--------------------------------------- Q[1]: What is Cantor set ? Author: Pavel.Pokorny@vscht.cz SA[1]: Short answer: Cantor set is a set of points on a line segment, whose triadic (using digits 0,1,2 only) record contains no digit 1. It is a simple example of a fractal. A[1]: Take the interval between 0 and 1. Remove the middle third. Two intervals remain, each one of length one third. From each remaining interval remove the middle third. Repeat the last step infinitely many times. What remains is called the Cantor set. It is one of the simplest examples of a fractal. The characteristic property of a fractal is *self-similarity*. That means that when observed with zoom new details appear on smaller scales. Other examples of fractals in mathematics are Sierpinski triangle, Koch snowflake, Peano curve, Mandelbrot set, and Lorenz attractor. Fractals also describe many real-world objects, such Examples of fractals in nature include clouds, mountains, turbulence, coastline, roots and branches of trees and veins and lungs of animals. Let us calculate the capacity (box counting) dimension of the Cantor set. Roughly speaking to cover a set by cubes with side e we need N(e) cubes. When e is small we need more cubes. For a 1-dim set N(e) is proportional to 1/e. For a 2-dim set N(e) is proportional to 1/e^2. The capacity dimension D is the exponent in N(e) \propto 1/e^D or D = -lim log(N(e)) / log(e) for e->0. For the Cantor set above, if e is 3 times smaller we need 2 times larger N (because holes do not need to be covered), thus D = log2 / log3 = 0.63.... Non-integer dimension is a typical property of fractals and fractal structure is typical for strange chaotic attractors. For more popular reading about fractals see the Usenet Newsgroup sci.fractals and sci.fractals.FAQ. See also http://www.shu.edu/html/teaching/math/reals/topo/defs/cantor.html http://ernie.bgsu.edu/~carother/cantor/Cantor1.html http://math.uw.bialystok.pl/~Form.Math/Vol7/html/cantor_1.html To read more about Georg Ferdinand Ludwig Philipp Cantor Born: 3 March 1845 in St Petersburg, Russia Died: 6 Jan 1918 in Halle, Germany visit http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cantor.html http://www.shu.edu/html/teaching/math/reals/history/cantor.html To read more about Cantor Function (function that is continuous, differentiable, increasing, non-constant, and the derivative is zero everywhere except at a set with length zero) visit http://www.shu.edu/~wachsmut/reals/cont/fp_cantr.html --------------------------------------- Q[2]: What is "reaction diffusion system" ? Short answer: RDS is a chemical, spatially distributed system, described by a set of partial differential equations. Imagine a medium (solid, liquid of gas), where various chemical species can react and diffuse. Then the vector c of concentrations c_i(x,t) of species i in location x and time t changes in time according to @c_i/@t = f_i(c) + D_i @2c_i/@x2 (@ denotes partial derivative). f(c) is vector function (nonlinear) describing local chemical reactions. f_i > 0 means chemical reactions produce species i, f_i < 0 means species i is consumed by chemical reactions as a whole. D_i is the diffusivity of species i. The corresponding mixed (or spatially homogeneous) system is dc/dt = f(c). Although the state space of a reaction diffusion system is infinite-dimensional, it may have a finite dimensional attractor (a stable steady point, a limit cycle, a torus or a low-dimensional strange chaotic attractor). --------------------------------------- Q[3]: What is Turing instability ? TI is the instability of a homogeneous stationary state of a reaction-diffusion system against a non-homogeneous perturbation, when the state is stable against a homogeneous perturbation. TI can occur only if diffusivities of various chemical species are not equal. TI is responsible for patterns formed spontaneously in chemical reaction-diffusion systems and possibly in biological systems. Turing,A.M., Phil.Trans.R.Soc.Lond. B 327,37-72 (1952) To read more about Alan Mathison Turing Born: 23 June 1912 in London, England Died: 7 June 1954 in Wilmslow, Cheshire, England http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Turing.html Author of "Turing machine" (1936). --------------------------------------- Q[4]: What is continuation ? Short answer: Continuation is a numerical method to find the solution of equation f(x) = 0 where f: R^{n+1} -> R^n, in the form of a 1-dim curve in R^{n+1}. Many problems