UTC-4, Phil Hobbs wrote:
> Fred Bloggs wrote:
> >
> > I'm mainly concerned with short term (3 days) weather prediction. The
> > European model streams real time weather data into the simulation
> > continuously, which I assume means the initial conditions are
> > constantly being reset. There use tens if not hundreds of thousands
> > of real time sensor inputs, terrestrial and satellite. If a butterfly
> > flaps its wings in the Himalayas, they know about it. The price they
> > pay is 24 hour turnaround, but they are consistently dead on
> > accurate. It's a gold standard of weather prediction right now.
> Okay, but your thread title is completely misleading--if a system is
> deterministic, it doesn't stop after three days.

You do say some of the strangest things.

--

Rick C.

- Get 1,000 miles of free Supercharging
- Tesla referral code - https://ts.la/richard11209

On Tue, 29 Jun 2021 12:10:45 -0700 (PDT), Fred Bloggs
<bloggs.fredbloggs.fred@gmail.com> wrote:

>Chaotic does not mean ultra-random as some people seem to think. Chaotic systems are fundamentally deterministic dynamical systems that have the property of exhibiting wildly different outcomes as a function of the initial conditions. Small errors in estimation/ measurement of initial conditions can result in unusably large differences in predicted results. If the prediction time scale is much smaller than the Lyapunov time, and initial conditions are measured with precision, predicted outcomes can have very good fidelity , and this is how they're tamed analytically.
>
>"Chaos theory concerns deterministic systems whose behavior can, in principle, be predicted. Chaotic systems are predictable for a while and then 'appear' to become random. The amount of time that the behavior of a chaotic system can be effectively predicted depends on three things: how much uncertainty can be tolerated in the forecast, how accurately its current state can be measured, and a time scale depending on the dynamics of the system, called the Lyapunov time. Some examples of Lyapunov times are: chaotic electrical circuits, about 1 millisecond; weather systems, a few days (unproven); the inner solar system, 4 to 5 million years.[19] In chaotic systems, the uncertainty in a forecast increases exponentially with elapsed time. Hence, mathematically, doubling the forecast time more than squares the proportional uncertainty in the forecast. This means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When
>meaningful predictions cannot be made, the system appears random."
>
>https://en.wikipedia.org/wiki/Chaos_theory

In real-life systems, social/economic/climactic, we have only crude
knowledge of the element behavior, the couplings, or the initial
state. People and clouds don't behave like planets.

People too used to power think they can understand, hence should
control, societies and economics. They usually screw things up.

--

John Larkin Highland Technology, Inc

The best designs are necessarily accidental.

On Tue, 29 Jun 2021 16:00:56 -0400, Phil Hobbs
<pcdhSpamMeSenseless@electrooptical.net> wrote:

>Fred Bloggs wrote:
>> Chaotic does not mean ultra-random as some people seem to think.
>> Chaotic systems are fundamentally deterministic dynamical systems
>> that have the property of exhibiting wildly different outcomes as a
>> function of the initial conditions. Small errors in estimation/
>> measurement of initial conditions can result in unusably large
>> differences in predicted results. If the prediction time scale is
>> much smaller than the Lyapunov time, and initial conditions are
>> measured with precision, predicted outcomes can have very good
>> fidelity , and this is how they're tamed analytically.
>>
>> "Chaos theory concerns deterministic systems whose behavior can, in
>> principle, be predicted. Chaotic systems are predictable for a while
>> and then 'appear' to become random. The amount of time that the
>> behavior of a chaotic system can be effectively predicted depends on
>> three things: how much uncertainty can be tolerated in the forecast,
>> how accurately its current state can be measured, and a time scale
>> depending on the dynamics of the system, called the Lyapunov time.
>> Some examples of Lyapunov times are: chaotic electrical circuits,
>> about 1 millisecond; weather systems, a few days (unproven); the
>> inner solar system, 4 to 5 million years.[19] In chaotic systems, the
>> uncertainty in a forecast increases exponentially with elapsed time.
>> Hence, mathematically, doubling the forecast time more than squares
>> the proportional uncertainty in the forecast. This means, in
>> practice, a meaningful prediction cannot be made over an interval of
>> more than two or three times the Lyapunov time. When meaningful
>> predictions cannot be made, the system appears random."
>>
>> https://en.wikipedia.org/wiki/Chaos_theory
>>
>
>The kicker is that chaotic systems are very powerful noise amplifiers,
>and there is a fundamental limit to the precision of the initial conditions.
>
>If you have a frictionless pool table with perfectly elastic collisions,
>and break the rack normally, the position uncertainties of the balls
>grow exponentially with time. That is, a small error in aiming makes
>both balls go off at a slightly incorrect angle. That angular error
>grows linearly as the balls roll, causing the next collisions to be
>further off in aim. That causes more angular error, which turns into
>more aiming error on the next collisions, and so on.
>
>I don't have it handy, but I recall reading a calculation that showed
>that after 30 seconds of this, the amplified Heisenberg uncertainty
>became larger than the diameter of a ball, meaning that you could no
>longer--even in principle--predict which balls would collide next.
>
>Information is appearing in the universe at a very high rate all the time.
>
>Like many chaotic systems, the motions of these ideal pool balls have
>constraints, such as that the total energy of the system is constant.
>The balls will bounce around and bounce around forever, but none of them
>can ever jump off the table and hit the ceiling.

