Dynamical Systems: An Introduction with Applications in Economics and Biology
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For example, enough periodicity turns out to imply zero entropy this shouldn't be surprising. If a system is periodic it means the past repeats regularly, so knowing the past we can predict the future perfectly; and we said this is the same as zero entropy. When we are observing an unknown system, can we determine its attributes by observing a sequence of measurements?
Notice that there is a difference between figuring them out from knowledge of the system and figuring them out by "participating" in the system. For example, if you know the law of gravity you can try to compute things about the solar system. But this is a different problem from that of an ancient astronomer who is ignorant of gravity but has at his disposal observations and measurements of the heavens. It turns out that even when we do not know the rules, we can sometimes estimate the entropy of a system, though not necessarily its periodicity! Given a concrete rule governing the dynamics of a system, can we use the rule i.
Surprisingly, this is sometimes impossible. There are very simple rules that can generate dynamics so complicated that we can actually prove that we cannot understand them completely. Do "most" systems have a certain attribute? This is a somewhat subtle question because it depends what you mean by "most", But it turns out that, rather generally, for a given dynamical property either most systems have it, or most don't it can't be split down the middle, so to speak. For example, most systems have zero entropy, but most systems are also aperiodic.
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Which real-world systems have which attributes? As we already mentioned, there are many real-world system that display chaotic behavior like positive entropy and mixing, but there is a long list about which we are still not sure, and it is a challenge to develop methods for deciding this. This is a problem at the interface of abstract and concrete dynamics: testing specific systems for abstract properties. There are also many interesting applications of dynamical systems theory to other areas of mathematics.
In particular, dynamical methods have had great success in combinatorics and number theory, and are fundamental to parts of information theory. I won't go into these.
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There are lots of places to get more information about dynamical systems theory. My first suggestion would be to ask a friendly grad student or professor. There's lots of literature available on the subject, from popular accounts to technical monographs. I highly recommend these books:. Dynamical Systems Theory: What in the World is it? Mike Hochman This page is a non-technical overview of some subjects related to my field of research. It's meant for freshmen students, my friends, my mom hi mom! I've tried not to lie too much, though sometimes I've bent the truth a little in the interest of clarity I hope, anyway.
If you would like more details take a look at the list of references at the end.
If you have any questions or suggestions I'd be glad to hear from you. What is dynamics about? Examples of such systems include: The solar system sun and planets , The weather, The motion of billiard balls on a billiard table, Sugar dissolving in a cup of coffee, The growth of crystals The stock market, The formation of traffic jams, The behavior of the decimal digits of the square root of 2; and so on.
Describing dynamical systems mathematically All these models can be unified conceptually in the mathematical notion of a dynamical system , which consists of two parts: the phase space and the dynamics. Abstraction Abstract dynamics is the study of dynamical systems based on the description explained above and discarding most specialized information about the system or the origin of the dynamics. What is abstraction good for? Classification of dynamical systems: philosophy and examples A very general problem in abstract dynamics is to understand when two systems are "the same", either precisely or in some fuzzier sense.
The dynamics of dissolving a drop of red ink in a cup of water is essentially "the same" as the process of a drop of black ink. In the ink example above, if we run time backwards we get a different dynamics than running it forward, because when time goes forward the ink becomes more and more spread out and dissolved, whereas if time goes backward it starts out dissolved and becomes more and more concentrated in a drop this observation is the essence of the second law of thermodynamics The dynamics of billiard balls moving without friction on a billiard table look the same if we run time forward or backward.
In fact, if I showed you a movie of the balls rolling, you couldn't tell with any certainty which way time was going.
Therefore the dynamics of billiard balls is essentially different from the dynamics of dissolving ink. Actually, I am lying here. Ink and water are made up of atoms that bounce around a lot like billiard balls. But at the time scales we can observe they certainly behave differently. Example: stationary vs. Example: chaotic vs. Example: invariants An invariant of a dynamical system is some quantity you can measure either in practice or in theory which comes out the same when you compute it for different systems that are "the same". Here are a few of my favorites: What are the relationships between the different attributes?
Other applications There are also many interesting applications of dynamical systems theory to other areas of mathematics. Further reading There are lots of places to get more information about dynamical systems theory. Here are a few online sources that I know of: Tomasz Downarowicz 's webpage contains a friendly and fairly nontechnical explanation, assuming basic undergraduate level mathematics Karl Petersen 's webpage contains some nice and fairly elementary lecture notes.
Steve Kalikows 's book on ergodic theory.
The book is online and free. It is meant as a graduate course but is fairly non-technical. I highly recommend these books: "Dynamical systems and ergodic theory", by M. Pollicott and M Yuri. Both are a great place to start and give a broad overview of the field. Pollocott and Yuri's book is more elementary and more thorough, though it assumes some knowledge of university-level mathematics.
Assessment Written examination combined with assignments given during the course. Syllabus Revisions Latest syllabus applies from week 30, Previous syllabus applies from week 34, Previous syllabus applies from week 36, Previous syllabus applies from week 35, Nonlinear dynamics and chaos : with applications to physics, biology, chemistry and engineering Reading, Mass. We illustrate our approach on the classic example of relating democracy and economic growth, identifying non-linear relationships between these two variables.
We show how multiple variables and variable lags can be accounted for and provide a toolbox in R to implement our approach.
Complex Dynamic Systems, a.a.
This is an open-access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist. Social science usually aims to explain macro-level phenomena, such as stratification, segregation, democratisation, economic development and changes in values.
From the vast number of examples studied across sociology, politics and economics, a few examples include: Does inequality decrease or increase ethnic segregation ?
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What are the causes of economic growth ? How does the structure of a social network affect opinion dynamics ? In these wide ranging questions, the macro-level variables can concern a variety of scales, from schools and neighbourhoods, up to companies and countries. The questions about them are similar: from observed macro-level patterns, can we work out the relationships that characterize these patterns . While it is widely recognized that understanding at the micro-level is the key to causal mechanisms in sociology  —  , it is possible to gain some understanding of social systems through macro-level patterns alone.
One example of macro-level inference is fitting sigmoidal and saturating curves to describe diffusion of innovations  , .
europeschool.com.ua/profiles/vasugisej/carta-natal-transitos.php In this case, the type of growth curve is hypothesized to differ depending on whether innovation is driven by endogenous or exogenous factors at the micro-level. Specifically, by fitting the growth curve 1 to the proportion of individuals adopting a particular activity over time , we can infer the relative importance of exogenous and endogenous social factors from the relative weights of the parameters and , respectively.
While having certain known limitations  ,  , this approach has been usefully applied in the context of, for example, medical innovations and radio airplay. Only a very small subset of social systems are characterized by sigmoidal growth curves. However, non-linear interactions between variables in social systems are common, and using differential equations to give an initial insight into macro-level relationships has a great deal of potential  — . Econometrics provides a starting point for such an analysis. For example, in growth econometrics cross-country data is used to find which factors promote economic growth  — .
However, growth econometric analyses usually focus on the rate of change of one variable as a function of many potential factors, rather than dynamic interactions between variables.
It is precisely these dynamic feedbacks which are of most interest in the social sciences and where reliable statistical approaches are required . In recent years, detailed data describing long term changes in social systems has become widely available. For example, a variety of indicators now measure changes in the economics  ,  , social development  , political systems  —  and cultural values  of different countries and local regions. Identifying relationships between these macro-level indicators poses new challenges, but also opens up new opportunities.
These challenges are not unique to between country comparisons, but arise in everything from social movements, workplaces, and neighbourhoods, down to modeling individual panel data on emotion dynamics  , .