okay
okay so good afternoon to everybody
i'm marco canersa and i am happy to
share this colloquium at the university
of
romator vergata by michael gill first
romator vergata by michael gill first of
all let me
remind you to mute your microphones
and if you have to ask question you
you can always send a note and there
will be
room for questions at the end of the
talk
let me just introduce our speaker who
was supposed in fact to give this talk
in
april and in rome and
then things unfortunately went wrong
very rapidly but we i'm glad we we
managed
to make up this way and of course we
will be happy to
restore the traditional invitation as
soon as possible
so michael gill if you look into
wikipedia
you will find of michael a definition
like
an american physicist focusing in
atmospheric
and oceanic sciences and climate
dynamics
well that may be correct but maybe
needs some elaboration deserves some
more
elaboration uh so michael um
graduated um he's american
he graduated from uh nyu
duran institute he was a peter luck
student
and he developed quickly deep
interest for models well in
climate and geosciences and in fact
all his career was um well
and a progress into this
kind of uh disciplines he is
currently distinguished professor at
ucla
and that i call at uh sp normal superior
in paris and uh until now i mean
long during his career he covered
an enormous number of uh important
responsibilities in uh directing
the sciences departments environmental
research
institutes um
all sorts of teams and committees
on climate uh is
also received many awards in his career
too many to to to to to list these
members of and
is a member of a number of uh registered
scientific
academies and he has an overwhelming
publication record uh both
for quantity and quality
but i like to think of michael introduce
him as the father of
modern mathematical modeling and
analysis
in climate sciences
so today we'll be happy to lost his um
his talk will be which will be on
non-autonomous and
random dynamical systems in the climate
sciences
please thank you very much pierre marco
for
leaving me a little bit of time to give
my talk
i apologize for all my friends and
former students who
haven't gotten around to expand that
wikipedia radical
you know that was written by somebody at
ucla that i don't know
but anyhow
so i just wanted to emphasize that while
normally in this microsoft team setting
questions have to be at the end because
it's difficult for me to interact with
the audience uh i would like
to ask
fabio to allow questions
because in the united states we
are used to being interrupted it gives
me a feeling
that not everybody is sleeping and i
always tell my students that there are
no stupid questions
only stupid answers so if you feel like
interrupting please interrupt
but now i will start sharing my screen
remember that i cannot see you
so the only thing that i see is
my own screen from this time on
okay so um
as pia michael kindly remarked i would
much rather be with you in home
you know both my wife and i uh love
italy in general and
home in particular dearly but i hope
that
things will get straightened out and
have been able will be able to catch up
and i can
maybe give you another talk on one of
the other topics that i've been working
on
anyhow this is one of my main
preoccupations these days
so uh the talk is based on joint work
with many students and colleagues
and it includes at least one person in
the audience
stefano galatolo and his colleagues
pisa angelo di garbo
and mohanjo and cedro but also people
in in buenos aires
uh in nice in wedding
in hokkaido it's and in israel
they must have oh yeah there's one from
university of nevada in reno and
somebody was a professor at mcgill
so uh i've been having a lot of fun with
this and i hope that you will too
so if you are interested in any of the
stuff aside from the recording
there are papers the really the best
place to give your perspective
is a website at ucla and then of course
there's the research gate where there's
much more
okay so the outline is
first i will tell you about intrinsic
versus force variability
and this will be essentially the
motivation
for dropping into this deep hole
of non-autonomous and random dynamical
systems
so i'll talk about multiple scales of
climate dynamics and climate sensitivity
both
the classical equilibrium climate
sensitivity or ecs
and the non-equilibrium one that will be
interested here
so when we talk about non-autonomous
random dynamical systems i'll
introduce you to pullback snapshots and
random attractors
and to tipping points and then the
climate applications will include
definitely the lorenz the famous
lorenz 1963 convection
convection model but with random forcing
so you know uh the lorenz random
attractor with lovely
some of the people that i listed on the
cover page called it lovingly laura
and then as time permits i'll talk about
applications through the armenian
southern oscillation
armored tongues and other things about
pullback attractors and
working progress would include
non-equilibrium climate sensitivity
and the algebraic topology or homology
groups of laura
and conclude with some remarks in
bibliography
so to proceed
the motivation is that the climate
system is highly nonlinear and quite
complex
which is sort of obvious but with what
we call in the united states motherhood
of
apple pie uh you know you find words
like this in the introduction
to the various ipcc reports
but then they're quickly forgotten as
you get into the chapters
so there's much more attention given to
the system's major
components the atmosphere ocean's ice
sheets which revolve around many times
in space scales
and which are modeled in great detail
but still the predictive understanding
of all of this has to rely
uh of course on the physical chemical
and biological modeling but
also on the thorough mathematical
analysis of the models as obtained
so it's what's sometimes called the
forest versus the trees
so we're going to be interested here in
the forest while the great majority of
people who work in the field
are interested in particular trees
branches
leaves or needles sometimes flowers
so because of the difficulty of covering
all the details on all these scales in
space and time
the approach which has been pursued in
the climate sciences
is one of hierarchical modeling which
allows them to give proper way to the
understanding provided by the models
versus their realism
so there has to be a back and forth
between what's sometimes
called toy models more
nicely put conceptual models and models
that those who
work on the details like to call
detailed model
but like to call excuse me realistic
but i put that into quotation marks
because
like i put the toy models
like i put the toy models because
the illusion given by the details of
being realistic
