[Music]
there was a numerical mathematician very
famous now no longer with us gene Golub
at Stanford we used to say that it's
obviously redundant to introduce the
speaker everybody's in the room because
they know who the speaker is so one
should introduce the audience know so
this has been a great audience so far
and I hope you'll stay that way okay
so yeah I can see there but I have to
paint it so let's see
but will this know this will not know
where where is yeah yeah I know I know I
know
very the monk mother stem oh oh okay
okay terrific
okay that so well I'll be talking a
little bit about one of the main reasons
for for having organized this entire
trimester which is my fundamental belief
that this is important for understanding
climate change by taking care correctly
of climate variability and if you're
interested in any of this you can find
more at these websites so actually the
full title that I had given was the
option of non autonomous and random
dynamical systems into the climate
sciences but actually yesterday we had
sequence of three talks by the group of
tomash tale that you know was an
important companion group in in this
eruption so I decided to shorten the
title a little bit so this is based on
joint work with many students and
colleagues to whom great thanks and they
actually quite a few of them in the room
I'm sorry that Mikhail couldn't be here
but I'm delighted that 50 gene is here
and Valerio and
deniz and of course user although he
didn't talk about what we are trying to
finish together and I don't know some of
these people are going to the I'm going
to show some slides are going to be next
week here if they are not here now okay
so yeah I'm sorry about not following
the transition of showing the portraits
of all of them but would that would take
a tie or sometime so the outliers is
trinsic versus forced variability
multiple scales of dynamics climate
sensitivity both in the usual
equilibrium version especially the less
usual but becoming more so non
equilibrium version a little bit about
non autonomous and random systems which
as I say have been covered by now but a
few other speakers pullback snapshot on
random attractors and tipping points in
this setting and then just two little
climate applications out of many so this
is a Lawrence without the T 1916
convection model at random forcing
affectionately called by its followers
Laura and then something about the El
Nino Southern Oscillation and Arnold
tongs and pullback attractors and then
some work in progress or non equilibrium
climate sensitivity and the topology of
law a concluding remarks on bibliography
okay so again motivation the climate
system highly nonlinear and quite
complex that's motherhood and apple pie
and they say in the US the system's
major components the atmosphere oceans
ice sheets etc evolved on many time and
space scales the predictive
understanding of this system has to rely
on the system's physical chemical and
biological modeling in other words on
all the details and all the boxes and
the diag diagram but also on the
thorough mathematical analysis of the
modes obtained the forests versus the
trees and I must say that you know that
another way of talking about this is
that you know rabbits I'm sorry they're
hedgehogs and foxes you know the
Hedgehog knows one thing how to all up
to defend himself and the Fox knows many
things so anyhow let's say that the more
holistic point of view and whether
mathematical and physical perspectives
has evolved quite a bit over the last
few decades but still the this
hierarchical modeling approach is not
followed quite consistently throughout
the community and I think it should be
so a little bit more it allows one to
give proper way to the understanding
provided by the models versus their
realism in other words back and forth
between toy or conceptual models and
detailed realistic models and between of
course models and data
I'm putting toy into quotation marks
because of course the people who do the
detailed one sort of look down their
noses at people who do the little models
remember very distinguished member of
our community who once told me at a
national high surge council meeting in
in Washington you know Michael you don't
want me to really take your little model
seriously you know she's been one of the
people who are really deserved the
distinction of the lab Nobel Peace Prize
so on the other hand I put here
realistic into quotation marks because
the people who do the big model think
that they are realistic but of course
you know while there are based on
fundamental principles of physics
conservation of mass energy momentum etc
they are not really entirely what's
called ab initio there are a lot of
so-called parametrizations and a lot of
semi empirical parameters which are
known within a factor of two or even
within the
ten so I don't want to insist on this
point but anyhow you know and should
give up this little somewhat aggressive
dialogue and collaborate more so how do
we disentangle natural variability from
the anthropogenic forcing should we
canwe and of course yes we can but just
to give you a very simple example you
know in the case of n so that for
instance if a gene here has made
important contributions to it's very
hard to say what's natural variability
and what is glow you know global warming
because a tropical Pacific depending on
exactly how you define it is about one
third of the surface of the globe so
