I'm sad to say that this is the end of my posting here. I have finished my Masters and will be graduating on Friday with a First Class Honours in Engineering Mathematics. Whoop! I don't think I could have done it in quite as much style if I hadn't had this blog to blow off some steam and help my thought processes out.
It may be the end of this blog but as I go out into the wide world of possibilities I do think that perhaps one day I will do a PhD and then a blog of similar form shall emerge to keep me going through that. So one day we will be reunited in the struggle to understand a bit more about the world around us in a wonderfully engineering mathematical way!
Thanks for watching!
If you would like to make contact in the future please use: bp3TT.7@gmail.com (the email for me here) or comment on a post and I will get back to you.
Tuesday 12 July 2011
Monday 2 May 2011
Finish line
Here we are. I'm done. My thesis is written. My poster is printed. My webpage is uploaded and ready for viewing. And all one day before the deadline too! This makes today notable not just for Osama Bin Laden's death. :)
It has been a slog as I am sure you could tell. I managed to plough some understanding out of the murky subject of two-dimensional noninvertible maps. And now I pass it to you.
My webpage is available for your viewing pleasure.
On it are links to my poster and thesis. (Ah, just tried to upload my thesis pdf to it and it didn't work...I'll try again tomorrow so check for it then!).
*dusts hands off* Right that's a third of this year and 16.667% of my degree done. Onwards to viva and exams!
It has been a slog as I am sure you could tell. I managed to plough some understanding out of the murky subject of two-dimensional noninvertible maps. And now I pass it to you.
My webpage is available for your viewing pleasure.
On it are links to my poster and thesis. (Ah, just tried to upload my thesis pdf to it and it didn't work...I'll try again tomorrow so check for it then!).
*dusts hands off* Right that's a third of this year and 16.667% of my degree done. Onwards to viva and exams!
Wednesday 27 April 2011
Poster!
So, long time, no post. However these last couple weeks have been crucial for my thesis and all the extras and the next week is not going to be fun either since I hand in everything on Tuesday. *sarcastic yay!*
However, I did say I would put up my poster. Enjoy. Let me know how cool you think it is on a scale from -10 to 10. :-)
Soon I will link this to the website I am producing for the project so you have the complete picture of my year's work.
However, I did say I would put up my poster. Enjoy. Let me know how cool you think it is on a scale from -10 to 10. :-)
Soon I will link this to the website I am producing for the project so you have the complete picture of my year's work.
Friday 8 April 2011
On a less serious note
So, I realised just a few minutes ago that I haven't written about chaotic attractors in all their splendour. An incredible oversight on my part. These objects are achingly beautiful and intricately complex in form. They arise in phase space (I believe I have covered this) in certain parameter regions of systems. Their fractal geometry means that dynamics is incredibly sensitive on them and while trajectories will follow the path of the attractor (duh! attracting :-) ), there is no predicting which exact part of the attractor it will land on and follow round or ultimately where it will end up. The attractor is fixed, the trajectories are not. A nice succinct explanation is available here.
The one that ties in closest to my work is the Lorenz attractor. The three dimensional system for describing turbulence gives birth to an amazing structure. This attractor is where the 'butterfly effect' expression originated. The attractor itself is geometrically between two and three dimensions and winds itself around two points.
The image (from wiki) shows exactly this. Where the trajectories 'cross' in this two-dimensional visualisation is actually where they layer over each other - manifolds cannot intersect (the unstable manifold exactly describes the attractor). The view is better in 3d. The fractal (Hausdorff) dimension of the attractor is about 2.06.
Other systems with strange attractors are the Rossler system and Henon map. This page gives a nice description and a few images.
As a side note: for those of you who read and enjoyed 'Harry Potter and the Methods of Rationality' or 'Luminosity' I would recommend this strangely named (but you'll understand all too soon) 'Baby Eating Aliens' also from the Less Wrong family and the (there is a theme here, no?) 'Harry Potter and the Wastelands of Time' if you like a bit of complex, epic writing with lots of action and magic!
Enjoy! Also, something to look forward to: I hope to have my poster available for you lot next time I post. :D
Doh! Almost forgot. One of the lecturers at my university (Hinke Osinga) actually crocheted the Lorenz stable manifold! If you want to know how, read this!
Incidentally, my project supervisor is the Bernd Krauskopf mentioned in the article. And yes, it is one of their Christmas decorations. ;)
The one that ties in closest to my work is the Lorenz attractor. The three dimensional system for describing turbulence gives birth to an amazing structure. This attractor is where the 'butterfly effect' expression originated. The attractor itself is geometrically between two and three dimensions and winds itself around two points.
The image (from wiki) shows exactly this. Where the trajectories 'cross' in this two-dimensional visualisation is actually where they layer over each other - manifolds cannot intersect (the unstable manifold exactly describes the attractor). The view is better in 3d. The fractal (Hausdorff) dimension of the attractor is about 2.06.
Other systems with strange attractors are the Rossler system and Henon map. This page gives a nice description and a few images.
