Disordered behavior can be stable. Non-periodic systems with the butterfly effect can have stable average properties. So the average or statistical properties of a system can be predictable, even if its details are not. Complex behavior can arise from simple rules. Simple dynamical systems do not necessarily lead to simple results. In particular, we will see that simple rules can produce patterns and structures of surprising complexity.
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Dynamical Systems - Examples of Complex Behaviour | Jürgen Jost | Springer
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Start now for free! Sign up. Overview Related Courses Reviews. Overview In this course you'll gain an introduction to the modern study of dynamical systems, the interdisciplinary field of applied mathematics that studies systems that change over time. The course will focus on some of the realizations from the study of dynamical systems that are of particular relevance to complex systems: 1.
Taught by David Feldman. Tags usa. Browse More Calculus courses. I appreciated the structured approach to the course material and the painstaking development of foundational concepts. Feldman presents the course in an informal, across-the-desk manner. Each lecture feels like you are experiencing an individual tutoring session during office hours. I recommend the course to students who struggle with math or computing anxiety; only mininal calculus is needed to understand and apply the material. Below is a topical overview of the 9-week course.
Lectures: What is a dynamical system? General properties - classification…. Iterated functions, orbit, itinerary; examples. Differential equations, examples; rule is indirect, involves rate of change of a variable. Chaos, examples; deterministic, orbits bounded, aperiodic, sensitive dependence on initial conditions. The butterfly effect.
Algorithmic randomness, incompressibility. The logistic equation. Time is continuous vs discrete time intervals; dependent variable is continuous vs discrete values.
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Bifurcation diagrams; examples. Period-doubling route to chaos. Universality in period doubling, Feigenbaum's constant. Universality in physical systems; examples. The phase plane. Stable and unstable fixed points, orbits can tend to infinity, limit cycles attracting cyclic behavior - but no chaos.
Phase space. Strange attractors: stable attractors but motion on the attractor is chaotic; examples. Stretching and folding in chaotic orbits. Strange attractors combine elements of order and disorder; motion is locally unstable, globally stable. Pattern formation in dynamical systems; examples. Reaction-diffusion systems. Simple, spatially-extended dynamical systems with local rules are capable of producing stable, global patterns and structures.
Interviews: Stephen Kellert, prof of philosophy at Hamline University. Chaos theory represents an evolution vs revolution , a new style of scientific reasoning or doing science. Represents a conceptual reconfiguration, gets rid of old dichotomies.
You can have conceptual change that's brought about through methodological challenges, not just through grand theoretical structures being changed. Chaos theory is a part of postmodernism - challenging of strict binaries. Stephen W. Morris prof of geophysics at Univ. Pattern formation in nature; examples and demonstrations. Conceptual - thematic summary Remarks.
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Was this review helpful to you? Dave Feldman is a great teacher, very dedicated to explaining the ideas in a simple way, accessible to most if not all. I was already acquainted with chaos, which I had studied at University So, I found the course fairly easy and didn't watch all videos.
Nevertheless, the material I reviewed was very relevant and interesting: the excellent refresher I was lookin…. Nevertheless, the material I reviewed was very relevant and interesting: the excellent refresher I was looking for to remind me of the good old days! I took the class parallel to re-reading the book "Chaos", by James Gleick, which Dave Feldman cleverly recommends as a complementary resource. The topic of chaos is coverred very progressively, to introduce the key ideas. At a fundamental level the world is governed by simple laws of physics, but out of these laws has emerged the immense complexity and beauty that we see around us.
Our Nonlinear and Complex systems researchers are engaged in the study of how such complexity and order arises from simple rules. The subject is vast, spanning many topics from the highly abstract to the intensely practical.
The symmetric barrier billiard system was first introduced by Wiersig, and is an example of a pseudo-integrable dynamical system: a system which consists of multiple copies of an integrable Hamiltonian system, but which itself is not integrable. The motion consists of a particle moving at constant speed inside a rectangle experiencing specular collisions with the boundary think of a screensaver , but with a partial barrier placed centrally which splits the configuration space into two chambers.
It can be shown that the system can be completely understood by studying the transitions between these two chambers, and this leads to analysis of the autocorrelation function.
For a particle moving at with slope equal to the golden mean, we have shown using a renormalization analysis that the correlations at Fibonacci times chaotically explore an invariant surface. Our current aim is to construct a model space which completely describes the behaviour of the renormalization operator which gives rise to this phenomena. Otherwise, its evolution with respect to time is generally unpredictable and very sensitive to initial conditions a phenomenon commonly known as chaos.
One concept is not intrinsically antonymous with the other, but there seems to be indeed an inverse correlation between the two in practical examples. A Gilbert tessellation is created when cracks begin to form at a set of points randomly spread throughout the plane. It is a simple model for the growth of needle shaped crystals, but its properties have remained resistant to mathematical analysis for decades. Recent work, in collaboration with Professor Richard Cowan University of Sydney , has focussed on a Gilbert Tessellation which we call the "half rectangular" model which has surprisingly similar properties to the full rectangular model.
Other editions. Enlarge cover. Error rating book. Refresh and try again. Open Preview See a Problem? Details if other :. Thanks for telling us about the problem. Return to Book Page. Breadth of scope is unique Author is a widely-known and successful textbook author Unlike many recent textbooks on chaotic systems that have superficial treatment, this book provides explanations of the deep underlying mathematical ideas No technical proofs, but an introduction to the whole field that is based on the specific analysis of carefully selected examples Includes a Breadth of scope is unique Author is a widely-known and successful textbook author Unlike many recent textbooks on chaotic systems that have superficial treatment, this book provides explanations of the deep underlying mathematical ideas No technical proofs, but an introduction to the whole field that is based on the specific analysis of carefully selected examples Includes a section on cellular automata Get A Copy.
Paperback , pages. Published August 1st by Springer first published January 1st More Details Original Title. Other Editions 1. Friend Reviews.