Science Series Part II: Synchronisation of Chaotic Networks

This is the second edition of my three-part series about the science behind my work at the University of Maryland TREND program (Training and Research Experience in Nonlinear Dynamics). I am currently working on the experimental side of the field of nonlinear dynamics and chaos. These are my posts:

Part 1: Nonlinear and chaotic systems

Part 2: Synchronisation of Chaotic Networks

Part 3: Nonlinear networks at my lab

In this post I would like to talk about the development of the field of research in synchronisation of chaotic systems. I would like to do this by talking about the work of Lou Pecora and highlighting some of his important papers.

Lou Pecora
Lou Pecora is a researcher at the NRL (Naval Research Laboratory) in Washington DC. He has done a lot of good work on synchronisation of chaotic systems and has worked a lot with Thomas Carroll (Also from NRL) and Francesco Sorrentino (University of New Mexico) on this subject. My Professors Thomas Murphy and Rajarshi Roy have also collaborated often with him and he even came to give a talk here during my program.

The nice thing about his work and the way that I will highlight it here, is the line followed of increasing generality and solving broader and broader problems. Of course these papers are not solely the result of his own work, but were in collaboration with others, but since his name is the only one appearing in all of these papers, it is reasonable to focus on him. Here is an overview of the papers I would like to expose:

• [1] "Synchronization in Chaotic Systems", Louis M. Pecora and Thomas L. Carroll (1990)
• [2] "Master Stability Functions for Synchronized Coupled Systems", Louis M. Pecora and Thomas L. Carroll (1997)
• [3] "Cluster synchronization and isolated desynchronization in complex networks with symmetries", Louis M. Pecora, Francesco Sorrentino, Aaron M. Hagerstrom, Thomas E. Murphy & Rajarshi Roy (2014)
• [4] "Complete characterization of the stability of cluster synchronization in complex dynamical networks", Francesco Sorrentino, Louis M. Pecora, Aaron M. Hagerstrom, Thomas E. Murphy, Rajarshi Roy (2016)
• [5] "Synchronization of chaotic systems", Louis M. Pecora, Thomas L. Carroll (2015) (Review Paper)

"Synchronization in Chaotic Systems" (1990)
This short three-page paper kicked off the field of study of synchronisation of chaotic systems. People (Russian and Japanese researchers) had actually looked at this before, as Pecora and Carroll admit in their 2015 review paper [5], but, as so often in science, the last person to discover something is usually the one who is remembered. What Pecora and Carroll did was investigate what happens if they were to take the well known chaotic Lorenz system, which has three x, y and z variables, and run one Lorenz system, then feed the x-variable of this system into the x variable of a second Lorenz system. It turned out that by driving the second system with only one variable of the three, the second system would synchronise!

Image I took from the 1990 paper [1]. It shows how a driven Lorenz system synchronises,
and approximately synchronises for small parameter mismatches.

Why is this result special? This is unexpected because chaotic systems are very sensitive to small differences, and so you would expect that small differences between the driven and driving system's conditions would always amplify and so prevent synchronisation. This synchronisation is like only telling someone your latitude during your trip, and them figuring out your longitude and exact route as well, and after that they are suddenly walking next to you, pushing you off the sidewalk! They did some mathematics on the equations underlying this, did a stability analysis, some simulations, and invented a thing called a sub-Lyapunov exponent. I mentioned Lyapunov exponents towards the end of my previous post. This sub-Lyapunov exponent quantifies the rate at which a state in the driven system will approach or go away from the nearby state of the driving system. If all sub-Lyapunov exponents are negative, the system will synchronise, if one is positive, then the systems will never synchronise since whenever their states get close, they will diverge again. Note that the maximal sub-Lyapunov exponent is approximately the steepness of the line in the figure I enclosed above. If it is positive, differences will increase, if it is negative, solutions converge.

I don't have many good graphs or pictures at hand of examples
of synchronising chaotic systems, but here is the best I have:
two oscilloscopes from my experiment displaying the outputs of two nodes.
As you can see, their signals have synchronised, although if you look
closely, you will notice that they are in antiphase on the lower oscilloscope.

Lou Pecora and Thomas Carroll were initially not just interested in this for purely scientific purposes, but they also justified their work with the idea of finding a new method of encoding messages in these chaotic signals for Military or Naval communication purposes. See a demonstration here (at approx 1:50). This turned out to be an over-complicated way of achieving this and there are now better ways to do this. An interesting thing about their encryption though is that it could be achieved with analogue circuits and since digital communications were still not common place at the time, this was a useful thing. In their paper they continued to demonstrate this synchronisation in a real experiment with two electronic circuits. This is very common in literature on synchronising chaotic systems: showing something works on paper and simulations, and then proving that it is a real-world effect by using an experiment, which is always has imperfections compared to theory.

"Master Stability Functions for Synchronized Coupled Systems" (1998)
Soon people began to consider more general problems. Where Pecora's 1990 paper initially considered one system synchronising to another, questions were asked such as "What happens if we couple a system both ways?", or "What if we couple multiple systems?", or "Can we build networks of systems?". From this all stemmed the question: how can we generally predict global synchronisation for a coupled system? Global synchronisation means that we consider a network of multiple chaotic systems, and seeing if by coupling them, we can get all systems to synchronise together, i.e. global synchrony. To get an idea of global synchrony, watch this video of metronomes synchronising. Pecora and Carroll's 1998 paper gives a definitive answer to this question. They considered the following general system:

Here x represents the ith node, F is a function indicating each node has the same internal dynamics, G indicates that the network is connected by a matrix, and H is the function that determines how each node influences the rest.