As a side note: for those of you who read and enjoyed 'Harry Potter and the Methods of Rationality' or 'Luminosity' I would recommend this strangely named (but you'll understand all too soon) 'Baby Eating Aliens' also from the Less Wrong family and the (there is a theme here, no?) 'Harry Potter and the Wastelands of Time' if you like a bit of complex, epic writing with lots of action and magic!
Enjoy! Also, something to look forward to: I hope to have my poster available for you lot next time I post. :D
Doh! Almost forgot. One of the lecturers at my university (Hinke Osinga) actually crocheted the Lorenz stable manifold! If you want to know how, read this!
Incidentally, my project supervisor is the Bernd Krauskopf mentioned in the article. And yes, it is one of their Christmas decorations. ;)
Wednesday 30 March 2011
Horseshoe Chaos
While I struggle with supposedly impossible inversions, I'd like to introduce you all to Smale's horseshoe. This object is where the definition of chaos originates yet it is a relatively simple idea.
You start with a square. Squish is down and pull it out to the sides then bend it at the halfway point so it looks like a horseshoe. Then replace it over the square so that the two lengths form vertical strips on the square. This is the basic transformation of the horseshoe map. It is also invertible. Take the bent strips, rotate back to horizontal, and unbend it then squish it in from the sides and pull from the top and bottom then you get the square again. If you do this inversion again you get strips intersection horizontally.
Repeatedly performing the forward transformations you get more and increasingly thin vertical strips: 2 strips become 4, 4 strips become 8 i.e. each strip gets two thinner strips within it at each iteration. Doing the same backwards, you get lots of increasingly thinner horizontal strips. The points in the strips are the points that remain in the set at that iteration. Most points leave after even just a few iterations in either direction.
If you overlay the two directions, the points at the intersections that remain in the square under all iterations (if you iterate infinitely in either direction). This is the invariant set. This set is a Cantor set (disconnected and unstable fractal).
If you labelled each point with 0 or 1 at each iteration you could describe the positions of all the points in the square e.g. .0101 means a point that is in the 6th vertical strip after four iterations (the right hand strip of the left hand strip of the right hand strip of the left hand strip in the first iteration) or
If you can see that...sorry if it's too small.
Since you can do this in both directions, the whole dynamics of any point in the square can be described by it's bi-infinite sequence of positions.
Knowing this we can construct any orbit we like and we know it will describe at least one point in the square.
So we can make a periodic point: ...010101.0101010... which alternates from the left to the right. Since we have infinite lengths in both direction, we can make (uncountably) infinitely many of these periodic points.
Or we can make a non-periodic point: ...010001101100000.1010100110111... by, for example, 'counting' in binary as above or just by adding the wrong value to a periodic point i.e. ...01010101.0101011...is non-periodic. There are (uncountably) infinite of these too.
Among this set of non-periodic orbits we can find at least one that comes arbitrarily close to the invariant set. This is trivial since you can take just part of the sequence that describes a point in the invariant set and it would come as close as you like!
I hope you are following since I have just demonstrated that Smale's horseshoe perfectly defines chaos.
Chaotic motion must consist of the following:
- infinitely many periodic orbits
- non-periodic orbits
- dense orbits - ones that come arbitrarily close to any point in the set.
There tends also to be the condition of sensitive dependence on initial conditions but this can be shown through the dense orbits - you can take two sequences that are the same for any arbitrary iterations and in the next they can end up in two different strips i.e. they may start very close but can end up very far apart.
Voila!
Strogatz does give a very basic introduction but for a more detailed approach read the first chapter of "Elements of Applied Bifurcation Theory" by Yuri Kuznetsov. Wiki also has its own explanation.
You start with a square. Squish is down and pull it out to the sides then bend it at the halfway point so it looks like a horseshoe. Then replace it over the square so that the two lengths form vertical strips on the square. This is the basic transformation of the horseshoe map. It is also invertible. Take the bent strips, rotate back to horizontal, and unbend it then squish it in from the sides and pull from the top and bottom then you get the square again. If you do this inversion again you get strips intersection horizontally.
Repeatedly performing the forward transformations you get more and increasingly thin vertical strips: 2 strips become 4, 4 strips become 8 i.e. each strip gets two thinner strips within it at each iteration. Doing the same backwards, you get lots of increasingly thinner horizontal strips. The points in the strips are the points that remain in the set at that iteration. Most points leave after even just a few iterations in either direction.
If you overlay the two directions, the points at the intersections that remain in the square under all iterations (if you iterate infinitely in either direction). This is the invariant set. This set is a Cantor set (disconnected and unstable fractal).
If you labelled each point with 0 or 1 at each iteration you could describe the positions of all the points in the square e.g. .0101 means a point that is in the 6th vertical strip after four iterations (the right hand strip of the left hand strip of the right hand strip of the left hand strip in the first iteration) or
Since you can do this in both directions, the whole dynamics of any point in the square can be described by it's bi-infinite sequence of positions.
Knowing this we can construct any orbit we like and we know it will describe at least one point in the square.
So we can make a periodic point: ...010101.0101010... which alternates from the left to the right. Since we have infinite lengths in both direction, we can make (uncountably) infinitely many of these periodic points.
