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A consequence of describing particles as waveforms rather than points is that it is mathematically impossible to calculate with precision both the position and momentum of a particle at a given point in time. This became known as the uncertainty principle, a concept first introduced by Werner Heisenberg in 1927. | 313 | History_of_atomic_theory | https://en.wikipedia.org/wiki/History_of_atomic_theory | 400 | 54 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Schrödinger's wave model for hydrogen replaced Bohr's model, with its neat, clearly defined circular orbits. The modern model of the atom describes the positions of electrons in an atom in terms of probabilities. An electron can potentially be found at any distance from the nucleus, but, depending on its energy level and angular momentum, exists more frequently in certain regions around the nucleus than others; this pattern is referred to as its atomic orbital. The orbitals come in a variety of shapes— sphere, dumbbell, torus, etc.—with the nucleus in the middle. The shapes of atomic orbitals are found by solving the Schrödinger equation. Analytic solutions of the Schrödinger equation are known for very few relatively simple model Hamiltonians including the hydrogen atom and the hydrogen molecular ion. Beginning with the helium atom—which contains just two electrons—numerical methods are used to solve the Schrödinger equation. | 940 | History_of_atomic_theory | https://en.wikipedia.org/wiki/History_of_atomic_theory | 401 | 55 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Qualitatively the shape of the atomic orbitals of multi-electron atoms resemble the states of the hydrogen atom. The Pauli principle requires the distribution of these electrons within the atomic orbitals such that no more than two electrons are assigned to any one orbital; this requirement profoundly affects the atomic properties and ultimately the bonding of atoms into molecules. | 384 | History_of_atomic_theory | https://en.wikipedia.org/wiki/History_of_atomic_theory | 402 | 56 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
In condensed matter physics and materials science, an amorphous solid (or non-crystalline solid) is a solid that lacks the long-range order that is characteristic of a crystal. The terms " glass " and "glassy solid" are sometimes used synonymously with amorphous solid; however, these terms refer specifically to amorphous materials that undergo a glass transition. Examples of amorphous solids include glasses, metallic glasses, and certain types of plastics and polymers. | 473 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 403 | 0 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
The term comes from the Greek a ("without"), and morphé ("shape, form"). | 72 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 404 | 1 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Amorphous materials have an internal structure of molecular-scale structural blocks that can be similar to the basic structural units in the crystalline phase of the same compound. Unlike in crystalline materials, however, no long-range regularity exists: amorphous materials cannot be described by the repetition of a finite unit cell. Statistical mesures, such as the atomic density function and radial distribution function, are more useful in describing the structure of amorphous solids. | 492 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 405 | 2 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Although amorphous materials lack long range order, they exhibit localized order on small length scales. By convention, short range order extends only to the nearest neighbor shell, typically only 1-2 atomic spacings. Medium range order may extend beyond the short range order by 1-2 nm. | 287 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 406 | 3 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
The freezing from liquid state to amorphous solid - glass transition - is considered one of the very important and unsolved problems of physics. | 144 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 407 | 4 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
At very low temperatures (below 1-10 K), large family of amorphous solids have various similar low-temperature properties. Although there are various theoretical models, neither glass transition nor low-temperature properties of glassy solids are well understood on the fundamental physics level. | 296 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 408 | 5 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Amorphous solids is an important area of condensed matter physics aiming to understand these substances at high temperatures of glass transition and at low temperatures towards absolute zero. From 1970s, low-temperature properties of amorphous solids were studied experimentally in great detail. For all of these substances, specific heat has a (nearly) linear dependence as a function of temperature, and thermal conductivity has nearly quadratic temperature dependence. These properties are conventionally called anomalous being very different from properties of crystalline solids. | 584 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 409 | 6 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
On the phenomenological level, many of these properties were described by a collection of tunneling two-level systems. Nevertheless, the microscopic theory of these properties is still missing after more than 50 years of the research. | 234 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 410 | 7 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Remarkably, a dimensionless quantity of internal friction is nearly universal in these materials. This quantity is a dimensionless ratio (up to a numerical constant) of the phonon wavelength to the phonon mean free path. Since the theory of tunneling two-level states (TLSs) does not address the origin of the density of TLSs, this theory cannot explain the universality of internal friction, which in turn is proportional to the density of scattering TLSs. The theoretical significance of this important and unsolved problem was highlighted by Anthony Leggett. | 561 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 411 | 8 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Amorphous