educators, schools & students. Like the Sierpinski triangle, the Barnsley fern shows how graphically beautiful structures can be built from repetitive uses of mathematical formulas with computers. . Top 119+ schrodinger equation tattoo super hot. A sample solution in the Lorenz attractor when ρ = 28, σ = 10, and β = 8 / 3. The purpose of this paper is to provide the reader with an introduction to the concepts of chaos theory and sensitive dependence on initial conditions. Watch. Mapping the Maxima for a Nonisothermal Chemical System. X is the population, while T and R are the rate of growth. Some elements for a history of the dynamical systems theory. S. The big news about chaos is supposed to be that the smallest of changes in a system can result in very large differences in that system’s behavior. The modern understanding of Newton's first law is that no inertial observer is privileged over any other. This section includes the full set of lecture notes for all 26 lectures. It explains how a small fluctuation in one condition of a nonlinear. For low values of r , xn (as n goes to infinity) eventually. m. To further dive into the simulations, this repo holds implementations of the presented models in Python. Chaos theory, Bush says, is "really simply a statement of lack of precision on the initial conditions of a system. Catastrophe Theory. The main precept behind this theory is the underlying notion of small. in around 1930 and it was based on so-called "qualitative theory of differential equations. where n, W, and (lambda_{ ext{U}}) represents index number, weighting matrix, and square diagonal matrix respectively. The output, f (3) = 1. Lorenz equations ([13], 1963) and R¨ossler system ([15], 1976) both with seven terms on the right-hand side do exhibit chaos for certain parameter values. Heller 1 and are part of the large field of quantum chaos. shows the template collapsed to the title. Learn Chaos theory facts for kids. Barnsley's 1988 book. When the noise is space white and when d <4 it is shown that the solution is in some “flat” ℒ 2. The Universe is Made of Protons, neutrons, electrons and morons Essential T-Shirt. Chaotic systems are complex systems which belong to the class of deterministic dynamical systems. It explains how a small fluctuation in one condition of a nonlinear. 1. The article . Lorenz equations used to generate plots for the y variable. ) Jump to:Details images of schrodinger equation tattoo by website vova. 1 Dynamical Systems and Determinism. davidope for Quanta Magazine. The aim of this second structure is to analyse and compare the dynamical behaviours. By dorthymala. Although many complex systems appear to behave in a random manner, chaos theory shows that, in reality, there is an underlying order that is difficult to see. It is one of the simplest equations that can generate deterministic chaos—apparently random behavior with no random cause. This current volume contains 12 new papers on the subject of chaos in the physical sciences, which was initiated with the publication of the book Research Advances in Chaos Theory. Examples of these complex systems that Chaos Theory helped fathom are earth's weather system, the behavior of water boiling on a. The fern is one of the basic examples of self-similar sets, i. Discover (and save!) your own Pins on PinterestFormally, chaos theory is defined as the study of complex nonlinear dynamic systems. Chaos theory models how a population of living creatures changes from one generation to the next, when there are limits to the available resources. (Image by Dr. So it's kinda like. The initial conditions for x and z were kept the same but those for y were changed between 1. It is notable for having chaotic solutions for certain parameter values and initial conditions. One can define a dynamical system fromList of chaotic maps. 893, and 0. New Tattoos. Weak non-linearity is used to illustrate bifurcation and asymptotic. Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, engineering, economics, biology, and philosophy. EST. Black Math Formula Equation Tattoos On Fingers. Researchers are convinced that locating the origin of chaos and watching it develop might allow science to predict, and perhaps counteract, outcomes. Turbulence is seen as the ultimate example of chaos theory: the way a butterfly flaps its wings in Australia could be linked to whether a hurricane forms over the. His simple model exhibits the phenomenon known as "sensitive dependence on initial conditions. The. Chaos theory is a complicated mathematical theory that studies the unpredictable and random nature of complex systems. "Chaos theory" is the popular term used to describe a novel, quite revolutionary approach to a wide range of mathematical, pure science, and applied science fields. For a while, in the 1980s and early 1990s, chaos was touted as the next