when you have an amino that has you know
is reflected in the global temperature
okay so we really need to understand
this okay
so intrinsic versus force availability
and multiple scales and climate
sensitivity okay so I think I don't know
where the fede Boucher is still here but
this is the quote that I was alluding to
when he showed us a certain slide this
is fair phrasing so this you know John
phenomen is of course very well known in
the climate in the weather in climate
community because of that famous first
experimental forecast trying to filter
for diamond you know recognizing that
numerical weather prediction is one of
the most important applications of of
peacetime computers but this is a
somewhat less well-known he also
gathered a meeting just before his
untimely death at Princeton and actually
as far as I know this book which was
only published after his death three
years after the meeting in 1960 has the
words dynamics of climate together for I
think for the first time okay so I said
the
value problem American weather
prediction is the easiest one
that's your Matt we know most about the
asymptotic problem long term climate a
little harder but the intermediate
problem which sometimes called
low-frequency variability in other words
it's low lower frequency or longer
periods that the ones of the most
intense motions on the timescale of
interest
involves you know we know now multiple
equilibria non-periodic considerations
intermittent seasonal transients and
what's now called used to be called
bifurcations and is now called more
alarmingly tipping points that's clearly
the hardest so you know that's the one
that we're trying to deal with so again
this is a very popular diagram there are
many versions this is out of
encyclopedia article of mine but it was
first was published by Mary Mitchell in
the quaternary science reviews in 1976
which coincidentally is the same year as
a famous husband on paper in tell us and
it's an artist rendition of the climate
spectrum so this is variability of
course there's no no single time serious
that extends to the entire length of the
history of Earth and has a resolution of
these or always these are super
positions of various pieces of
information but basically it's made up
of three portions one which is the
continuous part of the spectrum which is
really most of the variability and then
there are some lines so that's the point
spectrum okay that's rotation of the
earth around its axis and evolution
around the Sun or the center of the
solar system the massive and then there
bumps and the bumps are you know
controversial but I think that there are
reasons of hope because they reflect
underlying hope verifications that get
thickened you know widened by climbing
higher up in the back
country to more complicated types of
behavior but leaving a trace and
basically you know the whole communities
which are organized around elucidating
what's called in traditional or these
days as to L sub C is not to seasonal
variability or you know this sort of
stuff involves the ocean this one
involves Nino Southern Oscillation this
involves material quaternary glaciation
cycles and really typically what under
does when works on something like this
this one looks at the higher frequencies
he wants to the right of the piece that
were interested in as white noise or
what I like to call warm color typically
the assumption is red but essentially
there's more variance at lower
frequencies so that's what done to the
right and what's done to the left is
handled a slow or a physical angry
adiabatic evolution of parameters so
basically what we are talking about now
non-autonomous a random dynamical system
is embedded in the way that we typically
work on this diagram but what you are
trying to do is to do so in a more you
know self-possessed self-consistent
deeper way also why this is a citation a
literal citation out of Mary Mitchell's
paper no known source of deterministic
internal variability which was a state
of the field this was exactly one year
after my PhD at the Quran first occurred
okay so I think that looking back at
these decades it's what I've been trying
to disprove hopefully with a little bit
of success okay
so this comes out of that cyclopædia
paper and it reflects on the one hand
the so-called
equilibrium climate sensitivity point of
view or ECS
which maybe started with the report to
the same National Research Council by a
panel chaired by Drew Charney in 1979 so
that's exactly 40 years ago and the
answer was you know for a doubling of
co2 global term surface air temperatures
averaged over the seasons would increase
by one point five to four point five
degrees now that uncertainty has not
been diminished it is still with us and
it has rather been increasing and that
sort of obvious why because each time
you add another knob
to your machinery you know you expect us
to grow so the the the purpose should
not be to try to reduce it but to
understand it better and to still be
able to act with that uncertainty so
again I don't need to tell the front the
audience here that you know a scaler
línea de you know the person who told
that told me I mean we're good friends
this wasn't something personally hostile
but you know since I have to say she you
might guess who that was but he probably
deserved that Nobel Prize a lot more
than Al Gore did so you know this was
still the interpretation of all those
integrations with big GCMs by the time