materials will have some degree of short-range order at the atomic-length scale due to the nature of intermolecular chemical bonding. Furthermore, in very small crystals, short-range order encompasses a large fraction of the atoms ; nevertheless, relaxation at the surface, along with interfacial effects, distorts the atomic positions and decreases structural order. Even the most advanced structural characterization techniques, such as X-ray diffraction and transmission electron microscopy, can have difficulty distinguishing amorphous and crystalline structures at short size scales. | 598 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 412 | 9 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Due to the lack of long-range order, standard crystallographic techniques are often inadequate in determining the structure of amorphous solids. A variety of electron, X-ray, and computation-based techniques have been used to characterize amorphous materials. Multi-modal analysis is very common for amorphous materials. | 320 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 413 | 10 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Unlike crystalline materials which exhibit strong Bragg diffraction, the diffraction patterns of amorphous materials are characterized by broad and diffuse peaks. As a result, detailed analysis and complementary techniques are required to extract real space structural information from the diffraction patterns of amorphous materials. It is useful to obtain diffraction data from both X-ray and neutron sources as they have different scattering properties and provide complementary data. Pair distribution function analysis can be performed on diffraction data to determine the probability of finding a pair of atoms separated by a certain distance. Another type of analysis that is done with diffraction data of amorphous materials is radial distribution function analysis, which measures the number of atoms found at varying radial distances away from an arbitrary reference atom. From these techniques, the local order of an amorphous material can be elucidated. | 965 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 414 | 11 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
X-ray absorption fine-structure spectroscopy is an atomic scale probe making it useful for studying materials lacking in long range order. Spectra obtained using this method provide information on the oxidation state, coordination number, and species surrounding the atom in question as well as the distances at which they are found. | 333 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 415 | 12 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
The atomic electron tomography technique is performed in transmission electron microscopes capable of reaching sub-Angstrom resolution. A collection of 2D images taken at numerous different tilt angles is acquired from the sample in question, and then used to reconstruct a 3D image. After image acquisition, a significant amount of processing must be done to correct for issues such as drift, noise, and scan distortion. High quality analysis and processing using atomic electron tomography results in a 3D reconstruction of an amorphous material detailing the atomic positions of the different species that are present. | 621 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 416 | 13 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Fluctuation electron microscopy is another transmission electron microscopy based technique that is sensitive to the medium range order of amorphous materials. Structural fluctuations arising from different forms of medium range order can be detected with this method. Fluctuation electron microscopy experiments can be done in conventional or scanning transmission electron microscope mode. | 391 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 417 | 14 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Simulation and modeling techniques are often combined with experimental methods to characterize structures of amorphous materials. Commonly used computational techniques include density functional theory, molecular dynamics, and reverse Monte Carlo. | 249 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 418 | 15 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Amorphous phases are important constituents of thin films. Thin films are solid layers of a few nanometres to tens of micrometres thickness that are deposited onto a substrate. So-called structure zone models were developed to describe the microstructure of thin films as a function of the homologous temperature (T h), which is the ratio of deposition temperature to melting temperature. According to these models, a necessary condition for the occurrence of amorphous phases is that (T h) has to be smaller than 0.3. The deposition temperature must be below 30% of the melting temperature. | 591 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 419 | 16 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Regarding their applications, amorphous metallic layers played an important role in the discovery of superconductivity in amorphous metals made by Buckel and Hilsch. The superconductivity of amorphous metals, including amorphous metallic thin films, is now understood to be due to phonon -mediated Cooper pairing. The role of structural disorder can be rationalized based on the strong-coupling Eliashberg theory of superconductivity. | 434 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 420 | 17 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Amorphous solids typically exhibit higher localization of heat carriers compared to crystalline, giving rise to low thermal conductivity. Products for thermal protection, such as thermal barrier coatings and insulation, rely on materials with ultralow thermal conductivity. | 273 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 421 | 18 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Today, optical coatings made from TiO 2, SiO 2, Ta 2 O 5 etc. (and combinations of these) in most cases consist of amorphous phases of these compounds. Much research is carried out into thin amorphous films as a gas separating membrane layer. The technologically most important thin amorphous film is probably represented by a few nm thin SiO 2 layers serving as isolator above the conducting channel of a metal-oxide semiconductor field-effect transistor (MOSFET). Also, hydrogenated amorphous silicon (Si:H) is of technical significance for thin-film solar cells. | 565 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 422 | 19 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
