**Stellar pulsations** are caused by expansions and contractions in the outer layers as a star seeks to maintain equilibrium. These fluctuations in stellar radius cause corresponding changes in the luminosity of the star. Astronomers are able to deduce this mechanism by measuring the spectrum and observing the Doppler effect.^{[1]} Many intrinsic variable stars that pulsate with large amplitudes, such as the classical Cepheids, RR Lyrae stars and large-amplitude Delta Scuti stars show regular light curves.

This regular behavior is in contrast with the variability of stars that lie parallel to and to the high-luminosity/low-temperature side of the classical variable stars in the Hertzsprung-Russell diagram. These giant stars are observed to undergo pulsations ranging from weak irregularity, when one can still define an average cycling time or period, (as in most RV Tauri and semiregular variables) to the near absence of repetitiveness in the irregular variables. The W Virginis variables are at the interface; the short period ones are regular and the longer period ones show first relatively regular alternations in the pulsations
cycles, followed by the onset of mild irregularity as in the RV Tauri stars into which they gradually morph as their periods get longer.^{[2]}^{[3]} Stellar evolution and pulsation theories suggest that these irregular stars have a much higher luminosity to mass (L/M) ratios.

Many stars are non-radial pulsators, which have smaller fluctuations in brightness than those of regular variables used as standard candles.^{[4]}^{[5]}

A prerequisite for irregular variability is that the star be able to change its amplitude on the time scale of a period. In other words, the coupling between pulsation and heat flow must be sufficiently large to allow such changes. This coupling is measured by the relative linear growth- or decay rate κ (kappa) of the amplitude of a given normal mode in one pulsation cycle (period). For the regular variables (Cepheids, RR Lyrae, etc.) numerical stellar modeling and linear stability analysis show that κ is at most of the order of a couple of percent for the relevant, excited pulsation modes. On the other hand, the same type of analysis shows that for the high L/M models κ is considerably larger (30% or higher).

For the regular variables the small relative growth rates κ imply that there are two distinct time scales, namely the period of oscillation and the longer time associated with the amplitude variation. Mathematically speaking, the dynamics has a center manifold, or more precisely a near center manifold. In addition, it has been found that the stellar pulsations are only weakly nonlinear in the sense that their description can be limited powers of the pulsation amplitudes. These two properties are very general and occur for oscillatory systems in many other fields such as population dynamics, oceanography, plasma physics, etc.

The weak nonlinearity and the long time scale of the amplitude variation allows the temporal description of the pulsating system to be simplified to that of only the pulsation amplitudes, thus eliminating motion on the short time scale of the period. The result is a description of the system in terms of amplitude equations that are truncated to low powers of the amplitudes. Such amplitude equations have been derived by a variety of techniques, e.g. the Poincaré-Lindstedt method of elimination of secular terms, or the multi-time asymptotic perturbation method,^{[6]}^{[7]}^{[8]} and more generally, normal form theory.^{[9]}^{[10]}^{[11]}

For example, in the case of two non-resonant modes, a situation generally encountered in RR Lyrae variables, the temporal evolution of the amplitudes A_{1} and A_{2} of the two normal modes 1 and 2 is
governed by the following set of ordinary differential equations

where the Q_{ij} are the nonresonant coupling coefficients.^{[12]}^{[13]}

These amplitude equations have been limited to the lowest order nontrivial nonlinearities. The solutions of interest in stellar pulsation theory are the asymptotic solutions (as time tends towards infinity) because the time scale for the amplitude variations is generally very short compared to the evolution time scale of the star which is the nuclear burning time scale. The equations above have fixed point solutions with constant amplitudes, corresponding to single-mode
(A_{1} 0, A_{2} = 0) or
(A_{1} = 0, A_{2} 0) and double-mode
(A_{1} 0, A_{2}0)
solutions. These correspond to singly periodic and doubly periodic pulsations of the star. It is important to emphasize that no other asymptotic solution of the above equations exists for physical (i.e., negative) coupling coefficients.

For resonant modes the appropriate amplitude equations have additional terms that describe the resonant coupling among the modes. The Hertzsprung progression in the light curve morphology of classical (singly periodic) Cepheids is the result of a well-known 2:1 resonance among the fundamental pulsation mode and the second overtone mode.^{[14]} The amplitude equation can be further extended to nonradial stellar pulsations.^{[15]}^{[16]}

In the overall analysis of pulsating stars, the amplitude equations allow the bifurcation diagram between possible pulsational states to be mapped out. In this picture, the boundaries of the instability strip where pulsation sets in during the star's evolution correspond to a Hopf bifurcation.^{[17]}

The existence of a center manifold eliminates the possibility of chaotic (i.e. irregular) pulsations on the time scale of the period. Although resonant amplitude equations are sufficiently complex to also allow for chaotic solutions, this is a very different chaos because it is in the temporal variation of the amplitudes and occurs on a long time scale.

