††thanks: email<EMAIL_ADDRESS>
# Approximate Entropy Analysis for Nonlinear Beam Dynamics

Yongjun Li ID Brookhaven National Laboratory, Upton 11973, New York, USA

###### Abstract

In this paper, we apply approximate entropy (ApEn) analysis to the nonlinear
beam dynamics in circular accelerators. Due to the presence of strong
nonlinear magnets, chaos of beam motion gradually increases with amplitude.
Such chaos can be quantitatively characterized with ApEn of beam turn-by-turn
readings. Then ApEn, as a chaos indicator, can be used for nonlinear lattice
optimization and analysis.

##  I introduction

For circular particle accelerators, the nonlinearities of beam dynamics
confine long-term motions to be stable only within a limited region in
6-dimensional phase space, namely, dynamic aperture (DA) Chao  _et al._
(2023). Even within DA, particle motions could still be chaotic. It is
commonly believed that, for a given magnetic lattice, through suppressing
chaos, one can enlarge its DA and local momentum acceptance (LMA), and also
enhance its robustness to errors. Therefore, various chaos indicators have
been adopted to characterize the nonlinearities of beam motions Bazzani  _et
al._ (2023), such as the Lyapunov exponent (LE) Wolf  _et al._ (1985); Schmidt
_et al._ (1991); Habib and Ryne (1995), frequency map analysis (FMA) Laskar
(1999), forward-reversal integration (FRI) Panichi  _et al._ (2017); Li  _et
al._ (2021a); Borland (2000), data-driven chaos indicator Li  _et al._ (2022),
fluctuation of approximate invariant Li  _et al._ (2021b), etc. In this paper
we apply approximate entropy (ApEn) to analyze the nonlinear beam dynamics.

The concept of entropy has its origins in classical physics under the second
law of thermodynamics. In the context of nonlinear dynamics, information
entropy is central in quantifying the degree of uncertainty or information
gain, and is therefore widely used to explain complex nonlinear behavior in
real-world systems. Among many entropy analyses, ApEn conception was initially
developed by Pincus Pincus (1991) to analyze medical data, such as heart rate,
and later spread its applications in many other fields, such as finance Pincus
and Kalman (2004), and nonlinear dynamics Volos  _et al._ (2020), etc. It is a
technique used to quantify the amount of regularity and the unpredictability
of fluctuations particular for short and noise time-series data.

Due to the presence of strong nonlinear magnetic fields in circular
accelerators, the chaos of beam motions gradually increase with amplitude.
Such chaos is often visualized with the Poincaré map, i.e., the intersection
of a periodic orbit projected in a certain lower-dimensional subspace, usually
a 2-dimensional coordinate-momentum phase space. Experimentally it can also be
observed from turn-by-turn (TBT) data after beam is excited. ApEn can
quantitatively characterize the chaos of circulating beam TBT readings. Based
on that, we can determine if magnetic lattices are well configured or not.
Thus, ApEn, as a chaos indicator, can be used for nonlinear lattice
optimization at design stage, or online beam-based optimization as well Huang
(2018) if BPMs have TBT resolution.

The remainder of this paper is outlined as follows: Sect. II reviews the
definition of ApEn and explains its principle briefly. Then ApEn analysis is
applied to a Hénon map as a proof-of-principle in Sect. III. In Sect. IV,
ApEns observed in the transverse $x$–$y$ plane are used as minimization
objectives to optimize the National Synchrotron Light Source II (NSLS-II)
nonlinear lattice. In Sect. V, we implement a detailed ApEn analysis for an
elite candidates selected from the previous optimization, and interpret
physics information being conveyed by such analysis. A summary is given in
Sect. VI.