In the pharmaceutical industry, some amorphous drugs have been shown to offer higher bioavailability than their crystalline counterparts as a result of the higher solubility of the amorphous phase. However, certain compounds can undergo precipitation in their amorphous form in vivo, and can then decrease mutual bioavailability if administered together. | 354 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 423 | 20 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Amorphous materials in soil strongly influence bulk density, aggregate stability, plasticity, and water holding capacity of soils. The low bulk density and high void ratios are mostly due to glass shards and other porous minerals not becoming compacted. Andisol soils contain the highest amounts of amorphous materials. | 319 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 424 | 21 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
The occurrence of amorphous phases turned out to be a phenomenon of particular interest for the studying of thin-film growth. The growth of polycrystalline films is often used and preceded by an initial amorphous layer, the thickness of which may amount to only a few nm. The most investigated example is represented by the unoriented molecules of thin polycrystalline silicon films. Wedge-shaped polycrystals were identified by transmission electron microscopy to grow out of the amorphous phase only after the latter has exceeded a certain thickness, the precise value of which depends on deposition temperature, background pressure, and various other process parameters. The phenomenon has been interpreted in the framework of Ostwald's rule of stages that predicts the formation of phases to proceed with increasing condensation time towards increasing stability. | 867 | Amorphous_solid | https://en.wikipedia.org/wiki/Amorphous_solid | 425 | 22 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Bandwidth is the difference between the upper and lower frequencies in a continuous band of frequencies. It is typically measured in unit of hertz (symbol Hz). | 159 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 426 | 0 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
It may refer more specifically to two subcategories: Passband bandwidth is the difference between the upper and lower cutoff frequencies of, for example, a band-pass filter, a communication channel, or a signal spectrum. Baseband bandwidth is equal to the upper cutoff frequency of a low-pass filter or baseband signal, which includes a zero frequency. | 352 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 427 | 1 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Bandwidth in hertz is a central concept in many fields, including electronics, information theory, digital communications, radio communications, signal processing, and spectroscopy and is one of the determinants of the capacity of a given communication channel. | 261 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 428 | 2 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
A key characteristic of bandwidth is that any band of a given width can carry the same amount of information, regardless of where that band is located in the frequency spectrum. For example, a 3 kHz band can carry a telephone conversation whether that band is at baseband (as in a POTS telephone line) or modulated to some higher frequency. However, wide bandwidths are easier to obtain and process at higher frequencies because the § Fractional bandwidth is smaller. | 467 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 429 | 3 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Bandwidth is a key concept in many telecommunications applications. In radio communications, for example, bandwidth is the frequency range occupied by a modulated carrier signal. An FM radio receiver's tuner spans a limited range of frequencies. A government agency (such as the Federal Communications Commission in the United States) may apportion the regionally available bandwidth to broadcast license holders so that their signals do not mutually interfere. In this context, bandwidth is also known as channel spacing. | 522 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 430 | 4 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
For other applications, there are other definitions. One definition of bandwidth, for a system, could be the range of frequencies over which the system produces a specified level of performance. A less strict and more practically useful definition will refer to the frequencies beyond which performance is degraded. In the case of frequency response, degradation could, for example, mean more than 3 dB below the maximum value or it could mean below a certain absolute value. As with any definition of the width of a function, many definitions are suitable for different purposes. | 580 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 431 | 5 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
In the context of, for example, the sampling theorem and Nyquist sampling rate, bandwidth typically refers to baseband bandwidth. In the context of Nyquist symbol rate or Shannon-Hartley channel capacity for communication systems it refers to passband bandwidth. | 262 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 432 | 6 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
The Rayleigh bandwidth of a simple radar pulse is defined as the inverse of its duration. For example, a one-microsecond pulse has a Rayleigh bandwidth of one megahertz. | 169 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 433 | 7 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
The essential bandwidth is defined as the portion of a signal spectrum in the frequency domain which contains most of the energy of the signal. | 143 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 434 | 8 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