Or we can make a non-periodic point: ...010001101100000.1010100110111... by, for example, 'counting' in binary as above or just by adding the wrong value to a periodic point i.e. ...01010101.0101011...is non-periodic. There are (uncountably) infinite of these too.
Among this set of non-periodic orbits we can find at least one that comes arbitrarily close to the invariant set. This is trivial since you can take just part of the sequence that describes a point in the invariant set and it would come as close as you like!
I hope you are following since I have just demonstrated that Smale's horseshoe perfectly defines chaos.
Chaotic motion must consist of the following:
- infinitely many periodic orbits
- non-periodic orbits
- dense orbits - ones that come arbitrarily close to any point in the set.
There tends also to be the condition of sensitive dependence on initial conditions but this can be shown through the dense orbits - you can take two sequences that are the same for any arbitrary iterations and in the next they can end up in two different strips i.e. they may start very close but can end up very far apart.
Voila!
Strogatz does give a very basic introduction but for a more detailed approach read the first chapter of "Elements of Applied Bifurcation Theory" by Yuri Kuznetsov. Wiki also has its own explanation.
Monday 21 March 2011
How many drafts...?!
So here we are, nearing the end of my project. Well, actually there are six weeks left til I hand in my thesis but it doesn't feel like it. It feels more like two. :S
I've been asked to do two drafts already. A third one is coming up next week and I'll redo the current one into a fourth. After that they combine into a fifth. Then will follow a sixth from recommended improvements. I'll probably create another couple just for kicks! So that's at leasts one a week...a draft roughly for each month I've worked on it. I'm drowning in them!!
Ah well, at least it means I have shiny pictures for you this week. My draft task this week was to tell the story of one side of my research (decreasing lambda) as purely in images as possible. Obviously the details will be missing but I hope you enjoy the obvious progression and changes in images as lambda varies.
My image report draft is available here.
In other parts of my Masters world I am winding down on lectures and winding up for revision ready for exams in nine weeks. I want to try something new for my revision this year - sort of step it up a level. If anyone has any recommendations that would be very welcome. I have a few ideas but would like to augment these with other influences.
I also have to produce a poster and a website for my Masters project but I'll share them here first to get opinions! I will of course add a link to this blog in my website (cheeky I know!); maybe they'll give me extra marks for initiative and knowledge sharing never mind the pretty pictures!
I've been asked to do two drafts already. A third one is coming up next week and I'll redo the current one into a fourth. After that they combine into a fifth. Then will follow a sixth from recommended improvements. I'll probably create another couple just for kicks! So that's at leasts one a week...a draft roughly for each month I've worked on it. I'm drowning in them!!
Ah well, at least it means I have shiny pictures for you this week. My draft task this week was to tell the story of one side of my research (decreasing lambda) as purely in images as possible. Obviously the details will be missing but I hope you enjoy the obvious progression and changes in images as lambda varies.
My image report draft is available here.
In other parts of my Masters world I am winding down on lectures and winding up for revision ready for exams in nine weeks. I want to try something new for my revision this year - sort of step it up a level. If anyone has any recommendations that would be very welcome. I have a few ideas but would like to augment these with other influences.
I also have to produce a poster and a website for my Masters project but I'll share them here first to get opinions! I will of course add a link to this blog in my website (cheeky I know!); maybe they'll give me extra marks for initiative and knowledge sharing never mind the pretty pictures!
Tuesday 15 March 2011
"To err is human"
And guess what?! I'm human. My error this time was such a small thing but it could have had huge consequences if I had not caught it when I had. Actually it would have been far better to have caught it much much earlier but I had no reason to see problems until this point.
I had successfully combined my manifold examination with the Julia Set calculations into one succinct image for each value of the parameter lambda. Nothing twigged until I sat there showing them to my supervisor and he commented that surely the attractors should actually lie within the prisoner set (the points that remained after the iteration). My manifolds also did not line up with the areas they would be expected to. This meant one of two things. Either I had some scaling issue (quite possible with all the jiggling around I had done to get them on the same image) or something was wrong with either side of the analysis (the MATLAB or DSTool code).
Since the DSTool code had been written by someone far more experienced in the area, it was to my code that I turned my attention to find the problem. Took me a while but (as the error rule says) when I found it it was easy to fix. I like to program quite efficiently and as I was testing the absolute value of Z at each iteration I had set
absZ = abs(Z)*abs(Z);
Now instead of square rooting this after I had tested it I had left it which meant that instead of just dividing by the square of abs(Z) at each iteration, it was dividing by the fourth power of abs(Z). Doh!
Thus all my Julia Set images were actually wrong. Re-running the (easily) corrected code showed just how wrong it had been. But it is fixed now and since then I have been working like stink to analyse the correct progression of Julia Sets.
One of the areas that I have now been progressing into was a direct result of the correction. Once the correct maps were generated, I added a further capability to zoom the section around the interesting circle region. This showed that the points within the Julia Set boundary were actually behaving very much like the stable set patterns seen by the PhD student I have been working with. So I have been expanding my analysis to include calculation of the stable set within the Julia Set boundary. This is likely to contribute to connectedness and other behaviours of the map I have already examined.