##  II Approximate entropy

According to Pincus Pincus (1991), ApEn is defined as: Two input parameters,
$m$ and $r$, must be fixed to compute ApEn, $m$ is the “length” of compared
runs, and r is effectively a filter. Given N data point
$u={u(1),u(2),\dots,u(N)}$ form vector sequences $x(1)$ through $x(N-m+1)$,
with $m\leq N$ define by $x(i)={u(i),u(i+1),\dots,u(i+m-1)}$. These vectors
represent $m$ consecutive $u$ values, starting with the $i^{th}$ point. Define
the distance $d[x(i),x(j)]$ between $x(i)$ and $x(j)$ as the maximum
difference in their respective scalar components,
$d[x(i),x(j)]=\max_{k=1,2,\dots,m}(|u(i+k-1)-u(j+k-1)|)$. Then we calculate
the value $C^{m}_{i}(r)=(\text{number of}\,j\leq N-m+1\,\text{such
that}\,d[x(i),x(j)]\leq r)/(N-m+1)$. The numerator of $C^{m}_{i}$ counts,
within the tolerance $r$, the number of blocks of consecutive values of length
$m$ which are similar to the pattern of window length $m$. Then the
$C^{m}_{i}$’s measure the regularity, or frequency, of patterns similar to
this window. Define

$\varphi^{m}(r)=\frac{1}{N-m+1}\sum_{i=1}^{N-m+1}\log C^{m}_{i}(r),$ (1)

where $\log$ is the natural logarithm. The ApEn is defined as

$\text{ApEn}(m,r,N)(u)=\lim_{N\rightarrow\infty}\varphi^{m}(r)-\varphi^{m+1}(r),$
(2)

which is a measure of system complexity. Fundamentally, ApEn measures the
conditional probability that nearby pattern runs remain close in the next
incremental comparison. A positive ApEn usually indicates a chaos Eckmann and
Ruelle (1985) . The value of $N$, the number of input data points for ApEn
computations, is typically between 75 and 5,000. Based on the calculations
that included both theoretical analysis and clinical applications, Pincus
concluded that, for $m=1,2$, values of filter $r$ between 0.1 to 0.25 standard
deviation of the $u(i)$ data can produce good statistical validity Pincus
(1995). We found that such settings are also applicable to nonlinear beam
dynamics for circular accelerators.

ApEn can be computed directly based on the above definition. It requires
execution time analogous to the square of the size of the input signal. Fast
algorithms, such as Manis (2008), were proposed to speed up its computation.
In the meantime, well-developed and documented computation packages, such as
EntropuHub Flood and Grimm (2021) (used in this paper), are also available.

##  III Application to Hénon map

In this section, ApEn analysis is applied to a 1-dimensional Hénon’s quadratic
area-preserving map Hénon (1969),

$\left(\begin{array}[]{c}x\\\
p\end{array}\right)_{n}=\left(\begin{array}[]{cc}\cos\mu&\sin\mu\\\
-\sin\mu&\cos\mu\end{array}\right)\left(\begin{array}[]{c}x\\\ p-\lambda
x^{2}\end{array}\right)_{n-1}.$ (3)

From the view of beam dynamics, this discrete map represents a thin-lens
sextupole kick followed by a linear phase space rotation at a phase advance
$\mu=2\pi\nu$. Here, sextupole strength $\lambda=1$, and the linear tune is
chosen to be $\nu=0.205$ to observe the $5^{th}$-order resonance at certain
amplitudes. The map is iterated for 512 runs, then the ApEns of
$x$-coordinates are computed for each initial condition in the phase space
$x$-$p$ as illustrated in Fig. 1. Its ApEn (chaos) increase gradually with the
initial condition’s amplitude $\sqrt{x^{2}+p^{2}}$, except while crossing
stable resonances. The contours of FMA and FRI analyses are also computed for
the purpose of comparison.

Figure 1: (Colored) Comparison of the Apen (top), FMA (middle) and FRI
(bottom) analyses for a Hénon map with 512 iterative runs. The color-maps
represent the ApEn, tune diffusion and distance between forward and reversal
trajectories respectively at the locations of their initial conditions. The
blank area represents unbounded trajectories.