Couldn't that happen? Like five balls converge on one and transfer all
their momentum. That doesn't seem to violate any conservation rules.

--

John Larkin Highland Technology, Inc

The best designs are necessarily accidental.

On Tue, 29 Jun 2021 13:31:11 -0700 (PDT), Fred Bloggs
<bloggs.fredbloggs.fred@gmail.com> wrote:

>On Tuesday, June 29, 2021 at 4:01:05 PM UTC-4, Phil Hobbs wrote:
>> Fred Bloggs wrote:
>> > Chaotic does not mean ultra-random as some people seem to think.
>> > Chaotic systems are fundamentally deterministic dynamical systems
>> > that have the property of exhibiting wildly different outcomes as a
>> > function of the initial conditions. Small errors in estimation/
>> > measurement of initial conditions can result in unusably large
>> > differences in predicted results. If the prediction time scale is
>> > much smaller than the Lyapunov time, and initial conditions are
>> > measured with precision, predicted outcomes can have very good
>> > fidelity , and this is how they're tamed analytically.
>> >
>> > "Chaos theory concerns deterministic systems whose behavior can, in
>> > principle, be predicted. Chaotic systems are predictable for a while
>> > and then 'appear' to become random. The amount of time that the
>> > behavior of a chaotic system can be effectively predicted depends on
>> > three things: how much uncertainty can be tolerated in the forecast,
>> > how accurately its current state can be measured, and a time scale
>> > depending on the dynamics of the system, called the Lyapunov time.
>> > Some examples of Lyapunov times are: chaotic electrical circuits,
>> > about 1 millisecond; weather systems, a few days (unproven); the
>> > inner solar system, 4 to 5 million years.[19] In chaotic systems, the
>> > uncertainty in a forecast increases exponentially with elapsed time.
>> > Hence, mathematically, doubling the forecast time more than squares
>> > the proportional uncertainty in the forecast. This means, in
>> > practice, a meaningful prediction cannot be made over an interval of
>> > more than two or three times the Lyapunov time. When meaningful
>> > predictions cannot be made, the system appears random."
>> >
>> > https://en.wikipedia.org/wiki/Chaos_theory
>> >
>> The kicker is that chaotic systems are very powerful noise amplifiers,
>> and there is a fundamental limit to the precision of the initial conditions.
>>
>> If you have a frictionless pool table with perfectly elastic collisions,
>> and break the rack normally, the position uncertainties of the balls
>> grow exponentially with time. That is, a small error in aiming makes
>> both balls go off at a slightly incorrect angle. That angular error
>> grows linearly as the balls roll, causing the next collisions to be
>> further off in aim. That causes more angular error, which turns into
>> more aiming error on the next collisions, and so on.
>>
>> I don't have it handy, but I recall reading a calculation that showed
>> that after 30 seconds of this, the amplified Heisenberg uncertainty
>> became larger than the diameter of a ball, meaning that you could no
>> longer--even in principle--predict which balls would collide next.
>>
>> Information is appearing in the universe at a very high rate all the time.
>>
>> Like many chaotic systems, the motions of these ideal pool balls have
>> constraints, such as that the total energy of the system is constant.
>> The balls will bounce around and bounce around forever, but none of them
>> can ever jump off the table and hit the ceiling. As in epidemics,
>> exponential growth just applies to the early stages. ;)
>
>I'm mainly concerned with short term (3 days) weather prediction. The European model streams real time weather data into the simulation continuously, which I assume means the initial conditions are constantly being reset. There use tens if not hundreds of thousands of real time sensor inputs, terrestrial and satellite. If a butterfly flaps its wings in the Himalayas, they know about it. The price they pay is 24 hour turnaround, but they are consistently dead on accurate. It's a gold standard of weather prediction right now.
>

Sure. Look at which way the wind is blowing, check the temperature
upwind from here, predict.

