# A posteriori learning of quasi-geostrophic turbulence parametrization: an
experiment on integration steps

Hugo Frezat
Univ. Grenoble Alpes, CNRS UMR LEGI, Grenoble, France
Univ. Grenoble Alpes, CNRS UMR IGE, Grenoble, France
IMT Atlantique, CNRS UMR Lab-STICC, Brest, France
<EMAIL_ADDRESS>&Julien Le Sommer
Univ. Grenoble Alpes, CNRS UMR IGE, Grenoble, France &Ronan Fablet
IMT Atlantique, CNRS UMR Lab-STICC, Brest, France &Guillaume Balarac
Univ. Grenoble Alpes, CNRS UMR LEGI, Grenoble, France
Institut Universitaire de France (IUF), Paris, France &Redouane Lguensat
Learning, Data and Robotics Lab, ESIEA, Paris, France
LOCEAN-IPSL, Sorbonne Université, Institut Pierre Simon Laplace, Paris, France

###### Abstract

Modeling the subgrid-scale dynamics of reduced models is a long standing open
problem that finds application in ocean, atmosphere and climate predictions
where direct numerical simulation (DNS) is impossible. While neural networks
(NNs) have already been applied to a range of three-dimensional flows with
success, two dimensional flows are more challenging because of the backscatter
of energy from small to large scales. We show that learning a model jointly
with the dynamical solver and a meaningful a posteriori-based loss function
lead to stable and realistic simulations when applied to quasi-geostrophic
turbulence.

##  1 Introduction

Understanding and predicting the evolution of various natural systems would
not be possible without simulations of turbulent flows. However, solving all
the spatial and temporal scales of the associated partial differential
equation (PDE), i.e., the Navier-Stokes equations remains computationally
prohibitive. One popular solution (e.g. [14, 8, 4]) is to resolve only the
largest scales and use subgrid closures (or physical parametrizations) to
represent the smaller ones.

Recently, neural networks (NNs) have been proposed as a promising alternative
to algebraic parametrizations in three-dimensional incompressible turbulence
[6, 15, 5]. Two-dimensional problems, however, are more challenging due to the
inverse cascade of energy that leads to negative viscosities, and it has been
demonstrated that numerical stability of the trained model in decaying
turbulence requires either the removal of negative eddy viscosities [13] or a
large training dataset [9]. 2D turbulence is also a simple model that well
approximates more complex rotating stratified flows (found in atmosphere and
ocean dynamics).

We show that on a forced configuration of 2D turbulence, stability can be
obtained when the NN is trained end-to-end on a temporal horizon, i.e. jointly
with the forward solver (see Fig. 1). At equivalent computational complexity,
the standard learning strategy does not lead to a stable parametrization and
is thus not usable in practice. Additionally, the presented strategy allows us
to constrain the calibration of the model on a posteriori metrics, i.e. the
evolution of quantities of interest over time, which are the absolute
performance measures in turbulence modeling. Finally, we show that the number
of iterations performed during the training process have a significant impact
on the long-time statistics of the model.

${\mathbf{y}(t)}$${\mathbf{y}(t+\Delta t)}$${\mathbf{y}(t+N\Delta
t)}$${\mathbf{y}(t+(N+1)\Delta
t)}$${g(\bar{\mathbf{y}}(t))}$${g(\bar{\mathbf{y}}(t+N\Delta
t))}$${\bar{\mathbf{y}}(t)}$${+}$${\bar{\mathbf{y}}(t+\Delta
t)}$${\bar{\mathbf{y}}(t+N\Delta t)}$${+}$${\bar{\mathbf{y}}(t+(N+1)\Delta
t)}$${\mathcal{M}_{\mathrm{NN}}(\bar{\mathbf{y}}(t))}$${\mathcal{M}_{\mathrm{NN}}(\bar{\mathbf{y}}(t+N\Delta
t))}$${R(\bar{\mathbf{y}}(t))}$$\scriptstyle{f(\mathbf{y}(t))}$$\scriptstyle{\mathcal{T}(\mathbf{y}(t))}$${\cdots}$$\scriptstyle{f(\mathbf{y}(t+N\Delta
t))}$${\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\cdots}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{L}_{\mathrm{aposteriori}}}$${\cdots}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{L}_{\mathrm{aposteriori}}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{L}_{\mathrm{aposteriori}}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathcal{L}_{\mathrm{apriori}}}$
Figure 1: Sketch of one learning step for the presented strategies. The a
priori loss is computed at instantaneous time $t$ (dashed, red), while the a
posteriori loss is evaluated on states forward in time (dashed, blue).

##  2 A priori and a posteriori learning

In the machine learning community, it is well known that models can take
advantage of additional informations during training. For instance, the basic,
purely data-based approach has been extended to train models end-to-end using
an entire system. In practice, some examples of those systems include
autonomous vehicles [2], objects detection [19] and super-resolution [3]. This
idea has also recently found many applications in the physical sciences
community (e.g. [10, 11]).