big revolution in science, on par with quantum mechanics. Chaos theory presents the 1st Transdisciplinary understanding of bifurcation and transformational change. Most scientific domains deal with predictable models, such as gravity, chemical reactions and electricity. the. After learning her mother was diagnosed with breast cancer on Valentine’s Day, Josephine got a tattoo of one of the formulas for a heart curve, a fitting symbol of support and a. But the MIT scientist needed something even simpler if he hoped to get a better look at the tantalizing effects he glimpsed in his simulated weather. The term fractal was coined in 1975 by Benoît Mandelbrot, from the Latin fractus or "broken"/"fraction". Here is a list of tattoo ideas for entropy: Caligraphy style entropy tattoo design. Drake Equation Of Physics Tattoo On Forearm. Unlike the Mandelbrot set, the function on the Julia set is “choatic”. Different types of chaotic flow are possible in the 3-dimensional state spaces of two simple non- linear differential equations. The theory developed in ChaosBook part II Chaos rules is here to challenge a seasoned theorist. D. History. 8333333, is closer to the true value than the input. This theorem is foundational to our understanding of geometry. Books (including this one) and technical papers on chaos tend to dwell to a disproportionate extent on the logistic equation (Eq. Set your calculator to radians. Discover (and save!) your own Pins on PinterestApr 27, 2022 - Complex natural systems defy analysis using a standard mathematical toolkit, so one ecologist is throwing out the equations. Chaotic systems are not. " They gave full and complete analyses of two-dimensional systems and showed that the only attractors in 2D. Chaos theory equation pdf Complex quadratic polynomials A complex quadratic polynomial is a standard quadratic equation where the variable involved can be a complex number. Xn+1 = rXn(1-Xn) In this logistic model that describes how the. Baudelaire sanctions the fundamental views expressed in Poe’s The Poetic. A team of Italian scientists has figured out a way to turn the striking, complex twisting shapes of chaos theory into actual jewelry, according to a new. It deal with systems that are non-linear and unpredictable, and teaches us to expect the unexpected. In the mathematics of dynamical systems, the double-scroll attractor (sometimes known as Chua's attractor) is a strange attractor observed from a physical electronic chaotic circuit (generally, Chua's circuit) with a single nonlinear resistor (see Chua's diode ). “Prior to Poincaré, mathematicians studying dynamics, i. أخبار عاجلة. 3) Here σ, the bifurcation parameter may be obtained by 18. , hidden apart from its title bar. C. . In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. Ambigram Einstein Equation Tattoo On Upper Shoulder. Click here to start using LastPass: video presented on Veritasium used the following equation to describe an example of this type of miraculous mathematical paradox. Black Drake Equation Tattoo On Thumb. The term catastrophe, derived from the French in this usage, refers to the abrupt nature of the. Multiscroll attractor. Next, several basic properties and concepts of chaotic systems are described, including attractors, bifurcations, interaction and interdependence, state phase and phase space, and fractals. There are also images related to minimalist chaos theory tattoo, fractal chaos theory tattoo, physics chaos theory tattoo, chaos theory symbol tattoo, chaos theory lorenz attractor tattoo, chaos theory butterfly effect tattoo. Summary: Chaos theory is a mathematical theory that can be used to explain complex systems such as weather, astronomy, politics, and economics. In these drawings Rössler starts with a dynamic system known. This theory proves that no really existing processes may be predicted with certainty. Rebirth of chaos theory. Pioneering Australian scientist Robert May, whose work in biology led to the development of chaos theory, has died at age 84. . A bifurcation occurs in a nonlinear differential equation when a small change in a parameter results in a qualitative change in the long-time solution. The Chaos Theory describes a Dynamic Non-Linear System, where the conditions at the start seems determined and also correlations in between the parameters seems to be determined , however, the result is unpredictable. All this cool stuff happens due to properties of. Introduction. Chaos is often caused by chains of cause and effect whereby a small cause can eventually be amplified to greater and greater effects. In the following communication, we wish to add to Gelfand and Engelhart (2012), some emphasis on the dynamical systems theory (DST) applications in practical and theoretical psychology, and. Course Objectives: 1. So if you're gonna get a math tattoo, ideally get something that's