While long term irregular behavior in the temporal variations of the pulsation amplitudes is possible when amplitude equations apply, this is not the general situation. Indeed, for the majority of the observations and modeling, the pulsations of these stars occur with constant Fourier amplitudes, leading to regular pulsations that can be periodic or multi-periodic (quasi-periodic in the mathematical literature).

The light curves of intrinsic variable stars with large amplitudes have been known for centuries to exhibit behavior that goes from extreme regularity, as for the classical Cepheids and the RR Lyrae stars, to extreme irregularity, as for the so-called Irregular variables. In the Population II stars this irregularity gradually increases from the low period W Virginis variables through the RV Tauri variables into the regime of the semiregular variables. Low-dimensional chaos in stellar pulsations is the current interpretation of this established phenomenon.

The regular behavior of the Cepheids has been successfully modeled with numerical hydrodynamics since the 1960s,^{[18]}^{[19]} and from a theoretical point of view it is easily understood as due to the presence of center manifold which arises because of the weakly dissipative nature of the dynamical system.^{[20]} This, and the fact that the pulsations are weakly nonlinear,
allows a description of the system in terms of amplitude equations^{[21]}
^{[22]} and a construction of the bifurcation diagram (see also bifurcation theory) of the possible types of pulsation (or limit cycles), such fundamental mode pulsation, first or second overtone pulsation, or more complicated, double-mode pulsations in which several modes are excited with constant amplitudes. The boundaries of the instability strip where pulsation sets in during the star's evolution correspond to a Hopf bifurcation.

In contrast, the irregularity of the large amplitude Population II stars is more challenging to explain.
The variation of the pulsation amplitude over one period implies *large* dissipation, and therefore there exists no center manifold.
Various mechanisms have been proposed, but are found lacking. One, suggests the presence of several closely spaced pulsation frequencies that would beat
against each other, but no such frequencies exist in the appropriate stellar models. Another, more interesting suggestion is that the variations are of a stochastic nature,^{[23]} but no mechanism has been proposed or exists that could provide the energy for such large observed amplitude variations. It is now established that the mechanism behind the irregular light curves is an underlying low dimensional chaotic dynamics (see also Chaos theory). This conclusion is based on two types of studies.

The numerical computations of the pulsations of *sequences* of W Virginis stellar models exhibit two approaches to irregular behavior that are a clear signature of low dimensional chaos. The first indication comes from *first return maps* in which one plots one maximum radius, or any other suitable variable, versus the next one.
The sequence of models shows a period doubling bifurcation, or cascade, leading to chaos.
The near quadratic shape of the map is indicative of chaos and implies an underlying horseshoe map.^{[24]}^{[25]}^{[26]} Other sequences of models follow a somewhat different route, but also to chaos,
namely the Pommeau-Manneville or *tangent bifurcation* route.^{[27]}^{[28]}

The following shows a similar visualization of the period doubling cascade to chaos for a sequence of stellar models that differ by their average surface temperature T.
The graph shows triplets of values of the stellar radius (R_{i}, R_{i+1}, R_{i+2})
where the indices _{i},_{ i+1},_{ i+2} indicate successive time intervals.

P0 | P2 | P4 | P8 | Banded Chaos | FullChaos |

The presence of low dimensional chaos is also confirmed by another, more sophisticated, analysis of the model pulsations which extracts the lowest unstable periodic orbits and examines their topological organization (twisting). The underlying attractor is found to be banded like the Roessler attractor, with however an additional twist in the band.^{[29]}

The method of *global flow reconstruction*^{[30]} uses a single observed signal {s_{i}} to infer properties of the dynamical system that generated it. First N-dimensional 'vectors' **S**_{i}=(s_{i},s_{i-1},s_{i-2},...,s_{i-N+1}) are constructed.
The next step consists in finding an expression for the nonlinear evolution operator **M** that takes the system from time i to time i+1, i.e.
**S**_{i+1}= **M** (**S**_{i}).
Takens' theorem guarantees that under very general circumstances the topological properties of this reconstructed evolution operator are the same as that of the physical system, provided the embedding dimension N is large enough.
Thus *from the knowledge of a single observed variable* one can infer properties about the real physical system which is governed by a number of independent variables.