Note that, while crossing stable resonances, the behavior of FMA differs from
ApEn and FRI analyses. Fig. 2 shows four trajectories in the Poincére section
$x$–$p$, while a $5^{th}$ order resonance is being crossed driven by the
amplitude dependent detuning. When the tune is sufficiently close to 0.2, and
some trajectories are trapped by elliptic fixed points, five isolated islands
are formed and then gradually merge to five fixed points. Around those stable
fixed points, the fundamental tune diffusions are relatively large (at the
order of $10^{-5\sim 6}$). However, the dimension of islands is relatively
small, then time-series data composed of $x$ (or $p$) is highly regular with a
low periodicity, which yields near zero ApEn. Similar behaviors are observed
in the difference of FRIs as well Li  _et al._ (2021a). It could be explained
as, for an on-resonance but stable time-series data, its regularity is higher
in the time domain than in the frequency domain. Another observation from Fig.
1 is that, the FMA and ApEn could provide more rich information (having more
fine structures in their chaos maps) than the FRI analysis. It means they
could be more sensitive to the variations of chaos.

Figure 2: (Colored) Trajectories of the Hénon map in the $x$–$p$ phase space
while crossing the $5^{th}$ order resonance stably. Once tune is sufficiently
close to 0.2, and trajectories are trapped by elliptic fixed points, motions
in the time domain become highly regular with a low periodicity 5, then near-
zero ApEns are observed.

##  IV Application to nonlinear lattice optimization

In this section, we use ApEn as chaos indicator to optimize the nonlinear
lattice for the NSLS-II storage ring. In a linearly stable lattice, the motion
of particle, seen by a BPM at a certain location, is a periodical time-series
oscillating with a fixed frequency, known as the linear tune. Nonlinear
magnets, such as sextupoles for chromaticity correction, can perturb regular
motions. Thus, signals seen by the BPM now have fluctuations on top of regular
motions. The ApEn of TBT BPM readings reflects the likelihood that similar
patterns of the TBT readings will not be followed by additional similar
readings. Given a nonlinear lattice, if the ApEns of TBT readings are low, the
beam motion is less chaotic, and vice versa. By minimizing the ApEn of
different trajectories through tuning nonlinear knobs, the lattice could be
optimized. Usually, ApEn analysis can let chaos be visible from a short time-
series with only several tens of data. It means that, only short-term TBT data
is needed to drive optimizer. At the early stage of lattice design, such low
computation cost can efficiently narrow down the search range by ruling out
bad candidates.

Next, we explain the detailed optimization implementations using the NSLS-II
storage ring as an example. From a certain longitudinal observation location
(the injection point in this example), multiple initial conditions are
uniformly populated within a Region of Interest (RoI) with a transverse
dimension $x\in[-40,40]\,y\in[0,15]\,mm$ (Fig. 3). The RoI should be chosen
slightly larger than the desired dynamic aperture. Particle trajectories are
simulated with a symplectic integrator Yoshida (1990) implemented in the code
elegant Borland (2000). Each trajectory is simulated for $N=256$ turns if
survived, and TBT data at the observation point is recorded, then used to
compute its ApEn.

The goal of optimization is minimizing the ApEns for all initial conditions
within the RoI. It is neither practical nor necessary to minimize every
initial condition simultaneously, therefore, the RoI is divided into several
zones as shown in Fig. 3. For each zone, two objectives are the horizontal and
vertical ApEns averaged over all initial conditions if survived after $N=256$
turns. By tuning the sextupole knobs, we attempt to minimize each zone’s ApEns
simultaneously to suppress the overall chaos inside the whole RoI. The reason
of choosing $N=256$ turns is that such data-sets are already long enough to
measure the chaos with ApEn. In the meantime, the computation cost needed for
particle tracking simulation is relatively low. Usually, most light source
rings are composed of multiple identical cells, therefore, we can even use one
cell to optimize the DA for the ideal error-free lattice to narrow down the
search range quickly. Then in the next stage, a full ring lattice including
various imperfections is used to search robust solutions within a narrow
range.

Figure 3:  (Colored) Dividing the region of interest (RoI) into $2\times 3=6$
zones in the $x$–$y$ plane at the observation. In each zone, multiple initial
conditions (represented with same-colored dots) are uniformly populated. The
optimization objectives are the averaged ApEns of all initial conditions
within each individual zone. The solid line is the desired DA profile for the
NSLS-II storage ring.