of the third so-called Assessment Report
you know all those were interpreted in
terms of this linear scale ax o de and
actually the first assessment report of
the ipcc was had chapters pretty much
like this where people you know were
specialized in certain feedbacks and in
certain sources on things of all forcing
but of course what we need to consider
it is much more complicated picture and
that's what we are doing during this
workshop and this semester trimester
okay so again out of that actually
fourth assessment report is that
gram which shows quite clearly that for
each one of these scenarios you know the
uncertainty is huge and I don't want to
take too much time you know over the
last one the fifth one the situation is
still pretty much the same okay so
getting on to non equilibrium climate
sensitivity so this is sort of a
schematic diagram out of the you know
China Dell 79 and subsequent versions of
equilibrium sensitivity global
temperature in your head and co2
concentration in dashed blue you know it
jumps this clearly adjust to the new
level okay and you're done so you only
have to I greased exactly about what the
some of the cases and what the cues are
okay well what about non equilibrium
sensitivity so let's say for instance
that and so we're purely periodic which
it isn't it has a lot of the variance in
the so called quasi biennial and quasi
quadrennial or liquid low frequency
component a were purely periodic well
the average temperature is still going
to go up I mean you know in spite of
efforts of various deniers even of an
educated one to say otherwise the iris
effect and so on there's no way really
that the average will not grow up okay
but what can still change and what's
also interesting is a period amplitude
phase of the limit cycle and I don't
have to explain here what the limit
cycle is I usually have to and then what
I say here what about some real stuff in
other words chaotic plus random so again
the mean will go up but there's a lot of
other things that will change higher
moments there's the bution of extremes
etc okay yeah
no no we don't know I mean you know the
person I'm referring to is not here but
those who know know who has come up with
the iris effect and that has been
disproved conclusively okay so well
because it's you know it's like
gentlemen don't read other people's
letters you know you don't you don't
attack people who are not into who okay
but if you want the United States lens
and then dick lism doesn't only not
believe in global warming he also does
not believe in the results of studies on
the effects of smoking okay although he
has stopped smoking by now okay so okay
so just a little bit because we've heard
all these names pullback snapshot and
random attractors I will come a little
bit to that and then to tipping points
in this context in this broader context
of time dependent systems where it has a
meaning aside from the rhetorical one
okay so every once in a while I have to
do a beamer slide because there are too
many equations so the source of say the
the sources of non autonomous dynamics
so the main point is physically opened
as we heard the usual stuff physically
open versus closed systems so their
fluxes of mass momentum energy between
the system and its surroundings which
are present or not
and so the mathematical framework of non
autonomous dynamical system is
appropriate for physically open ions so
basically what I'm trying to emphasize
here is that there are multiple sources
in the matter
medical and physical literature of the
approach to this so the first one
chronologically is that the
deterministic one of George cell just
passed away recently of skew product
flows so you have X dot is equal to F
and X and Q and Q dot is a function of
only Q and not of X okay so so X is not
D and Q is can be another spacer and the
name used is the driving force for X
okay and then there are two
complementary sources of Kobach and
snapchat attractors is in mathematical
literature Frankel unduly was here for
the course that preceded this workshop
and give a really beautiful lecture on
why you have to take this pullback
approach why the usual approach of
autonomous differential dynamical
systems with forward asymptoting does
not work and I'm having a slide
immediately after this and then transfer
promotion they thought with the postdoc
Ramirez came up with something that is
it's pretty much the same idea so anyhow
here you have a stochastic differential
equation and again you can have this Q
from here and the beauty is a Brownian
motion in Rd and the reason you have to
write these separately guess everybody
in this room knows but not everybody
outside this room is that essentially DT
is proportional to DW squared okay as
you saw were showing you know we know
the Brownian motion has mean spread
that's square root of time so more
generally studying explicit time
dependence in forcing or in coefficients
so in the case of the atmosphere and
oceans you know it doesn't have to be an
external forcing the randomness can come
from sub grade scale parameterizations
basically you know you can't resolve
clouds
and you shouldn't probably and all of
their small-scale processes in the ocean
so you model anything that you're not
interested in in the details of you can
model a stochastic so anything that has
explicit time dependents for all
coefficients requires and the SS so the
term was a little bit confusing is a