That doesn't scale.

--

John Larkin Highland Technology, Inc

The best designs are necessarily accidental.

On Tuesday, June 29, 2021 at 3:24:11 PM UTC-7, jla...@highlandsniptechnology.com wrote:
> On Tue, 29 Jun 2021 12:10:45 -0700 (PDT), Fred Bloggs
> <bloggs.fred...@gmail.com> wrote:
>
> >Chaotic does not mean ultra-random as some people seem to think. Chaotic systems are fundamentally deterministic dynamical systems that have the property of exhibiting wildly different outcomes as a function of the initial conditions.

> In real-life systems, social/economic/climactic, we have only crude
> knowledge of the element behavior, the couplings, or the initial
> state. People and clouds don't behave like planets.

There's lots of real-life systems we DO have knowledge of. Nothing
about 'social/economic/climactic' subsets of real-life is distinctive in this respect.

> People too used to power think they can understand, hence should
> control, societies and economics. They usually screw things up.

So, you aren't a monarchist? Inherited entitlements of political power is, indeed,
not generally held in high regard nowadays.

On Tuesday, June 29, 2021 at 3:27:15 PM UTC-7, jla...@highlandsniptechnology.com wrote:
> On Tue, 29 Jun 2021 16:00:56 -0400, Phil Hobbs
> <pcdhSpamM...@electrooptical.net> wrote:

> >Like many chaotic systems, the motions of these ideal pool balls have
> >constraints, such as that the total energy of the system is constant.
> >The balls will bounce around and bounce around forever, but none of them
> >can ever jump off the table and hit the ceiling.

> Couldn't that happen? Like five balls converge on one and transfer all
> their momentum. That doesn't seem to violate any conservation rules.

For momentum, each component is separately conserved. Five balls moving
in the X-Y plane don't have Z-component momentum.

jlarkin@highlandsniptechnology.com wrote:
> On Tue, 29 Jun 2021 16:00:56 -0400, Phil Hobbs
> <pcdhSpamMeSenseless@electrooptical.net> wrote:
>
>> Fred Bloggs wrote:
>>> Chaotic does not mean ultra-random as some people seem to think.
>>> Chaotic systems are fundamentally deterministic dynamical systems
>>> that have the property of exhibiting wildly different outcomes as a
>>> function of the initial conditions. Small errors in estimation/
>>> measurement of initial conditions can result in unusably large
>>> differences in predicted results. If the prediction time scale is
>>> much smaller than the Lyapunov time, and initial conditions are
>>> measured with precision, predicted outcomes can have very good
>>> fidelity , and this is how they're tamed analytically.
>>>
>>> "Chaos theory concerns deterministic systems whose behavior can, in
>>> principle, be predicted. Chaotic systems are predictable for a while
>>> and then 'appear' to become random. The amount of time that the
>>> behavior of a chaotic system can be effectively predicted depends on
>>> three things: how much uncertainty can be tolerated in the forecast,
>>> how accurately its current state can be measured, and a time scale
>>> depending on the dynamics of the system, called the Lyapunov time.
>>> Some examples of Lyapunov times are: chaotic electrical circuits,
>>> about 1 millisecond; weather systems, a few days (unproven); the
>>> inner solar system, 4 to 5 million years.[19] In chaotic systems, the
>>> uncertainty in a forecast increases exponentially with elapsed time.
>>> Hence, mathematically, doubling the forecast time more than squares
>>> the proportional uncertainty in the forecast. This means, in
>>> practice, a meaningful prediction cannot be made over an interval of
>>> more than two or three times the Lyapunov time. When meaningful
>>> predictions cannot be made, the system appears random."
>>>
>>> https://en.wikipedia.org/wiki/Chaos_theory
>>>
>>
>> The kicker is that chaotic systems are very powerful noise amplifiers,
>> and there is a fundamental limit to the precision of the initial conditions.
>>
>> If you have a frictionless pool table with perfectly elastic collisions,
>> and break the rack normally, the position uncertainties of the balls
>> grow exponentially with time. That is, a small error in aiming makes
>> both balls go off at a slightly incorrect angle. That angular error
>> grows linearly as the balls roll, causing the next collisions to be
>> further off in aim. That causes more angular error, which turns into
>> more aiming error on the next collisions, and so on.
>>
>> I don't have it handy, but I recall reading a calculation that showed
>> that after 30 seconds of this, the amplified Heisenberg uncertainty
>> became larger than the diameter of a ball, meaning that you could no
>> longer--even in principle--predict which balls would collide next.
>>
>> Information is appearing in the universe at a very high rate all the time.
>>
>> Like many chaotic systems, the motions of these ideal pool balls have
>> constraints, such as that the total energy of the system is constant.
>> The balls will bounce around and bounce around forever, but none of them
>> can ever jump off the table and hit the ceiling.
>
> Couldn't that happen? Like five balls converge on one and transfer all
> their momentum. That doesn't seem to violate any conservation rules.

