Currently I'm reading a book on quantum physics ("In Search of Schrodinger's Cat" by John Gribbon) and I was struck by the similar history of development it has with chaos theory i.e. no-one wants to admit it's really true and almost no-one really understands it. It has taken almost 100 years for quantum theory to be discovered, accepted and still it is only fully taught at university level after student have already been ingrained into thinking that electrons whoosh around the nucleus. I remember each level of school (middle school, pre-GCSE, GCSE) I took chemistry, I was told that everything I had been taught previously was wrong and too simple and here is the real thing! It got to the point that when I was choosing my A levels I did not pick Chemistry. I figured they would just lie to me again and I was better off learning the truth out of university textbooks. So I took Philosophy instead. I wonder whether others felt the same. I just wished they could have taught it correctly from the beginning and get over the fact that most 10-16 year-olds might not get it. Some will! Maybe in another hundred years quantum will be understood by enough people *cough* politicians *cough* that it can be taught straight from the beginning and not require almost a reprogramming of minds. Perhaps the same progress will happen for chaos theory. One can only hope.
This week my paper is on interaction of populations of competing species. Available here. It's quite a nice paper (longer than the last one). The purpose was to analyse how three equally competing species interact to change each other and their own population size over time. The time series plots provide examples of how systems require a certain amount of time to settle down to the final state. The equilibria that I talk about in the paper are fixed points for the system. They are the stable behaviour of the species interactions. So the species may start at different values (with the same parameters) but, over time, they will settle down to a fixed point (or periodic behaviour, or chaotic behaviour).
Taking the example of the pendulum again, when you set the pendulum in motion it will oscillate a few times and then settle back down to the lower central point. This is because it is damped by drag. This behaviour can be seen mathematically as a perturbation of initial conditions away from the stable equilibrium but over time it will return to that stable equilibrium, that stable position where it can hang without oscillating quite happily until the end of time. This stable equilibrium attracts all motion of the pendulum towards it so it is called an attracting fixed point.
Now opposite that stable equilibrium there is an unstable equilibrium. Right at the top of the circle that the pendulum could swing through. At this point the complete opposite happens to the point at the bottom. It repels all motion from it. A pendulum cannot hold itself up until the end of time (not with gravity anyway). So at the top we have a repelling fixed point and at the bottom we have an attracting fixed point. Mathematically the system makes sense.
Right, so now the easy bit is done, I'll add something a little harder to the end. You might be about to comment, "So what is happening when the pendulum in a clock oscillates on and on". So I'll tell you...
That is when the system has a period two oscillation if you set drag to zero (it is not exactly possible in the real world because drag does affect it due to not quite being zero which is why you had to keep winding them up). This means that when you set the motion going, you can get the pendulum to swing for the rest of time side to side. The distance and time period of the swing will be exactly the same for every oscillation. This is a stable period two oscillation and has a pretty sine or cosine curve on the time series plot.
Pendulum's can display chaotic motion. Just check this out. And each time you set it in motion it will swing and flip in a completely different way thanks to the effect of chaos - sensitive dependence on initial conditions!
Friday 17 December 2010
Sunday 12 December 2010
A Biological Twist
Since I am an engineer you might wonder what I mean by the title. As I said previously, chaos theory is applicable in every science. Biological systems especially exhibit chaotic behaviour. The system under examination today involves DNA within a cell. Some background on DNA and the DNA - transcription - mRNA - translation - protein cycle is available in an interactive form here.
My document for this week is available here.
The Goodwin Oscillator describes a gene's behaviour when it is self-repressing i.e. as it is transcribed and translated it recognises if it is already present at a certain concentration and prevents any more being made. Exactly how this happens biologically I have very little idea. As an chaos mathematician and an engineer all I need is the equations used to mathematically describe the system. From there, as you will see in the paper, I can build a view of what the system does under different parameter values.
In the paper I have used two software packages: MATLAB and XPP. XPP is the best for newcomers to the chaos scene if you want to play with systems. A tutorial and the free download are available here. MATLAB is a (definitely not free), matrix based, mathematical tool provided by Mathworks. A lot of people don't like to use it but I have worked with is so long I can now avoid most of the annoyances and can get it to do what I want now. It is good at crunching lots of numbers for chaotic systems and displaying their behaviour diagrammatically.
In the paper I have focused on the bifurcations of the system. A bifurcation is a point where the system qualitatively changes behaviour dramatically and instantaneously. A steady system can suddenly become oscillatory under the right parameters. In the paper the system of gene self-repression in the cell is bistable - it has a period two oscillation; it oscillates between two values - for all parameter values that I could calculate. The unusual behaviour comes just before that - the behaviour before it settles down to bistable for small x/alpha values is rather strange even for chaotic system. If I could have furthered the paper I would have tested that area more thoroughly and worked out what happens there. However, I was limited by time and pages.