total non autonomous he used both for
the deterministic case following cell
and he actually call them skew product
because the skewness comes from the fact
that this is independent of X well X is
not independent of Q so and for a
unified perspective on the deterministic
and the random case there are some
technical difficulties in the two
separate cases but there's an attempt of
having a unified theory for instance
there's a book by Caballero and honey
which explicitly tries to do this so the
commonality between the two cases is
independence and I said of Q of W of X
and the semigroup property I'm going to
come to this two slides from now of the
driving force whether the security of W
sub T for instance the stochastic
Liberty is placed here other than there
so likewise pull back attractors is used
both for the deterministic in the random
case when the radical is more
specifically the phrase random attractor
is used okay so now more specifically
oops okay
much of the theoretical work on the
intrinsic variability of many things but
I give this off in the context of the
windows ocean circulation work which was
presented in some detail last week by
Stefano Pyare me who's still here and
that stuff is on his website but okay so
the proper framework is this so we do
here with a toy model given by the law
okay so this is a table that I will talk
but the main point here is that I wanted
to make
a very simple pullback attract us again
you know the X dot minus Alpha X when
Sigma equal to zero is what was in that
scalar linear equation that I showed you
and we all know that what the limit of
the attractor is X identical to zero but
what about this so in this case you can
actually compute it explicitly but you
know the point that Franco was trying to
make on the blackboard two weeks ago at
some length is it's not clear how you
know you say you're observing your
sister in your system with a certain
time and how do you describe what
happens at this time versus at this time
etc so basically what you have to do if
I follow the picture here is to pull
back to a compact set and let it rip and
it goes to these various things so let
me just if I managed to do this quickly
enough find the appropriate
okay so home who is also here from whom
I have this little this little video
okay so here you go again okay you pull
back pull back pull back and you see
clearly that that at each point okay at
each point it as you pull back to s2
it's infinity okay you converge more and
more to this so let me just do it one
more time okay you see how it converges
say at this point okay as you go back it
converges and the only difference
between the definition of snapshot
attractors in the in the physical
literature and the pull back I mean the
method
literature is a rigor with mathematical
rigor with which this pulling back is
carried out in practice fortunately you
only need to pull back for a finite time
and it depends essentially how
dissipative your system is okay so
that's that's what allows you to
actually compute these things
numerically okay so again here I just
have the references oops sorry I just
have the references for the various
things that I talked about okay some
general references and the application
they're just one paper each from each
one of the groups which has cultivated
these applications with a particular
dedication okay so we have in the room
and they gave talk yesterday all three
of of Gabor Tomas and Tomas okay and
then that's the paper that I mentioned
for snapshot attractors and of course
Hannes Paul wasn't here but Franco was
and as of course a book by Ludwig Arnold
which is not very digestible that's why
in appendices to our work we try to
explain to more physical audience you
know how to understand what's there and
what hands Carol and Franco's London he
did understand that they're both
postdocs of Ludwig and he's told us an
interesting story about how the famous
paper came about so handle dynamical
system this theory is a counterpart for
handling force dynamical systems of the
so called geometric theory of ordinary
differential equations it allows one to
treat stochastic differential equations
and more general systems given by noise
that I showed as flows in phase space
cross probability space okay so you know
these are the analogies between the
stochastic and ordinary differential
equations random and so-called
differentiable
different differentiable dynamical
systems but which implies autonomous and
Ludwig Arnold and VI have not no family
connection I try to emphasize the
difference by the usual transcription of
the Nucky's knock in Russian yeah okay
so the setting this is just I promised
you'd a little bit more on the semigroup
property so here it comes have a
facespace think of euclidean probability
space Omega so the sigma-algebra
of events the signals above sets is what
the events are in this axiomatic theory
of of probability and I just like to
mention your where the guests here not
just of the STTR upon K but within it of
the sometime in Borel you know biology
one who's guilty of the sigma-algebra of
events
okay so we've been a measure so a model
of the noise of preserved a measure
which is called the driving system
example vina process and a mapping which
in the random setting is called the co
cycle pole little map this is where the
time sits and this is where the
probability sits and this is where the
actual thing that were observing the X