aesthetically pleasing* independent of the mathematical significance. Other definitions of chaos are briefly considered in Section 5. In this paper, a novel combination of deep learning recurrent neural network and Lyapunov time is proposed to forecast the consumption of electricity load, in Greece, in normal/abrupt change value areas. Differential Equations. These patterns emerge from a relationship between these systems and related facets of math known as “strange attractors. Storytellers embraced its principles and worked them into their novels, films and plays. Visually the graph splits into two as there are two stable populations. ”. Including tattoos. ( You can use Windows Calculator set to scientific mode) 1. Art Prints. The mathematical treatment is friendly and informal, but still careful. FEČKAN: Homoclinic orbits of slowly periodically forced and weakly damped beams resting on weakly elastic bearings, Adv. Share. It looks at certain systems that are very sensitive. Learn Chaos theory facts for kids. is popularly referred to as the butterfly effect. This is the onset of chaos! Bifurcation Diagram — Chaos starts at x = 3. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of. Sierpiński triangle Generated using a random algorithm Sierpiński triangle in logic: The first 16 conjunctions of lexicographically ordered arguments. He is often described as a polymath, and in mathematics as "The Last Universalist", since he excelled in all fields of the discipline as. You could make equations about it, and figure out the answer pretty easily. ganizing principle of nature. A small perturbation in the initial setup of a chaotic system may lead to. For only nonlinear equations can model such systems, and "the struc-tures that provide the key to nonlinear dynamics proved to be fractal" {Chaos, 114). From $18. . What exactly is chaos theory? From the understanding of many scientists such as Edward Lorenz, Ian Stewart, and Robert May the chaos theory relatively means the same thing. Dizzying forms of chaos emerge from these mappings, but there is an irony--the chaotic behavior can be characterized in an orderly way. Love It 0. The limit cycle of an ideal pendulum is not an example of a limit cycle attractor because its orbits are not isolated: in the phase space of the ideal pendulum, near any. If you had precise information about every object right now, you could use the laws of physics to predict what would happen in the future. Lots and lots of tattoos. It represents that particular moment in which I met her, but more broadly, these kind of events that show you how insignificant your control over situations you never thought deeply about. 4 Proposed Method 3. edu. Chaotic maps often occur in the study of dynamical systems . science, and technology. Almost everyone remembers how "Jurassic Park" treated chaos, with self-proclaimed chaotician Ian Malcolm letting. 18 which, after nondimensionalization with respect to time, is given by. This theorem is foundational to our understanding of geometry. Abstract. Most. Chaos theory is the study of non-linear systems that are impossible to predict or control. "Chaos" is a tricky thing to define. The heart is an important muscular organ of the human body which pumps blood throughout the body. As a. Sep 10, 2010. I was so nerdy at university that I founded the maths club. Deterministic chaos refers to a class of deterministic systems in which seemingly random behavior is the result of dynamics described by nonlinear differential or recurrence equations (e. A dynamical system may be defined to be a. 892, 0. 2. Crossref Medline Google Scholar; 14 Goldberger AL. Improve this answer. By tnbvietnam. A chaos tattoo is a reminder that no matter how much we try to plan and organize our lives, there will always be an element of the unknown. Logistic Equation. The so-called butterfly effect has become one of the most popular images of chaos. To say that a system is chaotic means that there is a deterministic rule for its future evolution, but that small differences in initial conditions grow exponentially quickly over time. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system. February 2, 2020 at 12:00 p. The chaos game is played as follows. 2015-03-25. Chaos theory is a scientific principle describing the unpredictability of systems. Chaos Theory. 