The best thing that the system I examined in the paper shows is that even very simple systems can exhibit highly unusual behaviour as well as perfectly understandable behaviour. It also demonstrates how understanding the mathematics leads to better understanding of the biology. Even without knowing the biology or chemistry to flesh out the details I can tell you that a self-repressing gene will allow itself to be 'switched on' when the concentrations of the mRNA and protein reach certain low levels and will then 'switch off' itself when the concentrations reach certain high levels with the levels being determined by the particular physically characteristics of the system (parameter values). I can tell you that interesting research (in both the biological and the mathematical sense) would be in the area for very small parameter values. I can tell you that I feel that I now understand DNA transcription and translation behaviour better because of this research.
You might think that what I described happening is perfectly reasonable and of course is what happens! But the paper can specify which cells (if exact enough measurements can be taken) will display which behaviour for that gene. And it can specify which cells will not. And it can go further and build from the simple, constrained system to a more complex one taking the mathematical and biological understanding further. Biology is one field that is now accepting chaos theory with open arms because it opens new doors to research and furthers understanding of biological phenomenon by being able to generalise as well as specify. While the field may have arisen from Physics, it is Biology that is currently receiving most of the benefits. In my department, four out of fourteen lecturers have research specialisms in applying chaos to biological systems. That's 28.6% of the department - almost a third.
My final thought for you is for you to consider, if even genes behave chaotically, what other biological processes could do so also?
Further reading for bifurcations: Strogatz (if you have it) covers them really well in ch 3 (basics) and ch 8 (much more interesting!) also a good starting point is the NLDC web page under 'Nonlinear dynamics in action'.
My document for this week is available here.
The Goodwin Oscillator describes a gene's behaviour when it is self-repressing i.e. as it is transcribed and translated it recognises if it is already present at a certain concentration and prevents any more being made. Exactly how this happens biologically I have very little idea. As an chaos mathematician and an engineer all I need is the equations used to mathematically describe the system. From there, as you will see in the paper, I can build a view of what the system does under different parameter values.
In the paper I have used two software packages: MATLAB and XPP. XPP is the best for newcomers to the chaos scene if you want to play with systems. A tutorial and the free download are available here. MATLAB is a (definitely not free), matrix based, mathematical tool provided by Mathworks. A lot of people don't like to use it but I have worked with is so long I can now avoid most of the annoyances and can get it to do what I want now. It is good at crunching lots of numbers for chaotic systems and displaying their behaviour diagrammatically.
In the paper I have focused on the bifurcations of the system. A bifurcation is a point where the system qualitatively changes behaviour dramatically and instantaneously. A steady system can suddenly become oscillatory under the right parameters. In the paper the system of gene self-repression in the cell is bistable - it has a period two oscillation; it oscillates between two values - for all parameter values that I could calculate. The unusual behaviour comes just before that - the behaviour before it settles down to bistable for small x/alpha values is rather strange even for chaotic system. If I could have furthered the paper I would have tested that area more thoroughly and worked out what happens there. However, I was limited by time and pages.
The best thing that the system I examined in the paper shows is that even very simple systems can exhibit highly unusual behaviour as well as perfectly understandable behaviour. It also demonstrates how understanding the mathematics leads to better understanding of the biology. Even without knowing the biology or chemistry to flesh out the details I can tell you that a self-repressing gene will allow itself to be 'switched on' when the concentrations of the mRNA and protein reach certain low levels and will then 'switch off' itself when the concentrations reach certain high levels with the levels being determined by the particular physically characteristics of the system (parameter values). I can tell you that interesting research (in both the biological and the mathematical sense) would be in the area for very small parameter values. I can tell you that I feel that I now understand DNA transcription and translation behaviour better because of this research.
You might think that what I described happening is perfectly reasonable and of course is what happens! But the paper can specify which cells (if exact enough measurements can be taken) will display which behaviour for that gene. And it can specify which cells will not. And it can go further and build from the simple, constrained system to a more complex one taking the mathematical and biological understanding further. Biology is one field that is now accepting chaos theory with open arms because it opens new doors to research and furthers understanding of biological phenomenon by being able to generalise as well as specify. While the field may have arisen from Physics, it is Biology that is currently receiving most of the benefits. In my department, four out of fourteen lecturers have research specialisms in applying chaos to biological systems. That's 28.6% of the department - almost a third.
My final thought for you is for you to consider, if even genes behave chaotically, what other biological processes could do so also?
Further reading for bifurcations: Strogatz (if you have it) covers them really well in ch 3 (basics) and ch 8 (much more interesting!) also a good starting point is the NLDC web page under 'Nonlinear dynamics in action'.
Saturday 4 December 2010
The Beginning...
Everyone starts somewhere.
I intend for this blog to be an aid for my Masters year. I intend to post papers I produce, thoughts on subjects I am taking, links to things that interest me (and maybe you) and perhaps the occasional insight into my thought processes. But as everyone knows, the best laid plans of mice and men...