As far as I understand it, the stable set is the set of points that go to the attractor(s) via the stable manifold due to the existence of a saddle point. Some points, in the course of being iterated, will end up on the stable manifold or on the saddle itself before slowly reaching the attractor. The stable set is all the points back in time that forward in time end up at the stable manifold (thus more pre-image calculations are involved!).
I have calculated this for a few values and they show promising information - I would have inserted images here but the server is rejecting them...
They should help to determine what the hole actually does to the map at least in the limited sense that I am examining it.
Oh, also I would also have shown you the bifurcation progression through lambda but since the server is rejecting my images I won't be. I'll try next week. Sufficed to say that there is a lot going on. I'm not entirely sure I'll find it easy to stick to the 45 page limit. If anyone has any recommendations for that (things that they would find interesting so think should be included) please let me know. :-)
I had successfully combined my manifold examination with the Julia Set calculations into one succinct image for each value of the parameter lambda. Nothing twigged until I sat there showing them to my supervisor and he commented that surely the attractors should actually lie within the prisoner set (the points that remained after the iteration). My manifolds also did not line up with the areas they would be expected to. This meant one of two things. Either I had some scaling issue (quite possible with all the jiggling around I had done to get them on the same image) or something was wrong with either side of the analysis (the MATLAB or DSTool code).
Since the DSTool code had been written by someone far more experienced in the area, it was to my code that I turned my attention to find the problem. Took me a while but (as the error rule says) when I found it it was easy to fix. I like to program quite efficiently and as I was testing the absolute value of Z at each iteration I had set
absZ = abs(Z)*abs(Z);
Now instead of square rooting this after I had tested it I had left it which meant that instead of just dividing by the square of abs(Z) at each iteration, it was dividing by the fourth power of abs(Z). Doh!
Thus all my Julia Set images were actually wrong. Re-running the (easily) corrected code showed just how wrong it had been. But it is fixed now and since then I have been working like stink to analyse the correct progression of Julia Sets.
One of the areas that I have now been progressing into was a direct result of the correction. Once the correct maps were generated, I added a further capability to zoom the section around the interesting circle region. This showed that the points within the Julia Set boundary were actually behaving very much like the stable set patterns seen by the PhD student I have been working with. So I have been expanding my analysis to include calculation of the stable set within the Julia Set boundary. This is likely to contribute to connectedness and other behaviours of the map I have already examined.
As far as I understand it, the stable set is the set of points that go to the attractor(s) via the stable manifold due to the existence of a saddle point. Some points, in the course of being iterated, will end up on the stable manifold or on the saddle itself before slowly reaching the attractor. The stable set is all the points back in time that forward in time end up at the stable manifold (thus more pre-image calculations are involved!).
I have calculated this for a few values and they show promising information - I would have inserted images here but the server is rejecting them...
They should help to determine what the hole actually does to the map at least in the limited sense that I am examining it.
Oh, also I would also have shown you the bifurcation progression through lambda but since the server is rejecting my images I won't be. I'll try next week. Sufficed to say that there is a lot going on. I'm not entirely sure I'll find it easy to stick to the 45 page limit. If anyone has any recommendations for that (things that they would find interesting so think should be included) please let me know. :-)
Saturday 5 March 2011
Inconvenient Delays
So here we are a month since I last posted. I would explain but that would feel like giving excuses and I'm trying to get out of the habit of feeling guilty for things which were out of my hands. I do however have something for you to read and perhaps enjoy.
My most recent deadline has been for a draft masters thesis for my project. This is available for you to read via this link.
What I have written covers about a third of the area of work I have been doing. It is very minimalist and basic with very little of the style or continuity that I duly put in my work. My introduction and background are perhaps not as full as I would like but maybe you can provide some thoughts on that since it is aimed at bringing people of different ranges of knowledge on the subject up to speed before I share my analysis. If there is any point of logic missing please do let me know.
The version I am sharing is the version I would have like to share with the examiners but my supervisor had other ideas so I actually submitted a stripped down version that is even worse so I will not be sharing that.
One point which you should all love is all the images generated that I have included in the report. In my time I have analysed the Julia set changes, the stability changes, the manifolds and most recently the full bifurcations in one half with the other half due to be completed this week. I have finally reached the crux of my examination of the map which is a strange one. The circle hole in one half has an important bearing on the location of the attractor which "disappears" and yet still points remain in the set - this is only possible with an attractor still present yet at some point there doesn't even exist a higher period attractor, just saddles and repellors. Thus this week my question to answer is whether an attractor still appears to exist for the points under the mapping despite the properties attributed to it or if the boundary of the hole itself becomes the attractor or neither of these things (which would mean either a waste of half a year's work or something totally cool and new).
I'll let you know next week hopefully and also provide you will the full bifurcation diagram.
My most recent deadline has been for a draft masters thesis for my project. This is available for you to read via this link.