is some somewhat of an illusion because
the details are not necessarily based on
fundamental principles of physics
chemistry
etc there are a lot of so-called
semi-empirical
parameterizations where parametrization
is climate modeling jargon in other
words this is not like parameterizing
the curve
by arc length it is representing
in an integrated way so-called subject
scale effects
by some form of averaging at the scale
of the models
so there has to be a back and forth
between these toy and detailed models
and of course between the models and
data
so after saying all of this the question
is
can we disentangle natural variability
from the anthropogenic forcing which of
course we've been hearing more and more
because it has more and more of an
effect on the natural variability
so the rhetorical questions are
should we can we the obvious answer to
should we
is yes can we i hope i will convince you
that indeed we can
okay so intrinsic versus forced
variability
uh multiple states of dynamics and
climate sensitivity
i will start here by paraphrasing jon
von neumann
who probably many of you know
was co-author of the first paper that
demonstrated
the possibility of numerical weather
prediction
aside from enormously uh
[Music]
much mathematics and physics yes
it's a personal problem don't worry go
ahead okay all right
uh okay so as you know john from nyman
was actually
um you know it is le it is said that
was the last person who knew all of
mathematics
but it is also said that when john von
neumann was asked how much of
mathematics one person could know
he stopped for a second and he said oh
about two-thirds uh
well good luck these days with knowing
with being john von neumann and knowing
two-thirds of mathematics
so another reason for asking questions
and interacting
so uh aside from convening
the whole group of people in the 40s to
the institute for advanced studies
to carry out this experimental or proof
of concept
numerical weather prediction he also
was the motivator of a meeting
that took place very shortly before his
untimely death
in the 50s and
this the proceedings of that meeting
were published as a book
by programmer one in 1960 after von
neumann's death
by edited by richard pfeffer but
for neymar gave an introduction to these
proceedings
and this is where he said these things
he said the initial value problem in
other words the medical weather
prediction that's the easiest
and that's what most people who apply
mathematics
to the climate sciences know about the
asymptotic
problem in other words long term climate
is a little harder
but it is really the intermediate
problem which is
broadly known as low frequency
variability which includes things like
multiple equilibria
or if you wish multiple fixed points
long periodic oscillations limit cycles
intermittency
slow transients and what's called these
days tipping points
is the hardest so this that this is the
hardest problem
which we're trying to come to grapes
these days
okay so uh to give you an idea again
of the range of time scales
there is a synthetic
power spectrum or spectral density
which is really sort of an
artist's impression of the spectrum
because we do not have any time series
that extend over the age of the earth
and have a resolution
of days or hours so each one
of the bands in this in this
spectrum correspond to a certain way
of looking at phenomena and
a certain kind of measurements
over to the right of the spectrum we
have actually instrumental observations
but already in the middle of the
spectrum we have what's called proxy
records
there are various
oxygen and hydrogen isotopes in marine
sediments and
in ice scores and then as you move
further towards the left it becomes sort
of harder and harder
but what i would like you to notice in
this spectrum
is that of course most of the variance
is in the continuous spectrum
what we call in mathematically
absolutely continuous spectrum and then
there are lines
which correspond to the rotation of the
earth around its axis one day and the
evolution out of the earth around the
sun or more precisely around the mass
center
of the solar system which is not very
far from the center of the sun
and then there are bombs so
uh the usual idea that we get in
mathematics that it's
either absolutely continuous or aligned
spectrum and so both of them for
instance can be
encompassed by a studious integral
here what's really characteristic to the
climate
is that these two features of spectra
that we're familiar with
are also supplemented by these bumps
which are
uh in my view and by now that of
many of my former students postdocs and
other friends
uh the signs essentially of
uh hubs bifurcations which result in
limit cycles which become destabilized
and their lines associated with them
become broadened
and so that's where the low frequency
variability comes from
note that at the bottom of
the slide it says that it is the
original version of this
the spectrum goes back to mary mitchell
who published it in
quaternary science reviews in 1976
and it contained this version comes from
an
article in the encyclopedia of global
environmental change
but in the original article there was a
sentence
no known source of deterministic
internal variability
so this was the prevailing opinion
one year after i got my phd at ground
okay the idea was that everything
is either forcing which gives you lines
or noise which gives you the continuous
spectrum
okay so we run a little bit beyond that
i
since i took a little bit longer than
since i took a little bit longer than i
wanted with this introductory part so
you see here the reference at the
on this slide but basically uh
sorry the
really the concept at least in the first
few
assessment reports uh of the
international intergovernmental panel on
climate change
uh was that of equilibrium climate
sensitivity
and all those complicated computer lands
were
analyzed in terms of such simple
scalar linear ordinary differential
equations
and the chapters had to do a lot with
the presence of several
feedbacks positive and negative so there
were experts on each one of these
feedbacks
and the sources and sinks of
greenhouse gases and of aerosols
so the if this view had been correct
an exponential increase in greenhouse
gases
should have corresponded to a linear
increase
of temperatures all other things being
equal
which they never are so what we see is
not this red line
but this black line which says that we
need to consider instead
a system of non-linear partial
differential equations
that have various as i told you
semi-breaker parameters which are only
known within a factor of 2 or even of 10
and can be either additive or multiple
oh
sorry and either additive or
multiplicative forcing
in other words both deterministic and
stochastic effects
so this leads to an uncertainty
so this graph here is taken
out of an assessment report so