ApEn need be computed in the horizontal and vertical planes respectively, and
they are usually at different scales and need to be minimized separately. Then
the number of optimization objectives is two times the number of zones. The
tuning knobs in this example are the normalized gradients $K_{2}$ of six
harmonic sextupole families. The range of $K_{2}$ is within
$[-40,40]\,(m^{-3})$ limited by their power supply capacities and magnetic
saturation. These harmonic sextupoles doesn’t contribute to the linear
chromaticity, but can compensate the geometric and chromatic optics
abbreviations generated by chromaticity correction sextupoles.

This multi-objective optimization is solved with the widely used genetic
algorithm Deb  _et al._ (2002); Yang  _et al._ (2011); Li  _et al._ (2018). A
small population with 1,000 candidates evolves more than 30 generations, a
good convergence of the average ApEn has been reached (Fig. 4). The DAs of all
candidates in the $30^{th}$ generation are calculated for picking up some
elite candidates among them. The on-momentum DA profiles of top 20 elites are
illustrated in the top subplot of Fig. 5. It is interesting to observe that
the distributions of their six knobs (sextupole gradients $K_{2}$) also
converge to some small ranges as shown in the bottom subplot. The correlation
between the average ApEns and the area of DAs is illustrated in Fig. 6, which
confirms that suppressing the ApEns in both the horizontal and vertical planes
is essential in enlarging DA.

Figure 4: (Colored) Convergence of averaged ApEns in the genetic algorithm
optimization. Solid lines stand for the horizontal plane, and dashed-dotted
ones for the vertical plane. Lines with a same color are from the same zone.

Figure 5: (Colored) Top: on-momentum DA profiles for 20 elites selected from
the last generation of the genetic optimization. Bottom: distribution of six
sextupole knob configurations of these elites. They converge to some narrow
ranges as well. Figure 6: (Colored) Correlation between ApEn and DA. The
horizontal and vertical axes are the sums of six zone’s averaged ApEns. Each
dot represents one candidates in the $30^{th}$ generation of the genetic
optimization, colored with its area of on-momentum DA. The correlation
confirms that having small ApEns in both the horizontal and vertical planes is
essential to enlarge the NSLS-II ring DA.

In designing a nonlinear lattice, the DA and local momentum acceptance (LMA)
must be considered simultaneously to satisfy the requirements on the injection
efficiency and beam lifetime Borland  _et al._ (2011). A same strategy as ref.
Li  _et al._ (2021a) is used to include some off-momentum DAs as optimization
objectives. In this example, on $\delta=\pm 2.5\%$ off-momentum planes, ApEns
are added as the objectives, which are evaluated in the same way as Fig. 3.
Considering off-momentum DAs could be smaller than the on-momentum one,
slightly tight ROIs can be used. The on- and two off-momentum ($\delta=\pm
2.5\%$) DAs for a selected elite candidate are illustrated in Fig. 7, which
should be sufficient to achieve high-efficient injections and 3 hours Touschek
beam lifetime Bernardini  _et al._ (1963) by comparing with our current
operation lattice.

Figure 7: (Colored) On- and two off-momentum DAs for a selected elite
candidate.

##  V  Information learned from ApEn

Like FMA or FRI etc., ApEn can also provide detailed chaos information for a
given nonlinear lattice, such as the strength and location of resonances, and
robustness to errors. Below we use one of elite candidates (the same one as
shown in Fig. 7) as an example to implement a detailed ApEn analysis. To
achieve a more accurate result, for each initial condition, a $N=1024$ long
TBT data is obtained with the code elegant. In the meantime, high density
initial conditions are populated to produce a high-resolution DA profile to
identify resonance lines. As already observed in Fig. 4, the horizontal and
vertical ApEns are at different scales. Therefore, three ApEn profiles are
provided in Fig. 8: two separated ApEn maps observed in either the horizontal
or vertical plane solely, and one weighted map obtained by adding them after
normalizing with their maxima,

$\text{ApEn}_{x,y}=\frac{\text{ApEn}_{x}}{\max{\text{ApEn}_{x}}}+\frac{\text{ApEn}_{y}}{\max{\text{ApEn}_{y}}},$
(4)

where $\max{\text{ApEn}}_{x,y}$ are the maxima in each planes respectively.