What I have written covers about a third of the area of work I have been doing. It is very minimalist and basic with very little of the style or continuity that I duly put in my work. My introduction and background are perhaps not as full as I would like but maybe you can provide some thoughts on that since it is aimed at bringing people of different ranges of knowledge on the subject up to speed before I share my analysis. If there is any point of logic missing please do let me know.
The version I am sharing is the version I would have like to share with the examiners but my supervisor had other ideas so I actually submitted a stripped down version that is even worse so I will not be sharing that.
One point which you should all love is all the images generated that I have included in the report. In my time I have analysed the Julia set changes, the stability changes, the manifolds and most recently the full bifurcations in one half with the other half due to be completed this week. I have finally reached the crux of my examination of the map which is a strange one. The circle hole in one half has an important bearing on the location of the attractor which "disappears" and yet still points remain in the set - this is only possible with an attractor still present yet at some point there doesn't even exist a higher period attractor, just saddles and repellors. Thus this week my question to answer is whether an attractor still appears to exist for the points under the mapping despite the properties attributed to it or if the boundary of the hole itself becomes the attractor or neither of these things (which would mean either a waste of half a year's work or something totally cool and new).
I'll let you know next week hopefully and also provide you will the full bifurcation diagram.
Saturday 5 February 2011
Weird Manifolds
So this week, I've been using the new software (which is working well now) to calculate the invariant manifolds of the map I've been researching. The best way to think of manifolds are the shapes/curves that the system points hop around on as they iterate under the mapping. There has to be saddle points in the system in order that the stable and unstable manifolds can cross and then be calculated. The calculation to find the manifolds requires tracking backwards in time the path of the saddle point...which shouldn't be possible in a noninvertible system. This is where the weird stuff comes in especially in the case for my map.
For some points, there are no points to track back to or four and for all others there are two possible previous values. Currently the software can calculate two possible values and then for each carries on the path back taking either the 'positive' or 'negative' - whichever is the most likely to be the previous value. It does factor in the no pre-images case but not yet the four possible values. That is part of the work I still have to do this year.
It helps to see some images so here are some I calculated just this week!!
The red line is the unstable manifold and the blue is the stable. The point where they cross is a saddle point in the system and the other cross is the repelling point. I've only shown the positive manifolds (one side of the possible previous values) but the negative manifolds are just the reflection in the x-axis.
For some points, there are no points to track back to or four and for all others there are two possible previous values. Currently the software can calculate two possible values and then for each carries on the path back taking either the 'positive' or 'negative' - whichever is the most likely to be the previous value. It does factor in the no pre-images case but not yet the four possible values. That is part of the work I still have to do this year.
It helps to see some images so here are some I calculated just this week!!
The red line is the unstable manifold and the blue is the stable. The point where they cross is a saddle point in the system and the other cross is the repelling point. I've only shown the positive manifolds (one side of the possible previous values) but the negative manifolds are just the reflection in the x-axis.
Monday 31 January 2011
Given all the givens this might be relevant
After my post yesterday, I had a thought, "What about all those other areas of interest I mentioned I would talk about originally?" and since phone hacking has been on the news quite a lot, it seems pertinent to share a group project with you.
This year I studied a module called Information Security lectured by the Computer Science department. These lectures went alongside a group research project on a related subject of choice. Our group chose, "GSM Security". GSM is the global standard system for mobile communications and because of this the system's security is important to 90% of the population of the world (that means you!). GSM has been widely studied both academically and also practically (by ethical hackers).
Our task was to present an up-do-date image of the security measures, vulnerabilities and threats on GSM. The paper produced is available for you to read. It's not as detailed as we originally made it because we had a page limit (lame!) but it gets the main points across. We also got a fantastic mark for it!
After reading it you will have a better idea of how people (including journalists) can access personal information from your phone, listen into calls or prevent use of the phone.
Now, a disclaimer: I don't know if any of these methods presented were actually the ones thought to be used by the accused journalists or if they are guilty of phone hacking. I'm just presenting our research.
This year I studied a module called Information Security lectured by the Computer Science department. These lectures went alongside a group research project on a related subject of choice. Our group chose, "GSM Security". GSM is the global standard system for mobile communications and because of this the system's security is important to 90% of the population of the world (that means you!). GSM has been widely studied both academically and also practically (by ethical hackers).
Our task was to present an up-do-date image of the security measures, vulnerabilities and threats on GSM. The paper produced is available for you to read. It's not as detailed as we originally made it because we had a page limit (lame!) but it gets the main points across. We also got a fantastic mark for it!
After reading it you will have a better idea of how people (including journalists) can access personal information from your phone, listen into calls or prevent use of the phone.
Now, a disclaimer: I don't know if any of these methods presented were actually the ones thought to be used by the accused journalists or if they are guilty of phone hacking. I'm just presenting our research.
Sunday 30 January 2011
A Pause
There are three very good reasons I haven't really got anything good to talk about this week:
1. I have been wrestling with 32 bit compile on 64 bit architecture in order to get the essential program I need for the next stage of my project installed, compiled and working. It has only just been sorted with some fantastic work-arounds.