These equations describe how a system consisting of identical nodes, coupled through connections defined in the matrix G, behaves. The question is whether this can synchronise. It turns out you can predict this with theory. What Pecora and Carroll did is invent a thing called the Master Stability Function (MSF). This allows us to separately analyse the shape of the network and the chaotic system which the network nodes represent. By analysing the network we can calculate so-called eigenvalues of the network. By analysing the system which makes up each node, we can get the master stability function. If all the eigenvalues of our network are to be found in the area where the master stability function is negative, then all sub-Lyapunov exponents will be negative, and our system can synchronise.

Example of a star network, which may, or may not synchronise

Below you can see that all the round dots, which represent the eigenvalues of a circular network, lie in the area that the MSF is negative, so that system will synchronise. One of the stars which represent the eigenvalues of a star-shaped network lies in the positive part, and so that system will not synchronise. By changing the coupling strength we can get these values to lie in or outside these negative areas, and so by making a system couple too strongly, it will not synchronise, and by coupling it too weakly, it will not synchronise either. The coupling has to be just right.

Picture from 1998 paper [2]. It shows the graph of the Master Stability Function

Cluster synchronisation (2014 and 2016)
Lou Pecora went further, and wrote a series of papers with Francesco Sorrentino, going beyond general synchronisation. He considered: What if clusters within our network synchronise? When does that happen? This is detailed in two papers from 2014 and 2016, which I will admit, I did not fully understand, contrary to the previous papers, but I got the general point...!

It turns out that networks can support clusters of synchrony, or even clusters of synchrony with at the same time other sections being desynchronised! The conditions for this are determined by the symmetries of a network. To understand this better, you need to know some group theory. To put it simply, symmetries can induce so-called orbits in networks.

Orbits are collections of nodes that can be permuted to each other through a symmetry. These orbits can form groups that synchronise, because these nodes are essentially 'equivalent', since they are symmetric to each other. Although these networks in theory support synchrony, these synchronised solutions may be unstable. That is where the stability analysis from Lou's work comes in again, to show that these clusters can be stable. Here is a nice paper from Joe Hart, the graduate student at our lab who helped me so much, where he demonstrates all possible clusters in a four-node network!

Example of a network, where I gave the nodes in the same orbits, the same colours.
There are therefore 4 orbits. Can you find all 8 symmetries?

Weeks 6,7,8,9,9.5: Travelling around, but no wheels, Duncan comes

I am writing this post by my bedroom window which, despite the airconditioning in my flat, is wide open, so that I can hear and enjoy the outside summer noises. I have now been here for almost eight weeks and I am settling in here well, getting used to living in the USA. My research on the other hand, is going a bit pear-shaped. Duncan is coming on Tuesday and I will be heading off in two weeks time!
...time passes...
I am now writing this post on Wednesday evening, and Duncan has come. Things are speeding along and here is an overview of what I have been up to!

New York + Visiting Jutta
The week before last I headed off from work early on Friday, to Union Station in DC, took the "Washington Deluxe", a pretty standard but otherwise good bus, straight to New York. In New Jersey, as the sun set, the Big Apple started to loom in the distance, with my bus gliding towards the city, marked by the towers which, planted in the city, extended high into the night sky. After going through the tunnel to Manhattan, stepping out of the bus, I found myself on the corner of 36th St and 7th Ave, and made a dash to the Port Authority bus station. On the way I was impressed by the atmosphere of the old, brick Manhattan buildings, with modern cafés in between, and the eclectic business of 8th Avenue. I jumped into my commuter bus, swooping me out of the city, to Demarest, New Jersey, on the other side of the Hudson River, to see Jutta, my lovely host. As soon as I saw the sign "Demarest", I pressed the stop button, confirmed with the bus driver this was Demarest, and jumped out.

I discovered that although this was definitely Demarest, I was nowhere near the Duck Pond bus stop, where I was meeting Jutta-I had jumped out way too early. Some walking and guessing remedied this and soon I found her and we headed off to her place. Jutta is an old family friend. I last met her in 2010, when I travelled to New York for the first and last time since I was born there! She is from Westfalen, in Germany and emigrated to the USA in 1960, and has known my grandparents for a long, long time. My father was even sent to stay with her in 1978 as a fifteen-year old! She made some lovely dinner-sandwiches for me and supplied me with ample dessert. After having a lovely post-dinner chat with her, I headed to bed, and slept like a log.

This bowl I came across just comes to demonstrate that drawing technique has not changed
much over the course of 3000 years.