In some contexts, the signal bandwidth in hertz refers to the frequency range in which the signal's spectral density (in W/Hz or V /Hz) is nonzero or above a small threshold value. The threshold value is often defined relative to the maximum value, and is most commonly the 3 dB point, that is the point where the spectral density is half its maximum value (or the spectral amplitude, in V or V / H z , is 70.7% of its maximum). This figure, with a lower threshold value, can be used in calculations of the lowest sampling rate that will satisfy the sampling theorem. | 567 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 435 | 9 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
The bandwidth is also used to denote system bandwidth, for example in filter or communication channel systems. To say that a system has a certain bandwidth means that the system can process signals with that range of frequencies, or that the system reduces the bandwidth of a white noise input to that bandwidth. | 312 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 436 | 10 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
The 3 dB bandwidth of an electronic filter or communication channel is the part of the system's frequency response that lies within 3 dB of the response at its peak, which, in the passband filter case, is typically at or near its center frequency, and in the low-pass filter is at or near its cutoff frequency. If the maximum gain is 0 dB, the 3 dB bandwidth is the frequency range where attenuation is less than 3 dB. 3 dB attenuation is also where power is half its maximum. This same half-power gain convention is also used in spectral width, and more generally for the extent of functions as full width at half maximum (FWHM). | 630 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 437 | 11 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
In electronic filter design, a filter specification may require that within the filter passband, the gain is nominally 0 dB with a small variation, for example within the ±1 dB interval. In the stopband (s), the required attenuation in decibels is above a certain level, for example >100 dB. In a transition band the gain is not specified. In this case, the filter bandwidth corresponds to the passband width, which in this example is the 1 dB-bandwidth. If the filter shows amplitude ripple within the passband, the x dB point refers to the point where the gain is x dB below the nominal passband gain rather than x dB below the maximum gain. | 643 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 438 | 12 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
In signal processing and control theory the bandwidth is the frequency at which the closed-loop system gain drops 3 dB below peak. | 130 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 439 | 13 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
In communication systems, in calculations of the Shannon–Hartley channel capacity, bandwidth refers to the 3 dB-bandwidth. In calculations of the maximum symbol rate, the Nyquist sampling rate, and maximum bit rate according to the Hartley's law, the bandwidth refers to the frequency range within which the gain is non-zero. | 325 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 440 | 14 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
The fact that in equivalent baseband models of communication systems, the signal spectrum consists of both negative and positive frequencies, can lead to confusion about bandwidth since they are sometimes referred to only by the positive half, and one will occasionally see expressions such as B = 2 W , where B is the total bandwidth (i.e. the maximum passband bandwidth of the carrier-modulated RF signal and the minimum passband bandwidth of the physical passband channel), and W is the positive bandwidth (the baseband bandwidth of the equivalent channel model). For instance, the baseband model of the signal would require a low-pass filter with cutoff frequency of at least W to stay intact, and the physical passband channel would require a passband filter of at least B to stay intact. | 793 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 441 | 15 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
The absolute bandwidth is not always the most appropriate or useful measure of bandwidth. For instance, in the field of antennas the difficulty of constructing an antenna to meet a specified absolute bandwidth is easier at a higher frequency than at a lower frequency. For this reason, bandwidth is often quoted relative to the frequency of operation which gives a better indication of the structure and sophistication needed for the circuit or device under consideration. | 472 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 442 | 16 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
There are two different measures of relative bandwidth in common use: fractional bandwidth ( B F ) and ratio bandwidth ( B R ). In the following, the absolute bandwidth is defined as follows, B = Δ Δ f = f H − − f L where f H and f L are the upper and lower frequency limits respectively of the band in question. | 312 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 443 | 17 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Fractional bandwidth is defined as the absolute bandwidth divided by the center frequency ( f C ), B F = Δ Δ f f C . | 116 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 444 | 18 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
The center frequency is usually defined as the arithmetic mean of the upper and lower frequencies so that, f C = f H + f L 2 and B F = 2 ( f H − − f L ) f H + f L . | 164 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 445 | 19 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
However, the center frequency is sometimes defined as the geometric mean of the upper and lower frequencies, f C = f H f L and B F = f H − − f L f H f L . | 154 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 446 | 20 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