Figure 8: (Colored) FMA and ApEn analyses for a selected elite candidate. Left
column: Tune diffusion maps of FMA observed in the horizontal, vertical and
both planes; Right column: ApEn maps observed in the horizontal, vertical and
both planes.

In the horizontal $ApEn_{x}$ map, besides visible resonances around $x=\pm
20\,mm$, strong chaos also appears in the vicinity of y-axis ($x=0$),
particularly when $y\geq 10\,mm$. While in the vertical plane, strong chaos
shows up when $x\geq|20|\,mm$. Based on this observation, we can conclude
that, a strong nonlinear cross-talking must exist between two transverse
planes. Such cross-talking drives the vertical motion to be chaotic at large
horizontal amplitudes and vice versa. In the vicinity of y-axis, small
amplitude horizontal TBT data is polluted by the coupling from the vertical
plane, thus its signal-noise-ratio is low, which only cause some visual chaos
there. The real problem is that, such cross-talking would cause horizontal DA
to reduce significantly in the presence of vertical physical apertures and
errors. We use the $ApEn_{y}$ (see Fig. 8’s middle-right subplot) to further
illustrate this consequence. When particle’s horizontal amplitude excesses
$20\,mm$, although the horizontal motions remain regular (shown with “cold”
colors in the top-right subplot), their vertical motions become chaotic
dramatically and should have large fluctuations while observing their TBT
data. Particles can still survive in an open space. However, once small
vertical physical apertures and errors are in place, they can be scraped. Such
cross-talking has a practical effect for every light source rings, in which
in-vacuum undulators (IVU) usually have a few millimeters vertical apertures,
and cause a significant reduction of DA in the horizontal plane. Note that
most of light source rings need a sufficient horizontal DA for injection. As
shown in Fig. 9, after a $\pm 2.5\,mm$ vertical aperture is imposed and the
existing NSLS-II magnet systematic multipole errors are included, the
horizontal DA immediately reduces to the area with low $ApEn_{y}$. If only the
area of DA in free space is used as the objective of optimization, some
candidates might be estimated optimistically. Previously, such difficulty has
been overcome by including physical apertures and errors into the tracking
simulation and optimization Borland  _et al._ (2011), but it greatly slows
down the speed of optimization. One of the benefits of using ApEn as the chaos
indicator is that, even with error-free magnet models, the robustness of
lattice to errors and physical apertures could be visible and under
consideration to a certain extent. Therefore, ApEn could be particularly
useful in the early stage of lattice optimization to narrow down the search
range for robust solutions, even with lack of information on magnet errors.

Figure 9: (Colored) Reduction of DA when a vertical physical aperture (dash-
dot line) and multipole errors are imposed. The horizontal aperture is lost in
the area where the vertical ApEn is high. The survived DA (red solid line)
confirms that having low ApEn is essential to obtain robust solutions.

The standard FMA analyses are also illustrated in the left column of Fig. 8
for comparison. Three tune diffusion maps (in the horizontal, vertical and
both planes respectively) gradually increase with their initial amplitude, but
there is no obvious jump between the regular and chaotic regime as shown in
the subplot of $ApEn_{y}$ profile. However, FMA does provide a much clear view
on resonances.

##  VI Summary

We introduce the ApEn concept to analyze the chaos of nonlinear beam dynamics.
Using the TBT data observed by BPMs, nonlinear lattice configurations can be
optimized through suppressing their ApEns. Two advantages of using ApEn are
its low computational demand and robustness to noise Pincus (1991). ApEn can
be designed to work for small data samples. It might be particularly useful in
online beam-based optimization for electron storage rings. Because, valid TBT
data can last only a few hundreds, or even a few tens turns due to strong
radiation damping and beam decoherence. another advantage is, when data is
noisy, the ApEn measure can be compared to the noise level in the data to
determine what quality of true information may be present in the data. These
advantages might make ApEn be a suitable objective for not only lattice design
but also online optimization.

