Fractals And Chaos Pdf 16

Fractals And Chaos Pdf 16

Fractals And Chaos Pdf 16 - https://geags.com/2t7e6b

Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities.[1] Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization.[2] The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions).[3] A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas.[4][5][6]

Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in numerical computation, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general.[7] This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution[8] and is fully determined by their initial conditions, with no random elements involved.[9] In other words, the deterministic nature of these systems does not make them predictable.[10][11] This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:[12]

Chaotic behavior exists in many natural systems, including fluid flow, heartbeat irregularities, weather, and climate.[13][14][8] It also occurs spontaneously in some systems with artificial components, such as the road traffic.[2] This behavior can be studied through the analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. Chaos theory has applications in a variety of disciplines, including meteorology,[8] anthropology,[15] sociology, environmental science, computer science, engineering, economics, ecology, and pandemic crisis management.[16][17] The theory formed the basis for such fields of study as complex dynamical systems, edge of chaos theory, and self-assembly processes.

In common usage, "chaos" means "a state of disorder".[20][21] However, in chaos theory, the term is defined more precisely. Although no universally accepted mathematical definition of chaos exists, a commonly used definition, originally formulated by Robert L. Devaney, says that to classify a dynamical system as chaotic, it must have these properties:[22]

If attention is restricted to intervals, the second property implies the other two.[26] An alternative and a generally weaker definition of chaos uses only the first two properties in the above list.[27]

As suggested in Lorenz's book entitled "The Essence of Chaos", published in 1993,[5] "sensitive dependence can serve as an acceptable definition of chaos". In the same book, Lorenz defined the butterfly effect as: "The phenomenon that a small alteration in the state of a dynamical system will cause subsequent states to differ greatly from the states that would have followed without the alteration." The above definition is consistent with the sensitive dependence of solutions on initial conditions (SDIC). An idealized skiing model was developed to illustrate the sensitivity of time-varying paths to initial positions.[5] A predictability horizon can be determined before the onset of SDIC (i.e., prior to significant separations of initial nearby trajectories).[29]

Topological mixing is often omitted from popular accounts of chaos, which equate chaos with only sensitivity to initial conditions. However, sensitive dependence on initial conditions alone does not give chaos. For example, consider the simple dynamical system produced by repeatedly doubling an initial value. This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points eventually becomes widely separated. However, this example has no topological mixing, and therefore has no chaos. Indeed, it has extremely simple behavior: all points except 0 tend to positive or negative infinity.

where x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} make up the system state, t {\displaystyle t} is time, and σ {\displaystyle \sigma } , ρ {\displaystyle \rho } , β {\displaystyle \beta } are the system parameters. Five of the terms on the right hand side are linear, while two are quadratic; a total of seven terms. Another well-known chaotic attractor is generated by the Rössler equations, which have only one nonlinear term out of seven. Sprott[46] found a three-dimensional system with just five terms, that had only one nonlinear term, which exhibits chaos for certain parameter values. Zhang and Heidel[47][48] showed that, at least for dissipative and conservative quadratic systems, three-dimensional quadratic systems with only three or four terms on the right-hand side cannot exhibit chaotic behavior. The reason is, simply put, that solutions to such systems are asymptotic to a two-dimensional surface and therefore solutions are well behaved.

Under the right conditions, chaos spontaneously evolves into a lockstep pattern. In the Kuramoto model, four conditions suffice to produce synchronization in a chaotic system.Examples include the coupled oscillation of Christiaan Huygens' pendulums, fireflies, neurons, the London Millennium Bridge resonance, and large arrays of Josephson junctions.[62]

An early proponent of chaos theory was Henri Poincaré. In the 1880s, while studying the three-body problem, he found that there can be orbits that are nonperiodic, and yet not forever increasing nor approaching a fixed point.[63][64][65] In 1898, Jacques Hadamard published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature, called "Hadamard's billiards".[66] Hadamard was able to show that all trajectories are unstable, in that all particle trajectories diverge exponentially from one another, with a positive Lyapunov exponent.

Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after mid-century, when it first became evident to some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments like that of the logistic map. What had been attributed to measure imprecision and simple "noise" was considered by chaos theorists as a full component of the studied systems.

The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand. Electronic computers made these repeated calculations practical, while figures and images made it possible to visualize these systems. As a graduate student in Chihiro Hayashi's laboratory at Kyoto University, Yoshisuke Ueda was experimenting with analog computers and noticed, on November 27, 1961, what he called "randomly transitional phenomena". Yet his advisor did not agree with his conclusions at the time, and did not allow him to report his findings until 1970.[73][74]

Edward Lorenz was an early pioneer of the theory. His interest in chaos came about accidentally through his work on weather prediction in 1961.[13] Lorenz and his collaborator Ellen Fetter[75] were using a simple digital computer, a Royal McBee LGP-30, to run weather simulations. They wanted to see a sequence of data again, and to save time they started the simulation in the middle of its course. They did this by entering a printout of the data that corresponded to conditions in the middle of the original simulation. To their surprise, the weather the machine began to predict was completely different from the previous calculation. They tracked this down to the computer printout. The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number, so a value like 0.506127 printed as 0.506. This difference is tiny, and the consensus at the time would have been that it should have no practical effect. However, Lorenz discovered that small changes in initial conditions produced large changes in long-term outcome.[76] Lorenz's discovery, which gave its name to Lorenz attractors, showed that even detailed atmospheric modeling cannot, in general, make precise long-term weather predictions.

In December 1977, the New York Academy of Sciences organized the first symposium on chaos, attended by David Ruelle, Robert May, James A. Yorke (coiner of the term "chaos" as used in mathematics), Robert Shaw, and the meteorologist Edward Lorenz. The following year Pierre Coullet and Charles Tresser published "Itérations d'endomorphismes et groupe de renormalisation", and Mitchell Feigenbaum's article "Quantitative Universality for a Class of Nonlinear Transformations" finally appeared in a journal, after 3 years of referee rejections.[43][83] Thus Feigenbaum (1975) and Coullet & Tresser (1978) discovered the universality in chaos, permitting the application of chaos theory to many different phenomena.

In 1986, the New York Academy of Sciences co-organized with the National Institute of Mental Health and the Office of Naval Research the first important conference on chaos in biology and medicine. There, Bernardo Huberman presented a mathematical model of the eye tracking dysfunction among people with schizophrenia.[85] This led to a renewal of physiology in the 1980s through the application of chaos theory, for example, in the study of pathological cardiac cycles.

In the same year, James Gleick published Chaos: Making a New Science, which became a best-seller and introduced the general principles of chaos theory as well as its history to the broad public.[88] Initially the domain of a few, isolated individuals, chaos theory progressively emerged as a transdisciplinary and institutional discipline, mainly under the name of nonlinear systems analysis. Alluding to Thomas Kuhn's concept of a paradigm shift exposed in The Structure of Scientific Revolutions (1962), many "chaologists" (as some described themselves) claimed that this new theory was an example of such a shift, a thesis upheld by Gleick. 2b1af7f3a8