While the geometric mean is more rarely used than the arithmetic mean (and the latter can be assumed if not stated explicitly) the former is considered more mathematically rigorous. It more properly reflects the logarithmic relationship of fractional bandwidth with increasing frequency. For narrowband applications, there is only marginal difference between the two definitions. The geometric mean version is inconsequentially larger. For wideband applications they diverge substantially with the arithmetic mean version approaching 2 in the limit and the geometric mean version approaching infinity. | 601 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 447 | 21 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Fractional bandwidth is sometimes expressed as a percentage of the center frequency (percent bandwidth, % % B ), % % B F = 100 Δ Δ f f C . | 138 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 448 | 22 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Ratio bandwidth is defined as the ratio of the upper and lower limits of the band, B R = f H f L . | 98 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 449 | 23 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Ratio bandwidth may be notated as B R : 1 . The relationship between ratio bandwidth and fractional bandwidth is given by, B F = 2 B R − − 1 B R + 1 and B R = 2 + B F 2 − − B F . | 178 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 450 | 24 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Percent bandwidth is a less meaningful measure in wideband applications. A percent bandwidth of 100% corresponds to a ratio bandwidth of 3:1. All higher ratios up to infinity are compressed into the range 100–200%. | 214 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 451 | 25 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
Ratio bandwidth is often expressed in octaves (i.e., as a frequency level) for wideband applications. An octave is a frequency ratio of 2:1 leading to this expression for the number of octaves, log 2 ( B R ) . | 213 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 452 | 26 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
The noise equivalent bandwidth (or equivalent noise bandwidth (enbw)) of a system of frequency response H ( f ) is the bandwidth of an ideal filter with rectangular frequency response centered on the system's central frequency that produces the same average power outgoing H ( f ) when both systems are excited with a white noise source. The value of the noise equivalent bandwidth depends on the ideal filter reference gain used. Typically, this gain equals | H ( f ) | at its center frequency, but it can also equal the peak value of | H ( f ) | . | 549 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 453 | 27 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
The noise equivalent bandwidth B n can be calculated in the frequency domain using H ( f ) or in the time domain by exploiting the Parseval's theorem with the system impulse response h ( t ) . If H ( f ) is a lowpass system with zero central frequency and the filter reference gain is referred to this frequency, then: | 318 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 454 | 28 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
B n = ∫ ∫ − − ∞ ∞ ∞ ∞ | H ( f ) | 2 d f 2 | H ( 0 ) | 2 = ∫ ∫ − − ∞ ∞ ∞ ∞ | h ( t ) | 2 d t 2 | ∫ ∫ − − ∞ ∞ ∞ ∞ h ( t ) d t | 2 . | 129 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 455 | 29 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
The same expression can be applied to bandpass systems by substituting the equivalent baseband frequency response for H ( f ) . | 127 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 456 | 30 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
The noise equivalent bandwidth is widely used to simplify the analysis of telecommunication systems in the presence of noise. | 125 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 457 | 31 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
In photonics, the term bandwidth carries a variety of meanings: | 63 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 458 | 32 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
A related concept is the spectral linewidth of the radiation emitted by excited atoms. | 86 | Bandwidth_(signal_processing) | https://en.wikipedia.org/wiki/Bandwidth_(signal_processing) | 459 | 33 | 2,024 | 8 | 10 | 0 | 30 | 59 | 0 | |
In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state. | 210 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 460 | 0 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
The term is closely associated with the work of the mathematician and meteorologist Edward Norton Lorenz. He noted that the butterfly effect is derived from the metaphorical example of the details of a tornado (the exact time of formation, the exact path taken) being influenced by minor perturbations such as a distant butterfly flapping its wings several weeks earlier. Lorenz originally used a seagull causing a storm but was persuaded to make it more poetic with the use of a butterfly and tornado by 1972. He discovered the effect when he observed runs of his weather model with initial condition data that were rounded in a seemingly inconsequential manner. He noted that the weather model would fail to reproduce the results of runs with the unrounded initial condition data. A very small change in initial conditions had created a significantly different outcome. | 871 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 461 | 1 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
The idea that small causes may have large effects in weather was earlier acknowledged by the French mathematician and physicist Henri Poincaré. The American mathematician and philosopher Norbert Wiener also contributed to this theory. Lorenz's work placed the concept of instability of the Earth's atmosphere onto a quantitative base and linked the concept of instability to the properties of large classes of dynamic systems which are undergoing nonlinear dynamics and deterministic chaos. | 490 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 462 | 2 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
The concept of the butterfly effect has since been used outside the context of weather science as a broad term for any situation where a small change is supposed to be the cause of larger consequences. | 201 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 463 | 3 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
In The Vocation of Man (1800), Johann Gottlieb Fichte says "you could not remove a single grain of sand from its place without thereby... changing something throughout all parts of the immeasurable whole". | 205 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 464 | 4 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
