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Stellar pulsations are caused by expansions and contractions in the outer layers as a star seeks to maintainequilibrium. These fluctuations instellar radius cause corresponding changes in theluminosity of the star. Astronomers are able to deduce this mechanism by measuring thespectrum and observing theDoppler effect.^[1] Many intrinsicvariable stars that pulsate with largeamplitudes, such as the classicalCepheids,RR Lyrae stars and large-amplitudeDelta Scuti stars show regularlight 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 theHertzsprung–Russell diagram. These giant stars are observed to undergo pulsations ranging from weak irregularity, when one can still define an average cycling time orperiod, (as in mostRV Tauri andsemiregular variables) to the near absence of repetitiveness in theirregular variables. TheW Virginis variables are at the interface; the short period ones are regular and the longer period ones show first relatively regular alternations in the pulsationscycles, 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 givennormal mode in one pulsation cycle (period). For the regular variables (Cepheids, RR Lyrae, etc.) numerical stellar modeling and linearstability 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 acenter 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 foroscillatory systems in many other fields such aspopulation 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. thePoincaré–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 A₁ and A₂ of the twonormal modes 1 and 2 isgoverned by the following set ofordinary differential equationswhere theQ_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 thenuclear burning time scale. The equations above havefixed point solutions with constant amplitudes, corresponding to single-mode(A₁${\displaystyle \ne }$ 0, A₂ = 0) or(A₁ = 0, A₂${\displaystyle \ne }$ 0) and double-mode(A₁${\displaystyle \ne }$ 0, A₂${\displaystyle \ne }$0)solutions. These correspond to singly periodic and doubly periodic pulsations of the star. No other asymptotic solution of the above equations exists for physical (i.e., negative) coupling coefficients.

Forresonant 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 secondovertone 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 thebifurcation diagram between possible pulsational states to be mapped out. In this picture, the boundaries of theinstability strip where pulsation sets in during the star's evolution correspond to aHopf 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).

Thelight curves of intrinsicvariable stars with large amplitudes have been known for centuries to exhibit behavior that goes from extreme regularity, as for the classicalCepheids and theRR Lyrae stars, to extreme irregularity, as for the so-calledIrregular variables. In thePopulation II stars this irregularity gradually increases from the low periodW Virginis variables through theRV Tauri variables into the regime of thesemiregular 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 ofcenter manifold which arises because of the weakly dissipative nature of thedynamical 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 alsobifurcation theory) of the possible types of pulsation (orlimit cycles), suchfundamental mode pulsation, first or secondovertone pulsation, or more complicated, double-mode pulsations in which several modes are excited with constant amplitudes. The boundaries of theinstability strip where pulsation sets in during the star's evolution correspond to aHopf 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 implieslarge 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 beatagainst 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 alsoChaos theory). This conclusion is based on two types of studies.

Thecomputational fluid dynamics numerical forecasts for the pulsations ofsequences of W Virginis stellar models exhibit two approaches to irregular behavior that are a clear signature of low dimensionalchaos. The first indication comes fromfirst return maps in which one plots one maximum radius, or any other suitable variable, versus the next one.The sequence of models shows aperiod doubling bifurcation, or cascade, leading to chaos.The near quadratic shape of the map is indicative of chaos and implies an underlyinghorseshoe map.^[24][25] Other sequences of models follow a somewhat different route, but also to chaos,namely the Pommeau–Manneville ortangent bifurcation route.^[26][27]

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 underlyingattractor is found to be banded like theRoessler attractor, with however an additional twist in the band.^[28]

The method ofglobal flow reconstruction^[29] uses a single observed signal {s_i} to infer properties of the dynamical system that generated it. First N-dimensional 'vectors'${\displaystyle S_i=s_i,s_{i-1},s_{i-2},...s_{i-N+1}}$ are constructed.The next step consists in finding an expression for the nonlinearevolution operator${\displaystyle M}$ that takes the system from time${\displaystyle i}$ to time${\displaystyle i+1}$, i.e.,${\displaystyle S_{i+1}=M(S_i)}$.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 theembedding dimension N is large enough.Thusfrom 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 theAAVSO data for the starR Scuti^[30][31]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 offalse nearest neighbors corroborates an embedding dimension of 4.Thefractal dimension of the dynamics of R Scuti as inferred from the computedLyapunov exponents lies between 3.1 and 3.2.

From an analysis of thefixed points of the evolution operator a nice physical picture can be inferred, namely thatthe pulsations arise from the excitation of an unstable pulsation mode that couples nonlinearly to a second, stable pulsation mode which is in a 2:1resonance with the first one, a scenario described by the Shilnikov theorem.^[32]

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.^[33]

• Asteroseismology
• Hybrid pulsator
• Pulsating white dwarf
• Pulsating Radio Source

• Variable stars
• Astrophysics

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