On Saturday I headed into town, sauntered around on the busy 7th avenue, took a subway to get to the Metropolitan Museum of Art, accidentally went 50 streets too far, made my way back, traversed Central Park, and entered this museum with a member's ticket under the guise of Jutta Greweldinger, a patroness of the Met since 1978. Apparently very convincing, because I got in! They had many interesting things to mention, a vast collection, but two things I remember well was their section with Cypriot artifacts from pre-hellenic times, which I was very fond of, and their portraits and landscapes from Dutch artists in the 14th, 15th and 16th centuries. Even though the Dutch golden age was in the 17th century, these paintings were already very refined and impressive.

View from Brooklyn Bridge

After this, I headed down to southern Manhattan, to where the Staten Island Ferry leaves, and had a good look across the sound bay, then went on, marched across the Brooklyn bridge, and went to a small little barge by the water, which was currently being shouted over by some extremely loud concert nearby. There I attended the a chamber music performance at Bargemusic, with the performers being the director of the place, Peskov himself, a Ukrainian violinst, and a piano player. Together the played various duets. Peskov had quite an informal, jokey manner, and even needed to check the program to see what he was going to play 'Let's see what I will play today...'. I then headed off to Port Authority, got very lost in the metro along the way, and came back to Jutta.

Sunday
That morning I got up and watched the football world cup final with Jutta. Good game, pity France won, we were both routing for Croatia! Afterwards I said goodbye, went back to NY, and thoroughly investigated Times Square, which turned out to be quite a small square, with way too many tourists. Not much there, definitely not the center of the world. I then visited St. Patrick's Cathedral, which was quite stunning. St. Patrick's Cathedral is where I was baptised in 1998! It was really cool being there. They happened to be playing an organ concert at that moment.

St. Patrick's Cathedral from the inside

After this I passed the Empire State Building, got some reeeally cheap pizza slices, scoffed it down, and bussed back down to DC. 't Was a good weekend. I much prefer NY to DC. The people are more warm there, although they can also be outright. Better to show what you feel, than be cold I think! I could imagine returning to work in the future! On top this, I also think that the New Yorker accent is adorable. Once back at home I said farewell to John, my good flatmate, who was off to Alaska and finishing up in DC. He was thrilled to hear about the huge buildings in NY and even more thrilled to go shoot guns and see his girlfriend again!

Farewell to the Empire State of Mind

Disaster strikes: Bicycle Dismembered

Things were going all well the following week...until Tuesday morning I discovered someone had taken my bleeding bike wheels!! Both bloody wheels!! Parked right outside my bleeding bloody door! I took it as it came and the next day, I found myself doing the walk of shame with my considerably lighter bike across campus to the bike shop to explain what happened. They replaced my bike and only charged 80$ so good on them, thanks for the kindness!

The last thing you want to see on your Tuesday morning

Weekend with Verwiel Family
The weekend after this, I headed off to Baltimore Airport, where I met Frank Verwiel, an old friend of my father's, who was picking up his daughter Laurine. Together we headed off to their stunning weekend house at the east shore of Chesapeake Bay in Royal Oak, MD. Over there I met his wife Francine. Unfortunately, it was rainy the whole weekend and so we weren't able to enjoy the outdoors too much. I had a pleasant time talking to them. Even though Frank and Francine emigrated to the USA a long time ago, they have managed to completely raise their children in Dutch and so I actually only spoke Dutch the whole weekend!

Seeing Cecil
I have an old friend. Quite an old friend. To be exactly, I met Cecil in Brazil when I was 7 :-p. Since then we have seen each other a few times. He happens to study at Georgetown University in DC and was in the area, and so we met up, we met up twice! We had a good conversation, caught up, reminded each other of our parents' names etc. and generally had a good time inspecting Georgetown during my first visit, and DC monuments the second time.

Some people stand on shoulders of giants.......I sit on their lap [Einstein Monument]

Cecil-Francis and I at the WWII monument

Cycle to DC and Greek cooking
Next weekend I cycled to Washington DC on my proud refitted bike which had wheels again! I took a long bicycle trail which leads all the way from where I live to the East of DC and from there, headed on to the capitol. There I hung around, had a pleasant time, got a milk shake, ascertained that I was satisfied, and headed back. It's an 11.8 mile cycle there. At home, I baked Cypriot Halloumi bread according to the recipe that my friend Christos Kourris from Edinburgh gave me: mix halloumi cheese with dough and onions, and bake it. It was pretty good, although there is space for improvement, which is a good thing!

Ma' dow

Ma' loaf

On Sunday I also went to the National Shrine of the Immaculate Conception in Washington DC, which is a long name for the biggest Cathedral in North America! It was a very impressive church and the organ so loud, and the setup so overwhelming, that mass almost felt like a movie, or a show!

Outside

Inside

Chapel dedicated by Irish to St. Patrick

Duncan visits
Last week Tuesday, Duncan came to visit me! My phone had just broken down, and so all the coordination from then on had to be done with old fashioned e-mail and co-ordination. We looked at monuments in DC, went swimming at the pool, enjoyed diving, in the weekend looked at DC again, swam and dove more, and had a good time in each other's company. I even took him to an astronomy talk and he saw my lab and met some of my research mates! Monday morning he left off and I will be seeing him Sunday, when I return to Ireland, which I am really looking forward to. Since my phone broke down, I have no pictures as evidence. Duncan came just at the right moment because that weekend, my good flatmates Nishad (from Caltech) and Josh (Alaska/Arizona) left off, as well as Jessie (Alaska/Washington). Now it's just Kwazi and me camping out in our flat.