in the numerical study of dynamical systems lead to the system of n equations for n+1 unknowns, written as f(x) = 0 where x \in R^{n+1} and f(x) \in R^n. One simple example is to find a stationary point (stable or unstable) y0 of ODE with 1 real parameter p. dy/dt = f(y,p) The stationary point y0 is the solution of the equation f(y0,p) = 0. The brute-force attack is to solve this equation for various values of p. The points x=(y0,p) will (under general conditions on f) lie on a 1-dim curve in R^{n+1}. This curve may form a S-shaped meander giving more than one y for certain p. Continuation is a good method to follow this curve. Continuation is a predictor-corrector method. Given one point x0 on the curve we may predict the next point x1 = x0 + h * u where h is the step size and u is a unit vector in the desired direction. One simple prediction may be the (vector) difference of the last 2 points. Then the corrector is used to locate the next point x on the curve precisely. One simple corrector may be to solve the set of equations f(x) = 0 (x-x1).(x1-x0) = 0 where . is the dot product of two vectors in R^{n+1}. The second equation means that we want the correction to be perpendicular to the prediction. These corrector equations can be solved numerically using the Newton's method. It is a good practice to adjust the step size according to how complicated the solution is. A simple step size controller may be based on the number of Newton's iterations necessary to achieve a prescribed precision. If the number of iterations is small, meaning the solution is easy, then enlarge the step size; if the number of iterations is large, then shorten the step size. See [1] Eusebius Doedel, Xianjun Wang and Thomas Fairgrieve. AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations, 1994. California Institute of Technology. CONT by Igor Schreiber (schrig@tiger.vscht.cz) Marek and Schreiber: Chaotic Behavior of Deterministic Dissipative Systems, Cambridge University Press, Great Britain, 1991 LOCBIF version 2 Interactive LOCal BIFurcation Analyzer. A. Khibnik, Y.A. Kuznetsov, V.Levitin and E.V. Nikolaev. Distributed by CAN Expertise Centre,Kruislaan 413, 1098 SJ Amsterdam, the Netherlands. email: can@can.nl. R.Seydel: Tutorial on Continuation; Int.J.Bif.Chaos Vol.1,No.1 (1991) 3-11 --------------------------------------- Q[5]: What is Singular Value Decomposition ? Author: Pavel.Pokorny@vscht.cz Short answer: SVD is a technique to find singular vectors and singular values of a matrix. For a trajectory matrix (its rows are successive vectors along a trajectory in the phase space) singular vectors define directions in the phase space and singular values measure how much the attractor is spanned in these directions. If we take a unit sphere in R^n and multiply each vector in it by a m x n matrix A we get an ellipsoid in R^m. Singular values of A (they are square roots of eigenvalues of the matrix A^T.A) give the lengths of principal axes of the resulting ellipsoid. Singular vectors of A (eigenvectors of A^T.A) are mapped to these principal axes. For A a real matrix, A^T.A is real and symmetric, hence the eigenvectors of A^T.A and the singular vectors of A form an orthogonal set. See Broomhead D.S. and King P.: Extracting Qualitative Dynamics from Experimental Data Physica 20 D (1986) 217-236 --------------------------------------- Q[6]: What is Poincare section of the second order ? Poincare section (of the first order) converts a n-dim continuous-time dynamical system to a (n-1)-dim discrete-time dynamical system by mapping every trajectory intersection with a suitable surface in the state space to the next intersection. These is no exact way how to apply Poincare section to a discrete-time dynamical system. Poincare section of the second order is an approximate conversion of a n-dim discrete-time dynamical system to a (n-1)-dim discrete-time dynamical system. Instead of a surface we take a thin layer (of thickness e) in the state space and we select only those points of the trajectory that lie inside this layer. Small e means small error, but with decreasing e the portion of points which are used also decreases. Poincare section of the second order converts a torus to a circle, which is easily detectable. To read more about Henri Poincare see http://www-chaos.umd.edu/misc/poincare.html http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Poincare.html --------------------------------------- Q[7]: What are Cellular Automata ? Short answer: Cellular automaton is a dynamical system with discrete time, discrete state space and discrete geometrical space. CA is a discrete counterpart of partial differential equation. The following table shows the 8 possibilities of taking time, state space or geometrical space either continuous (C) of discrete (D): time state space geometrical space PDE C C C ODE C C D C D C C D D D C C CML D C D D D C CA D D D CML is coupled-map-lattice. The most famous CA is the game life. CA evolves according to a deterministic rule which gives the state of a site in the next generation as a function of the states of neighboring sites in the present generation. This rule is applied to all sites. For further reading see Stephen Wolfram (yes, the author of Mathematica): Theory and Application of Cellular Automata World Scientific Singapore 1986 Physica 10 D (1984) the entire volume --------------------------------------- Q[8]: What is Excitable ? Excitability is a special property of neural and muscle cells to damp low (sub-threshold) input signal and to emit a large response to a super-threshold input signal. Certain chemical, physical and mathematical systems exhibit similar behavior. Excitability often occurs near oscillatory region in parameter space. See CHAOS, SOLITONS AND FRACTALS Volume 5, Issue 3-4 Special Issue: Nonlinear Phenomena in Excitable Physiological System, 4 March 1995 Tables of Contents http://www.elsevier.nl:80/cas/estoc/contents/SAJ/09600779/SZ954080.html Excitable media http://www-im.lcs.mit.edu/broch/fir1.html NEUROBIOLOGY OF EXCITABLE CELLS http://www.bmg.uab.edu/2nd_fac/brown/brown.html The Excitable Cell Membrane - A Biological Battery The Electro-Chemical basis of bioelectricity and related topics http://www.physiol.arizona.edu/CELL/Instruct/BodyElect/ExcitMemb.html To read more about self-organization and pattern formation in biology by Alexander V. Spirov (St. Petersburg, Russia) see http://avs.iephb.ru/resinter.htm > But what do you want to say about Excitable? Do you mean the spiral > wave patterns and such in excitable media? Yes, that is the most famous form of excitable systems. One striking (and not yet explained) difference between a periodically forced oscillatory and a periodically forced excitable system is that the Devil staircase in the "firing ratio versus forcing period" diagram is double-sided for oscillatory system and only single-sided for excitable system. Some plateaus are missing ! Another striking phenomenon is propagation failure in a system of 2 coupled excitable ODE systems. For certain parameter values, when one of the 2 systems is forced, the excitation propagates to the second system. And if you slightly INCREASE the forcing amplitude the propagation FAILS to propagate. This has been observed in a chemical experiment, modelled numerically and explained by Juraj Kosek (jkk@tiger.vscht.cz) now in Maddison, Wisconsin. I find these two phenomena very exciting. Unfortunately it is very difficult to explain the basic notion of excitability without pictures, just in ASCII text. --------------------------------------- Q[9]: What is Brusselator ? Brusselator is an imaginary chemical system designed by Prigogine and Nicolis is Brussels, Belgium. Nicolis, Gregoire & Prigogine, Ilya: EXPLORING COMPLEXITY: An Introduction W. H. FREEMAN 1989 ISBN: 0-7167-1859-6 http://www.opampbooks.com/SCI_CHAO/23.html It has been used as a standard model system to study nonlinear phenomena in both mixed and spatially distributed systems. Marek, Milos & Schreiber, Igor: Chaotic Behaviour of Deterministic Dissipative Systems Cambridge Univ. Press 1995 ISBN 0-521-43830-6 http://www.cup.org/Titles/43/0521438306.html Initial components A and B are transformed into products D and E via the reaction intermediates X and Y: A -> X B + X -> Y + D 2 X + Y -> 3 X X -> E For A and B constant (supplied from outside) and in dimensionless concentrations (to include the rate constants) the equations read: dX/dt = A - (B+1) X + X^2 Y dY/dt = B X - X^2 Y To read more about Ilya Prigogine see http://www.che.utexas.edu/grad-brochure/prigogine.html http://ttt.inferentia.it/west/project/10nob93/prigb.htm G.Nicolis and I. Prigogine Self - Organization in Nonequilibrium Systems, John Wiley & sons , 1977. Prigogine, Lefever : J. Chem.Phys. 48, (1968),1695. Anatol M. Zhabotinsky: A History of Chemical Oscillations and Waves Chaos Vol. 1 (No. 4) 1991 ISSN: 1054-1500 pp.379-386 ---------------------------------------