sets and well example the solution
operator of an SD and so
the co cycle property which is called in
general the semigroup property just says
okay so he runs your realization so this
approach to handle Ness is different
from the usual forward plank
approach okay
it's called sometimes to distinguish it
the pathways approach the particular
realization is running here long in
omega and these are the fibroids of the
bundles in other words they're
realizations of your face space for a
particular value of your random process
and the same ego property just says if
you go from here to there and from there
to further it's the same as going
directly from here to there which is
written down okay
and so this is a more rigorous language
what so in that very simple example the
pullback attractor was just a straight
line which you can compute analytically
you can Franco took a very simple
example of a sinusoidal driving and this
is a more general attractor and again at
your shows that you pull back in order
to see what happens at the certain
cross-section which we called in our
earliest paper not knowing about
although by personally neutrals or
abortion they thought I didn't know that
paper so we call these things snapshots
which is a natural thing okay so for
each one of these snapshots you have to
pull back let it rip and it converges
onto it and it does so basically in
finite time rather than having to go all
the way to minus infinity okay so now
what I was saying before I have to
finish so you know tipping points are
first very effective rhetorical device
you know we've had bifurcation since
Euler who did the buckling problem okay
but bifurcations you know it's a
Latinate words as you say in English
and anything that this polysyllabic and
Latinate doesn't sound threatening
enough
well tipping points does its anglo-saxon
you know so but the there is actually a
meaning which is very nicely described
by Peter Ashwin and colleagues in this
paper in which what we were just talking
about as being the adiabatic change in a
parameter can be handled in a more again
self consistent way and there are three
kinds of this so bifurcation do tipping
or slow changing apparently to assistant
passage to a classical bifurcation
noise-induced again random fluctuations
little system crossing an attractor
basin boundary and finally rating goose
tipping which is one that maybe was
least well-known before rapid changes
lead to the system's losing track of a
sort of changing its attractors so all
three types of tipping involve an open
system so we start okay so this is what
I do in general then I have more time
okay climate applications Lorentz law ha
and then and so okay so when we became
interested in spite of the fact that
this literature on random attractors
goes back a couple decades okay the best
realization of a snapshot of that first
of all there was only as now it's not
shot by small fools insight on forgiving
time at MIT which was even much older
than this one so what we're talking
about is a snapshot of the random
attractor for the Rhoyne system with
multiplicative noise at the standard
parameter values of the butterfly
computer a fixed time and for the same
realization omega made of points as I
said transported far away and even
computed path wisest object that
supports meaningful statistics so this
is at a much higher resolution and again
this was originally computed by a team
on a sitting here and has been shown
since many times so we compute the
probability measure at some fixed time
for fixed realizations we show a
projection so this is Z and this is X
so the DW gets multiplied by a
coefficient alpha and XY and Z in each
one of the equations so this is
unfortunately there was a different
choice of colors and a phrase show but
the important thing is that this is not
a small perturbation theory alpha here
is 0.5 okay so in this direction the
Lorentz subtract is a little bit bigger
but in this direction it is order 1 so
this is not negligible and what you see
is that unlike in the cases that were
analyzed before for instance by Yuri
Kiefer where additive noise will just
smear out the contour set structure of
drilling through the onion skin you have
here the theory says and you can verify
numerically that invariant measure that
is shown here is singularly supported on
the unstable manifold of the
deterministic attract essentially ok so
this is just a snapshot out of the
supplementary information in this paper
but Mikhail has now put it as a V mo
movie on his website so I you know many
of you who have sat for some of these
before have seen it but it is still
pretty spectacular ok so let me show it
to you
ok I think the nice I mean sort of
Denisha Mahalo is giving the last stop
on Friday and she will talk more about
the topology of this stuff so you see
that on the one hand you do see still
the underlined the Lorenz butterfly so
the but the noise interact so there's a
certain jitter in the movie which is
just the effect of the noise but then
the coupling between the noise and the
null
dynamics is entirely non-trivial and
very exciting and very much still an
open question okay so we're trying to
make a little bit of of headway on okay
so sorry I think that I went too far
okay yeah yeah sorry okay I so I don't
know why this jumped but okay
so here we are so now and so so again
physically this is what it is change in
sea surface temperatures in the tropical
Pacific and the think a change of depth