How have I been lately
I have been pretty good. On the one hand I am settling in here and finding my rhythm even better, on the other hand I am really looking forward to having some holiday and seeing family & friends again. I suppose that is double good! My work has been progressing steadily and I have pretty much rounded it up by now. I will be presenting on Friday at a Symposium with the other REU's.

Last Friday I started thinking about how symmetries in dynamical networks induce symmetries in their statistics, and how looking at the statistics allows us to infer symmetries! I have been thinking about it and I think it is pretty exciting. It needs a lot more work but it would be a new way of finding something out about the shapes of a network by looking at the nodes. I am not sure if other people have done this before though, I need to verify that. My professor seemed to be pretty interested. I have also made a research poster about my work, find it below. I posted it on a separate website so you can zoom in.

Also I just found a gem of a music video. It is the song "Freight Train", written and played by Elizabeth Cotton. This song has been covered many times by country music players. She wrote it around 1906 and in this video she is actually 90+ and performing in her own home! Take note of the fact that she is left-handed and so is playing this right-handed guitar upside down, plucking the basses with her fingers!

Science Series Part I: Synchronisation of Chaotic Networks

I am going to run a three-part series about the science and mathematics underlying my summer research at the University of Maryland TREND program (Training and Research Experience in Nonlinear Dynamics). I am currently working on the experimental side of the field of nonlinear dynamics and chaos. I will explain this is as the series progresses. The posts will build up so that when I talk about my work in the third post, I can reference to the earlier ones. I will talk about the following subjects:

Part 1: Nonlinear and chaotic systems

Part 2: Synchronisation Dynamical Networks

Part 3: Nonlinear networks at my lab

What are Models?

Physics is somewhat different to other sciences: instead of seeking to categorise and inventorise the natural world, it seeks to reduce natural phenomena to their essence and describe the behaviour of these processes. Physics mainly proceeds with models. These are, in the most general sense, (quantitative) assumptions, which combined with mathematics, can be used to model these behaviours and make predictions. Often the hardest part is to find the right assumptions for one's model, and then the mathematics falls into place. Other problems have simple assumptions, but are harder on the mathematical side to solve. The hardest problems in general, are those with non-obvious, complicated effects combined with tricky mathematics.

Differential Equations

Arguably the most important tool in physics is the differential equation: this is an equation that describes how, based on the assumptions of a model, the different involved quantities (numbers) influence and change each other. A simple example is the following: the rate of cooling of a hot object is proportional to the difference in temperature between its own and that of its surroundings. That is to say, when a stone is 200 degrees hotter than its surroundings, it will cool twice as fast as if it were only 100 degrees hotter. This is perhaps obvious. What is however not so obvious is that the time for the stone to cool from 200C to 100C is much shorter than the time it takes the stone to cool the last 100 degrees, since when it is hot, it loses heat faster, and so the hotter stone will need less than twice the time to cool down fully. Differential equations are necessary to make this reasoning precise. They can describe almost anything: from falling, to the pressure in the atmosphere, to the neurons driving your heart.

An important part of differential equations are the so-called initial conditions. These are the quantities with which we program our system to start off with. For example, the location and velocity of a skydiver, or the initial temperature of our stone. Setting initial conditions and evolving the system permits us to make predictions about what happens later.

The differential equation describing the cooling down of a hot stone.
On the left the rate of change of the temperature is proportional to the negative of the temperature on the right.
So a hot object loses heat more quickly!

Types of Differential Equations

Differential Equations pop up everywhere, and often, their shape recurs. The DE (differential equation) describing the cooling down of an object, for example, is the same as the DE that describes the pressure in the atmosphere. The DE that describes falling, also describes braking of a car! These are examples of so-called linear differential equations: they have a property called linearity that makes them relatively easy to solve and handle. All those outside of this category are called nonlinear. The DE describing the firing of your neurons is nonlinear. Physicists like to approximate everything to be linear for simplicity, but truth be told, most things are not linear. That is why studying these nonlinear systems is so important. The field I am doing research in does exactly that. Nonlinear systems can show all sorts of interesting behaviour that linear ones cannot: they can blow-up: i.e. go to infinity in a finite amount of time, they can have different equilibriums and they may even exhibit chaos.

The class of Linear differential equations is a tiny subset of the whole

Chaos
Chaos is a phenomenon that pops up in some nonlinear systems. To put it briefly, a system is chaotic when it is generally fiddly. More precisely, chaos boils down to sensitive dependence on initial conditions (Strogatz). A hot soup will take approximately the same amount of time to become luke warm if it were 90C instead of 91C. This system is not chaotic. Florida being 27C rather than 26.9C in April can however matter very much. It can totally change the weather, perhaps even be the difference between a hurricane and a good summer's day a month's down the line. This is called the butterfly effect: given enough time for the effect to propagate, a butterfly flapping its wings somewhere may induce a storm in another place! Weather is very sensitive on initial conditions and so is definitely an example of a chaotic system. A fascinating example of chaos is called the double pendulum.