again since these intergovernmental
panel on climate change or ipcc
reports involved by now hundreds of
people there are second order
acronyms so this is says assessment
report number four
and this comes out of the summary for
policy makers
of working group one the one that deals
with physical effects if you are
confused
you are not the only one because
actually somebody that
i i'm sure many of you know uriel fresh
uh when he had this particular summary
for policymakers said to me
well i can't understand this i don't i
don't
think that any policy maker
okay
sorry okay
okay so um pretty soon we are going to
be ready to do some business but this is
really pretty important
so i mentioned to you the so-called
equilibrium climate sensitivity which
has its own acronym
ecs and so the picture that i was
showing you
here okay gives you
something like this plot at the top in
other words this being
time and this being either global
temperature averaged over a year
or co2 so the temperature is a dashboard
sorry the temperature is ahead solid
line
and the co2 concentration is this dash
dotted blue line so the way that this
was defined
in 1979 by a report to the national
academy of sciences of the united states
it says you double co2 what happens to
temperature
well what happens to temperature is of
course this we know how to integrate
scala
these that are linear but
so you know climate is a stable
equilibrium fixed point if so mean
temperature will just shift gradually
to its new equilibrium value
and the delta t over delta co2
is this equilibrium climate sensitivity
now on the other hand let's say that
climate were purely periodic which would
be the case
if the el nino southern oscillation for
instance
were purely periodic as opposed to just
quasi-periodic
so that then the mean temperature will
still go up
but at the same time also
the limit cycle associated with this
stable periperiodic oscillation could
change
in period amplitude and things okay
and finally what i call this uh
what about some real stuff both chaotic
and random
well the mean will still go up nobody
has really come up with any good
way for it to do anything else but
a lot of other things will be changing
for instance the distribution of
extremes etc
so we want to proceed from
from this picture which dominated the
thinking
uh for the first couple decades of these
ipcc exercises from 1990 to about 2010
uh to the more modern view of this
so this is excuse me michael
when you say real stuff you mean
coming from observation and collected
data as you explained at the beginning
or what else
it is what i it is exactly what you said
if you look at observations um well
you know there are no pure observations
any observations get massaged one way or
the other
but something that is as close as
possible
to the observations and contains as
little prejudice of the observer in
other words of
models that are not oversimplified will
give you things that will have
essentially this mix of spectral
features
that i was indicating here okay
so you clearly see the seasonal cycle
okay
you clearly see the diurnal cycle you
know it's colder at night than
on during the day and it's colder in
winter
than summer that's sort of obvious but
you also see
a lot of stuff that irregular
and there are some disagreements about
whether the regular stuff
comes from deterministic chaos
random processes or both um
well i have a maybe a reserve slide at
the end which says nature is not
deterministic or
or random you know it depends
how much you want to know about the
phenomena whether you approximate them
by a deterministic model or by one that
contains stochastic processes okay
if you take a stochastic point of view
you renounce
some details okay you concentrate on
mean variance maybe a few other moments
etc
so uh chaotic versus random is to some
extent in the eye of the beholder
but i hope you'll convince you at the
end that taking into account both of
them is really useful
did i answer your question yeah thank
you
okay so we move now to i actually
realized that i forgot the word
dynamical here but it's non-autonomous
and random dynamical systems
so again aqua means nds and rds and
we're going to talk about
uh the particular attractors that need
to
how you need to generalize attractors
how you need to generalize attractors to
this situation
and about how you need to generalize
bifurcation points of autonomous systems
to something that's called today tipping
points
okay so here i have a slide because
there's a little bit of an argument
i'll i'll tell you about essentially the
two groups which introduced these
concepts
into the climate sciences but
uh since a little bit of argument
between physicists and mathematicians
what is what and who did what i had this
little slide to try to clarify this
the sources of not of non-autonomous
dynamics but of the theory of
non-autonomous dynamics
so basically you are dealing with
physically open versus closed systems
autonomous dynamics is particularly good
at closed systems okay
there is also an issue about whether
coefficients depend on time or not
but let's say that they do not depend on
time and the things depend largely
on whether the forcing depends on time
or not so in a closed system
it's closed it doesn't but of course the
climate system or
more generally the earth system which
also includes a biota etc is an open
system
the fluxes of mass momentum and energy
between the system
and its surroundings we get energy from
the sun
and we send it back to space and
we exchange momentum with the moon etc
so the mathematical framework of
non-autonomous dynamical system is
appropriate for physically open ones so
basically
it goes back to skew products skew
product flows of george cell who passed
away a few years ago
but i had the privilege to meet and in
this case
it looks like this you have something
that you call x and something that you
call
q and the x dot depends on both
x and q and q does not depend on x
that's the skewness in the
in the skew product so x is in rd and q
isn't saying some other space
and so q is called the driving force for
x
and then after this so i will give the
references
this goes back to the 60s and 70s and
sometime after that
we had these two concepts of pullback
attractors
due to well franco flundly
and crowell and ludwig arnold and on the
other hand to chelsea
beaujee and ed ott
so here we have a
stochastic differential equation
i'm using on purpose another letter than
here
and of course you have in the most
general case
a variance that depends also on the
state
so wt is a brownian motion and
i don't need to tell it to this audience
but i have this for people who haven't
seen
a stochastic differential equation in
their lives so basically the difficulty
is that dt is proportional to dw
squared and this goes back to einstein
1905.