In the context of partial differential equation parametrization, we are given
with a high-resolution system $f(\mathbf{y})$ and a low resolution system
$g(\mathbf{\bar{y}})$ describing the evolution of high and low resolution
variables $\mathbf{y}(t)$ and $\bar{\mathbf{y}}(t)$ respectively by,

$\displaystyle\begin{cases}\dfrac{\partial\mathbf{y}}{\partial
t}=f(\mathbf{y}),\hskip 14.22636pt&\mathbf{y}\in\Omega\\\\[8.61108pt]
\dfrac{\partial\bar{\mathbf{y}}}{\partial
t}=g(\bar{\mathbf{y}})+R(\bar{\mathbf{y}}),\hskip
14.22636pt&\bar{\mathbf{y}}\in\bar{\Omega}\\\\[8.61108pt]
\mathcal{T}(\mathbf{y})=\bar{\mathbf{y}}\\\ \end{cases}$ (1)

where $\bar{\Omega}$ is a smaller domain than $\Omega$ and $\mathcal{T}$ is a
known projection operator that goes from high to low resolution variables.

In this framework, we are interested in modeling the residual term, or
parametrization $R(\bar{\mathbf{y}})$ with a neural network
$\mathcal{M}_{\mathrm{NN}}(\bar{\mathbf{y}})$, based on high resolution data
coming from a direct numerical simulation of $f(\mathbf{y})$, collected in a
dataset

$\mathcal{D}:=\\{\bar{\mathbf{y}}(t)\\}\rightarrow\\{R(\bar{\mathbf{y}}(t))\\}.$
(2)

If we connect the machine learning naming to fluid dynamics concepts, we could
say that the data-based approach only gives the ability to optimize a model on
a priori metrics, i.e., instantaneous predictions of the targeted term. With
the end-to-end approach, however, we have access to simulation quantities over
time, which can be used to train the model on a posteriori metrics (See [17]
for more details). Using the data $\mathcal{D}$, a standard learning strategy
based on the direct prediction of the residual term can be written as
$\operatorname*{arg\,min}_{\theta}\mathcal{L}(R(\bar{\mathbf{y}}),\mathcal{M}_{\mathrm{NN}}(\bar{\mathbf{y}}|\theta))$,
which has already been applied to fluid dynamics parametrization, and
corresponds to an a priori minimization problem. Now, if we have a
differentiable solver for $g(\bar{\textbf{y}})$, the standard minimization
problem performed during learning can be rewritten as a series of a posteriori
losses on low resolution variables that are advanced in time, such that

$\operatorname*{arg\,min}_{\theta}\mathcal{L}(\mathbf{y}(t),\bar{\mathbf{y}}(t_{0})+\Phi_{\theta}(\bar{\mathbf{y}}(t_{0}),t_{0},t)),t\in[0,T]$
(3)

where $\mathcal{L}$ is the loss function, $[0,T]$ is the temporal horizon on
which the forward solver is integrated and $\Phi$ the flow operator that
advances the reduced model in time, expressed as

$\Phi_{\theta}(\bar{\mathbf{y}},t_{0},t)=\int_{t_{0}}^{t}g(\bar{\mathbf{y}}(\tau))+\mathcal{M}_{\mathrm{NN}}(\bar{\mathbf{y}}(\tau)|\theta)\,\mathrm{d}\tau.$
(4)

In practice, the forward solver performs $N$ discrete timesteps, which defines
the temporal horizon $T=N\Delta t$. A sketch of the a posteriori learning
strategy is shown in Fig. 1.

##  3 Application to quasi-geostrophic turbulence parametrization

The described strategies are applied to a classical two-dimensional single-
layer barotropic quasi-geostrophic (QG) potential vorticity equation [12],
defined by

$\partial_{t}\omega+J(\psi,\omega)=\nu\nabla^{2}\omega-\mu\omega+F$ (5)

with

$\displaystyle\mathbf{u}$ $\displaystyle=(-\partial_{y}\psi,\partial_{x}\psi)$
(6) $\displaystyle\omega$ $\displaystyle=\nabla^{2}\psi$ (7)

where $\omega$ is the vorticity, $\psi$ the stream function, $\mathbf{u}$ the
velocity vector and
$J(\psi,\omega)=\partial_{x}\psi\partial_{y}\omega-\partial_{y}\psi\partial_{x}\omega$
is the Jacobian operator. The model is parametrized by a viscosity $\nu$, a
linear drag $\mu$ and a source term $F$.

Projecting to the reduced system. In turbulence modeling, a common approach is
to suppose that the reduced system has been filtered so that the smallest
scales are removed, and instead predicted by a model. In practice, the
projection operator $\mathcal{T}$ is a spatial filter $\Lambda_{\delta}(x,y)$
at spatial scale $\delta>0$ defined such that

$\mathcal{T}(\omega):=\omega*\Lambda_{\delta}=\bar{\omega}(x,y,t)$ (8)

and the reduced system becomes

$\partial_{t}\bar{\omega}+J(\bar{\psi},\bar{\omega})=\nu\nabla^{2}\bar{\omega}-\mu\bar{\omega}+\bar{F}+\underbrace{J(\bar{\psi},\bar{\omega})-\overline{J(\psi,\omega)}}_{R(\psi,\omega)}$
(9)

where $R(\psi,\omega)$ is the subgrid-scale (SGS) term or parametrization that
needs to be modeled.