Poincaré: Stability of the Solar System
The first time that the subject of chaos and sensitive dependence on initial conditions came up, was when the Swedish King Oscar II initiated a mathematics competition to prove that the solar system is stable. Poincaré's partial solution won, but on revising his work, he discovered a mistake. He realised that in the long run, tiny deviations in the orbits of the planets, will cause two seemingly similar orbits to completely diverge. Since we cannot know the positions of the planets exactly, we will never be able to predict their movements into the far future. To this day, the stability of our solar system has not been mathematically proven! You can read more about this story and the work of other astronomers in "Newton's Clock: Chaos in the Solar System" by Ivars Peterson.

Three-body problem
The simple case that Poincaré examined is called the three-body problem: here we have three bodies, exerting gravity on each other, influencing each others movements. This is a famous Physics problem upon which many excellent mathematicians ranging from Newton to Lagrange have broken their teeth. Poincaré only examined the case of the so-called restricted three-body problem: here the third mass is a so-called 'test mass' moving in the same plane, meaning that it exerts no pull on the other bodies, and is completely at the mercy of their gravity, like a tiny satellite so-to-speak! As stated earlier, he found that tiny differences in the initial conditions would have huge differences down the line.

Restricted Three Body Problem simulation
I have made a Python program that simulates this problem, and exhibits chaos. I published the code here. It gives very interesting and attractive images! In my program, I placed two test masses very near each other. As the simulation progresses, the test masses diverge and the tiny initial distance grows exponentially, as can be seen on the graph. Once the distance between them reaches the top of the graph, they have completely separated and have started to move independently. This demonstrates sensitive dependence on initial conditions.

Here the Lyapunov exponent lambda is approximately 10/25s = 0.4 s-1 i.e. 50% divergence per second!

These divergence times may vary, for some orbits may be more sensitive to chaos than others. Below I have made a second simulation, with different starting conditions, where you can see that divergence can in fact happen a lot faster. Certain types of chaotic systems have the property that they diverge at an, on average, constant rate. These systems have a so-called Lyapunov Exponent, λ, which quantifies this rate. It is essentially equal to the steepness of the line on the logarithmic divergence plot. When a system diverges at lower than exponential speed, it has a Lyapunov Exponent of zero. This is an alternative definition for a non-chaotic system, i.e. a chaotic system must have a positive exponent.

Here the Lyapunov exponent lambda is approximately 10/100s = 0.1 s-1 i.e. 10% divergence per second!

Chaos as a buzzword

The term ' Chaos'  gets thrown around a great deal. It is important to stress that a system being messy or complex does not necessarily meet the definition of chaos. Chaos is about very tiny differences in situations being amplified to totally different behaviours in the long run. This amplification must be exponentially fast. Below you can see the exponential equation giving the average growth of the difference between a system's solutions. If λ is negative, the solutions will converge, meaning that they will go to some equilibrium together, if λ is positive, the solutions will diverge, regardless of how close they are.

This equation describes the approximate exponential growth between two initially close solutions, where λ is the Lyapunov exponent

Weeks 3, 4 and 5: Fireworks going up, boats going down

This weekly posting business hasn't really worked out so I will cease assigning blogs to fixed weeks. Below is a selection of interesting things that I have been up to lately. In other news, Tatiana, my colleague, has lent me a steel strings acoustic guitar, meaning that I have now naturally commenced studying 'Otherside' from the Red Hot Chilli Peppers. If anyone else has good song/piece ideas, please inform me.

During a visit to the cinema watching Deadpool 2 with John (front), Josh (middle), Nishad (rear),
I discoverd that American cinemas have lazy recliner seats for napping!

I have also just finished reading the book "The Name of the Rose" by the Italian author Umberto Eco. It is a grizzly and fascinating book about monastic life, the eternal battle against sin, the power of the medieval Catholic Church, whether Jesus and his disciples owned anything, and the English Franciscan William solving the murder mystery of a series of sinister killings of monks. I would recommend it with the reservation that it takes a while to get interesting and you have to be prepared to flip back and forth to the back to find the translations of many Latin quotes inserted in the text. If anyone has read the book and is interested in talking about it, please message me.

Petrushka by Symphonic Orchestra Institute

I bought tickets for a concert on the 24th of June in memory of the 100th birthday of the great American composer and conductor Bernstein. Upon arriving at the UMD Clarice hall were the concert would take place, I discovered that I was totally wrong, no-one was there, and I in fact bought tickets for a different concert, on the 30th. Oh well...

Orchestra performing Dukas' piece.
They projected video on the large decorated screen during Petrushka

One week later I found myself at the same place, same time, and now for the piece Petrushka, by Stravinsky. First the orchestra played "The sorcerer's apprentice" by the French composer Paul Dukas, followed by Symphonic Dances by Sergei Rachmaninoff, and then we came to the comedic piece Petrushka, the meat of the night. This was originally performed as a ballet and is about the plight of the Russian puppet 'Petrushka' at a carnaval. It is funny and grim at the same time. This time, the orchestra neither performed it as a ballet, or as a mere music piece. They got three puppeteers to play the puppets, combined this with smart use of live filming and pre-recorded video, and also used members of the orchestra when they did not have to play, using very witty props, hats and little moves to embellish the piece!