Not if the cue ball didn't start out with enough energy to do that.

Other chaotic systems, such as globular star clusters, can eject parts
at any speed.

Cheers

Phil Hobbs

--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510

http://electrooptical.net
http://hobbs-eo.com

whit3rd wrote:
> On Tuesday, June 29, 2021 at 3:27:15 PM UTC-7, jla...@highlandsniptechnology.com wrote:
>> On Tue, 29 Jun 2021 16:00:56 -0400, Phil Hobbs
>> <pcdhSpamM...@electrooptical.net> wrote:
>
>>> Like many chaotic systems, the motions of these ideal pool balls have
>>> constraints, such as that the total energy of the system is constant.
>>> The balls will bounce around and bounce around forever, but none of them
>>> can ever jump off the table and hit the ceiling.
>
>> Couldn't that happen? Like five balls converge on one and transfer all
>> their momentum. That doesn't seem to violate any conservation rules.
>
> For momentum, each component is separately conserved. Five balls moving
> in the X-Y plane don't have Z-component momentum.

'T'aint a momentum issue. The table can supply any Z-component
required, and of course the initial momentum of the cue ball is enough
to make it bounce--as any beginning pool player knows. ;)

Cheers

Phil Hobbs

--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510

http://electrooptical.net
http://hobbs-eo.com

On Tuesday, June 29, 2021 at 4:41:59 PM UTC-7, Phil Hobbs wrote:
> whit3rd wrote:
> > On Tuesday, June 29, 2021 at 3:27:15 PM UTC-7, jla...@highlandsniptechnology.com wrote:
> >> On Tue, 29 Jun 2021 16:00:56 -0400, Phil Hobbs
> >> <pcdhSpamM...@electrooptical.net> wrote:

> >>> The balls will bounce around and bounce around forever, but none of them
> >>> can ever jump off the table and hit the ceiling.
> >
> >> Couldn't that happen? Like five balls converge on one and transfer all
> >> their momentum. That doesn't seem to violate any conservation rules.
> >
> > For momentum, each component is separately conserved. Five balls moving
> > in the X-Y plane don't have Z-component momentum

> 'T'aint a momentum issue. The table can supply any Z-component
> required, and of course the initial momentum of the cue ball is enough
> to make it bounce--as any beginning pool player knows. ;)

If you use a cue ball with a different diameter than the other balls (not uncommon in
quarter-to-play tables) then the table can supply momentum. Otherwise, it takes a fierce
spin transfer at the cue or some kind of miracle to make that bounce.

whit3rd wrote:
> On Tuesday, June 29, 2021 at 4:41:59 PM UTC-7, Phil Hobbs wrote:
>> whit3rd wrote:
>>> On Tuesday, June 29, 2021 at 3:27:15 PM UTC-7, jla...@highlandsniptechnology.com wrote:
>>>> On Tue, 29 Jun 2021 16:00:56 -0400, Phil Hobbs
>>>> <pcdhSpamM...@electrooptical.net> wrote:
>
>>>>> The balls will bounce around and bounce around forever, but none of them
>>>>> can ever jump off the table and hit the ceiling.
>>>
>>>> Couldn't that happen? Like five balls converge on one and transfer all
>>>> their momentum. That doesn't seem to violate any conservation rules.
>>>
>>> For momentum, each component is separately conserved. Five balls moving
>>> in the X-Y plane don't have Z-component momentum
>
>> 'T'aint a momentum issue. The table can supply any Z-component
>> required, and of course the initial momentum of the cue ball is enough
>> to make it bounce--as any beginning pool player knows. ;)
>
> If you use a cue ball with a different diameter than the other balls (not uncommon in
> quarter-to-play tables) then the table can supply momentum. Otherwise, it takes a fierce
> spin transfer at the cue or some kind of miracle to make that bounce.
>

Soooooo? It has an infinite number of chances, and the compliance of
the felt isn't necessarily zero.