Configuration. In our experiment, we discretize the full solutions on
$2048\times 2048$ grid points using a pseudo-spectral method and use
$\delta=16$ with a spectral cut-off filter $\Lambda_{\delta}(k)=0,\forall
k>\pi/\delta\Delta$ where $\Delta$ is the filter width so that our reduced
solution lies on a $128\times 128$ grid. The source term $F$ is a time-
dependent wind-forcing acting at large scales $k=4$,

$F=C_{F}(t)\left[\cos(4y+\pi\sin(1.4t))-\cos(4x+\pi\sin(1.5t))\right]$ (10)

with steady enstrophy rate $C_{F}$ such that $\langle F^{2}\rangle/2=3$.
Following the procedure given by [8], we spin-up multiple simulations with
$1024\times 1024$ grid initialized with random modes at $k=4$ for
approximately $1300$ days. The simulations are then integrated for $35$ days
on the high-resolution grid to obtain a turbulent state that will be used as
initial conditions for training and testing. Finally, training data is
extracted from 10 independent $\approx 65$ days long ($48000$ iterations)
trajectories. The parameters of the simulations are summarized in dimensional
and non-dimensional units in Table 1.

Table 1: Parameters of the forced configuration in dimensional and non-dimensional units, with $L_{d}(x)=1.2\times 10^{6}s$ and $T_{d}(x)=504\times 10^{4}/\pi m$. Parameter | Dimensional | Numerical
---|---|---
Domain length ($L$) |  $10^{4}$ km | $2\pi$
Spatial resolution ($\Delta x$) |  $10$ km | $3.00\times 10^{-3}$
Time step ($\Delta t$) |  $120$ s | $1.00\times 10^{-4}$
Linear drag ($\mu$) |  $1.25\times 10^{-8}$ m-1 | $2.00\times 10^{-2}$
Viscosity ($\nu$) |  $22.0$ m2 s-1 | $1.02\times 10^{-5}$
Reynolds number ($Re$) | $\approx 220000$

Our model 111https://github.com/hrkz/torchqg is implemented with PyTorch [16]
in order to benefit from automatic differentiation, which is required when
computing the gradient of the flow operator (7).

Learning components. The neural network $\mathcal{M}_{\mathrm{NN}}$ used in
this experiment for both a priori and a posteriori strategies is realized with
a fully convolutional architecture. It consist of 10 convolutional layers with
64 features each, interspresed with ReLU activation functions using kernel
sizes of $5\times 5$. The parameters are optimized for a fixed number of
epochs with Adam and a learning rate of $10^{-4}$.

In particular, the a posteriori version is trained for a different number of
iterations $N=\\{1,5,30\\}$, with a loss function that minimizes the mean
squared error (MSE) of vorticity $\bar{\omega}$ over time, such that

$\mathcal{L}_{\mathrm{aposteriori}}:=\frac{1}{N}\sum_{i=1}^{N}(\omega(i\Delta
t)*\Lambda_{\delta}-\bar{\omega}(i\Delta t))^{2}$ (11)

Results. We run a new simulation at $\delta=16$, i.e. on a $128\times 128$
grid, comparing the NNs to the reference direct numerical simulation
$\overline{\mathrm{DNS}}$ and the dynamic Smagorinsky model [18, 7], a purely
dissipative algebraic model which is a common baseline in the fluid dynamics
community. The run is performed for 3000 and 48000 iterations of the low
resolution and direct numerical simulation systems, respectively, since the
timestep for low resolution systems can be $\delta$ times larger than the DNS
$\Delta t$.

First, we can see in Fig. 2 the vorticity fields $\omega$ at the end of the
new simulation. As expected, the dynamic Smagorinsky model is only dissipative
and thus the smallest scales of the simulation are not visible anymore. On the
other side, the model learnt from the a priori strategy accumulated too much
energy on the smallest scales which led to numerical instabilities, as
expected from similar studies [13, 9]. The 3 models trained with the a
posteriori strategy remain stable at the end of the simulation. Moreover,
these models are preserving the small-scale features of the flow. Now, we look
at quadratic invariants of 2D turbulence, which develops as a double cascade
in statistically stationary conditions [1]. In particular, the enstrophy
wavenumber spectrum $Z(k)$ and flux $\Pi_{Z}(k)$ are relevant measures of the
success of turbulence parametrizations, as proposed by [8].

In Fig. 3 (left), the enstrophy spectra confirm a large deviation at the
smallest scales (highest wavenumbers $k$) for the Smagorinsky model, and a
visible deviation at the largest scales (smallest wavenumbers $k$) for the a
priori model and a posteriori model trained with $N=1$. Overall statistical
performance of the model is verified by the enstrophy flux in Fig. 3 (right),
where the NN trained with the a posteriori strategy at $N=30$ is the closest
to the exact flux predicted by the DNS.