This approach has been applied to the AAVSO data for the star R Scuti^{[31]}^{[32]}
It could be inferred that the irregular pulsations of this star arise from an underlying 4-dimensional dynamics. Phrased differently this says that from any 4 neighboring observations one can predict the next one. From a physical point of view it says that there are 4 independent variables that describe the dynamic of the system. The method of false nearest neighbors corroborates an embedding dimension of 4.
The fractal dimension of the dynamics of R Scuti as inferred from the computed Lyapunov exponents lies between 3.1 and 3.2.

From an analysis of the fixed points of the evolution operator a nice physical picture can be inferred, namely that *the pulsations arise from the excitation of an unstable pulsation mode that couples nonlinearly to a second, stable pulsation mode which is in a 2:1 resonance with the first one*, a scenario described by the Shilnikov theorem.^{[33]}

This resonance mechanism is not limited to R Scuti, but has been found to hold for several other stars for which the observational data are sufficiently good.^{[34]}

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BL Boötis (abbreviated to BL Boo) is a pulsating variable star in the constellation Boötes. It varies from magnitude 14.45 to 15.10 over 0.82 days. It is located 4 arcminutes from the centre of (and assumed to be a member star of) the globular cluster NGC 5466. Its variability was first noted in 1961 by Russian astronomer Nikolaĭ Efimovich Kurochkin, who gave it the variable star designation BL Boötis. However, he thought it was an eclipsing binary. It was subsequently thought to be an RR Lyrae variable by T.I. Gryzunova in 1971.Robert Zinn confirmed it was a member of the globular cluster and found it was too blue to be an RR Lyrae variable. He gave it the name V19 within the cluster. He calculated its mass to be around 1.56 times and its luminosity to be around 278 times that of the Sun; its absolute magnitude is -1.27.BL Boötis has been designated the prototype of a rare class of variable star known as an anomalous Cepheid or BL Boötis variable. These stars are somewhat similar to Cepheid variables, but they do not have the same relationship between their period and luminosity. Their periods are similar to the ab subtypes of RR Lyrae variables; however, they are far brighter than these stars. Anomalous Cepheids are metal poor and have masses not much larger than the Sun's, on average, 1.5 solar masses. The origin of these stars is uncertain, but thought to possibly be from the merger of two stars. Detailed examination of the spectrum of BL Boötis with the Keck-1 telescope at the W. M. Keck Observatory showed that its effective (surface) temperature is around 6450 K at minimum light. It also showed that the chemical composition was consistent with ageing metal-poor (Population II) stars and hence cast doubt on the origin as a result of a stellar merger. The radial velocity is slower than would be expected if it were from a stellar merger.

Cepheid variableA Cepheid variable () is a type of star that pulsates radially, varying in both diameter and temperature and producing changes in brightness with a well-defined stable period and amplitude.

A strong direct relationship between a Cepheid variable's luminosity and pulsation period established Cepheids as important indicators of cosmic benchmarks for scaling galactic and extragalactic distances. This robust characteristic of classical Cepheids was discovered in 1908 by Henrietta Swan Leavitt after studying thousands of variable stars in the Magellanic Clouds. This discovery allows one to know the true luminosity of a Cepheid by simply observing its pulsation period. This in turn allows one to determine the distance to the star, by comparing its known luminosity to its observed brightness.

The term Cepheid originates from Delta Cephei in the constellation Cepheus, identified by John Goodricke in 1784, the first of its type to be so identified.

Classical Cepheid variableClassical Cepheids (also known as Population I Cepheids, Type I Cepheids, or Delta Cepheid variables) are a type of Cepheid variable star. They are population I variable stars that exhibit regular radial pulsations with periods of a few days to a few weeks and visual amplitudes from a few tenths of a magnitude to about 2 magnitudes.

There exists a well-defined relationship between a classical Cepheid variable's luminosity and pulsation period, securing Cepheids as viable standard candles for establishing the galactic and extragalactic distance scales. Hubble Space Telescope (HST) observations of classical Cepheid variables have enabled firmer constraints on Hubble's law. Classical Cepheids have also been used to clarify many characteristics of our galaxy, such as the Sun's height above the galactic plane and the Galaxy's local spiral structure.Around 800 classical Cepheids are known in the Milky Way Galaxy, out of an expected total of over 6,000. Several thousand more are known in the Magellanic Clouds, with more known in other galaxies. The Hubble Space Telescope has identified classical Cepheids in NGC 4603, which is 100 million light years distant.