Cheers

Phil Hobbs

--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510

http://electrooptical.net
http://hobbs-eo.com

On Tue, 29 Jun 2021 19:38:31 -0400, Phil Hobbs
<pcdhSpamMeSenseless@electrooptical.net> wrote:

>jlarkin@highlandsniptechnology.com wrote:
>> On Tue, 29 Jun 2021 16:00:56 -0400, Phil Hobbs
>> <pcdhSpamMeSenseless@electrooptical.net> wrote:
>>
>>> Fred Bloggs wrote:
>>>> Chaotic does not mean ultra-random as some people seem to think.
>>>> Chaotic systems are fundamentally deterministic dynamical systems
>>>> that have the property of exhibiting wildly different outcomes as a
>>>> function of the initial conditions. Small errors in estimation/
>>>> measurement of initial conditions can result in unusably large
>>>> differences in predicted results. If the prediction time scale is
>>>> much smaller than the Lyapunov time, and initial conditions are
>>>> measured with precision, predicted outcomes can have very good
>>>> fidelity , and this is how they're tamed analytically.
>>>>
>>>> "Chaos theory concerns deterministic systems whose behavior can, in
>>>> principle, be predicted. Chaotic systems are predictable for a while
>>>> and then 'appear' to become random. The amount of time that the
>>>> behavior of a chaotic system can be effectively predicted depends on
>>>> three things: how much uncertainty can be tolerated in the forecast,
>>>> how accurately its current state can be measured, and a time scale
>>>> depending on the dynamics of the system, called the Lyapunov time.
>>>> Some examples of Lyapunov times are: chaotic electrical circuits,
>>>> about 1 millisecond; weather systems, a few days (unproven); the
>>>> inner solar system, 4 to 5 million years.[19] In chaotic systems, the
>>>> uncertainty in a forecast increases exponentially with elapsed time.
>>>> Hence, mathematically, doubling the forecast time more than squares
>>>> the proportional uncertainty in the forecast. This means, in
>>>> practice, a meaningful prediction cannot be made over an interval of
>>>> more than two or three times the Lyapunov time. When meaningful
>>>> predictions cannot be made, the system appears random."
>>>>
>>>> https://en.wikipedia.org/wiki/Chaos_theory
>>>>
>>>
>>> The kicker is that chaotic systems are very powerful noise amplifiers,
>>> and there is a fundamental limit to the precision of the initial conditions.
>>>
>>> If you have a frictionless pool table with perfectly elastic collisions,
>>> and break the rack normally, the position uncertainties of the balls
>>> grow exponentially with time. That is, a small error in aiming makes
>>> both balls go off at a slightly incorrect angle. That angular error
>>> grows linearly as the balls roll, causing the next collisions to be
>>> further off in aim. That causes more angular error, which turns into
>>> more aiming error on the next collisions, and so on.
>>>
>>> I don't have it handy, but I recall reading a calculation that showed
>>> that after 30 seconds of this, the amplified Heisenberg uncertainty
>>> became larger than the diameter of a ball, meaning that you could no
>>> longer--even in principle--predict which balls would collide next.
>>>
>>> Information is appearing in the universe at a very high rate all the time.
>>>
>>> Like many chaotic systems, the motions of these ideal pool balls have
>>> constraints, such as that the total energy of the system is constant.
>>> The balls will bounce around and bounce around forever, but none of them
>>> can ever jump off the table and hit the ceiling.
>>
>> Couldn't that happen? Like five balls converge on one and transfer all
>> their momentum. That doesn't seem to violate any conservation rules.
>
>Not if the cue ball didn't start out with enough energy to do that.

Oh don't get technical. I hate it when people get technical.

--

John Larkin Highland Technology, Inc

The best designs are necessarily accidental.