Figure 2: Vorticity fields at the end of the testing simulation with the
filtered DNS (leftmost) and the discussed models. Note that the model trained
with a priori learning was numerically unstable and its states diverged to
infinity. Figure 3: Enstrophy spectrum (left) and time-averaged enstrophy flux
(right) of the different models w.r.t the reference DNS (dashed). For the NNs
models, only those trained with the a posteriori strategy are stable and thus
have valid statistics.

##  4 Conclusion

We show that a NN trained jointly with the forward solver on a posteriori
informations lead to stable simulations and outperforms the classical data-
driven strategy and algebraic models based on different statistical metrics
for the prediction of quasi-geostrophic turbulence parametrization. However,
it is important to consider that this requires a differentiable solver for the
reduced order system, which is not the case for most global circulation models
developed nowadays.

### Acknowledgments

The authors thank Laure Zanna, Olivier Pannekoucke and Corentin Lapeyre for
the helpful discussions. This research was supported by the CNRS through the
80 PRIME project and the ANR through the Melody, OceaniX, and HRMES projects.
Additional support was provided by Schmidt Futures, a philanthropic initiative
founded by Eric and Wendy Schmidt, as part of its Virtual Earth System
Research Institute (VESRI). Computations were performed using GPU resources
from GENCI-IDRIS.

## References

  * Boffetta [2007] Guido Boffetta.  Energy and enstrophy fluxes in the double cascade of two-dimensional turbulence.  _Journal of Fluid Mechanics_ , 589:253–260, 2007.
  * Bojarski et al. [2016] Mariusz Bojarski, Davide Del Testa, Daniel Dworakowski, Bernhard Firner, Beat Flepp, Prasoon Goyal, Lawrence D Jackel, Mathew Monfort, Urs Muller, Jiakai Zhang, et al.  End to end learning for self-driving cars.  _arXiv preprint arXiv:1604.07316_ , 2016.
  * Chen et al. [2018] Yu Chen, Ying Tai, Xiaoming Liu, Chunhua Shen, and Jian Yang.  Fsrnet: End-to-end learning face super-resolution with facial priors.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , pages 2492–2501, 2018.
  * Dipankar et al. [2015] Anurag Dipankar, Bjorn Stevens, Rieke Heinze, Christopher Moseley, Günther Zängl, Marco Giorgetta, and Slavko Brdar.  Large eddy simulation using the general circulation model icon.  _Journal of Advances in Modeling Earth Systems_ , 7(3):963–986, 2015.
  * Frezat et al. [2021] Hugo Frezat, Guillaume Balarac, Julien Le Sommer, Ronan Fablet, and Redouane Lguensat.  Physical invariance in neural networks for subgrid-scale scalar flux modeling.  _Physical Review Fluids_ , 6(2):024607, 2021.
  * Gamahara and Hattori [2017] Masataka Gamahara and Yuji Hattori.  Searching for turbulence models by artificial neural network.  _Physical Review Fluids_ , 2(5):054604, 2017.
  * Germano et al. [1991] Massimo Germano, Ugo Piomelli, Parviz Moin, and William H Cabot.  A dynamic subgrid-scale eddy viscosity model.  _Physics of Fluids A: Fluid Dynamics_ , 3(7):1760–1765, 1991.
  * Graham and Ringler [2013] Jonathan Pietarila Graham and Todd Ringler.  A framework for the evaluation of turbulence closures used in mesoscale ocean large-eddy simulations.  _Ocean Modelling_ , 65:25–39, 2013.
  * Guan et al. [2021] Yifei Guan, Ashesh Chattopadhyay, Adam Subel, and Pedram Hassanzadeh.  Stable a posteriori les of 2d turbulence using convolutional neural networks: Backscattering analysis and generalization to higher re via transfer learning.  _arXiv preprint arXiv:2102.11400_ , 2021.
  * Holl et al. [2019] Philipp Holl, Nils Thuerey, and Vladlen Koltun.  Learning to control pdes with differentiable physics.  In _International Conference on Learning Representations_ , 2019.
  * Kochkov et al. [2021] Dmitrii Kochkov, Jamie A Smith, Ayya Alieva, Qing Wang, Michael P Brenner, and Stephan Hoyer.  Machine learning–accelerated computational fluid dynamics.  _Proceedings of the National Academy of Sciences_ , 118(21), 2021.
  * Majda and Wang [2006] Andrew Majda and Xiaoming Wang.  _Nonlinear dynamics and statistical theories for basic geophysical flows_.  Cambridge University Press, 2006.
  * Maulik et al. [2019] Romit Maulik, Omer San, Adil Rasheed, and Prakash Vedula.  Subgrid modelling for two-dimensional turbulence using neural networks.  _Journal of Fluid Mechanics_ , 858:122–144, 2019.
  * Meneveau and Katz [2000] Charles Meneveau and Joseph Katz.  Scale-invariance and turbulence models for large-eddy simulation.  _Annual Review of Fluid Mechanics_ , 32(1):1–32, 2000.
  * Park and Choi [2021] Jonghwan Park and Haecheon Choi.  Toward neural-network-based large eddy simulation: application to turbulent channel flow.  _Journal of Fluid Mechanics_ , 914, 2021.
  * Paszke et al. [2019] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al.  Pytorch: An imperative style, high-performance deep learning library.  In _Advances in Neural Information Processing Systems_ , volume 32, pages 8026–8037, 2019.
  * Pope [2001] Stephen B Pope.  Turbulent flows, 2001.
  * Smagorinsky [1963] Joseph Smagorinsky.  General circulation experiments with the primitive equations: I. the basic experiment.  _Monthly Weather Review_ , 91(3):99–164, 1963\.
  * Zhou and Tuzel [2018] Yin Zhou and Oncel Tuzel.  Voxelnet: End-to-end learning for point cloud based 3d object detection.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , pages 4490–4499, 2018.