CoRoTCoRoT (French: Convection, Rotation et Transits planétaires; English: Convection, Rotation and planetary Transits) was a space telescope mission which operated from 2006 to 2013. The mission's two objectives were to search for extrasolar planets with short orbital periods, particularly those of large terrestrial size, and to perform asteroseismology by measuring solar-like oscillations in stars. The mission was led by the French Space Agency (CNES) in conjunction with the European Space Agency (ESA) and other international partners.

Among the notable discoveries was COROT-7b, discovered in 2009 which became the first exoplanet shown to have a rock or metal-dominated composition.

CoRoT was launched at 14:28:00 UTC on 27 December 2006, atop a Soyuz 2.1b rocket, reporting first light on 18 January 2007. Subsequently, the probe started to collect science data on 2 February 2007. CoRoT was the first spacecraft dedicated to the detection of transiting extrasolar planets, opening the way for more advanced probes such as Kepler and TESS. It detected its first extrasolar planet, COROT-1b, in May 2007, just 3 months after the start of the observations. Mission flight operations were originally scheduled to end 2.5 years from launch but operations were extended to 2013. On 2 November 2012, CoRoT suffered a computer failure that made it impossible to retrieve any data from its telescope. Repair attempts were unsuccessful, so on 24 June 2013 it was announced that CoRoT has been retired and would be decommissioned; lowered in orbit to allow it to burn up in the atmosphere.

ESO 3.6 m TelescopeThe ESO 3.6 m Telescope is an optical reflecting telescope run by the European Southern Observatory at La Silla Observatory, Chile since 1977, with a clear aperture of about 3.6 metres (140 in) and 8.6 m2 (93 sq ft) area.

The telescopes uses the HARPS instrument and has discovered more than 130 exoplanets. In 2012, it discovered Alpha Centauri Bb, a now-disproven possible planet in the Alpha Centauri system only 4.4 light-years away.It saw first light in 1976 and entered full operations in 1977. It received an overhaul in 1999 and a new secondary in 2004. When completed in the late 1970s, it was one of the world's largest optical telescopes. The ESO 3.6-metre Telescope has supported many scientific achievements and presented ADONIS, one of the first adaptive optics system available to the astronomical community in the 1980s.

Gamma Doradus variableGamma Doradus variables are variable stars which display variations in luminosity due to non-radial pulsations of their surface. The stars are typically young, early F or late A type main sequence stars, and typical brightness fluctuations are 0.1 magnitudes with periods on the order of one day. This class of variable stars is relatively new, having been first characterized in the second half of the 1990s, and details on the underlying physical cause of the variations remains under investigation.

The star 9 Aurigae was first noticed to be variable in 1990. However, none of the currently-accepted explanations were adequate: it pulsated too slowly and was outside of the Delta Scuti instability strip, and there was no evidence for any eclipsing material, although Gamma Doradus and HD 96008 were noted to be similar. These three stars, as well as HD 224638, were soon hypothesized to belong to a new class of variable stars in which variability was produced by g-mode pulsations rather than the p-mode pulsations of Delta Scuti variables. HD 224945 and HD 164615 were noticed to be similar as well, while HD 96008 was ruled out on the basis of its more regular period. Eclipses and starspots were soon ruled out as the cause of the Gamma Doradus' variability, and the variability of 9 Aurigae was confirmed to be caused by g-mode pulsations a year later, thus confirming the stars as the prototypes of a new class of variable stars. Over ten more candidates were quickly found, and the discovers dubbed the group the Gamma Doradus stars, after the brightest member and the first member found to be variable.

Gamma LeonisGamma Leonis (γ Leonis, abbreviated Gamma Leo, γ Leo), formally named Algieba , is a binary star system in the constellation of Leo. In 2009, a planetary companion around the primary was announced.

Gamma Leonis bGamma Leonis b is an extrasolar planet located 125.5 light years away in the constellation Leo, orbiting the giant star Gamma Leonis.

List of largest starsBelow is an ordered list of the largest stars currently known by radius. The unit of measurement used is the radius of the Sun (approximately 695,700 km; 432,288 mi).

The exact order of this list is very incomplete, as great uncertainties currently remain, especially when deriving various important parameters used in calculations, such as stellar luminosity and effective temperature. Often stellar radii can only be expressed as an average or within a large range of values. Values for stellar radii vary significantly in sources and throughout the literature, mostly as the boundary of the very tenuous atmosphere (opacity) greatly differs depending on the wavelength of light in which the star is observed.