okay so that's not new
so more generally studying explicit time
depends on enforcing our coefficients
requires ndss
the term non-autonomous is used both for
the deterministic
case and you know there's a lot of
terminology here which is
a little bit confusing it is used both
for the deterministic case
and for a unified perspective on the
deterministic and the random case
and i maybe come back to that later
the commonality between the two cases is
the independence of the driving force
and the semi-group property of the
driving force whether it's q of t
or w of t okay um
so this is what i didn't mention here
but i'm going to show it on the
following transparency
likewise pullback attractor is used both
for the deterministic and the random
case while in the latter case
one uses more specifically the phrase
random attract
okay so sorry this slide i forgot to
change
it but it comes from a talk where i
specifically apply this
to uh the winter in ocean circulation as
an example of a climate system
so in general the work with theoretical
mathematical
work up to about a dozen years ago
was done at least conceptually it's not
that
people didn't use a seasonal cycle or
anthropogenic forcing
but they did it like mr jordan you know
in moliere
who the bourgeois who
takes all these lessons in dancing and
so on
and uh also in literature
and so his professor tries to
teach him the difference between prose
and poetry
and uh the reaction of mr gerardo is
oh so you mean i've been speaking prose
all my life okay so likewise
when people use seasonal forcing or
something else anthropogenic
effects they didn't did it without
realizing what they were doing
okay so to address truly coupled ocean
atmosphere behavior and climate change
an important step is to examine time
dependent wind stress or time dependent
force in general
proper framework for doing so the theory
theory of non-autonomous and random
dynamical systems
we do here with a toy model if i have
the time i'll actually show you the toy
model from
that that is referred to in uh in this
slide
and the forcing is deterministic
aperiodic and dominated by
multi-indicator variability
in that case but here i'm just giving
the example of the pullback attract of a
linear scalar
ode okay so we all know how to solve
this
if sigma is equal to zero it's just
exponentially
uh tending to zero but if sigma
is not uh equal to zero and in
particular it's positive
actually in this simple case you can
compute the so-called
pullback attractor but i would like to
show you
if you haven't seen this before a little
uh
[Music]
little video which
tells you where the word pullback comes
from
so the idea is that you would like you
see this is a sort of
stuff that carol and flandoli had to
overcome when they started thinking
about non-autonomous dynamical systems
as post-docs of ludwig arnold in bremen
and i've just heard this from franco the
story during this
uh uh trimester at the eye of the
institute
in poincare in paris pia marco also
attended at stefano
so uh uh you know it's not obvious
what a proper generalization of a fixed
point should be
in this time dependent context so let's
say that you're
at time zero and you want to know what's
attractive
what is it uh oh so i forgot of course
to tell you that it's really important
that alpha be positive
so that you have dissipation you cannot
define these things in hamiltonian
systems
so we're in this effective system but
that not big a big bottle in climate
because there's enough dissipation in it
to take care of a lot of stuff
so what what will you observe
well if you do pull back to
in principle to minus infinity but
fortunately that's not really necessary
okay here you pull back and you observe
that this stuff
as you pull back further and further
converges to a point here
okay you want to do the same thing over
here
again you start you know you start
pulling back
and you know this that you converge to
this point here
so in general you cannot get this
equation
explicitly but you can observe this
behavior
okay so we'll go back
to the slides
okay pullback's natural kind of
attractor so again i'm repeating here
the framework of physical
mathematical autonomousness given by
skew products and deterministic
systems and in the math literature
okay so i should mention here also carl
but anyhow
um the
the proper definition of the sort of
stuff i showed you in the little movie
was pullback attractors because of the
fact
that instead of dealing with a
single parameter semigroup as you do
in deterministic systems where you just
go to plus infinity
here you have a two parameter semi group
where you have
also a parameter that goes back to minus
infinity
in the physical literature they have
been called stock
uh snapshot attractors so here are the
references
okay carl and flandoli tech reports
volume
superior in a publication in
a refereed
journal in 1994
snapshot attractors this paper by
romeros was a postdoc of baboji and art
and they coined this word snapshot
attractors
and when my students and i started
working on this in other words mikhail
shakron
who was doing his phd with me at normal
and the ixi monet whom i had co-advised
first with roger mom for his phd and
then was also a
post-doc with me at ucla we started
looking at this we're not
aware of this work of
although i knew both chelsea and ed and
so we were aware of the mathematical
edition
okay so one of the references to
georgetown sellings 1967
and another one which i like very much
on
pullback attractors is lysan young
what are srb measures and that's 2002 so
about 20 years ago
okay so a little bit more on the theory
uh so basically the theory now of random
dynamical systems is a counterpart for
randomly forced dynamical systems
of the geo it's a geometric theory it's
like a geometric theory of odes
and it allows one to treat stochastic
differential equations
sds and more general systems driven by
noise as flows in phase space
cross probability state space so this
is just memo technical the analogies are
between sds and ods between rds's and
dds stands usually for differentiable
dynamical systems but
these what are meant are really the
autonomous ones
and ludwig arnold no pun intended
vir mode okay although this one is much
more readable than this one
i remember being invited by ludwig whom
i had met at the current institute
because everybody came through at the
grant institute
where i stayed a little bit after the
phd
that michael mentioned so he invited me
to a meeting
in in germany when he got an award for
studying stochastic partial differential
equations
in the context of climate and other
applications
and although i knew his little book on
linear
sds and loved it
i found his talk really pretty
incomprehensible and not terribly useful
and the consolation is that when lysan
young
invited me to this meeting sponsored by
indum in coronaldo on hyperbolic
dynamical systems
which for the first time was inviting
people to talk also about applications
so i gave
something like a much earlier version of
this and stefano was there
and he will lent me his car to get to
urbino thank you again stefano
pleasant memory uh you know my son
sunset to me over
lunch michael you have to understand
that even people who are on top of
hyperbolic dynamical systems you know
don't
really understand these pullback
attractors so
you know i felt much better because i