# Amazing behavior and transition to chaos of some sequences using Collatz
like problems and Quibic duffing ††thanks: Citation: Authors. Title. Pages….
DOI:000000/11111.

Zeraoulia Rafik
University of batna2.Algeria
Departement of mathematics
yabous ,khenchela
<EMAIL_ADDRESS>

###### Abstract

In this paper we shall show amazing behavior of some discrete maps using
Collatze like problems and some advanced theories in analytic number theory
and dynamical system,we have investigated the driven cubic-quintic Duffing
equation such that we were able to predict the number of limit cycles around
the equilibrium and to develop a theoretical approach to chaos suppression in
damped driven systems using Collatze like problem sequences , some new results
regarding behavior of that sequence are presented.

_K_ eywords Collatze like problem $\cdot$ irrationality $\cdot$ sequences

##  1 Introduction

The Collatz conjecture is one of the most famous unsolved problems in
mathematics.[13],[7] The conjecture asks whether repeating two simple
arithmetic operations will eventually transform every positive integer into 1.
It concerns sequences of integers in which each term is obtained from the
previous term as follows: if the previous term is even, the next term is one
half of the previous term. If the previous term is odd, the next term is 3
times the previous term plus 1. The conjecture is that these sequences always
reach 1, no matter which positive integer is chosen to start the sequence
,That the conjecture has been tested via computers for numbers up to $\approx
5.48\cdot 10^{18}$, which is quite impressive, although we was thinking ahead
of time it would have been tested by this time for incredibly large integers
given we’ve had computers for more than 50 years now.

Newly discovered fundamental theories (meta mathematics) of integer numbers
may be used to formalise and formulate a new theoretical number system from
which other formal analytical frameworks may be discovered ([18]), primed and
developed. The proposed number system ([15]), as well as its most general
framework which is based on the modelling results derived from an
investigation of the Collatz conjecture ([17]) (i.e., the 3x+1 problem), has
emerged as an effective exploratory tool for visualising, mining and
extracting new knowledge about quite a number of mathematical theorems and
conjectures, including the Collatz conjecture . an increased interest has been
witnessed in studying the theory of discrete dynamical systems including
Collatz , specifically of their associated difference equations. Sizable
number of works on the behavior and properties of pertaining solutions
boundedness and unboundedness of sequences which are derived From Collatz
problems have been published in various areas of applied mathematics and
physics. ([1],[3],[4] ,[14],[5])

###### Definition 1.1

The (borderline) Collatz-like problems: A map $f:\mathbb{N}\to\mathbb{N}$ will
be called a Collatz-like map if

$0\neq\lim_{n\to\infty}\left(\prod_{r=1}^{n}\frac{f(r)}{r}\right)^{1/n}\leq 1$
(1)

If the inequality (1) is an equality then the map $f$ will be called a
borderline Collatz-like map. For each (borderline) Collatz-like map $f$, we
have the (borderline) Collatz-like problem asking whether its iterations
diverges nowhere to infinity, i.e.

$\forall n>0,\ \exists m,r>0\text{ with }f^{\circ(m+r)}(n)=f^{\circ m}(n).$

If the answer is yes, then let us call $f$ an acceptable (borderline) Collatz-
like map.

This really focus on a specific family of borderline Collatz-like problems:

For any given $\alpha>0$, let us consider the following map [10]

$f_{\alpha}:n\mapsto\left\\{\begin{array}[]{ll}\left\lfloor{n\alpha}\right\rfloor&\text{
if }n\text{ even,}\\\ \left\lfloor{n/\alpha}\right\rfloor&\text{ if }n\text{
odd.}\end{array}\right.$ (2)

The map $f_{\alpha}$ in (2) is borderline Collatz-like. Let $S$ be the set of
$\alpha>0$ for which $f_{\alpha}$ is acceptable.