In the meantime, some limitations also exist in the ApEn analysis Richman and
Moorman (2000). The algorithm counts each sequence as matching itself to avoid
the occurrence of $\log(0)$ in the calculations. This step might introduce
bias in ApEn, which causes ApEn to have two poor properties in practice: ApEn
is heavily dependent on the record length and is uniformly lower than expected
for short records. The ApEn analysis sometimes lacks relative consistency.
That is, if ApEn of one data set is lower than that of another, it should, but
does not, remain lower for all conditions tested. Particularly, in the
vicinity of low order resonances, although TBT readings are regular due to a
low periodicity, such motions are vulnerable to errors. Therefore, it might be
better to use ApEn along with other chaos maps, such as FMA and FRI, to
identify the chaos from different aspects in the nonlinear beam dynamics. All
these chaos maps share same times-series (TBT data) and can be implemented in
parallel.

Beside ApEn, many other entropy algorithms, such as Kolmogorov-Sinai entropy
Shiryayev (1993); Sinai (2009), sample entropy Richman and Moorman (2000),
Fuzzy entropy De Luca and Termini (1993), etc., are also available for
analyzing nonlinear systems. Each algorithm has its pros and cons in
quantifying chaos. Some further exploration on applying entropy analysis to
beam dynamics might be interesting and fruitful.

###### Acknowledgements.

This research used resources of the National Synchrotron Light Source II, a
U.S. Department of Energy (DOE) Office of Science User Facility operated for
the DOE Office of Science by Brookhaven National Laboratory under Contract No.
DE-SC0012704.