Chaos theory and the sensitive dependence on initial conditions were described in numerous forms of literature. This is evidenced by the case of the three-body problem by Poincaré in 1890. He later proposed that such phenomena could be common, for example, in meteorology. | 272 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 465 | 5 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
In 1898, Jacques Hadamard noted general divergence of trajectories in spaces of negative curvature. Pierre Duhem discussed the possible general significance of this in 1908. | 173 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 466 | 6 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
In 1950, Alan Turing noted: "The displacement of a single electron by a billionth of a centimetre at one moment might make the difference between a man being killed by an avalanche a year later, or escaping." | 208 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 467 | 7 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
The idea that the death of one butterfly could eventually have a far-reaching ripple effect on subsequent historical events made its earliest known appearance in " A Sound of Thunder ", a 1952 short story by Ray Bradbury. "A Sound of Thunder" features time travel. | 264 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 468 | 8 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
More precisely, though, almost the exact idea and the exact phrasing —of a tiny insect's wing affecting the entire atmosphere's winds— was published in a children's book which became extremely successful and well-known globally in 1962, the year before Lorenz published: | 270 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 469 | 9 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
"...whatever we do affects everything and everyone else, if even in the tiniest way. Why, when a housefly flaps his wings, a breeze goes round the world." | 154 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 470 | 10 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
-- The Princess of Pure Reason | 30 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 471 | 11 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
In 1961, Lorenz was running a numerical computer model to redo a weather prediction from the middle of the previous run as a shortcut. He entered the initial condition 0.506 from the printout instead of entering the full precision 0.506127 value. The result was a completely different weather scenario. | 302 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 472 | 12 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
Lorenz wrote: | 13 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 473 | 13 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
At one point I decided to repeat some of the computations in order to examine what was happening in greater detail. I stopped the computer, typed in a line of numbers that it had printed out a while earlier, and set it running again. I went down the hall for a cup of coffee and returned after about an hour, during which time the computer had simulated about two months of weather. The numbers being printed were nothing like the old ones. I immediately suspected a weak vacuum tube or some other computer trouble, which was not uncommon, but before calling for service I decided to see just where the mistake had occurred, knowing that this could speed up the servicing process. Instead of a sudden break, I found that the new values at first repeated the old ones, but soon afterward differed by one and then several units in the last [decimal] place, and then began to differ in the next to the last place and then in the place before that. In fact, the differences more or less steadily doubled in size every four days or so, until all resemblance with the original output disappeared somewhere in the second month. This was enough to tell me what had happened: the numbers that I had typed in were not the exact original numbers, but were the rounded-off values that had appeared in the original printout. The initial round-off errors were the culprits; they were steadily amplifying until they dominated the solution. | 1,424 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 474 | 14 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
In 1963, Lorenz published a theoretical study of this effect in a highly cited, seminal paper called Deterministic Nonperiodic Flow (the calculations were performed on a Royal McBee LGP-30 computer). Elsewhere he stated: | 220 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 475 | 15 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
One meteorologist remarked that if the theory were correct, one flap of a sea gull's wings would be enough to alter the course of the weather forever. The controversy has not yet been settled, but the most recent evidence seems to favor the sea gulls. | 251 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 476 | 16 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
Following proposals from colleagues, in later speeches and papers, Lorenz used the more poetic butterfly. According to Lorenz, when he failed to provide a title for a talk he was to present at the 139th meeting of the American Association for the Advancement of Science in 1972, Philip Merilees concocted Does the flap of a butterfly's wings in Brazil set off a tornado in Texas? as a title. Although a butterfly flapping its wings has remained constant in the expression of this concept, the location of the butterfly, the consequences, and the location of the consequences have varied widely. | 594 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 477 | 17 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
The phrase refers to the idea that a butterfly's wings might create tiny changes in the atmosphere that may ultimately alter the path of a tornado or delay, accelerate, or even prevent the occurrence of a tornado in another location. The butterfly does not power or directly create the tornado, but the term is intended to imply that the flap of the butterfly's wings can cause the tornado: in the sense that the flap of the wings is a part of the initial conditions of an interconnected complex web; one set of conditions leads to a tornado, while the other set of conditions doesn't. The flapping wing represents a small change in the initial condition of the system, which cascades to large-scale alterations of events (compare: domino effect). Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different—but it's also equally possible that the set of conditions without the butterfly flapping its wings is the set that leads to a tornado. | 985 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 478 | 18 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
