Low dimensional chaos in stellar pulsations
Encyclopedia
Low dimensional chaos in stellar pulsations is the current interpretation of an established phenomenon. 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 variable
s through the RV Tauri
variables into the regime of the Semiregular variables.
The regular behavior of the Cepheids has been successfully modeled with numerical hydrodynamics since the 1960s
, 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 . This, and the fact that the pulsations are weakly nonlinear,
allows a description of the system in terms of amplitude equations
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 ,
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:
Numerical hydrodynamics simulations of the pulsations of stellar models
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
.
Other sequences of models follow a somewhat different route, but also to chaos,
namely the Pommeau-Manneville or tangent bifurcation route.
.
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 (Ri, Ri+1, Ri+2)
where the indices i, i+1, i+2 indicate successive time intervals.
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.
.
Global flow reconstruction from observed light curves
The method of global flow reconstruction
uses a single observed signal {si} to infer properties of the dynamical
system that generated it.
First N-dimensional 'vectors'
Si=(si,si-1,si-2,...,si-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.
Si+1= M (Si).
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
.
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 exponent
s 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 "http://www.scholarpedia.org/article/Shilnikov_bifurcation".
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
as for the classical Cepheids
Cepheid variable
A Cepheid is a member of a class of very luminous variable stars. The strong direct relationship between a Cepheid variable's luminosity and pulsation period, secures for Cepheids their status as important standard candles for establishing the Galactic and extragalactic distance scales.Cepheid...
and the RR Lyrae
RR Lyrae
RR Lyrae is a variable star in the Lyra constellation. It is the prototype of the RR Lyrae variable class of stars. It has a period of about 13 hours, and oscillates between apparent magnitudes 7 and 8. Its variable nature was discovered by the Scottish astronomer Williamina Fleming at Harvard...
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 variable
W Virginis variable
W Virginis variables are a subclass of Type II Cepheids which exhibit pulsation periods between 10–20 days, and are of spectral class F6 – K2.They were first recognized as being distinct from classical Cepheids by Walter Baade in 1942, in a study of Cepheids in the Andromeda Galaxy that proposed...
s through the RV Tauri
RV Tauri
RV Tauri is a star in the constellation Taurus. It is a yellow supergiant and is the prototype of a class of pulsating variables known as RV Tauri variables.RV Tau gives a better idea of the lives and deaths of stars like our Sun...
variables into the regime of the Semiregular variables.
The regular behavior of the Cepheids has been successfully modeled with numerical hydrodynamics since the 1960s
, and from a theoretical point of view it is easily understood as due to the presence of center manifold
Center manifold
In mathematics, the center manifold of an equilibrium point of a dynamical system consists of orbits whose behavior around the equilibrium point is not controlled by either the attraction of the stable manifold or the repulsion of the unstable manifold. The first step when studying equilibrium...
which arises because of the weakly dissipative nature of the dynamical system . This, and the fact that the pulsations are weakly nonlinear,
allows a description of the system in terms of amplitude equations
and a construction of the bifurcation diagram (see also bifurcation theory
Bifurcation theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations...
) of the possible types of pulsation (or limit cycles), such fundamental mode pulsation, first or second overtone
Overtone
An overtone is any frequency higher than the fundamental frequency of a sound. The fundamental and the overtones together are called partials. Harmonics are partials whose frequencies are whole number multiples of the fundamental These overlapping terms are variously used when discussing the...
pulsation, or more complicated, double-mode pulsations in which several modes are excited with constant amplitudes. The boundaries of the instability strip
Instability strip
The Instability strip is a nearly vertical region in the Hertzsprung–Russell diagram which is occupied by pulsating variable stars .The instability strip intersects the main sequence in the region of A...
where pulsation sets in during the star's evolution correspond to a Hopf bifurcation
Hopf bifurcation
In the mathematical theory of bifurcations, a Hopf or Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Eberhard Hopf, and Aleksandr Andronov, is a local bifurcation in which a fixed point of a dynamical system loses stability as a pair of complex conjugate eigenvalues of...
.
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 ,
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
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
). This conclusion is based on two types of studies:
Numerical hydrodynamics simulations of the pulsations of stellar models
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
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
. 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
Horseshoe map
In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavior of the orbits of the van der Pol oscillator...
.
Other sequences of models follow a somewhat different route, but also to chaos,
namely the Pommeau-Manneville or tangent bifurcation route.
.
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 (Ri, Ri+1, Ri+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
Attractor
An attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed...
is
found to be banded like the Roessler attractor, with however an additional twist in the band.
.
Global flow reconstruction from observed light curves
The method of global flow reconstruction
uses a single observed signal {si} to infer properties of the dynamical
system that generated it.
First N-dimensional 'vectors'
Si=(si,si-1,si-2,...,si-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.
Si+1= M (Si).
Takens' theorem
Takens' theorem
In mathematics, a delay embedding theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of a dynamical system...
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
.
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
Fractal dimension
In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...
of the
dynamics of R Scuti as inferred from the computed Lyapunov exponent
Lyapunov exponent
In mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories...
s 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
Resonance
In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...
with the first one,
a scenario described by the Shilnikov
theorem "http://www.scholarpedia.org/article/Shilnikov_bifurcation".
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