On Tuesday, June 29, 2021 at 4:01:05 PM UTC-4, Phil Hobbs wrote:
> Fred Bloggs wrote:
> > Chaotic does not mean ultra-random as some people seem to think.
> > Chaotic systems are fundamentally deterministic dynamical systems
> > that have the property of exhibiting wildly different outcomes as a
> > function of the initial conditions. Small errors in estimation/
> > measurement of initial conditions can result in unusably large
> > differences in predicted results. If the prediction time scale is
> > much smaller than the Lyapunov time, and initial conditions are
> > measured with precision, predicted outcomes can have very good
> > fidelity , and this is how they're tamed analytically.
> >
> > "Chaos theory concerns deterministic systems whose behavior can, in
> > principle, be predicted. Chaotic systems are predictable for a while
> > and then 'appear' to become random. The amount of time that the
> > behavior of a chaotic system can be effectively predicted depends on
> > three things: how much uncertainty can be tolerated in the forecast,
> > how accurately its current state can be measured, and a time scale
> > depending on the dynamics of the system, called the Lyapunov time.
> > Some examples of Lyapunov times are: chaotic electrical circuits,
> > about 1 millisecond; weather systems, a few days (unproven); the
> > inner solar system, 4 to 5 million years.[19] In chaotic systems, the
> > uncertainty in a forecast increases exponentially with elapsed time.
> > Hence, mathematically, doubling the forecast time more than squares
> > the proportional uncertainty in the forecast. This means, in
> > practice, a meaningful prediction cannot be made over an interval of
> > more than two or three times the Lyapunov time. When meaningful
> > predictions cannot be made, the system appears random."
> >
> > https://en.wikipedia.org/wiki/Chaos_theory
> >
> The kicker is that chaotic systems are very powerful noise amplifiers,
> and there is a fundamental limit to the precision of the initial conditions.
>
> If you have a frictionless pool table with perfectly elastic collisions,
> and break the rack normally, the position uncertainties of the balls
> grow exponentially with time. That is, a small error in aiming makes
> both balls go off at a slightly incorrect angle. That angular error
> grows linearly as the balls roll, causing the next collisions to be
> further off in aim. That causes more angular error, which turns into
> more aiming error on the next collisions, and so on.
>
> I don't have it handy, but I recall reading a calculation that showed
> that after 30 seconds of this, the amplified Heisenberg uncertainty
> became larger than the diameter of a ball, meaning that you could no
> longer--even in principle--predict which balls would collide next.
>
> Information is appearing in the universe at a very high rate all the time.
>
> Like many chaotic systems, the motions of these ideal pool balls have
> constraints, such as that the total energy of the system is constant.
> The balls will bounce around and bounce around forever, but none of them
> can ever jump off the table and hit the ceiling. As in epidemics,
> exponential growth just applies to the early stages. ;)
>
> Cheers
>
> Phil Hobbs
>
> --
> Dr Philip C D Hobbs
> Principal Consultant
> ElectroOptical Innovations LLC / Hobbs ElectroOptics
> Optics, Electro-optics, Photonics, Analog Electronics
> Briarcliff Manor NY 10510
>
> http://electrooptical.net
> http://hobbs-eo.com
Right. I once made this analog chaos circuit. (copied from P. Horowitz.. Lorenz butterfly thingie)
with things set 'optimally' you could sometimes see a third / fourth period doubling. Till the noise in
the circuit (mostly from multipliers) washed out the signal. With a computer you can watch period doubling
to whatever accuracy you like.

George H.

Fred Bloggs wrote:
> Chaotic does not mean ultra-random as some people seem to think.
> Chaotic systems are fundamentally deterministic dynamical systems
> that have the property of exhibiting wildly different outcomes as a
> function of the initial conditions. Small errors in estimation/
> measurement of initial conditions can result in unusably large
> differences in predicted results. If the prediction time scale is
> much smaller than the Lyapunov time, and initial conditions are
> measured with precision, predicted outcomes can have very good
> fidelity , and this is how they're tamed analytically.
>
> "Chaos theory concerns deterministic systems whose behavior can, in
> principle, be predicted. Chaotic systems are predictable for a while
> and then 'appear' to become random. The amount of time that the
> behavior of a chaotic system can be effectively predicted depends on
> three things: how much uncertainty can be tolerated in the forecast,
> how accurately its current state can be measured, and a time scale
> depending on the dynamics of the system, called the Lyapunov time.
> Some examples of Lyapunov times are: chaotic electrical circuits,
> about 1 millisecond; weather systems, a few days (unproven); the
> inner solar system, 4 to 5 million years.[19] In chaotic systems, the
> uncertainty in a forecast increases exponentially with elapsed time.
> Hence, mathematically, doubling the forecast time more than squares
> the proportional uncertainty in the forecast. This means, in
> practice, a meaningful prediction cannot be made over an interval of
> more than two or three times the Lyapunov time. When meaningful
> predictions cannot be made, the system appears random."
>
> https://en.wikipedia.org/wiki/Chaos_theory

I was attracted by the subject, but I find it strange that this is
considered something new.

--
Defund the Thought Police

On 29/06/2021 23:29, jlarkin@highlandsniptechnology.com wrote:

<snip>
>
> Sure. Look at which way the wind is blowing, check the temperature
> upwind from here, predict.

Here, during the Wimbledon tennis tournament, the main court has a
sliding roof which is closed when rain approaches, a procedure which
takes a few minutes.

