In mathematics, chaos theory describes the behavior of some dynamic systems-that is, systems whose states evolve with time-which may show dynamic behaviors that are highly sensitive to initial conditions (usually called butter fly effect). This sensitivity is manifested in the exponential growth of disturbances under initial conditions, so the behavior of chaotic systems seems random. Even if these systems are deterministic, that is to say, their future dynamics are completely determined by initial conditions and do not contain any random factors, this situation will still happen. This behavior is called deterministic chaos, or chaos for short.
Chaotic behavior can also be observed in natural systems, such as weather. This can be explained by chaos theory analysis of the mathematical model of such a system, which embodies the physical laws related to natural systems.
general survey
Chaotic behavior has been observed in various systems in the laboratory, including circuits, lasers, oscillating chemical reactions, fluid dynamics and mechanical and magneto-mechanical devices. The observation of chaotic behavior in nature includes the dynamics of satellites in the solar system, the temporal evolution of celestial magnetic field, the population growth in ecology, the dynamics of action potentials in neurons and molecular vibration. Everyday examples of chaotic systems include weather and climate. [1] There are some disputes about the existence of chaotic dynamics in plate tectonics and economics. [2][3][4]
The system showing mathematical chaos is deterministic, so it is orderly in a sense; This technical usage of the word chaos is inconsistent with the common saying, which implies complete chaos. A related field of physics called quantum chaos theory studies systems that follow the laws of quantum mechanics. Recently, another field, the so-called relativistic chaos [5], has emerged to describe the system following the law of general relativity.
In addition to being orderly in a deterministic sense, chaotic systems usually have clearly defined statistical data. [Need to quote] For example, the Lorenz system in the figure is chaotic, but it has a clearly defined structure. Bounded chaos is a useful term to describe disorder model.
history
The first discoverer of chaos was Henry Poincare. In 1890, while studying three bodies, he found that there can be aperiodic orbits, but they will not increase forever or approach a fixed point. [6] In 1898, Jacques Hadamard published an influential paper, which studied the chaotic motion of free particles sliding without friction on a surface with constant negative curvature. [7] In the studied system "Hadamard's Billiards", Hadamard can prove that all trajectories are unstable, because all particle trajectories diverge exponentially from each other and have a positive Lyapunov exponent.
Many early theories were almost entirely developed by mathematicians in the name of ergodic theory. Later, G.D. Birkhoff, [8] A. N. Kolmogorov, [9] [10] [1] M.L. Cartwright and J.E. Littlewood, [1]. [13] Except Smale, these studies are directly inspired by physics: boekhoff's three-body, turbulence and astronomical problems in Kolmogorov, and radio engineering in Cartwright and Littlewood. Although chaotic planetary motion has not been observed, experimenters have encountered turbulence in fluid motion and aperiodic oscillation in radio circuits, and there is no theory to explain what they have seen.
Although there was a preliminary view in the first half of the 20th century, the chaos theory was formally formed only after the middle of the 20th century, when some scientists first discovered that the dominant system theory-linear theory-could not explain some behaviors observed in experiments, such as logical mapping. What was previously excluded from the measurement accuracy and simple "noise" is considered by chaos theory to be an integral part of the studied system.
The main catalyst for the development of chaos theory is the electronic computer. Most of the mathematics of chaos theory involves repeated iterations of simple mathematical formulas, so it is unrealistic to do it by hand. Electronic computers make these repeated calculations feasible, while numbers and images make it possible to visualize these systems. ENIAC, one of the earliest electronic digital computers, was used to run simple weather forecast models.
The early pioneer of this theory was edward lorenz, and his interest in chaos came about by accident through his weather forecast work in 196654 38+0. Lorenz uses a simple digital computer Royal McBee LGP-30 to run his weather simulation. He wanted to see a series of data again. In order to save time, he started the simulation halfway. He can do this by inputting a printout of the data corresponding to the conditions in the last calculated simulation process.
To his surprise, the weather predicted by the machine was completely different from the weather calculated before. Lorenz tracked the computer printout. The computer works with a precision of 6 digits, but the printout rounds the variable to 3 digits, so a value like 0.506 127 is printed as 0.506. This difference is very small, and the consensus at that time was that it actually had no impact. However, Lorenz found that small changes in initial conditions will lead to great changes in long-term results. [15] Lorenz's discovery (named after Lorenz attractor) proves that meteorology can't reasonably predict the weather with more than one cycle (at most).