didn't know how
well my talk went over okay so
uh if you still don't understand after
this talk
uh well maybe try to go to one of the
papers because we have made a big effort
in our publications with mikhail and
eric
to explain it uh in as
simple language as possible but not
any simpler again paraphrasing einstein
so the setting is
a phase space x is for instance
euclidean if you still deal with
differential equations a probability
space
for instance the one you're most
familiar with the venous space
within a measure p and a model of the
noise
okay that preserves the measure
and is called the driving system so this
is
the one that has to have the semi-group
property that i was mentioning okay
so as you shift by t
uh you know i'll have a picture
which comes out of one of the appendices
of those early papers
it starts the noise at s instead of a t
okay
and then a mapping that has a so-called
co-cycle property
e g the solution operator of an sde
so here comes the picture okay which
uh we didn't find anywhere certainly not
in ludwig's book
okay a geometric view of sde
so the idea is that these sales
okay this and this and that are fibers
in other words this is x cross
omega and this is the one that happens
after you've advanced by
s time units and this is the one that
after you advise
advanced by s plus t units and the co
cycle property is that going from here
to here and from here to there
is the same as going from here to there
okay
so in technical language phi
is a random dynamical system or a co
cycle
and this is a flow on this bundle made
up of these copies of
x okay so random attractor
so omega is a variable in the cap omega
is a it's a label of the realization
so it is both invariant and pullback
attracting
invariant is says this
okay and attracting is that if you start
as i did in the little video from some
compact set
in the past okay and you're looking now
at the
so the yellow stuff is the
random attractor and this is a cross
section
which we called a snapshot of the
pullback attractor
not knowing that ed art and chelsea boy
bergey
had called the whole attractors natural
detractor but there's no contradiction
they didn't worry about going all the
way to back to minus infinity and
actually they were right
because you can show that you only have
to go back and practice numerically so
far
anyhow you start in a compact subset of
your
ex and you let it rip you go forward
and things will converge to this cross
section this snapshot of the
random attractor okay
so i promise to say just a little bit
about tipping points because at the
beginning they got on my nerves
you know i was wondering what have these
guys
gotten enthused about if some social
scientist
came up with this word when we've been
doing bifurcations all the time
but it turns out that it makes some
sense to use this more general term
in the context of non-autonomous or
random dynamical systems and there's
actually a little classification
of some of the ways that
the old-fashioned bifurcation points
behave depending on
the kind of process that induces
the uh the
critical transition through that point
so this is a paper by peter ashwin and
co-authors
in the field of the philosophical
transactions oops there's one zero too
many
uh so there's b tipping or bifurcation
to tipping
so slow change in the parameter leads to
the system passes to a classical
bifurcation
that's what we knew all the time and
then there's noise-induced tipping
random fluctuations leads to a system
crossing attractive basin boundary
well we sort of also knew this about
double well potentials and so on
but one that's really interesting that
that people haven't thought a lot about
before
is our tipping was overrated skipping in
which rapid changes leads to the
system's
losing track of a slow change in its
attractors
so it is different from this one
so all three types involve an open
system
and we start with a closed system we'll
study the modification factor then you
proceed to the open ones and see how
that changes things
okay so now we're at the climate
applications uh
i will now proceed as one of my teachers
at the current institute used to
there's a very good expositor joe keller
unfortunately no longer with us
either but he was a frequent visitor in
italy as well
so joe said when you give a colloquium
talk
you start very very slowly and so that
you still have all the audience with you
for about 20 minutes
and then you accelerate uh until you've
lost everybody by the end
okay so now we're at the point of
acceleration
and i hope not to lose all of you at the
end
so we'll start with the known uh
lorenz model sorry was there a question
no no i was wondering whether anyone
will still be with you at the end
but you are aiming to still have someone
with you good yeah i will i would hope
so
especially people who've heard this
already like stefano
and who's actually an expert and who
probably knows more about it than i do
and he's also a colleague of franco
flanders so you know
uh anyhow so um
here we go with laurence so
a random attractor of the stochastically
perturbed marine system
so the interesting thing is that there
was a pretty
hefty literature on these non-autonomous
and random dynamical systems when we
became interested so our first
publication
at the economy was
in 2008 and uh
at that time there was i showed you the
earlier references so certainly
like the early 90s uh the concept of a
pullback attract or
you know the existence of uh
of a global throwback attract under
certain conditions so as proved by carol
fandoli and there was a book by ludwig
in 1998 etc
so there was a lot of literature and a
lot of it was illegible
but there was a single numerical picture
and it was not as good as this one it is
it was due to
small foos in the cycle
and i forget the exact year so what
we're looking at here
is a snapshot in other words it's one of
those
cross sections like this one or this one
okay it's a snapshot of the random
attractor
computed at the fixed time t
and for the same realization omega
l using different initial data
and having pulled back okay for each you
know having pulled back like i said here
and you take a lot of initial data and
you let it rip
okay so and so
we use uh so we're made up of points
transported by the stochastic
flow from the remote task where t is
much larger than one
one being the characteristic time of the
system
we use multiplicative noise in the
deterministic lines model with a
classical parameter
values that give you give you the
strength attractor the deterministic
chaos
so the point here is you have to
understand that it is very different
from doing planck and doing
you know forward convergence or
probability density function
averaged over the realizations you have
a single realization omega
and it's still very interesting so
uh this is
the projection of the invariant
probability measure
at some fixed time t factorization omega
so we show a projection like this okay
and there is a multiplicative factor so
this is why it's multiplicative because
there's the x i
in front of the brownian motion and
there's an extra coefficient
alpha which is not small so this is not
a perturbative analysis uh you'll see
alpha can be 0.3 or 0.5 which is
large for the lowering system this one
has 10 million of initial points
okay and what you see here this is where
the origin lies