One of the most important topic in analytic number theory ([12]) , which has
attracted attention of researchers in the field, is irrationality measure of
transcendental numbers like $\sqrt{2},\pi,e,\cdots$ ,([16],[17]). In this
paper we shall give amazing and surprising behavior of the following discrete
map which is deduced from (2) taking $\alpha=\sqrt{2}$ then ,let us consider
[22] :

$f:n\mapsto\left\\{\begin{array}[]{ll}\left\lfloor{n/\sqrt{2}}\right\rfloor&\text{
if }n\text{ even,}\\\ \left\lfloor{n\sqrt{2}}\right\rfloor&\text{ if }n\text{
odd.}\end{array}\right.$

such that it involves $\sqrt{2},\pi$ and parity and so on .

##  2 Analysis and discussion

Consider the following map:

$f:n\mapsto\left\\{\begin{array}[]{ll}\left\lfloor{n/\sqrt{2}}\right\rfloor&\text{
if }n\text{ even,}\\\ \left\lfloor{n\sqrt{2}}\right\rfloor&\text{ if }n\text{
odd.}\end{array}\right.$

Let $f^{\circ(r+1)}:=f\circ f^{\circ r}$, consider the orbit of $n=73$ for
iterations of $f$, i.e. the sequence $f^{\circ r}(73)$:

$73,103,145,205,289,408,288,203,287,405,572,404,285,403,569,804,568,401,\dots$

It seems that this sequence diverges to infinity exponentially, and in
particular, never reaches a cycle. Let illustrate that with the following
picture of $(f^{\circ r}(73))^{1/r}$, with $200<r<20000$ .See Figure 1:

Figure 1: The iterated map for 73 times ($(f^{\circ r}(73))^{1/r}$, with
$200<r<20000$)

According to Figure1, it seems that $f^{\circ r}(73)\sim\delta^{r}$ with
$\delta\sim 1.02$.

Now consider the probability ([9]) of the $m$ first terms of the sequence
$f^{\circ r}(73)$ to be even:

$p_{0}(m):=\frac{|\\{r<m\ |\ f^{\circ r}(73)\text{ is even}\\}|}{m}.$

Then $p_{1}(m):=1-p_{0}(m)$ is the probability of the $m$ first terms of
$f^{\circ r}(73)$ to be odd.

If we compute the values of $p_{i}(m)$ for $m=10^{\ell}$, $\ell=1,\dots,5$, we
get something unexpected:

It is unexpected because it seems that $p_{0}(m)$ does not converge to $1/2$,
but to $\alpha\sim 0.465$. It matches with the above observation because

$\delta\sim 1.02\sim\sqrt{2}^{(0.535-0.465)}=\sqrt{2}^{(1-2\times
0.465)}\sim\sqrt{2}^{(1-2\alpha)}.$

$\begin{matrix}\ell&p_{0}(10^{\ell})&p_{1}(10^{\ell})\\\ 1&0.2&0.8\\\
2&0.45&0.55\\\ 3&0.467&0.533\\\ 4&0.4700&0.5300\\\ 5&0.46410&0.53590\\\
6&0.465476&0.534524\end{matrix}$

The line for $\ell=6$ was computed Using Pari/GP and setting internal
precision to $15000$ decimal digits we were able to get
$p_{1}(10^{6})=0.534524$,The value of $f^{1e6}(73)$ is about $3.89439e10394$,
(thus one needs such a big precision) and the $\log_{73}()$ of it is
$5578.52$. Now one can ask is it true that $f^{\circ r}(73)$ never reach a
cycle, that $(f^{\circ r}(73))^{1/r}$ converges to $\delta\sim 1.02$, that
$p_{0}(m)$ converges to $\alpha\sim 0.465$, and that $\delta^{2}4^{\alpha}=2$?
What are the exact values of $\delta$ and $\alpha$? namely, better
approximations? The following Figure provides the values of $p_{0}(m)$ for
$100<m<20000$,See Figure2

Figure 2:  values of $p_{0}(m)$ for $100<m<20000$

Note that this phenomenon is not specific to $n=73$, but seems to happen as
frequently as $n$ is big, and then, the analogous probability seems to
converge to the same $\alpha$. If $n<100$, then it happens for $n=73$ only,
but for $n<200$, it happens for $n=73,103,104,105,107,141,145,146,147,$
$148,149,151,152,153,155,161,175,199$; and for $10000\leq n<11000$, to exactly
$954$ ones.

Below is the picture shown by Figure3 as Figure2 but for $n=123456789$:

Figure 3:  values of $p_{0}(m)$ for $100<m<20000$, $n=123456789$

One can ask Is it true that the set of $n$ for which the above phenomenon
happens has natural density one? Is it cofinite? When it happens, does it
involves the same constant $\alpha$?

There are exactly $1535$ numbers $n<10000$ for which the above phenomenon does
not happen. The next Figure,namely ,Figure 4 displays for such $n$ the minimal
$m$ (in blue) such that $f^{\circ m}(n)=f^{\circ(m+r)}(n)$ for some $r>0$,
together with the miniman such $r$ (in red):

Figure 4:  The minimal $m$ for which $f^{\circ m}(n)=f^{\circ(m+r)}(n)$ for
some $r>0$

In fact all these numbers (as first terms) reach the following cycle of length
$33$:

$(15,21,29,41,57,80,56,39,55,77,108,76,53,74,52,36,25,35,49,69,97,137,193,272,192,135,190,134,94,66,46,32,22)$

except the following ones:

$7,8,9,10,12,13,14,18,19,20,26,27,28,38,40,54,$

which reach $(5,7,9,12,8)$, and that ones $1,2,3,4,6$ which reach $(1)$, and
$f(0)=0$.