A year ago, Benoit Mandelbrot found a recurring pattern on every scale of cotton price data. [16] Before that, he studied the information theory and concluded that the noise pattern is similar to Cantor set: at any scale, the ratio of noise-containing period to error-free period is constant, so the error is inevitable and must be planned by introducing redundancy. [17] Mandelbrot described the "Noah effect" (in which sudden discontinuous changes may occur, for example, the price of a stock after bad news, thus challenging the normal distribution theory in statistics, also known as the bell curve) and the "Joseph effect" (in which the persistence of a value may appear for a period of time, but then suddenly changes). [18] [19]1967, he published How long is the coast of Britain? Statistical self-similarity and fractal dimension "shows that the length of coastline varies with the scale of measuring instruments, and it is similar to itself in all scales, and the length is infinite for infinitely small measuring equipment." [20] He thinks that when viewed from a distance, a mass of hemp thread looks like a point (0 dimension), and when viewed from a close distance, it looks like a ball (3 dimension) or a curve (1 dimension). He thinks that the dimension of an object is relative to the observer and can be a fraction. Objects whose irregularities remain unchanged at different scales ("self-similarity") are fractal (for example, Koch curve or "snowflake", which is infinitely long, but contains a limited space, and its fractal dimension is about equal to 1.26 19, mengle sponge and Sierpiński washer). 1975 Mandelbrot published Fractal Geometry of Nature, which became a classic of chaos theory. Branches of biological systems, such as circulatory system and bronchial system, have been proved to conform to fractal model.
Chaos was observed by many experimenters before it was recognized; For example, van der Pol[2 1] 1927 and R.L. Ives 1958. [22][23] However, it seems that Yoshisuke Ueda was the first experimenter to confirm the chaotic phenomenon by using an analog computer on 1 1 July 27th. Chaos shown by analog computer is a real phenomenon. Compared with chaos calculated by digital computer, chaos calculated by digital computer has a different limitation in accuracy. Hayashi, Ueda's guidance professor, did not believe in chaos, so he banned Ueda from publishing his findings until 1970. [24]
19771February, the New York Academy of Sciences organized the first seminar on chaos. The participants included David Reuel, Robert May, James York (the creator of the word "chaos" in mathematics) and robert shaw (a physicist, who participated in Eudaemons with J Donne Farmer and Norman Packard.
The following year, Mitchell Feigenbaum published a famous article "Quantitative Universality of a Class of Nonlinear Transformations", in which he described logical mapping. [25] feigenbaum once applied fractal geometry to the study of natural forms such as coastline. Feigenbaum remarkably discovered the universality of chaos, allowing chaos theory to be applied to many different phenomena.
In 1979, Albert J. Libchaber showed his experimental observation on bifurcation cascade leading to chaos and turbulence in convective Rayleigh-Benard system at a seminar organized by Pierre Hohenberg in Aspen. Together with Mitchell J. feigenbaum, he won the Wolf Prize in Physics in 1986 in recognition of his "outstanding experimental demonstration of the transition to turbulence and chaos in dynamic systems". [26]
The New York Academy of Sciences subsequently organized the first important conference on chaos in biology and medicine in 1986 with the National Institute of Mental Health and the Office of Naval Research. Bernard Huberman put forward a mathematical model of eye tracking disorder in schizophrenic patients. [27] Chaos theory revived physiology in1980s, for example, in the study of pathological cardiac cycle.
In 1987, Per Bak, Chao Tang and Kurt Wiesenfeld published a paper in Physical Review Letters[28], describing self-organized criticality (SOC) for the first time, which is considered as one of the mechanisms of complexity in nature. Except Bak-Tang-wiesen feld sa nd pile and other methods which are mainly based on laboratory, many other researches focus on large-scale natural or social systems which are known (or suspected) to exhibit scale-invariant behaviors. Although these methods are not always welcomed by experts (at least initially), SOC has become a powerful candidate to explain many natural phenomena, including: earthquakes (long before SOC was discovered, earthquakes were considered as the source of scale-invariant behaviors, such as Gutenberg-Richter Law, which describes the statistical distribution of earthquake scale, and Otsu Law, which describes the frequency of aftershocks [29]); Solar flare; Fluctuations in economic systems such as financial markets (it is common to mention SOC in economic physics); Landscape formation; Forest fire; Landslide; Epidemics; And biological evolution (SOC is cited, for example, as the dynamic mechanism behind the "punctuated equilibrium" theory put forward by niles Eldridge and Stephen Jay Gould). Worryingly, in view of the meaning of the scale-free distribution of events, some researchers suggest that another phenomenon that should be regarded as an example of SOC is the occurrence of war. These "application" studies of SOC include modeling attempts (developing new models or modifying existing models according to the specific conditions of a given natural system) and extensive data analysis to determine the existence and/or characteristics of natural scaling laws.
In the same year, james gleick published Chaos: Creating a New Science, which became a bestseller and introduced the general principle and history of chaos theory to the general public. At first, chaos theory was the work field of a few isolated individuals, and later it gradually became an interdisciplinary institutional discipline, mainly in the name of nonlinear system analysis. Referring to the concept of paradigm shift revealed by thomas kuhn in The Structure of Scientific Revolution (1962), many "chaos scientists" (some claim to be) claim that this new theory is an example of "transformation" supported by J Gerecke.
The appearance of cheaper and more powerful computers broadens the application scope of chaos theory. At present, chaos theory is still a very active research field, involving many different disciplines (mathematics, topology, physics, population biology, biology, meteorology, astrophysics, information theory, etc. ).
[Edit] Chaos Dynamics
To classify a dynamic system as chaos, it must have the following characteristics: [30]
It must be sensitive to initial conditions,
It must be topologically mixed, and
Its periodic orbit must be dense.