2. I have had every afternoon in the last two weeks filled with back to back lectures, mornings with exams and a presentation (which I mentioned last time) as well as work for those said lectures.
3. I have managed to slice open (a bit) the knuckle on my right hand thumb on Wednesday and thus typing, writing and pretty much anything is pretty hard now that I have to do it all left-handed.
Poor excuses, however next week I will dazzle you with fantastic images relating to invariant manifolds. You can read about them if you would like something to do that is more productive than reading this.
1. I have been wrestling with 32 bit compile on 64 bit architecture in order to get the essential program I need for the next stage of my project installed, compiled and working. It has only just been sorted with some fantastic work-arounds.
2. I have had every afternoon in the last two weeks filled with back to back lectures, mornings with exams and a presentation (which I mentioned last time) as well as work for those said lectures.
3. I have managed to slice open (a bit) the knuckle on my right hand thumb on Wednesday and thus typing, writing and pretty much anything is pretty hard now that I have to do it all left-handed.
Poor excuses, however next week I will dazzle you with fantastic images relating to invariant manifolds. You can read about them if you would like something to do that is more productive than reading this.
Sunday 23 January 2011
Pwetty Pictures and a Presentation
So, this week when I thought about what to blog I left it pretty last minute. This means that instead of an (usually very interesting) insight into chaos I will share with you the presentation I gave on Friday for my project to students and lecturers. I have left the notes attached so you can follow what I spoke about as well.
One moment from someone else's presentation I would like to share is this:
Brilliant!!
The presentation itself was about business intelligence which does not really interest me but the image brought some light humour to the talk.
One moment from someone else's presentation I would like to share is this:
Brilliant!!
The presentation itself was about business intelligence which does not really interest me but the image brought some light humour to the talk.
Saturday 15 January 2011
Nonlinearity gets all the fun
Nonlinearity is inherent in most systems before you make an assumption and reduce the beautiful complexity down to a boring bog standard linear system. There is a section in Strogatz that shows probably the most interesting thing you can do with linear systems (Chapter 5 section 3 page 138 onwards). And I only read it once because linear systems were dull and pretty repetitive long before A level.
Nonlinearity is pretty easy to understand. Take for example square numbers. You can make 4 by either multiplying 2 and 2 or -2 and -2. This means that, under the square operator, 4 has two (well, four really, but two are identical to the other two) possible previous values. This makes squaring a number nonlinear because tracking back you do not know whether two 2s or two -2s made the 4 you see at the moment. This particular brand of nonlinearity is called noninvertibility. You still following me?
Good. Right, to take this further we call the previous values of each number pre-images and the operation, a mapping. Now the quadratic (squaring) mapping that I am examining is:
what happens to z when a=2 and I vary lambda. This mapping does funny things depending on what I set lambda to. If it is between 0 and 1 then it behaves just like the squaring operator - each value of z has two pre-images as I iterate the map but in a circle radius=1-lambda around C the values have no pre-images. Now if I make lambda greater than 1, some values have two pre-images as before but the z values in the circle now have four pre-images.
Still with me? We can take this further but placing this property on curves not just points. A curve is just a line on a graph (an infinite set of points). Instead of looking backwards as we did with the points, we instead go forwards and see how the curve changes. It's first image is a circle, say. This curve then becomes the pre-image of the set of points of the circle under the mapping. And so on, forwards in time. It helps to look at the image below:
You can see the curves collecting on (heading towards) one curve called a manifold (it's attracting). Now consider the relationship of the points and the curves. Because a curve is just a set of points. With noninvertibility inherent in the system, what happens it you take the manifold and reverse the mappings for all the points? Wild Chaos! Tracking the possible points and curves and how and why they interact is the next stage of my project - at least for certain ranges of values for C, a and lambda. This limitation is purely because my project has a time limit of a year.
I am currently working with a PhD student, Stefanie Hittmeyer (who incidentally produced the image above) who is working on the full problem in all its scope. She has been focused on the manifold part of it for the moment while I have been examining the fractal structure side. My work nicely compliments hers. This work is on the fore-front of chaos research. Now it may seem that all this has little value in the real world but the great thing about nonlinear and chaotic systems is that it can accurately mimic real world systems. This is a major advantage over linear systems because when taking a real world problem you have to reduce the picture you are looking at so you can describe it using the simplest tools. With this kind of research, no such reduction is necessary and you can understand the system in it's full complexity. So maybe it'll turn out that neurological signals in the brain will turn out to have similar properties to the mapping that I am examining and my research will contribute to a fuller understanding of how we came to have consciousness and help neurosurgeons get better at fixing the biochemical circuitry that makes us who we are. Or maybe, and more likely since this mapping is a reduction of the five dimensional Lorenz model, it will just help meteorologists get really, really good at predicting weather patterns.
Who knows? It's sort of like the laser - a solution created with no problem that has now become integral to our lifestyle. Just check out how many things it is used in! I hope you managed to follow all that to the end. If you have any specific questions please feel free to email me but I would much rather you follow that urge to take a deeper look yourself (even if that is just reading all the wiki pages). It's way more fun!