I must confess, I had started to fall asleep during the two earlier pieces but I was wide awake paying good attention during the last one because it was so intriguing. It is hard to explain all the individual parts that made the whole great but as an example I will give the random groups of musicians standing up during the carnaval at the start drinking wodka, one of the bassists coming forward and juggling with handkerchiefs outrageously for a few minutes later in the piece, and the funny dances that involved the entire orchestra standing up and changing seats in seconds!

4th of July
On the 4th we had the day off-quite relaxing. In the evening I went with Nishad, John and Josh to see the fireworks in Washington DC. It was quite exciting. After some initial delays to do with getting hamburgers, we made our way to the mall, where they barricaded the entire area around the Lincoln and Washington monument. This forced us to go through a narrow security check area, where hundreds of people were. Fortunately it went quickly! We sat beside the Washington monument, the large obelisk, looking right towards the Lincoln Memorial, where all the fireworks were going to be set off, it was very busy and a mighty event was about to start.

Panaroama view of where we sat

Fire 'em up

9pm came and dusk, then night came upon Washington DC. As soon as this had completed, around 9.15pm, faint sounds of a national anthem were to be heard from speakers far away, I was told by others, but they did not reach us. The first rocket went into the sky and exploded, illuminating the whole area and leaving a trail of smoke behind. Many others followed over the following period. I believe I even saw a hamburger, and a baseball shaped firework! At the end I believe I saw the letters USA, but because we were looking at it from the side, it was pretty much invisible..! The fireworks were really cool and it was a non-stop barrage. Then the end came. There was a dramatic outpour of people and we managed to scramble out of this mess which felt like an apocalyptic scene from a zombie movie, into the metro station Archives, where we met three Dutch girls from Maastricht, who were not so talkative, and then took off on a different train soon afterwards ;-p!

The star-spangled banner on our way out

Windtunnel
Last week we visited the local UMD windtunnel. First built in 1948, this thing has been in continuous use since then, and was even used to test the original Ford GT! The same technology from 1948 is still used to drive the wind in the tunnel. The scale which they use in the tunnel to measure forces on the test objects is also still the same to this day, with minor modifications. Unfortunately they were not operating the wind tunnel that day... Despite this, the windtunnel visit blew our previous campus nuclear reactor visit out of the water.

The enormous fan used the same propellor as a B-29 Superfortress,
used to drop the atomic bomb. 

The rear of the fan smooths the air movements out.

The actual windtunnel section was smaller because the large air currents
had to be compressed together for higher wind speed.

Nuclear Reactor
Earlier, we had the change to go to the campus nuclear reactor but that was a big let down. It was kept in the nuclear engineering and materials science building and was in fact not used to generate power. In fact, there is no longer nuclear engineering department! Changing the name is just too costly. I asked our guide what it was used for and it turns out almost no research is done there, it is mainly used for giving tours at and it is nice because in the winter months it keeps the building a bit warmer due to the heat it gives off.

Large reactors have large water cooling towers. This one was so small (250 kW), that it did not need cooling. The operator, an undergraduate student, seemed to be boring his brains out. I was left with the feeling that the reason the reactor is still operational is because tearing it down would be too costly although I was too polite and refrained from that question. On the plus side, we did see Cherenkov radiation, the green-bluish tinge given off by reactor fuel rods. We were forbidden to take pictures so I have nothing to show for it.

Canoe Chaos
This Sunday I went out to Little Seneca Lake together with my flatmate Nishad and colleague Keshav. We drove out to Black Hills regional park and went for a short hike, followed by canoeing. We briefly considered taking a more stable rowing boat but then decided to be bad asses and went with the canoe. I distinctly remember the guy saying "People usually don't capsize". I also distinctly remember the rentals man saying to the other "Maybe you should have explained how it works" the moment that we set off. You can guess where this is leading to. Jiggling around, often turning full 360-degree circles due to our poor boat control and co-ordination, we steadily made our way around the lake, with capsizing always a possibility.  On the last half-mile stretch back to our boat house all was going well, and then we flipped. I basically had no say in the matter. I have no idea what caused it but some tiny shift caused us all three to be in the water two seconds later with no consent!

An image I nicked of an ideal canoe experience on
the fateful Little Seneca Lake

Is there a way to right a canoe?
Turns out there is no good way to empty a canoe and get going again! General practice is to swim to the shore and then empty it there. If we had only known that! We were in a bit of a pickle, although the warm water offset this slightly and made capsizing quite comfortable. Luckily we had left our phones and wallets in the car but this still meant that we had no way of contacting the boat people to get some help, of which we were in dire need. Fortunately this Chinese family consisting of two parents and a daughter with no English at all, and a teenage son, with some English came along on a rowing boat towards our frantic waving.

"911 my boat sank, fish me out here!"
We told them to call the people from whom we rented the boats but, partially due to the language barrier, partially due to the confusion, they ended up calling 911. This ended up with Nishad, whilst still swimming, talking out of the water into their phone, confusing them, and being confused himself, not realising that he was speaking to 911. These darned people kept on offering medical assistance and to send officers when all we needed was someone to bloody well drag us out the water! In the meanwhile the Chinese dad had gotten to work and slowly but surely, managed to find a way to drag our canoe out of the water and empty it. This permitted me to climb in. Nishad made it into their rowing boat, but Keshav was still stuck and couldn't manage to hoist himself in/get lifted into the boat. Him entering the boat required manoeuvring to the side of the lake with much difficulty, permitting him to step in.