of the thermal clients or general since
I'm nearing the end I won't go into the
details these are two times here is that
characterize the state of the tropical
Pacific one is this average of sea
surface temperatures of a certain area
in the eastern tropical Pacific the
other indices three point four etc
never mind the SOI is the seesaw of sea
level pressures Tahiti and Darwin and
these two things are clearly just
incredibly incredibly anti-correlated
so it is very clear although there were
silly attempts when deterministic chaos
became popular to declare that all sorts
of climate things were actually a
fractional dimensional and low
dimensional you know that's clearly not
so but what you see here is that clearly
a lot of the variance is just in some
large scale stuff right they say okay
so if you just try to model the large
scale stuff okay this is again out of
well at the time we couldn't publish
really this color picture but
it's out of the vanillin Gill paper in
physical D 96 and basically this is a
parameter that has to do that
essentially fixes the periodicity of the
intrinsic period and this one is a
coupling coefficient between the
atmosphere and oceans so this is a
well-known quasi periodic scenario of
transition to chaos so the various
colors represent period you know period
1 so just a seasonal period 2 3 4 and 5
and where you have overlap you have
chaos and then so this was the the first
one and then in this one we use the
standard circle map or idled map I know
that amok is not well you see The
Devil's Staircase and you see the effect
of noise so in other words the the PDF
which is clearly you know at this level
of malini arity is discrete you know
there there are separate peaks because
the the PDF becomes multimodal but
connected and now in this work we are
trying to look into the details of the
the effect of noise on the on the Arnold
map with for instance changes in the
fact that the devil staircase is no
longer monotonic okay and so you know to
follow up on this if you have two
coupled Arnold Maps it looks even more
exciting and again the effect of noise
we never got to publish this but the
effect on noise on this is also smearing
out basically what's going on okay so
this comes I mean the quotation mark
comes from the fact that I got a certain
award of the EGU of the European
geosciences Union and the press agent
told me I have to attach some big words
to it for the press conference so here
it goes
you know classical strange attractor
physically closed system model
mathematical autonomous neither
deterministic more random forcing to you
can think if you can think of humanity
as being deterministic well you can
associate that with the warming trend
no random things like volcanic eruptions
or the effect of clouds the attractor is
strange but still fixed in time so again
you know people don't like things are
not familiar with so it's strange for
the same region that an irrational
number is irrational it's a it's a curse
world okay
so climate sensitivity changing the
average value first moment of the
coordinates X or Z or the parameter
lambda changes well this is out of the
cyclopædia radical of Lenny Smith you
know you have in in systems of interest
in in climate you have course raining
that has been discussed here basically
in this case you have at least long
transients of law of coarse graining you
start with the PDF concentrated at the
origin and you move in time like this
and eventually it will be you know
unimodal here but for a long time it's
bimodal yes okay so you know now come
the snapshots of what I showed you the
attractor is now pull back and evolves
in time
okay environment sensitive changing the
statistical properties first and high
order moments of the attractor as one or
more parameters change okay and so here
you have you know Lawrence's climate is
deterministic and autonomous but highly
nonlinear hustle man says climate is
stochastic but quite linear trajectories
decay back to the mean and you have full
synchronization grand unification
climate is deterministic and stochastic
as well as highly nonlinear internal
variability interact and forcing
interact strongly changing sensitivity
refer to both mean and higher moment
distribution of extreme etc so I will
have to skip this but promise you that
the needs will tell you much more about
Louis topology the you know I don't want
to steal her thunder but just to keep
you very related to your seeds till the
end of the workshop on Friday you know
the answer is that it's the of course as
you might have suspected from the from
the video
the topology is different from the
classical one of the butterfly and the
changes in time okay so you have what
you saw called topological bifurcations
okay
so sorry and the dependence on
parameters is not smooth okay so there
are other ways and this is just to say
that within a global pullback attractor
deterministic asia can be local ones
with very different properties and
probably with the fractal Basin
boundaries we've heard about such things
yesterday and you can look it up on
Stefano's website the full talk that he
gave here last week and he's still here
to discuss it with him okay and this is
just the two slides out of what the
needs will show you so this is just the
definition it turns out again that
Poincare a contributed to the two
important aspects here one is creating
algebraic topology and the other one is