and this is the neighborhood so that's
the conv
a conductive equilibrium okay
uh when the
rayleigh number is is very small of zero
and these are the convective equilibria
which are destabilized and which
in the deterministic case have limit
cycles around them
and then other things happen and help
breaks loose if you see a similarity
with the cat map as far as i know it's
fortuitous and in any case we don't know
yet the reason for it and i think it's
an interesting question
if anybody likes this stuff
and here again apologies
for the different aspect ratio and
different
color scheme but here we have
one billion initial data and now you can
see
the beast looks even more ferocious okay
but aside from that what you do
see is something very different from
what
i or anybody else thought about the
effect of noise
on the strength of the butterfly of the
lawrence system before
if you use additive forcing okay
uh you expect essentially
the cantor set structure of the strange
attractor to be smeared out
by the noise and the resolution
to be lost here what you see is there
is cantaset structure and this is for
very high resolution and explored
mathematically
which is what the theory tells you it
says the invariant measure
is supported uh singularly
on the uh unstable manifolds
of the attractor and so we've tested
this very carefully
here's another one again different color
scheme but
in addition i will show you now another
video
which is somewhat more spectacular
than the previous one okay
so uh
uh again what is plotted
is the density oh i forgot to tell you
that uh in most of the literature this
invariant measure
that is observed this way is called the
disintegrated measure
which sounds very bad and very long
and we prefer the word that lysan
uses for it which is sampling measure
okay
so what you're looking at is that this
invariant sample measure
and you are going to see how it evolves
in time
this is pretty dramatic okay
you see essentially two things one is
that the measure is transported as you
see it now
by the deterministic portion of the flow
you see a jiggle which is a pure
stochastic stuff
and then you see things which are in
neither which are clearly the
non-trivial interaction
between the noise and the chaotic
dynamics
so i'm very sad to have to stop this in
the interests of time
but you can see it in two places
okay you can see it
either in a low resolution version as
a supplementary information in this
paper published in physica d
in 2011 so that was our second paper or
as a vimeo movie on the website of
mikhail shakrum
at ucla
okay so i hope that you find this
uh pretty fascinating and i still have a
few minutes
so i will try to show it for another
model
because actually the lawrence model is
not a terribly good model to think about
climate
because its spectrum is purely
absolutely continuous and there are no
lines nor
bumps so you have low frequency
variability
in the el nino southern oscillation
and that has a lot to do with what
happens with the atmosphere and oceano
tropical specific
i'll skip the explanation again if you
can find these pictures on the web
but there are two scalar indices
which capture the large cave behavior of
enso
one is a so-called southern oscillation
index
which is essentially a different of sea
level
pressures between the western and the
eastern boundary
of the uh tropical pacific or
uh by the so-called vineyard iii index
which is the temperatures sea surface
temperatures averaged over some
area around here and you see that these
two things the nino3 and the
soi are very highly anticoagulated
in spite of the fact that they're very
different animals one is the atmosphere
the other one is the ocean
they have very different spatial
properties so this really suggests
not that enzo is low low dimensional
as some people naively asserted
also about the climate system in general
but there is a lot of
variance in the large scales
so um enteroid prongs in noise
so the key idea is ancient angularity is
due to the interaction of intrinsic
periodicity there is a limit cycle that
would exist there in the absence of
seasonal forcing
but that interacts with the seasonal
cycle and with the noise
so this picture comes from essentially
a pde model i don't have the time to go
into the details
but it's a pde model
and this diagram shows
is what happens the behavior of
this model with respect to two
parameters
one of which is the coupling parameter
between the atmosphere and the ocean
and this is a parameter that has to do
with the ocean
and essentially gives you the
periodicity of the intrinsic variability
and the colors here indicate the
periodicity of the solution
at a particular point in this diagram
so black means period one in other words
you only see the seasonal cycle
green means period two
this is frequency ratio 0.5 means period
two
and then period three and uh period
four and period five and when
the system is uh
confused between two periods essentially
this is a generalization of the
well-known devil staircase
we called it the devil's bleachers of
devil's terraces
and then the system goes chaotic so what
you expect
is something like the imo tongues that
are highs or in the material equation or
something like that
and this is the effect of the noise on
the pdf of the animal tongue at one
third so one of these
sections to these lines in other words
there's a certain smearing out
and this stuff was looked at in much
greater detail in
recent people with stefano and with his
colleagues
in and around pisa which was published
last
year in the journal of statistical
physics
and this is something that maybe
stefano will tell me uh during the
session uh question session
be whether you've started working on
this because we had this picture
which we had not published but which
which
is what happens when you have two
um uh standard maps
uh coupled to each other and you have uh
this very fine structure of uh
you know crosswise resonances and then
you can look likewise to what happens to
that in the presence of noise
okay we started a little bit late so i
still have a few minutes i think
uh i'll try to go through the grand
unification and i think i'm going to
have to skip
the topological stuff which is
fascinating
anyhow the grand unification is here's a
classical strange attractor
so this is the ed lawrence view of
climate physically closed system model
mathematical is autonomous
neither deterministic if you want to
think of the anthropogenic forcing as
being deterministic nor
random in other words natural forcing
clearly volcanic eruptions are
intermittently occur at random times
cloud phenomena which are separate scale
can be very well represented as
stochastic
because we're really interested in the
net effect on the larger scales etc
so the attractor in this case is strange
but still fixed in time
so you know it's a analogous to an
irrational number
people don't think don't like things
that they're not familiar with
so the name irrational was given because
they didn't know about them before
so in this context of the law and
subtract the climate sensitivity is
still the change in the average value
first moment of the coordinates x y z as
a parameter lambda of the system changes
you have actually numerically what you
have
is exponential divergence versus course
gradient so class of your dynamic system
here is
when you have a positive epinephrine