All's well that ends well
A lake patrol boat came in the end to pick them up and bring them back. I in the meanwhile had managed to retrieve a lost ore and had started to head back solo as there was nothing more I could do to fix the state of affairs. The two of them were dropped off at the harbour, awaited by three police cars, one ambulance, a fire truck and much hilarity. I subsequently graciously glided into the harbour with a slightly water logged canoe. In Nishad's words, I was the one who had "managed to save most face, because at least you looked like you knew what you were doing on the boat". We were told by the boat rental people that this was their most over-the-top capsizing ever, since no-one every calls 911. In the end we were all fine and made it off only being the subject of laughter of all, although with Nishad unfortunately losing his glasses.

Week 2 - American Museums, Trump Hotels and Reservoirs

Now that my third week here in the Maryland is almost over, I ought really to be embarrassed about the lateness of posting about my second week, but instead, I will be shamelessly unapologetic. I am going to be following up with another blogpost soon about my third week here. I was set back by accidentally deleting all my previous work, which led in part to this post to being delayed.

My workweek wasn't to eventful, although I would like to note that I have discovered the beautiful pool here. At the recreation center they have both a 25m and 50m pool. The 50m pool is not only enormous in length, but also so deep, that I am worried of being gobbled up by a shark from the depths, sharing the fate of a poor seal in one of David Attenborough's documentaries.

Visit to DC
Saturday was my 20th Birthday. During the day I went in to Washington DC to visit the museum of American history. I was impressed by their exhibition on the internment of Japanese-Americans in camps in the western USA during the Second World War. Basically, all Japanese in the West, including citizens, were put into prison camps, because they were mistrusted, in total disregard of their rights!

Flowers from the exhibition

On the way back we stumbled on a familiar and nowadays quite notable hotel... Looking at it you could really see that the owner must be an educated man with a refined taste..!

My Alaskan flatmates John and Josh

We also passed the National Gallery sculpture garden, with a cooling fountain and interesting sculptures.

My all time favourite sculpture which tricked you to think it was a 3D house from all angles!

A different take on Rodin's The Thinker

Upon arriving back home I found a package awaiting me. It was the kind of package that every person in the world dreams of getting when they arrive home, and I was the lucky man that day. It contained a birthday cake from Timea! Which leads me straight onto the topic of my birthday.

The mysterious package

Birthday
That evening we celebrated my birthday with people from the TREND program and my flatmates. It was good fun. Being my third 20th birthday party in a third country I thought it was a fitting end to the festivities. On top of this, I have received many lovely cards. I now have a stunning collection plastering my desk!

Moments before digging into the cake

I got two 60th birthday cards!

Reservoir Computing
Up until now I had been working on a thing called reservoir computing. I have now changed my topic to Dynamical Systems, more on that later. I want to try to explain this technology anyway. It is really amazing. One of the postgraduate students here, Jaideep Pathak, someone who I have worked with, has written a very interesting and readable paper on the subject.

From Lu, Pathak et al. (2017)

Let me give an example of a problem that it can solve and how a reservoir computer solves this. The double pendulum is a very chaotic, unpredictable, crazy system. Say I were to give you the angle at every moment in time of this system and asked 'can you give me the positions of every part of this pendulum at every time, using just this one value?'. This is possible with a good dosis of Maths and Physics knowledge, plus a computer.

A reservoir however can solve this problem without knowing anything about the system. The 'nodes' in the reservoir have values and at each time step they receive a new input from our input value and simultaneously affect each others values in a randomly preset way. So far this network is just a jumble of values, continuously shifting around, and it is not useful in anyway. The magic comes in the output layer: over a certain amount of time we measure the values that we want our network to predict. We then determine the optimal way of adding together the values of the nodes at each moment in time, to approximate the output as closely as possible. We can then continue to apply this trick for times where we do not know the output. It turns out that this works very well. Look at the image below, where a reservoir can predict the y and z values of a chaotic Lorenz system using just the x value.

From Lu, Pathak et al. (2017)

This is quite amazing and the reservoirs perform very well. The question is if they can now be applied to other deterministic, yet hard to predict systems, such as the weather. Both the weather and the Lorenz system display the so-called 'butterfly effect', where a small initial difference can totally change the future. That is why there is the hope that this can be applied to such tricky to solve systems!

Week 1-Touchdown in College Park

I am going to initially be running this blog over the course of my 10-week TREND Research Programme (Training and Research Experience in Non-Linear Dynamics) at the University of Maryland for the interest of and to stay in contact with my family and friends abroad. UMD is in College Park, a suburb of Washington DC.
Het spijt me dat deze blog niet in het Nederlands kan; dat is nou eenmaal nodig om het merendeel van mijn lezerspubliek te bereiken.

Schiphol Airport
On Sunday the 3rd of June, I said goodbye to my family at Schiphol airport. Having just been there returning from Edinburgh on Thursday, this would almost give the impression that I was trying to flee my family as soon as possible. I can assure you that this is not true! If that was the case, I would have gone straight from Edinburgh. My only two full days at home in Bussum were mainly filled with celebrating birthdays: that of myself and Isabel, two weeks ahead of schedule, and that of my pal August. Not too bad.