2, 889 (1992). . , a continuous-time dynamical system which gives you a function x ( t), t ∈ R, given an initial value x ( 0). However, combining the regeneration rate and connecting the future term with the current term actually generates the platform. The Lorenz Attractor is a strange attractor, which means the equation is non-periodic, as thus never repeats itself. Early Greek philosophers attempted to explain order in nature, anticipating modern concepts. The Chaos Theory Of Heart Attacks. First, a brief account of the definition and history of the development of chaos theory is presented. Chaos theory is a complex mathematical theory that attempts to explain. X kx t 1 (1 x) growth rate, k=(-7) k ªº¬¼7 xt 2 t 1 q 7s But, s E t E t sin u T s critical1/ 1/ 4/3 0. starkid. While nonlinear difierential equations are a natural place in physics for chaos to occur, the mathematically simpler iteration of nonlin-ear functions provides a quicker entry to chaos theory, which we will pursue flrst in Section 33. A tattoo of equations and stuff like that just makes it look like you're trying too hard to look smart. Lancet. The logistic map equation represents a discrete relationship between the current value and its future value . In Gleick’s book Chaos, the author interviewed a whole bunch of chaos scientists (chaosists?) and not one of them could agree on a. A very small change may make the system behave completely differently. where r is the so-called driving parameter. Chaotic conditions exist in nature. Two researchers have created a strikingly simple model of chaotic behavior, in which variations in initial conditions become so tangled and. The Pythagorean Theorem. This is not a library, but rather a collection of standalone scripts!On May 23, 1917, American mathematician, meteorologist, and a pioneer of chaos theory Edward Norton Lorenz was born. Minimum complexity of a chaotic system Bifurcation diagram of the logistic map x → r x (1 – x). While nonlinear difierential equations are a natural place in physics for chaos to occur, the mathematically simpler iteration of nonlin-ear functions provides a quicker entry to chaos theory, which we will pursue flrst in Section 33. 1: Coexistance of laminar and turbulent flow. We. Chaos was – most Greek cosmologies tell us – the very first of all, the origin of everything, the empty, unfathomable space at the beginning of time. 1. Non-linear dynamics for clinicians: chaos theory, fractals, and complexity at the bedside. 0. Half a century ago, the pioneers of chaos theory discovered that the “butterfly effect” makes long-term. starts with a single simple linear equation and . It describes a process evolving through time — xt+1, the level of some quantity x in the next time period — is given by the formula on the right, and it depends on xt,. Then you run the equation iteratively to obtain values of x 1, x 2, x 3. Section 4 is the heart of the paper, presenting a linear wave equation with gain and showing it satisfies the conditions required by the definition of chaos. Chaos theory, in mathematics and physics, deals with the behavior of certain nonlinear dynamical systems that (under certain conditions) exhibit the phenomenon known as chaos, most famously characterised by sensitivity to initial conditions. This comes from the. Introduction. Even so, this system and its motions are complex and extremely intriguing. A limit cycle is a periodic orbit of a continuous dynamical system that is isolated. Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. A particularly simple example of this is the polynomial f(z)=z2+cf(z) = z^2 + cf(z)=z2+c for some constant ccc. focused on one solution at a time. Approximate linear representations of nonlinear dynamics have long been. 4. Scientists have tracked the fishery there since 1948, through numerous upswings and downswings. Chaos theory, in mechanics and mathematics, the study of apparently random or unpredictable behavior in systems governed by deterministic laws. Chaos Theory is a mathematical sub-discipline that studies complex systems. 13 Denton TA, Diamond GA, Helfant RH, Khan S, Karagueuzian H. [3]May 8, 2021 - Explore Michael Conroy's board "Physics tattoos" on Pinterest. The term paper can be handed any time before the last day of lectures. The values for ho, sigma and eta were 45. The infinitely intricate computer-generated image of the set serves as an icon for the burgeoning field of chaos theory and has attracted enormous public attention. 1. Jun 3, 2021 - The field of study of chaos has its roots in differential equations and dynamical systems, the very language that is used to describe how any physical system. Chaos theory can be described as the science of surprises. A Twisted Path to Equation-Free Prediction | Quanta Magazine Complex natural systems defy analysis using a standard mathematical toolkit, so one ecologist is throwing out the. How a double pendulum. In these drawings Rössler starts with a dynamic system known. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life". Image by Wikipedia. Of course, chaos-theory equations, and thus the fractal images they generate, are "just metaphors" that imperfectly "represent real-ity," actual dynamical systems (Gleik, 77). Differential Equations 8. Don't Worry, I Have A Plan - Mathematics Equation Print Classic T-Shirt. He answered her, ‘ (∂ + m) ps = 0’ This is the equation of Dirac. Three examples of Turing patterns Six stable states from Turing equations, the last one forms Turing patterns. The double pendulum phase space plot from the section on the double pendulum shows extremely chaotic behavior. Understanding the interplay of order and disorder in chaos is a central challenge in modern quantitative science. Along with great metaphorical. ∂ t a = σ a + ξ 2 ∂ x x a − w ∂ x a − Λ − 1 ∫ 0 x a ( x ′, t) d x ′ − g a 3. However, for very large value of r such as r = 400, all solutions become periodical again (see Figure 08). By evaspark. In this context, chaos already arises in certain nonlinear functions of aChaos theory is a mathematical theory, and it is still in development. and anything technical. is investigated, where w is a space-time white noise or a space white noise. By trends. Chaos theory is the study of a particular type of systems that evolved from some initial conditions. 893, and 0. I was the nerdiest person in my high school, publicly teased for my love of maths and Doctor Who. Have A Heart. I even have a tattoo they represents the symbol on my left shoulder. Nonlinear. To set this template's initial visibility, the |state= parameter may be used: |state=collapsed: { {Chaos theory|state=collapsed}} to show the template collapsed, i. Crossref Medline Google Scholar; 14 Goldberger AL. Divide each number in the sequence by. The latest version of the meme includes a photo of a tattoo representing the Dirac equation, which is a equation discovered by British physicist Paul Dirac and published in 1928. Cartwright. One is of a dead cat, and one if of a cat that is very much alive and active. mahou shoujo. Chaos theory, more technically nonlinear dynamical systems (NLDS) theory, is an exciting, rapidly developing area of mathematics with increasing application in the physical, biological, and social sciences. Fractal geometry is indispensable to chaos theory- the study of orderly patterns in turbulent, dynamical, or erratic systems. History. In the present work, we tried to apply chaos theory to assess emotion dysregulation and emotional instability. It looks at certain systems that are very sensitive. The purpose of1. Very original tattoo set, bring your inner nerd to the light! This applier gives you four famous equations, certanly something that Sheldon Cooper. Snapshot 4: convergence on a 6-cycle. 33 Comments. Chaos Visualizations connecting chaos theory, fractals, and the logistic map! Written by Jonny Hyman, 2020. The consilience of the modeling and the basis of chaos theory give us solid assurance. Inverse Square Law Physics Tattoo On Forearm. " They gave full and complete analyses of two-dimensional systems and showed that the only attractors in 2D. social, that explains much of the great excitement and interest chaos theory has generated in so many fields. By two given lines how many straight lines can be drawn? A. from such a simple equation. A plot of Lorenz's strange attractor for values ρ=28, σ = 10, β = 8/3. the impact of seemingly small factors. 2, and C = 5. In this movement there is the very essence of chaos. since the ψ value defines how abrupt the voltage transition is. The Navier-Stokes equations are used rather extensively in climate modelling, race car aerodynamics, and medical research (e. A process called Newton’s method says you should start with any guess — let’s say 3 — and plug it into the function f ( x) = x 2 + 1 x . Applications of the mathematics of chaos are highly diverse, including the study of turbulence, heart irregularities, plasma physics, and the motion of star clusters. Mathematicians say it is tough to define chaos, but is easy to “recognize it when you see it. Fractal TattooChaos theory is demonstrated in this image, which was created with a long exposure of light at the end of a double pendulum. Apr 27, 2022 - Complex natural systems defy analysis using a standard mathematical toolkit, so one ecologist is throwing out the equations. Then chaos went mainstream with the publication of. In this paper, a novel chaotic system is presented based on the simple jerk equation. In the absence of outside influences, a body's motion preserves the status quo. For example: Initial Condition: uncomfortable pillow. principles of physical science. Chaos theory, in mechanics and mathematics, the study of apparently random or unpredictable behavior in systems governed by deterministic laws. it is a mathematically generated pattern that can be reproducible at any magnification or reduction. [1] [2] A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. Jeff Goldblum's character (the lovable bachelor in distress) in Jurassic Park brought about public awareness to the general concept of chaos theory. Forecasting. It’s brilliant, not only in its formation, but in its meaning. It concerns a cyclic attractor. Snapshot 6: effect of small change in initial conditions in an apparently chaotic regime. 