experimental trajectories all the
trajectories diverge exponentially
but the presence of multiple regimes so
in other words two
red eyes of the beast
the two wings of the butterfly applies a
much more structured behavior in face
space
so this slide is from uh lenny smith
at the london school of economics and uh
and oxford you start with a pdf which is
concentrated at the origin
and you move between the wings of the
butterfly
but then every once in a while you know
there's a splitting
as the stuff comes down the z-axis and
decides whether to go
left or right and so in the intermediate
range
citing again john von neumann as i did
at the beginning
what you will see is multi-modal pdfs
although eventually they will converge
to a single bump
at the origin now random attractor
there's a physical open system model
mathematically as non-autonomous etc the
attractors so these are just
four snapshots from that video that i
showed you
the attractors pull back and divulge in
time so it's worse than your rational
it's like an imaginary complex number
climate sensitivity
is a change in the statistical
properties first and higher or the
moments
including actually distribution of
extremes at the attractor
as one or more parameters change so
again this is
comes from a picture also from that uh
this other encyclopedia where lenny
stuff was also published
so um i have to tell you that uh you
know grad unification is tongue in cheek
it was suggested to me by some
uh quote unquote press agent of some
one one of those uh
professional societies that pierre marco
mentioned that gave me some
some medal so they said you're going to
meet the press you have to tell them
something that
is going to catch their attention so i
think it's actually she who came up with
this term so anyhow what's the picture
there's a view of florence which i
already explained climate is
deterministic in autonomous but highly
non-linear
trajectories diverge exponentially
forward that's asymptotically
pdf is multimodal at least in the
intermediate range
klaus hussermann telus 1976 so this is
the same year
in which that murray mitchell paper was
published climate is stochastic and
noise driven but quite linear
so trajectories might be irregular but
they decay back to the mean
forward asymptotic pdf is multimodal
oops
right don't forget this grand
unification climate is deterministic
plus stochastic as well as highly
non-linear
internal variability and forcing
interacts strongly
change in sensitivity refer to both mean
and higher moments
what you're looking at here is also
coming from an enso model that i won't
have the time to show to you
but i can share the slides with fabio if
you like
this is again one snapshot in
in the
embedding space of a variable of the
model the depth of the so-called
seasonal thermocline
h of t and h of t plus one and so again
there is this filamental structure
and very regular distribution and
these things change either in time or in
with parameter or both so i'm going to
have to skip the work in progress
and i'll be happy to answer questions
i'll also have to uh this is you know
dedicated to
poincare after whom the institute where
we had that nice meeting were
held and i'm going to have to skip this
into interest on time
and again just pass through the
concluding remarks and the bibliography
so summary a change of paradigm from
closed autonomous systems to open
non-autonomous ones random attractors
are spectacular
i hope you'll agree after seeing that
uh useful i think we're trying starting
to show
that they they help really shed light on
the
underlying phenomena in the presence of
time dependence
and still just starting to be explored
for climate applications
work in progress study the effect of
specific stochastic parameterizations i
told you that the subgrid scale
phenomena
should really be parameterized
stochastically but until recently that
wasn't the case
and they were really parametrized
deterministically and now that that
parameter is stochastically still nobody
looks
people look at the effect that it has on
the simulations so the large case
simulations of the big model
but not how the behavior of the big
model is
modified systematically by such
things okay specifically on model
robustness applications and intermediate
models and general circulation models
implications for climate sensitivity for
predictability that again i didn't have
the time to discuss with you
they'll just leave you with some general
references
so uh one of the slides was taken from
this work which is done with stefano
pierini at napoli and
so this was a paper dedicated to the
90th birthday of peter lax
applying dynamic system theory to
climate problem and
dcds and this is the paper that stefano
gelato was asking me
earlier so this is really impressed it
should come out
uh any day or at least within a week or
two we've gone already for two hours of
proofs
and here are some of the other things
that i mentioned
um so
you have at least something to dip into
the literature with
so this is the
paper with valerio and
i'm done thank you very much
thank you
get out of this so that
i can see
okay okay thank you michael for this
nice talk
very stimulating and let's see if there
is uh
anybody who has questions or comments
maybe some questions
i'm just asking uh
you said somewhere that there is an
action
on the time maybe on the
sample space probably space
as a measure preserving transformation
is
such an action assuming to be ergodic
well it will not it will not be ergotic
if
okay that's a very good question thank
you very much
we we were able in the case of that
highly simplified wind driven ocean
circulation model
to prove that there is a global
uh compact attractor there's
disruptivity and there's a global
compact attractor
but numerically they are observed
very clearly the coexistence
of two local attractors with very
different stability properties in other
words
one is quiescent i this is going to take
time because
should i share again the screen to show
you that because i have the slide
okay uh
sorry
okay
right i have to find it
uh here you go okay
okay so uh
what is plotted here is not invariant
measure it's a measure of divergence of
trajectories
for each initial point in
a plane off
uh this four-dimensional space in which
uh
this system evolves and you see that
there is
uh there are points the clearly the warm
colors correspond to agitated behavior
and
the cold colors correspond to calm
behavior
okay and the two plots are
for two parameter values which measure
essentially the time constant component
of the forcing
so in the absence of the time dependent
forcing that i was mentioning
on one of the slides basically these two
panels
lie on either side of a classical
bifurcation a classical bifurcation
that occurs between a small
limit cycle and a large relaxation
oscillation
in the presence of the noise what you
see is that the space
occupied and the nature of the agitation
you know increases as you go through
that thing which becomes
a tipping point okay
francesco so you are saying that uh to
assume ergodisit is not a good thing
well it depends for what you know
i'm sure it's an excellent thing for a
lot of applications
including possibly climate not
everything in climate
is is bi or multimodal you know but
it's assuming it a priority
for a model that you don't know very