Goodbye to my beloved sister Marguerite!

Cycling in East Lothian with my girlfriend Timea before leaving Edinbugh

Arrival
After an 8 hour flight beside a lovely elderly Californian Lady, I arrived in a distinctly wet Dulles Airport, DC. I got out of the airport fast, skipping the long queue filled with waiting Europeans by using my American passport for the first time. I must say that I felt slightly indignant about the fact that the border guard did not give me the privilege of hassling me with the usual questions about who I thought I was, what I was doing there and why I would even consider going to the US! It seems that there are also disadvantages to travelling with an American passport.

Now getting to College Park was a bit more of a challenge due to the pouring rain and floods which had greeted me. I managed to hop onto an Uber outside the Terminal. You would think that having arrived in the USA no-one would know where this little country called the Netherlands is. Well my driver Barnabas damn well showed the contrary: this Ghanean driver was married to an Amsterdam lady! After an informative exchange regarding were to go and not to go in DC, and the finer points of Modern Physics and Relativity Theory, Barnabas managed to plough our way through the desperate weather conditions and to drop me off at my destination.

I immediately proceeded to address the matter of greatest importance in life: basic sustenance. Well, it turns out that College Park is a so-called 'food-desert' and finding a grocery store near a campus with more than 39,000 students is apparently tricky business. I proceeded to walk 1.5 miles in the rain to the nearest Whole Foods.

Having captured the most important items, I returned to find that there was a campus power cut, and so abandoned all plans, and got a huge meaty, 1200-Calorie Burrito and a big soft-drink at Chipotle, in true American-style.

Jetlag=Great
6 hours of Jetlag, is great. Waking up at 5.00 am means that you can run, explore, eat breakfast, call home, and read a book, and that all before work. It turns out there are some really nice running trails here in the area, provided they are not flooded... I followed this rhythm for the first few days of the week, although now I am once again finding myself struggling out of bed, and can confirm that the jetlag has disappeared. Jetlag may be great on the one hand, time difference is not. Being six hours behind means that I can either call in the morning, when I am groggy, or after work, by which time it is past 11pm in Europe, and those back home are getting groggy. Don't even bother calling in the evening...

Myself and the TREND 2018 Participants

Flatmates & Colleagues
There are 11 others in the TREND program. Most are living together. I on the other hand, am living with one other Physics student and two Alaskans interning at a government contractor. The Leonardtown Community, where I am staying is only 15 minutes walk from work. They are pretty good to live with, and I must say, I have learnt a lot about guns from one of them.

I was quite surprised at the start of the week that after work, everyone here seemed so mellow and content with just doing their thing and going to bed, really. I was getting so bored by Wednesday I practically forced others to socialize... I think I was being a bit impatient. People are settling in now and we have gotten to know each other a bit better. On Friday we had a potluck dinner, plenty of fun and good craic. I made pancakes, which everyone enjoyed, despite them calling them "crey-pes". My bacon and cheese pancakes where found to be slightly too unorthodox for their liking. This evening ended on the high note of me stepping on glass :-p ! Thank God, my foot was fine dragging me around DC the next day.

Visit to DC
Saturday afternoon I headed down to Washington DC by metro, to the so-called "Mall". The Mall is a huge park in central DC where many famous monuments and museums may be found. After the leaving the metro station Archives I immediately found myself, yes, before the impressive Archives of the United States, containing, among others, the declaration of independence, the constitution and the bill of rights.

USA Archives

Abraham Lincoln

We visited all the classics: we passed the Smithsonian museums, the Lincoln Monument and the Washington monument. Washington himself expressly disapproved of honouring leaders too much and yet they managed to build a 169m obelisk for him!

Special note for the Korean war memorial, where each statue of a soldier had such realistic and stunning facial expressions. I would also like to mention the beautiful speech from Lincoln's second inauguration carved on the inside of the Lincoln memorial.

The Korean War monument

The Washington momument is a lot, lot bigger than me in reality.

Later that day Kyle, one of my fellow TREND students, and I passed through the Hirshhorn museum and then got dinner at Chinatown. It struck me how the center of Washington was so full of expensive, and beautiful buildings but that once we crossed a street leading to Chinatown, the buildings suddenly became shabby and many homeless looking people appeared. I imagine that they are kept out of the city center somehow. The difference on both sides of the street was marked and like day and night!

Chinatown

On the way back from DC we met many people returning from the Washington DC Pride Parade. Many were dressed flamboyantly as you can imagine, while others were barely dressed at all!

Concert
On Sunday there was a concert given by the National Youth Orchestra at the local Episcopal church. It is quite a cute typical church building. I got to listen to an organist and many different woodwind and brass instrument players. Afterwards there was a free dinner and I even got to talk to two of the players over there, who invited me to come see their next concert Saturday!

One of the different groups under leadership of conductor Tiffany Lu

This post was a bit long and winding summation but there will be more structure in the next ones as things fall into place. I will also later tell more about my work, my colleagues and what I have been getting up to in and outside the lab. Saturday is my 20th birthday and hopefully that should be good as well. Just message me if you want my address.