Nonlinearity is pretty easy to understand. Take for example square numbers. You can make 4 by either multiplying 2 and 2 or -2 and -2. This means that, under the square operator, 4 has two (well, four really, but two are identical to the other two) possible previous values. This makes squaring a number nonlinear because tracking back you do not know whether two 2s or two -2s made the 4 you see at the moment. This particular brand of nonlinearity is called noninvertibility. You still following me?
Good. Right, to take this further we call the previous values of each number pre-images and the operation, a mapping. Now the quadratic (squaring) mapping that I am examining is:
what happens to z when a=2 and I vary lambda. This mapping does funny things depending on what I set lambda to. If it is between 0 and 1 then it behaves just like the squaring operator - each value of z has two pre-images as I iterate the map but in a circle radius=1-lambda around C the values have no pre-images. Now if I make lambda greater than 1, some values have two pre-images as before but the z values in the circle now have four pre-images.
Still with me? We can take this further but placing this property on curves not just points. A curve is just a line on a graph (an infinite set of points). Instead of looking backwards as we did with the points, we instead go forwards and see how the curve changes. It's first image is a circle, say. This curve then becomes the pre-image of the set of points of the circle under the mapping. And so on, forwards in time. It helps to look at the image below:
I am currently working with a PhD student, Stefanie Hittmeyer (who incidentally produced the image above) who is working on the full problem in all its scope. She has been focused on the manifold part of it for the moment while I have been examining the fractal structure side. My work nicely compliments hers. This work is on the fore-front of chaos research. Now it may seem that all this has little value in the real world but the great thing about nonlinear and chaotic systems is that it can accurately mimic real world systems. This is a major advantage over linear systems because when taking a real world problem you have to reduce the picture you are looking at so you can describe it using the simplest tools. With this kind of research, no such reduction is necessary and you can understand the system in it's full complexity. So maybe it'll turn out that neurological signals in the brain will turn out to have similar properties to the mapping that I am examining and my research will contribute to a fuller understanding of how we came to have consciousness and help neurosurgeons get better at fixing the biochemical circuitry that makes us who we are. Or maybe, and more likely since this mapping is a reduction of the five dimensional Lorenz model, it will just help meteorologists get really, really good at predicting weather patterns.
Who knows? It's sort of like the laser - a solution created with no problem that has now become integral to our lifestyle. Just check out how many things it is used in! I hope you managed to follow all that to the end. If you have any specific questions please feel free to email me but I would much rather you follow that urge to take a deeper look yourself (even if that is just reading all the wiki pages). It's way more fun!
Saturday 8 January 2011
Oh, a promise
I also promised a friend I would post a picture up this week so here it is:
This is from the edge of the Mandelbrot set - originally from Wikipedia
And an example from my own research will be uploaded next week.
And an example from my own research will be uploaded next week.
Rationalising Chaos
Since I have been on my Christmas Break, I have only just started getting back into the thick of my research. Thus, this week's post will be on a related interest of mine. Rationality and luminosity.
I have been attempting to find order from chaos within. I have always known myself quite well but there are times when, and I'm sure you will have had a similar moment, I have thought "What was I thinking?!" having just done or remembered something incredibly stupid or damaging. I have often wondered when I would get around to finding out. I started my path to luminosity through rationality when I began reading "Harry Potter and the Methods of Rationality" by Less Wrong who took the pen name from the website which is "a collaborative blog devoted to improving the art of human rationality". From that website I discovered Alicorn who had written a series of blog posts about luminosity and a subsequent rational fanfiction of Twilight called "Luminosity" .
These encouraged me to really being to explore who I am and how I do things. To examine my motives and actions and be able to explain to anyone why I do the things I do and think the things I think. Living this way prevents one from lying to oneself in order to imagine they or the world around them is they way they want instead of the way it is. If you too want to explore this, I would suggest reading the stories first to understand what I means to live like this - to stand out because you refuse to gloss over the parts of reality you don't want. The world is chaotic but not really as irrational as it seems.
By saying "gloss over reality" here I don't necessarily mean in a politically active sense in fact I tend to ignore politics because every side glosses over their reality and never say what they really mean - even taking an opposition to politics means you a glossing over the reality that democracy has to please the masses; I try to take each case individually on merit instead of generalising. It could be as simple as packing a first aid kit in the car, not because you know you will need it on that trip but because you know that things happen and it is better to be prepared for the worst instead of saying to yourself "by packing the first aid kit I'm tempting fate and someone will get hurt so I shouldn't pack it" just like thinking that if you take an umbrella with you, it is more likely to rain, or by not going to the doctor, you're not really sick and the problem will go away by itself - twisted and irrational logic.
By accepting the reality that everyone gets sick at some time or another, that it will rain unpredictably in England and that people can get hurt doing the safest things, you can prepare for these realities and maybe avoid the worst.