I'm currently almost 'half' way through but I'll do my best to get you up to speed as best I can. My hope is to post once a week but don't hate me if I don't always manage it- I'll do my best to make up for it when possible.
So to give an overview...I am a final year student at the University of Bristol reading (as we say in the UK) Engineering Mathematics. We are currently 8 weeks into the year and so far I have written two papers, about to finish a third (this one is a group one) and none are getting published but they will all contribute to the degree I get out at the end. I have weekly labs, many hours of lectures and there is always work to be done in the form of my Final Year Project. Yep, I know, capitals seem silly here but the doom that surrounds it requires them really - it is, after all, contributing to 16.7% of my final degree classification which is more than my whole second year is worth. Oh, not forgetting the constant reading of required texts plus extensive reading around the subject in areas of interest.
As you can see, the workload is extensive and I bet you are wondering how I possibly have time to maintain this blog. Well, you see I hope that the small amount of extra work this may be will allow others to get a glimpse into an area of research that very few know about and even fewer really understand - Nonlinear Dynamics and Chaos - and how it fits in with engineering. Most people have heard of the Butterfly Effect but they do not really understand it - how exactly can a bat of the wings of a small butterfly change the weather on the other side of the world? Well, chaos theory explains phenomenon like this through understanding systems with "sensitive dependence on initial conditions".
Nonlinear dynamics and chaos is a rich field of research and crosses many other areas of science. I will not be spending this whole blog paraphrasing many better introductions and explanations. Instead, I will be discussing my studies directly and in each case will point the reader in the direction of preceding and/or related work. If nothing else, Wikipedia provides a great intro to the area. The best book for anyone of a mathematical persuasion is "Nonlinear Dynamics and Chaos" by Steven Strogatz. Any for those who are not, "Chaos" by James Gleick is a great read.
The relevance of this field to all science cannot be underestimated. Most engineers only look at systems in their most simplistic using assumptions that mean the system can be understood by classical mechanics. Those engineers trained in chaos can explore all the possibilities in the system. The best example of this is a system which most people have come across at some point: the pendulum. Remember the step the teachers did in class where they assumed that sin x = x for small angles and from then on the maths was really easy and you could understand how clocks work? Well, I know what happens when the pendulum swings over a big angle and it is really very elegant. Do you want to know too?
I intend for this blog to be an aid for my Masters year. I intend to post papers I produce, thoughts on subjects I am taking, links to things that interest me (and maybe you) and perhaps the occasional insight into my thought processes. But as everyone knows, the best laid plans of mice and men...
I'm currently almost 'half' way through but I'll do my best to get you up to speed as best I can. My hope is to post once a week but don't hate me if I don't always manage it- I'll do my best to make up for it when possible.
So to give an overview...I am a final year student at the University of Bristol reading (as we say in the UK) Engineering Mathematics. We are currently 8 weeks into the year and so far I have written two papers, about to finish a third (this one is a group one) and none are getting published but they will all contribute to the degree I get out at the end. I have weekly labs, many hours of lectures and there is always work to be done in the form of my Final Year Project. Yep, I know, capitals seem silly here but the doom that surrounds it requires them really - it is, after all, contributing to 16.7% of my final degree classification which is more than my whole second year is worth. Oh, not forgetting the constant reading of required texts plus extensive reading around the subject in areas of interest.
As you can see, the workload is extensive and I bet you are wondering how I possibly have time to maintain this blog. Well, you see I hope that the small amount of extra work this may be will allow others to get a glimpse into an area of research that very few know about and even fewer really understand - Nonlinear Dynamics and Chaos - and how it fits in with engineering. Most people have heard of the Butterfly Effect but they do not really understand it - how exactly can a bat of the wings of a small butterfly change the weather on the other side of the world? Well, chaos theory explains phenomenon like this through understanding systems with "sensitive dependence on initial conditions".
Nonlinear dynamics and chaos is a rich field of research and crosses many other areas of science. I will not be spending this whole blog paraphrasing many better introductions and explanations. Instead, I will be discussing my studies directly and in each case will point the reader in the direction of preceding and/or related work. If nothing else, Wikipedia provides a great intro to the area. The best book for anyone of a mathematical persuasion is "Nonlinear Dynamics and Chaos" by Steven Strogatz. Any for those who are not, "Chaos" by James Gleick is a great read.
The relevance of this field to all science cannot be underestimated. Most engineers only look at systems in their most simplistic using assumptions that mean the system can be understood by classical mechanics. Those engineers trained in chaos can explore all the possibilities in the system. The best example of this is a system which most people have come across at some point: the pendulum. Remember the step the teachers did in class where they assumed that sin x = x for small angles and from then on the maths was really easy and you could understand how clocks work? Well, I know what happens when the pendulum swings over a big angle and it is really very elegant. Do you want to know too?
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