The butterfly effect presents an obvious challenge to prediction, since initial conditions for a system such as the weather can never be known to complete accuracy. This problem motivated the development of ensemble forecasting, in which a number of forecasts are made from perturbed initial conditions. | 303 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 479 | 19 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
Some scientists have since argued that the weather system is not as sensitive to initial conditions as previously believed. David Orrell argues that the major contributor to weather forecast error is model error, with sensitivity to initial conditions playing a relatively small role. Stephen Wolfram also notes that the Lorenz equations are highly simplified and do not contain terms that represent viscous effects; he believes that these terms would tend to damp out small perturbations. Recent studies using generalized Lorenz models that included additional dissipative terms and nonlinearity suggested that a larger heating parameter is required for the onset of chaos. | 674 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 480 | 20 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
While the "butterfly effect" is often explained as being synonymous with sensitive dependence on initial conditions of the kind described by Lorenz in his 1963 paper (and previously observed by Poincaré), the butterfly metaphor was originally applied to work he published in 1969 which took the idea a step further. Lorenz proposed a mathematical model for how tiny motions in the atmosphere scale up to affect larger systems. He found that the systems in that model could only be predicted up to a specific point in the future, and beyond that, reducing the error in the initial conditions would not increase the predictability (as long as the error is not zero). This demonstrated that a deterministic system could be "observationally indistinguishable" from a non-deterministic one in terms of predictability. Recent re-examinations of this paper suggest that it offered a significant challenge to the idea that our universe is deterministic, comparable to the challenges offered by quantum physics. | 1,002 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 481 | 21 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
In the book entitled The Essence of Chaos published in 1993, Lorenz defined 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." This feature is the same as sensitive dependence of solutions on initial conditions (SDIC) in. In the same book, Lorenz applied the activity of skiing and developed an idealized skiing model for revealing the sensitivity of time-varying paths to initial positions. A predictability horizon is determined before the onset of SDIC. | 614 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 482 | 22 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
Recurrence, the approximate return of a system toward its initial conditions, together with sensitive dependence on initial conditions, are the two main ingredients for chaotic motion. They have the practical consequence of making complex systems, such as the weather, difficult to predict past a certain time range (approximately a week in the case of weather) since it is impossible to measure the starting atmospheric conditions completely accurately. | 454 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 483 | 23 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
A dynamical system displays sensitive dependence on initial conditions if points arbitrarily close together separate over time at an exponential rate. The definition is not topological, but essentially metrical. Lorenz defined sensitive dependence as follows: | 259 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 484 | 24 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
The property characterizing an orbit (i.e., a solution) if most other orbits that pass close to it at some point do not remain close to it as time advances. | 156 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 485 | 25 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
If M is the state space for the map f t , then f t displays sensitive dependence to initial conditions if for any x in M and any δ > 0, there are y in M, with distance d (.,.) such that 0 < d ( x , y ) < δ δ and such that | 221 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 486 | 26 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
for some positive parameter a. The definition does not require that all points from a neighborhood separate from the base point x, but it requires one positive Lyapunov exponent. In addition to a positive Lyapunov exponent, boundedness is another major feature within chaotic systems. | 284 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 487 | 27 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
The simplest mathematical framework exhibiting sensitive dependence on initial conditions is provided by a particular parametrization of the logistic map : | 155 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 488 | 28 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
which, unlike most chaotic maps, has a closed-form solution : | 61 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 489 | 29 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
where the initial condition parameter θ θ is given by θ θ = 1 π π sin − − 1 ( x 0 1 / 2 ) . For rational θ θ , after a finite number of iterations x n maps into a periodic sequence. But almost all θ θ are irrational, and, for irrational θ θ , x n never repeats itself – it is non-periodic. This solution equation clearly demonstrates the two key features of chaos – stretching and folding: the factor 2 shows the exponential growth of stretching, which results in sensitive dependence on initial conditions (the butterfly effect), while the squared sine function keeps x n folded within the range [0, 1]. | 608 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 490 | 30 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