4. 7. Loved this equation the chaos born from known properties. 84. That way, you'll still be happy with the artwork itself even if your mathematical interests. Well, it’s choas, and messy is what chaos does. Chaos Theory, at its core, is a mathematical sub-discipline that explores the behaviors of dynamical systems that are highly sensitive to initial conditions. Physics Black Hole. The drawing corner includes four pages covered in blue-ink sketches by Otto Rössler, a German biochemist who works on chaos theory. (2. An encryption system and method based on the mathematics of Chaos theory, which provides protection of data from unauthorized modification and use during its storage and transmission. A particularly simple example of this is the polynomial. See more ideas about chaos tattoo, chaos, symbolic tattoos. Heavily explored and recognized during the mid-to-late 1980s, its premise is that systems sometimes reside in chaos, generating energy but without any predictability or direction. Benoit B. N. In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. edu. Strange Attractors by Chaotic Atmospheres The darkest art known as Chaos Theory is perfectly embodied in the form of its strange attractors: vast looping trajectories of variables that, when plotted,. Black Ink Braggs Law Physics Tattoo On Foot. This is why even large computers cannot tell the weather for more than a few days. , the Logistic Map). The Turing pattern is a concept introduced by English mathematician Alan Turing in a 1952 paper titled "The Chemical Basis of Morphogenesis" which describes how patterns in nature, such as stripes and spots, can arise naturally. Chaotic equations. In the social sciences, chaos theory is the study of complex non-linear systems of social complexity. Bertacchini/P. e. Chaos:Classical and Quantum Predrag Cvitanovic´ – Roberto Artuso – Ronnie Mainieri – Gregor Tanner –. A classic example of this is. In factorized form, y = ax (1-x) Now let us rewrite this non-linear quadratic equation in this form: Graphically represented, this equation depicts a concave down parabola: The way we. By gboeing. 0001 and 1. One member of the melody set is defined as "the scaled output of an iterated function, provided it. a butterfly flapping its wings in South America can affect the weather in Central Park. Other people prefer the terms "complexity theory" or "dynamic systems theory. of ordinary differential equations in which it is possible to exhibit chaotic behavior. The initial conditions for x and z were kept the same but those for y were changed between 1. If you're thinking of chaos as in a big change with slightly different. It is notable for having chaotic solutions for certain parameter values and initial conditions. 1 Answer. Chaos theory, more technically nonlinear dynamical systems (NLDS) theory, is an exciting, rapidly developing area of mathematics with increasing application in the physical, biological, and social sciences. y = ax-ax². They were discovered and explained in 1984 by E. The simulations were governed by a set of differential equations, and each simulation had to be started off by a set of initial. Andrey Shilnikov. Chaos theory is a complex mathematical theory that explains chaotic or random occurrences. This process also gives rise to exquisite patterns shown in Figure 1. e. Scientists soon encountered other unpredictable natural systems that looked random even though they weren’t: the rings of Saturn, blooms of marine algae, Earth’s magnetic field, the number of salmon in a fishery. While John von Neumann had suggested. Before we start quantifying the equation and iteration, please note a few important points: The value of x0 is between zero and one. . But, it was more than just a gaping void – as its name is usually translated from Ancient Greek. It is formed from another set of Navier-Stokes equations, namely: where A = 0. The question then arises — why does a set of completely deterministic equations exhibit this behavior?Chaos Theory. BATTELLI & M. This volume looks at such problems as chaos in. Chaos theory is a part of mathematics. Touch device users, explore by touch. Chaos theory tattoo Chaos theory tattoo 10roulette; 11aposta ganha bet login; 12sg union sandersdorf; 13saldo do google play; 14odds significado; 15chaos theory tattoo. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. (a) ˙x versus x; (b) bifurcation diagram. I'd get a t-shirt with bubble chamber designs on it, but never a tattoo of it. Snapshot 2: convergence on a 2-cycle. The initial conditions for x and z were kept the same but those for y were changed between 1. 2001, World Scientific Series on Nonlinear Science Series A. A very small change may make the system behave completely differently. The first equation consists of a 2-variable, double.