Radii of several stars can be directly obtained by stellar interferometry. Other methods can use lunar occultations or from eclipsing binaries, which can be used to test other indirect methods of finding true stellar size. Only a few useful supergiant stars can be occulted by the Moon, including Antares and Aldebaran. Examples of eclipsing binaries include Epsilon Aurigae, VV Cephei, and HR 5171.

List of nonlinear ordinary differential equationsSee also List of nonlinear partial differential equations.

Neutron-star oscillationAsteroseismology studies the internal structure of our Sun and other stars using oscillations. These can be studied by interpreting the temporal frequency spectrum acquired through observations. In the same way, the more extreme neutron stars might be studied and hopefully give us a better understanding of neutron-star interiors, and help in determining the equation of state for matter at nuclear densities. Scientists also hope to prove, or discard, the existence of so-called quark stars, or strange stars, through these studies.

S DoradusS Doradus (also known as S Dor) is located 160,000 light-years away, and is one of the brightest stars in the Large Magellanic Cloud (LMC), a satellite of the Milky Way. It is a luminous blue variable and one of the most luminous stars known, but so far away that it is invisible to the naked eye.

SigSpecSigSpec is an acronym of "SIGnificance SPECtrum" and addresses a statistical technique to provide the reliability of periodicities in a measured (noisy and not necessarily equidistant) time series. It relies on the amplitude spectrum obtained by the Discrete Fourier transform (DFT) and assigns a quantity called the spectral significance (frequently abbreviated by “sig”) to each amplitude. This quantity is a logarithmic measure of the probability that the given amplitude level is due to white noise, in the sense of a type I error. It represents the answer to the question, “What would be the chance to obtain an amplitude like the measured one or higher, if the analysed time series were random?”

SigSpec may be considered a formal extension to the Lomb-Scargle periodogram, appropriately incorporating a time series to be averaged to zero before applying the DFT, which is done in many practical applications. When a zero-mean corrected dataset has to be statistically compared to a random sample, the sample mean (rather than the population mean only) has to be zero.

Slowly pulsating B-type starA slowly pulsating B-type star (SPB), formerly known as a 53 Persei variable, is a type of pulsating variable star. As the name implies, they are main-sequence stars of spectral type B2 to B9 (3 to 9 times as massive as the Sun) that pulsate with periods between approximately half a day and five days, however within this most member stars have been found to have multiple periods of oscillations. They display variability both in their light emission and in their spectral line profile. The variations in magnitude are generally smaller than 0.1 magnitudes, making it quite hard to observe variability with the naked eye in most cases. The variability increases with decreasing wavelength, thus they are more obviously variable in ultraviolet spectrum than visible light. Their pulsations are non-radial, that is, they vary in shape rather than volume; different parts of the star are expanding and contracting simultaneously.These stars were first identified as a group and named by astronomers Christoffel Waelkens and Fredy Rufener in 1985 while looking for and analysing variability in hot blue stars. Improvements in photometry had made finding smaller changes in magnitude easier, and they had found that a high percentage of hot stars were intrinsically variable. They referred to them as 53 Persei stars after the prototype 53 Persei. Ten had been discovered by 1993, though Waelkens was unsure if the prototype was actually a member and recommended referring to the group as slowly pulsating B (SPB) stars. The General Catalogue of Variable Stars uses the acronym LPB for "comparatively long-period pulsating B stars (periods exceeding one day)", although this terminology is rarely seen elsewhere.The similar Beta Cephei variables have shorter periods and have p-mode pulsations, while the SPB stars show g-mode pulsations. By 2007, 51 SPB stars had been confirmed with another 65 stars possible members. Six stars, namely Iota Herculis, 53 Piscium, Nu Eridani, Gamma Pegasi, HD 13745 (V354 Persei) and 53 Arietis had been found to exhibit both Beta Cephei and SPB variability. Further examples of slowly pulsating B-type stars include V539 Arae, and Gamma Muscae.

V529 AndromedaeV529 Andromedae, also known as HD 8801, is a variable star in the constellation of Andromeda. It has a 13th magnitude visual companion star 15" away, which is just a distant star on the same line of sight.

It is also an Am star with a spectral classification Am(kA5/hF1/mF2), meaning that it has the calcium K line of a star with spectral type A5, the Balmer series of a F1 star, and metallic lines of an F2 star.

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