much about
is not a good idea
thank you okay
let's see if there is anybody else
yes can i make a question
sure please yes my name is alessandro
chiliti
so thank you very much for your very
nice talk
um let me say that i work on uh
dissipative systems since uh several
years but more on the field of celestial
mechanics
precisely on satellite subjects or to
tidal torques
the also from the theoretical point of
view
we did uh several words with raphael de
la llaver renato calleja
on a subset of dissipative systems which
are called conformal sympletic systems
we don't have stochastic we don't have
noise we have deterministic
dissipative systems and they wonder if
uh
um you can answer some questions which
uh trouble me since several
years let's say um so one of the major
problems i
find especially when doing um
numerical computations is to evaluate
the transition time
to land on the tractor so you shoot some
snapshots
i repeat transition time that i didn't
hear the words after them to land on the
tractor to arrive on the tractor
because you show in your in your talk
you showed
some snapshots at a given time
but in fact that do you know whether the
system
is already it's a let's say close to its
asymptotic behavior
that is it is close to the attractor or
can you measure
can you give a measure of the transition
time that you need
to go to the attractor this is
especially important when you are in the
sympathetic limit in the the weekly
distributive regime let's say
uh the second question is that you
mentioned that are not tongues
which i expect you computed
in the stochastic system so in a system
adding noise
and can you specify me whether it is uh
you mentioned the two couple the math
search
are they standard maps or arnold maps
so in the the
that picture which reminds me some
somewhat of what is called
recall i used to show it uh
many years ago when we were working on
that early work
uh along with a picture of a french
garden you know with the nice alleys
which intersect that
yeah straight angles and so on you know
including diagonals and so on
yeah those were uh those were standard
circle maps
and i don't know
stefano you want to say something about
whether there is some work now
proceeding on that
or
you mean about what
this is a this is a system which
comes from the interaction of these
circle maps and
noise and at the end is a
is a map of the circle plus noise
so at the very end this is a simple
system but still showing
oh we talked stefan and we talked with
you and angelo
about the possibility of uh you know at
some point
uh doing really two coupled uh
two couple circle maps which are
both active it's not that you're
averaging over one of them
okay so now this is technically possible
but we did not do it that
is a like mathematical result to get
mathematical results
yeah i know i think it's possible but
but
it's a bit more complicated i think it's
not trivial if you increase the
dimension of the face phase
let me then answer your question uh more
directly
i would appreciate it if you sent me an
email
because i've actually done work going
back
to 1992 published in psyop
with uh one of my last students at the
current institute gershwin volonsky who
has been appreciated i know it's
another paper very well it is one of my
favorites so
i'm so delighted i didn't think that uh
anybody
uh you know uh is still looking at that
but um so i know a little bit about the
separativity you know that the standard
assumption for people who haven't looked
at it
is of course the celestial mechanics is
why hamiltonian dynamics was invented
but uh there is small dissipation and
and so on
and uh we've done a lot of follow-up
work with
gershon and with uh with franz vardy who
unfortunately passed away some years ago
fairly young of a heart attack who was
both my student and my postdoc at ucla
uh on both on protoplanetary disks and a
lot of other stuff
and with bill cowler was one of the
greats of the field
on the on the great inequality
and so on so i don't know whether you
know that work not sure
yes um but uh
your immediate question we are working
we have a paper under review in
physics review letters with yuzuru sato
as lead author
and mikhail
the third member of the team not a lot
of importance
um in which we look exactly at this
question
you know i've been preoccupied by that i
mean numerically
we of course had to cope with that in
the first papers
uh with uh mikhail and with eric
2008 and 2011 where we dealt with the
circle map
and then with the lawrence with laura so
but in this paper with yuzuru
who happened to be a visitor at imperial
college when i was occupying
a visiting position in the math
department there
so to determine more precisely
how to know in advance how far you have
to pull back
so don't hold your breath you might go
blue in the face
because this thing has been going on
with yuzuru for a long time but i
and we were really supposed to do the
revised
version last month but you know what
happened
uh between march and last month so
it's not done yet but please stay tuned
okay thank you
thank you very much so i would send you
an email by the way
okay and tell me what you don't know
because
it looks like you know already some of
these papers so
okay thank you okay
thank you may had a a very small
comment on these uh which is related
also to
the work that michael uh
mentioned before on the end so
on the enso model
what happened and what generates this uh
irregularity
like arnold tongs and so on and so on is
that
in that paper we are looking to
invariant measure
stationary measure for the random sphere
system
and what happened is that
the convergence to equilibrium to the
system of course
if you look at the transfer operator
which is associated
the speed of convergence to equilibrium
depends on
the spectrum so and
what happened is that there are
eigenvalues coming to
radius one according to to
if the arithmetical properties of the
system
approach a a periodic one
so if you want to to measure the speed
of convergence to
the equilibrium you can look at the
transfer operator and
and see the eigenvalues but this is
maybe as too generally stupid to remark
but uh
this is the source of uh what you see in
this picture
yeah stefan sorry
this is clear to me um when you have a
circle map or even two
couple circle maps uh i have problems in
celestial mechanics which are more
dimensional and in that case it is not
trivial at all to measure
even the the opponent of exponents and
the transient time
because especially because you have a
fractal basis of attraction for example
and
uh a periodic orbits in the small
dissipation limit become extremely
entangled and so it is difficult to
to give a a reliable measure of this
transient time
yeah uh actually the big surprise in
that paper
with geshen walansky was that on the
intermediate time scale
we were getting a strange attractor
so that is essentially
related to what you are talking about
okay well uh let's see if there is
anybody else who wants to ask a question
otherwise i think we have been
well going on long enough
today and so i'm glad we managed to have
this
event and i thank again michael for his
talk
all of you for uh attending the
the colloquium and maybe hopefully we
can meet again
for a talk on this topic
in presence in a in a year or so
okay thank you very much for inviting me
okay
bye bye everybody