If the pattern continues like above up to infinity, they must have infinity
many such $n$. We may need to ask if there infinitely many $n$ reaching a
cycle? Do they all reach the above cycle of length $33$ (except the few ones
mentioned above)? What is the formula of these numbers $n$?

Below in Figure5 is their counting function (it looks logarithmic):

Figure 5:  counting function of the happned phenomena

##  3 Main result

  * •

1) A number $m$ admits no predecessor iff the interval
$[m\sqrt{2},(m+1)\sqrt{2}]$ admits no even number and the interval
$[m/\sqrt{2},(m+1)/\sqrt{2}]$ admits no odd number. There are exactly
$r_{\ell}$ such numbers $m<10^{\ell}$ with $\ell=1,2,3,4,5,6$ and
$r_{\ell}=2,29,292,2928,29289,292893$ Strangely, for $\ell\leq 6$ we observe
that $r_{\ell-1}=\lfloor r_{\ell}/10\lfloor$

  * •

2) Numbers without predecessors those of the form $\lfloor
n(2+\sqrt{2})\rfloor$ Moreover , numbers with one predecessor are those of the
form $\lfloor 2k\sqrt{2}\rfloor$ and numbers with two predecessors those of
the form $\lfloor(2k-1)\sqrt{2}\rfloor$

  * •

3)The homoclinic orbit of unperturbed system for Quibic duffing oscillators
using Collatz sequences separates the phase plane into two areas. Inside the
separatrix curve the orbits are around one of the centers, and outside the
separatrix curve the orbits surround both the centers and the saddle point

##  4 Analysis of the first result

We take pairs of $(m,n)$ ([2]) for consecutive $m$ and their 1-step
predecessors $n$ such that $f(n)=m$. The value $n=0$ indicates, that $m$ has
no predecessor. I didn’t reflect, that one $m$ can have two predecessors, but
if $n/2$ is odd, then $n/2$ is a second predecessor ([7]).(This makes the
table more interesting, because all odd predecessors $n$ are overwritten by
the even predecessors $2n$…

Moreover, a nearly periodic structure occurs. We tried to resemble this by the
arrangement of three or four columns of $(m,n)$ such that the first column
contains all $m$ which have no predecessor. The basic pattern is not really
periodic, but has super-patterns which again seem to be periodic but actually
aren’t. This pattern-superpattern-structure is also recursive. It reminds me
of a similar structure when I looked at $\beta=\log_{2}(3)$ and found a
similar style of pattern-superpattern-supersuperpattern-… and is there related
to the continued fraction of $\beta$. So We think we’ll get no nice
description for the cases $m$ which have no predecessor

Figure 6:  Interpretation table for predecessors of the happened phenomena

Some more explanation on the idea of "recursive aperiodic pattern". If we list
the values $m$ which have no predecessor, we get

    m_k:   3, 6,10,13, 17,20,23,27,30,...

Writing the differences (We have prepended a zero-value to the above list of
$m_{k}$)

    ,3,3,4  ,3,4  ,3,3,4  ,3,4  ,3,3,4  ,3,4  ,3,4  ,3,3,4 , ...

We note, that we have a pattern of two different words: ‘3,3,4‘ and ‘3,4‘
repeating, but aperiodical. Let’s denote the longer one with the capital ‘A‘
and the shorter one with the small ‘a‘ (and ‘A‘ means a difference of 10 and
‘a‘ of 7). We get

         Aa Aa Aaa
         Aa Aaa
         Aa Aa Aaa
         Aa Aaa
         Aa Aa Aaa
         Aa Aaa
         Aa ...

Again we find only two kind of "words". Let’s them shorten by ‘Aaa‘=‘B‘ and
‘Aa‘=‘b‘. ‘B‘ means now a difference of 24, ‘b‘ of 17. Then we get

           bbB bB
           bbB bB
           bbB bB bB
           bbB bB
           bbB bB bB
           bbB bB
           bbB bB
           bbB bB bB
           ...