Every year, someone writes in to a paper complaining that the
authorities have access to much better weather forecasts than are
available to the general public (or the man on the Clapham Omnibus, the
156 in this case).

--
Cheers
Clive

On Thursday, July 1, 2021 at 12:23:02 AM UTC+10, Clive Arthur wrote:
> On 29/06/2021 23:29, jla...@highlandsniptechnology.com wrote:
>
> <snip>
> >
> > Sure. Look at which way the wind is blowing, check the temperature
> > upwind from here, predict.
> Here, during the Wimbledon tennis tournament, the main court has a
> sliding roof which is closed when rain approaches, a procedure which
> takes a few minutes.
>
> Every year, someone writes in to a paper complaining that the
> authorities have access to much better weather forecasts than are
> available to the general public (or the man on the Clapham Omnibus, the
> 156 in this case).

That's a bit silly. The Wimbledon centre court is specific and relatively small area. Weather radar can seem rainstorms moving across the city and provide a forecast of when the next rainstorm is going to hit the centre-court at Wimbleton.

The general public wants to know when a rainstorm is going to hit them, wherever they are.

Mobile phone areas in cities are pretty compact - if each cell tower pushed out it's own forecast to all the phones it was serving at a particular instant, it could do almost as well.

--
Bill Sloman, Sydney

On Wed, 30 Jun 2021 15:22:55 +0100, Clive Arthur
<clive@nowaytoday.co.uk> wrote:

>On 29/06/2021 23:29, jlarkin@highlandsniptechnology.com wrote:
>
><snip>
>>
>> Sure. Look at which way the wind is blowing, check the temperature
>> upwind from here, predict.
>
>Here, during the Wimbledon tennis tournament, the main court has a
>sliding roof which is closed when rain approaches, a procedure which
>takes a few minutes.
>
>Every year, someone writes in to a paper complaining that the
>authorities have access to much better weather forecasts than are
>available to the general public (or the man on the Clapham Omnibus, the
>156 in this case).

We have open-top streetcars too.

https://www.dropbox.com/s/aj91qofbd4fk5cf/Streetcar.jpg?raw=1

I don't know who that Hayes guy is.

--

John Larkin Highland Technology, Inc

The best designs are necessarily accidental.

On 30/06/2021 15:44, Bill Sloman wrote:
> On Thursday, July 1, 2021 at 12:23:02 AM UTC+10, Clive Arthur wrote:
>> On 29/06/2021 23:29, jla...@highlandsniptechnology.com wrote:
>>
>> <snip>
>>>
>>> Sure. Look at which way the wind is blowing, check the temperature
>>> upwind from here, predict.
>> Here, during the Wimbledon tennis tournament, the main court has a
>> sliding roof which is closed when rain approaches, a procedure which
>> takes a few minutes.
>>
>> Every year, someone writes in to a paper complaining that the
>> authorities have access to much better weather forecasts than are
>> available to the general public (or the man on the Clapham Omnibus, the
>> 156 in this case).
>
> That's a bit silly. The Wimbledon centre court is specific and relatively small area. Weather radar can seem rainstorms moving across the city and provide a forecast of when the next rainstorm is going to hit the centre-court at Wimbleton.
>
> The general public wants to know when a rainstorm is going to hit them, wherever they are.
>
> Mobile phone areas in cities are pretty compact - if each cell tower pushed out it's own forecast to all the phones it was serving at a particular instant, it could do almost as well.

That's a pretty good idea - add a simple weather station atop each mast
and you could warn people to get the washing in a little in advance.

--
Cheers
Clive

On Wed, 30 Jun 2021 15:54:16 +0100, Clive Arthur
<clive@nowaytoday.co.uk> wrote:

>On 30/06/2021 15:44, Bill Sloman wrote:
>> On Thursday, July 1, 2021 at 12:23:02 AM UTC+10, Clive Arthur wrote:
>>> On 29/06/2021 23:29, jla...@highlandsniptechnology.com wrote:
>>>
>>> <snip>
>>>>
>>>> Sure. Look at which way the wind is blowing, check the temperature
>>>> upwind from here, predict.
>>> Here, during the Wimbledon tennis tournament, the main court has a
>>> sliding roof which is closed when rain approaches, a procedure which
>>> takes a few minutes.
>>>
>>> Every year, someone writes in to a paper complaining that the
>>> authorities have access to much better weather forecasts than are
>>> available to the general public (or the man on the Clapham Omnibus, the
>>> 156 in this case).
>>
>> That's a bit silly. The Wimbledon centre court is specific and relatively small area. Weather radar can seem rainstorms moving across the city and provide a forecast of when the next rainstorm is going to hit the centre-court at Wimbleton.
>>
>> The general public wants to know when a rainstorm is going to hit them, wherever they are.
>>
>> Mobile phone areas in cities are pretty compact - if each cell tower pushed out it's own forecast to all the phones it was serving at a particular instant, it could do almost as well.
>
>That's a pretty good idea - add a simple weather station atop each mast
>and you could warn people to get the washing in a little in advance.