figuring out chaos in the context of
social mechanics and even stating that
probably the same cases the same cases
for weather okay so these are just you
know two different snapshots of the
topology okay so concluding remarks
change of paradigm from closed
autonomous to open non autonomous system
and them attractors are spectacular
I think they're useful and they're
certainly just starting to be explored
for classic applications although with a
lot of enthusiasm here's some of the
work in progress study the effect of so
I study the effect of usually when you
have a stochastic parameterization like
previously deterministic ones people
just test because whether it does what
you wanted it to do modify a particular
aspect of our general circulation models
behavior but people haven't looked at
the effect on a lot of the other things
which is not always favorable
applications to intermediate
and high-end models via and fatal model
reduction and such methods of what's
called now deep learning but again it's
something we've had around for a while
patience for climate sensitivity and
even for predictability which I didn't
have the time to talk about and so this
is some of the references with those I'd
like you to see to look at more closely
emphasized and of course he wondered
whether you already showed yesterday and
which you know appropriately was
deposited on our chief just a few days
ago okay thank you very much
[Applause]
well it doesn't have to be when I showed
you the equation you know I did put I
think Sigma of X in front of the DW but
it doesn't have to be well you know the
properties might be different I mean
clearly the properties of you know the
interaction that you've seen between the
dynamics and the noise has a lot to do
with but it can be interesting even
otherwise okay and I think that actually
at least some of the examples that user
who showed you you know the noise was
not multiplicative right yeah so the
short answer is yes
the multiplicative noise is everywhere
people deal with additive noise because
a lamppost syndrome because it's much
easier to deal with
I mean noise is not you know first of
all I have another slide I kept just a
few reserved slides you know but here
nature is not deterministic or
stochastic all depends on what we can
need and want to know more or less
detail with greater or lesser accuracy
well typically larger scales more
accurately smaller scales also but we
need both descriptions okay so you know
the multiplicative is not special
additive is special but it's under the
lamppost okay yep
yeah we should we should use this you
have to now run after turn so I just
wanted to mention this name Sasi lead
since we have a long list of references
about non autonomous systems there is a
paper in 1978 actually in in nature who
is it from Boulder oh okay yeah okay
well he's defining the de climate in
terms of ensemble by statistics well but
that's not the same thing you know
they're always precursors but just doing
if I you know just running an ensemble
forward in time I mean again I've told
already this joke a few times since that
when Esther started but there is the
play by Moliere you know which is called
the warranty on the you know the burglar
who wanted to be an aristocrat and who
gets a lot of teachers he wants to learn
dancing and one of the teachers is a
teacher of literature and tries to
explain to him the difference between
prose and poetry
so mr. schirra dongho was oh so I've
been speaking prose my entire life you
know so yes people do well Samba runs
and but they do not necessarily quite
understand you know I'm sorry you know
the fact that you need two parameters
for the non autonomous situation in
other words if you have in the
autonomous situation if you have
uniqueness you know you can just do
forward asymptotics but here you have
need this other parameter which takes
you back in time and you really need to
define the convergence of what you
observe
with respect to that again and again you
don't just run the ensemble forward in
order to properly define the pull back
you know you stop at the time you know
when you want to observe it you have to
pull back fortunately just a finite time
and not all the way to minus infinity
and you have to go forward again to
really define the object and again I'm
sorry I didn't have the time here and I
don't want to you know and by there you
won't let me to cut too much into the
coffee break but for instance the work
that we did with Ilias of the opinion
which was published in an aju monograph
is we took another and somewhat also one
of these delay differential equation
models and we well he computed mikhail
was also no actually he he felt he
hadn't done enough to be actually a
co-author but anyhow there's part of the
paper that was the chapter that's due to
him so the Ilia computed the pullback
attractor and it teaches you a lot more
about the stuff than just looking at at
a video of a forward run okay so i still
would like the reference if you can send
me an email and you know it's there's a
real difference among the standing
between and i don't care at this point
whether your snapshot or pullback or
random or whatever but the the
understanding is that in order to to
really define what's going on you have
to follow the evolution of the attractor
and not just of a trajectory or even of
an ensemble of trajectories okay thanks
okay let's stop here thanks again to the
michael
[Applause]
you