Sensitivity to initial conditions means that every point in such a system can be arbitrarily approximated by other points with very different future trajectories. Therefore, any small disturbance of the current trajectory may lead to obviously different future behaviors.
Sensitivity to initial conditions is often called "butterfly effect" because Edward Lorenz submitted an article entitled "Predictability: Will Brazilian butterflies flap their wings to cause tornadoes in Texas?" to the American Association for the Advancement of Science in Washington, D.C. in 1972. The paper. Flapping represents a small change in the initial conditions of the system, which causes a series of events and leads to large-scale phenomena. If the butterfly didn't flap its wings, the trajectory of this system might be very different.
In the popular saying, sensitivity to initial conditions is often confused with chaos. It may also be a subtle property because it depends on the choice of measurement or the concept of distance in the phase space of the system. For example, consider a simple dynamical system generated by repeatedly doubling an initial value (defined by iterative mapping on the solid line of mapping x t o 2x). This system is sensitive to the initial conditions anywhere, because any pair of adjacent points will eventually become far apart. However, it has extremely simple behavior, because all points except 0 tend to infinity. On the contrary, if we use bounded metric on the line obtained by adding points at infinity and regard the result as a circle, the system is no longer sensitive to initial conditions. For this reason, when defining chaos, attention is usually limited to systems with bounded metrics or closed and bounded invariant subsets of unbounded systems.
Even for bounded systems, the sensitivity to initial conditions is not equal to chaos. For example, consider a two-dimensional torus described by a pair of angles (x, y), each ranging from 0 to 2π. Define a mapping from any point (x, y) to (2x, y+a), where a is any number that makes a/2π irrational. Due to the doubling in the first coordinate, the mapping shows a sensitive dependence on the initial conditions. However, due to unreasonable rotation in the second coordinate, there is no periodic orbit, so according to the above definition, the mapping is not chaotic.
Topological mixing means that the system will evolve with time, so the open set of any given region or its phase space will eventually overlap with any other given region. Here, "mixing" actually means a standard intuition: the mixing of colored dyes or liquids is an example of a chaotic system.
Linear systems will never be chaotic; For a dynamical system showing chaotic behavior, it must be nonlinear. Similarly, according to Poincare-Bendickson theorem, a continuous dynamic system on a plane cannot be chaotic; In continuous systems, only those systems whose phase space is non-planar (at least three-dimensional, or with non-Euclidean geometry) can show chaotic behavior. However, a discrete dynamic system (such as logical mapping) can exhibit chaotic behavior in one or two-dimensional phase space.
[edit] aspirator
Some dynamical systems are chaotic everywhere (see Anosov differential homeomorphism, for example), but in many cases, chaotic behavior is only found in a subset of phase space. When chaotic behavior occurs on the attractor, the most interesting situation appears, because at that time a large set of initial conditions will cause the orbit to converge to this chaotic region.
A simple way to visualize a chaotic attractor is to start from a point in the attractor's basin and then simply draw its subsequent orbit. Due to topological transitivity conditions, this is likely to produce an image of the entire final attractor.
For example, in a system describing a pendulum, the phase space may be two-dimensional, consisting of position and velocity information. People can draw the relationship between the position and speed of a pendulum. The static pendulum will be drawn as a point, while the periodic pendulum will be drawn as a simple closed curve. When such a graph forms a closed curve, the curve is called an orbit. Our pendulum has an infinite number of such orbits, forming a bunch of nested ellipses around the origin.
[Editor] Strange attractor
Although most of the motion types mentioned above produce very simple attractors, such as points and quasi-circular curves called limit cycles, chaotic motion produces so-called strange attractors, which can have great details and complexity. For example, a simple three-dimensional model of the Lorenz weather system produced the famous Lorenz attractor. Lorenz attractor is perhaps one of the most famous chaotic system diagrams, perhaps because it is not only one of the earliest chaotic system diagrams, but also one of the most complex chaotic system diagrams, thus producing a very interesting pattern that looks like a butterfly wing. Another such attractor is r? Ssler map, which experienced the second multiplication period leading to chaos, just like logistic map.
Strange attractor exists in both continuous dynamic systems (such as Lorenz system) and some discrete systems (such as Hénon mapping). Other discrete dynamical systems have a repulsive structure called Julia set, which is formed on the boundary between attractive basins of fixed points-Julia set can be regarded as a strange repulsive object. Strange attractor and Julia sets usually have fractal structures.
Poincare-bendiksen theorem shows that a strange attractor can only appear in a continuous dynamic system if it has three or more dimensions. However, this restriction does not apply to discrete systems, which can show strange attractors in two-dimensional or even one-dimensional systems.
Three or more objects can be arranged to produce chaotic motion through the initial conditions of gravitational interaction (see N-body problem).
Minimum complexity of chaotic system
Simple systems can produce chaos without relying on differential equations. An example is a logical map, which is a difference equation (recursive relationship) describing the population growth over time. Another example is the Rick model of population dynamics.
Even the evolution of simple discrete systems, such as cellular automata, may depend heavily on initial conditions. Stephen wolfram studied cellular automata with this characteristic, which he called Rule 30.
Arnold's cat map provides the minimum model of conservative (reversible) chaotic behavior.