And we'd have so many more temperature records.

--

John Larkin Highland Technology, Inc

The best designs are necessarily accidental.

George Herold wrote:
> On Tuesday, June 29, 2021 at 4:01:05 PM UTC-4, Phil Hobbs wrote:
>> Fred Bloggs wrote:
>>> Chaotic does not mean ultra-random as some people seem to think.
>>> Chaotic systems are fundamentally deterministic dynamical systems
>>> that have the property of exhibiting wildly different outcomes as a
>>> function of the initial conditions. Small errors in estimation/
>>> measurement of initial conditions can result in unusably large
>>> differences in predicted results. If the prediction time scale is
>>> much smaller than the Lyapunov time, and initial conditions are
>>> measured with precision, predicted outcomes can have very good
>>> fidelity , and this is how they're tamed analytically.
>>>
>>> "Chaos theory concerns deterministic systems whose behavior can, in
>>> principle, be predicted. Chaotic systems are predictable for a while
>>> and then 'appear' to become random. The amount of time that the
>>> behavior of a chaotic system can be effectively predicted depends on
>>> three things: how much uncertainty can be tolerated in the forecast,
>>> how accurately its current state can be measured, and a time scale
>>> depending on the dynamics of the system, called the Lyapunov time.
>>> Some examples of Lyapunov times are: chaotic electrical circuits,
>>> about 1 millisecond; weather systems, a few days (unproven); the
>>> inner solar system, 4 to 5 million years.[19] In chaotic systems, the
>>> uncertainty in a forecast increases exponentially with elapsed time.
>>> Hence, mathematically, doubling the forecast time more than squares
>>> the proportional uncertainty in the forecast. This means, in
>>> practice, a meaningful prediction cannot be made over an interval of
>>> more than two or three times the Lyapunov time. When meaningful
>>> predictions cannot be made, the system appears random."
>>>
>>> https://en.wikipedia.org/wiki/Chaos_theory
>>>
>> The kicker is that chaotic systems are very powerful noise amplifiers,
>> and there is a fundamental limit to the precision of the initial conditions.
>>
>> If you have a frictionless pool table with perfectly elastic collisions,
>> and break the rack normally, the position uncertainties of the balls
>> grow exponentially with time. That is, a small error in aiming makes
>> both balls go off at a slightly incorrect angle. That angular error
>> grows linearly as the balls roll, causing the next collisions to be
>> further off in aim. That causes more angular error, which turns into
>> more aiming error on the next collisions, and so on.
>>
>> I don't have it handy, but I recall reading a calculation that showed
>> that after 30 seconds of this, the amplified Heisenberg uncertainty
>> became larger than the diameter of a ball, meaning that you could no
>> longer--even in principle--predict which balls would collide next.
>>
>> Information is appearing in the universe at a very high rate all the time.
>>
>> Like many chaotic systems, the motions of these ideal pool balls have
>> constraints, such as that the total energy of the system is constant.
>> The balls will bounce around and bounce around forever, but none of them
>> can ever jump off the table and hit the ceiling. As in epidemics,
>> exponential growth just applies to the early stages. ;)
>>
>> Cheers
>>
>> Phil Hobbs
>>
>> --
>> Dr Philip C D Hobbs
>> Principal Consultant
>> ElectroOptical Innovations LLC / Hobbs ElectroOptics
>> Optics, Electro-optics, Photonics, Analog Electronics
>> Briarcliff Manor NY 10510
>>
>> http://electrooptical.net
>> http://hobbs-eo.com
> Right. I once made this analog chaos circuit. (copied from P. Horowitz.. Lorenz butterfly thingie)
> with things set 'optimally' you could sometimes see a third / fourth period doubling. Till the noise in
> the circuit (mostly from multipliers) washed out the signal. With a computer you can watch period doubling
> to whatever accuracy you like.
>

But you gain forecasting time only logarithmically as you improve
precision, and the initial conditions have quantum indeterminacy.

Cheers

Phil Hobbs

--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510

http://electrooptical.net
http://hobbs-eo.com