## References

  * Chao  _et al._ (2023) A. Chao, M. Tigner, H. Weise, and F. Zimmermann,  _Handbook of accelerator physics and engineering_ , 3rd ed. (World Scientific, 2023) Chap. 2.3.7, p. 140.
  * Bazzani  _et al._ (2023) A. Bazzani, M. Giovannozzi, C. E. Montanari, and G. Turchetti, “Performance analysis of indicators of chaos for nonlinear dynamical systems,” Phys. Rev. E 107, 064209 (2023).
  * Wolf  _et al._ (1985) A. Wolf, J. Swift, H. Swinney, and J. Vastano, “Determining lyapunov exponents from a time series,” Physica D: nonlinear phenomena 16, 285–317 (1985).
  * Schmidt  _et al._ (1991) F. Schmidt, F. Zimmermann, and F. Willeke, “Comparison of methods to determine long-term stability in proton storage rings,” Part. Accel. 35, 249–256 (1991).
  * Habib and Ryne (1995) S. Habib and R. Ryne, “Symplectic calculation of lyapunov exponents,” Phys. Rev. Lett. 74, 70–73 (1995).
  * Laskar (1999) J. Laskar, “Introduction to frequency map analysis,” Hamiltonian systems with three or more degrees of freedom , 134–150 (1999).
  * Panichi  _et al._ (2017) F. Panichi, K. Goździewski, and G. Turchetti, “The reversibility error method (REM): a new, dynamical fast indicator for planetary dynamics,” Monthly Notices of the Royal Astronomical Society 468, 469–491 (2017).
  * Li  _et al._ (2021a) Y. Li, Y. Hao, K. Hwang, R. Rainer, A. He, and A. Liu, “Fast dynamic aperture optimization with forward-reversal integration,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 988, 164936 (2021a).
  * Borland (2000) M. Borland,  _Elegant: A flexible SDDS-compliant code for accelerator simulation_ , Tech. Rep. Advanced Photon Source LS-287 (Argonne National Lab., IL (US), 2000).
  * Li  _et al._ (2022) Y. Li, J. Wan, A. Liu, Y. Jiao, and R. Rainer, “Data-driven chaos indicator for nonlinear dynamics and applications on ssftorage ring lattice design,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 1024, 166060 (2022).
  * Li  _et al._ (2021b) Y. Li, K. Hwang, C. Mitchell, R. Rainer, R. Ryne, and V. Smaluk, “Design of double-bend and multibend achromat lattices with large dynamic aperture and approximate invariants,” Physical Review Accelerators and Beams 24, 124001 (2021b).
  * Pincus (1991) S. Pincus, “Approximate entropy as a measure of system complexity.” Proceedings of the National Academy of Sciences 88, 2297–2301 (1991).
  * Pincus and Kalman (2004) S. Pincus and R. Kalman, “Irregularity, volatility, risk, and financial market time series,” Proceedings of the National Academy of Sciences 101, 13709–13714 (2004).
  * Volos  _et al._ (2020) C. Volos, S. Jafari, J. Munoz-Pacheco, J. Kengne, and K. Rajagopal, “Nonlinear dynamics and entropy of complex systems with hidden and self-excited attractors II,” (2020).
  * Huang (2018) X. Huang, “Robust simplex algorithm for online optimization,” Physical Review Accelerators and Beams 21, 104601 (2018).
  * Eckmann and Ruelle (1985) J. Eckmann and D. Ruelle, “Ergodic theory of chaos and strange attractors,” Reviews of modern physics 57, 617 (1985).
  * Pincus (1995) S. Pincus, “Approximate entropy (apen) as a complexity measure,” Chaos: An Interdisciplinary Journal of Nonlinear Science 5, 110–117 (1995).
  * Manis (2008) G. Manis, “Fast computation of approximate entropy,” Computer methods and programs in biomedicine 91, 48–54 (2008).
  * Flood and Grimm (2021) M. Flood and B. Grimm, “Entropyhub: An open-source toolkit for entropic time series analysis,” PloS one 16, e0259448 (2021).
  * Hénon (1969) M. Hénon, “Numerical study of quadratic area-preserving mappings,” Quarterly of applied mathematics , 291–312 (1969).
  * Yoshida (1990) H. Yoshida, “Construction of higher order symplectic integrators,” Phys. Lett. A150, 262–268 (1990).
  * Deb  _et al._ (2002) K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: Nsga-ii,” IEEE transactions on evolutionary computation 6, 182–197 (2002).
  * Yang  _et al._ (2011) L. Yang, Y. Li, W. Guo, and S. Krinsky, “Multiobjective optimization of dynamic aperture,” Phys. Rev. ST Accel. Beams 14, 054001 (2011).
  * Li  _et al._ (2018) Y. Li, W. Cheng, L-H. Yu, and R. Rainer, “Genetic algorithm enhanced by machine learning in dynamic aperture optimization,” Phys. Rev. Accel. Beams 21, 054601 (2018).
  * Borland  _et al._ (2011) M. Borland, L. Emery, V. Sajaev, and A. Xiao, “Multi-objective Optimization of a Lattice for Potential Upgrade of the Advanced Photon Source,” Conf.Proc. C110328, 2354–2356 (2011).
  * Bernardini  _et al._ (1963) C. Bernardini, G. Corazza, G. Di Giugno, G. Ghigo, J. Haissinski, P. Marin, R. Querzoli, and B. Touschek, “Lifetime and beam size in a storage ring,” Physical Review Letters 10, 407 (1963).
  * Richman and Moorman (2000) J. Richman and J. Moorman, “Physiological time-series analysis using approximate entropy and sample entropy,” American journal of physiology-heart and circulatory physiology (2000).
  * Shiryayev (1993) A. Shiryayev, “New metric invariant of transitive dynamical systems and automorphisms of lebesgue spaces,” Selected Works of AN Kolmogorov: Volume III: Information Theory and the Theory of Algorithms , 57–61 (1993).
  * Sinai (2009) Y. Sinai, “Kolmogorov-sinai entropy,” Scholarpedia 4, 2034 (2009).
  * De Luca and Termini (1993) A. De Luca and S. Termini, “A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory,” in  _Readings in Fuzzy Sets for Intelligent Systems_ (Elsevier, 1993) pp. 197–202.