The butterfly effect is most familiar in terms of weather; it can easily be demonstrated in standard weather prediction models, for example. The climate scientists James Annan and William Connolley explain that chaos is important in the development of weather prediction methods; models are sensitive to initial conditions. They add the caveat: "Of course the existence of an unknown butterfly flapping its wings has no direct bearing on weather forecasts, since it will take far too long for such a small perturbation to grow to a significant size, and we have many more immediate uncertainties to worry about. So the direct impact of this phenomenon on weather prediction is often somewhat wrong." | 699 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 491 | 31 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
The concept of the butterfly effect encompasses several phenomena. The two kinds of butterfly effects, including the sensitive dependence on initial conditions, and the ability of a tiny perturbation to create an organized circulation at large distances, are not exactly the same. In Palmer et al., a new type of butterfly effect is introduced, highlighting the potential impact of small-scale processes on finite predictability within the Lorenz 1969 model. Additionally, the identification of ill-conditioned aspects of the Lorenz 1969 model points to a practical form of finite predictability. These two distinct mechanisms suggesting finite predictability in the Lorenz 1969 model are collectively referred to as the third kind of butterfly effect. The authors in have considered Palmer et al.'s suggestions and have aimed to present their perspective without raising specific contentions. | 893 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 492 | 32 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
The third kind of butterfly effect with finite predictability, as discussed in, was primarily proposed based on a convergent geometric series, known as Lorenz's and Lilly's formulas. Ongoing discussions are addressing the validity of these two formulas for estimating predictability limits in. | 293 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 493 | 33 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
A comparison of the two kinds of butterfly effects and the third kind of butterfly effect has been documented. In recent studies, it was reported that both meteorological and non-meteorological linear models have shown that instability plays a role in producing a butterfly effect, which is characterized by brief but significant exponential growth resulting from a small disturbance. | 384 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 494 | 34 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
The first kind of butterfly effect, known as SDIC (Sensitive Dependence on Initial Conditions), is widely recognized and demonstrated through idealized chaotic models. However, opinions differ regarding the second kind of butterfly effect, specifically the impact of a butterfly flapping its wings on tornado formation, as indicated in two 2024 articles. For the third kind of butterfly effect, the limited predictability within the Lorenz 1969 model is explained by scale interactions in one article and by system ill-conditioning in another more recent study. | 561 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 495 | 35 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
According to Lighthill (1986), the presence of SDIC (commonly known as the butterfly effect) implies that chaotic systems have a finite predictability limit. In a literature review, it was found that Lorenz's perspective on the predictability limit can be condensed into the following statement: | 295 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 496 | 36 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
Recently, a short video has been created to present Lorenz's perspective on predictability limit. | 97 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 497 | 37 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
A recent study refers to the two-week predictability limit, initially calculated in the 1960s with the Mintz-Arakawa model's five-day doubling time, as the "Predictability Limit Hypothesis." Inspired by Moore’s Law, this term acknowledges the collaborative contributions of Lorenz, Mintz, and Arakawa under Charney’s leadership. The hypothesis supports the investigation into extended-range predictions using both partial differential equation (PDE)-based physics methods and Artificial Intelligence (AI) techniques. | 516 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 498 | 38 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 | |
By revealing coexisting chaotic and non-chaotic attractors within Lorenz models, Shen and his colleagues proposed a revised view that "weather possesses chaos and order", in contrast to the conventional view of "weather is chaotic". As a result, sensitive dependence on initial conditions (SDIC) does not always appear. Namely, SDIC appears when two orbits (i.e., solutions) become the chaotic attractor; it does not appear when two orbits move toward the same point attractor. The above animation for double pendulum motion provides an analogy. For large angles of swing the motion of the pendulum is often chaotic. By comparison, for small angles of swing, motions are non-chaotic. Multistability is defined when a system (e.g., the double pendulum system) contains more than one bounded attractor that depends only on initial conditions. The multistability was illustrated using kayaking in Figure on the right side (i.e., Figure 1 of ) where the appearance of strong currents and a stagnant area suggests instability and local stability, respectively. As a result, when two kayaks move along strong currents, their paths display SDIC. On the other hand, when two kayaks move into a stagnant area, they become trapped, showing no typical SDIC (although a chaotic transient may occur). Such features of SDIC or no SDIC suggest two types of solutions and illustrate the nature of multistability. | 1,396 | Butterfly_effect | https://en.wikipedia.org/wiki/Butterfly_effect | 499 | 39 | 2,024 | 8 | 10 | 0 | 31 | 0 | 0 |