I have been attempting to find order from chaos within. I have always known myself quite well but there are times when, and I'm sure you will have had a similar moment, I have thought "What was I thinking?!" having just done or remembered something incredibly stupid or damaging. I have often wondered when I would get around to finding out. I started my path to luminosity through rationality when I began reading "Harry Potter and the Methods of Rationality" by Less Wrong who took the pen name from the website which is "a collaborative blog devoted to improving the art of human rationality". From that website I discovered Alicorn who had written a series of blog posts about luminosity and a subsequent rational fanfiction of Twilight called "Luminosity" .
These encouraged me to really being to explore who I am and how I do things. To examine my motives and actions and be able to explain to anyone why I do the things I do and think the things I think. Living this way prevents one from lying to oneself in order to imagine they or the world around them is they way they want instead of the way it is. If you too want to explore this, I would suggest reading the stories first to understand what I means to live like this - to stand out because you refuse to gloss over the parts of reality you don't want. The world is chaotic but not really as irrational as it seems.
By saying "gloss over reality" here I don't necessarily mean in a politically active sense in fact I tend to ignore politics because every side glosses over their reality and never say what they really mean - even taking an opposition to politics means you a glossing over the reality that democracy has to please the masses; I try to take each case individually on merit instead of generalising. It could be as simple as packing a first aid kit in the car, not because you know you will need it on that trip but because you know that things happen and it is better to be prepared for the worst instead of saying to yourself "by packing the first aid kit I'm tempting fate and someone will get hurt so I shouldn't pack it" just like thinking that if you take an umbrella with you, it is more likely to rain, or by not going to the doctor, you're not really sick and the problem will go away by itself - twisted and irrational logic.
By accepting the reality that everyone gets sick at some time or another, that it will rain unpredictably in England and that people can get hurt doing the safest things, you can prepare for these realities and maybe avoid the worst.
Saturday 1 January 2011
Happy New Year!
This post I want to introduce my main project. The title is "Dynamics and Bifurcations of Two-Dimensional Noninvertible Maps". My plan for the project is available here to give you an idea of what I would like to achieve with it. Don't worry about all the scary words in it - the main idea is to find out what happens if I slightly permute the Mandelbrot set (a two dimensional map of Julia sets represented as a number set). Currently I'm focused on looking at how the Julia sets change. So far I have made lots of pretty pictures! as well as some observations on what happens.
Fractals are the most beautiful objects in chaos theory (wiki has a great introduction to them here). Simple definition is that fractals are shapes that are self-similar at all levels of magnitude - you cannot tell how zoomed in or out you are. The Mandelbrot is the most famous fractal and I love it because it is defined so simply but produces infinite complexity. My project considers the Mandelbrot as a single instance of a wider group of two-dimensional non-invertible maps. Noninvertible here means a many-to-one mapping forward in time i.e. if you back track there are many points that could have produced your original point.
Fractals were first observed in nature in the form of leaves on the trees. Ferns are made up of smaller ferns which are made up of even smaller ferns. The simplicity of defining a fractal has suggested that the information contained within the DNA of a plant to tell it what shape to grow into is in fact encoding fractal information. Another example would be one used recently in one of the Royal Institution Christmas Lectures - lungs. They translate a large volume of air onto a large surface area of blood vessels and they do it by being fractal. And recently there has been research that has found fractal patterns in semiconductor material at the quantum scale.
Fractals can be produced by nonlinear dynamics. This returns us back to the Lorenz equations and the butterfly attractor. The attractor found to describe the behaviour of the system is a fractal. Such a combination of parts of chaos theory is typical and continuing to find more links is very exciting. I have been finding bifurcation points and their stability in the fractal Julia sets as I permute the Mandelbrot set. Nonlinear dynamics can be produced by fractals. Thus my research is far reaching across the whole field of study. A field of study that is far reaching, interlinked and incredibly beautiful.
Fractals are the most beautiful objects in chaos theory (wiki has a great introduction to them here). Simple definition is that fractals are shapes that are self-similar at all levels of magnitude - you cannot tell how zoomed in or out you are. The Mandelbrot is the most famous fractal and I love it because it is defined so simply but produces infinite complexity. My project considers the Mandelbrot as a single instance of a wider group of two-dimensional non-invertible maps. Noninvertible here means a many-to-one mapping forward in time i.e. if you back track there are many points that could have produced your original point.
Fractals were first observed in nature in the form of leaves on the trees. Ferns are made up of smaller ferns which are made up of even smaller ferns. The simplicity of defining a fractal has suggested that the information contained within the DNA of a plant to tell it what shape to grow into is in fact encoding fractal information. Another example would be one used recently in one of the Royal Institution Christmas Lectures - lungs. They translate a large volume of air onto a large surface area of blood vessels and they do it by being fractal. And recently there has been research that has found fractal patterns in semiconductor material at the quantum scale.
Fractals can be produced by nonlinear dynamics. This returns us back to the Lorenz equations and the butterfly attractor. The attractor found to describe the behaviour of the system is a fractal. Such a combination of parts of chaos theory is typical and continuing to find more links is very exciting. I have been finding bifurcation points and their stability in the fractal Julia sets as I permute the Mandelbrot set. Nonlinear dynamics can be produced by fractals. Thus my research is far reaching across the whole field of study. A field of study that is far reaching, interlinked and incredibly beautiful.
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