Next obvious step gives

           Cc Cc Ccc
           Cc Ccc
           Cc Cc Ccc
           Cc Ccc
           Cc Cc Ccc
           Cc Ccc
           ...

with ‘c‘ representing a difference of$17+24=41$ and $C$of $17+17+24=58$. And
so on. If We recall correctly, then with the mentioned case of working with
$\beta=\log_{2}(3)$ the same style of recursive pattern reflected the
convergents of the continued fractions of $\beta$. The first few differences
here match the convergents of the continued fraction of $\sqrt{2}$ so far:

                    a    b    c                    short patterns
         -------------------------------------
        [1  1  3    7    17   41  99   239   577  ...  ]  convergents of contfrac(sqrt(2))
        [0  1  2    5    12   29  70   169   408  ...  ]
         -------------------------------------...
                  A/2   B/2   C/2                  long patterns

The above can be explained by the following:

  * •

1) a number of the form $\lfloor 2k\sqrt{2}\rfloor$ has exactly one
predecessor $4k$;

  * •

2)a number of the form $\lfloor(2k-1)\sqrt{2}\rfloor$ has exactly two
predecessors $2k-1$ and $4k-2$;

  * •

3) a number has no predecessors iff it has form $\lfloor
n(2+\sqrt{2})\rfloor$.

Using a back-step algorithm (recursive) it seems we’ve got the predecessing
tree of $m=73$. If no bugs, then this tree would also be complete. *(But my
routine may still be buggy, please check the results!)

The back-steps go from top-right south-west (antidiagonal) downwards. When
there are two possible predecessors, they occur in the same column, but on
separate rows. If there is a predecessor without further predecessor, a short
line (‘—‘) is printed.

                                            73   <--- start
                                        104
                                    148
                                105 ---

                                210
                            149
                        212
                    300 ---

                            298
                        211 ---

                        422
                    299
                424
            600 ---

                    598
                423 ---

                846 ---
            ---------------------------- tree seems to be complete (please check for errors!)

We may give heuristic proof for our main results using Beatty theorem [19]
which it states that:

###### Theorem 1

given an irrational number $r>1$ there exists $s>1$ so that the Beatty
sequences $\bm{B_{r}}$ and $\bm{B_{s}}$ partition the set of positive
integers: each positive integer belongs to exactly one of the two sequences

###### Proof 4.1

We can say with Beatty theorem(Theorem 1) : $A=\\{E(n(\sqrt{2}+2))\text{ ;
}n\in\mathbb{N}^{*}\\}$ and $B=\\{E(n\sqrt{2});n\in\mathbb{N}^{*}\\}$ is a
partition of $\mathbb{N}^{*}$ And we have $E(n(\sqrt{2}+2))=2n+E(n\sqrt{2})$
with $E$ is the function integer part ,and this proves the partial of result 2
(Form of numbers without predecessor).

For the First result , let assume that the probabilty for an integer $n$ to be
odd is $\frac{1}{2}$, and that the probabilty for $f(n)$ to be odd when $n$ is
even (resp. odd) is also $\frac{1}{2}$. We will observe that (surprisingly) it
is no more $\frac{1}{2}$ for $f^{\circ r}(n)$ when $r\geq 2$ (in some sense,
the probability does not commute with the composition of $f$ with itself).

  * •

1) if $n$ and $m=f(n)$ are even: note that $\frac{n}{\sqrt{2}}=m+\theta$ (with
$0<\theta<1$) so that $m=\frac{n}{\sqrt{2}}-\theta$, then

$f^{\circ
2}(n)=f(m)=\left\lfloor{\frac{m}{\sqrt{2}}}\right\rfloor=\left\lfloor{\frac{\frac{n}{\sqrt{2}}-\theta}{\sqrt{2}}}\right\rfloor=\left\lfloor\frac{n}{2}-\frac{\theta}{\sqrt{2}}\right\rfloor$

but $\frac{n}{2}$ is even with probability $\frac{1}{2}$, so in this case,
$f^{\circ 2}(n)$ is odd with probability $\frac{1}{2}$.

  * •

2) if $n$ is even and $m=f(n)$ is odd:

$f^{\circ
2}(n)=f(m)=\left\lfloor\sqrt{2}m\right\rfloor=\left\lfloor\sqrt{2}(\frac{n}{\sqrt{2}}-\theta)\right\rfloor=\left\lfloor
n-\sqrt{2}\theta)\right\rfloor$

but $n$ is even and the probability for $0<\sqrt{2}\theta<1$ is
$\frac{\sqrt{2}}{2}$ (because $\theta$ is assumed statistically
equidistributed [9] on the open interval $(0,1)$), so $f^{\circ 2}(n)$ is odd
with probability $\frac{\sqrt{2}}{2}$.

  * •

3) if $n$ is odd and $m=f(n)$ is even:

$f^{\circ
2}(n)=f(m)=\left\lfloor{\frac{m}{\sqrt{2}}}\right\rfloor=\left\lfloor{\frac{\sqrt{2}n-\theta}{\sqrt{2}}}\right\rfloor=\left\lfloor
n-\frac{\theta}{\sqrt{2}}\right\rfloor$

but $n$ is odd and $0<\frac{\theta}{\sqrt{2}}<1$, so $f^{\circ 2}(n)$ is even.

  * •

4) if $n$ is odd and $m=f(n)$ is odd:

$f^{\circ
2}(n)=f(m)=\left\lfloor\sqrt{2}m\right\rfloor=\left\lfloor\sqrt{2}(\sqrt{2}n-\theta)\right\rfloor=\left\lfloor
2n-\sqrt{2}\theta\right\rfloor$

but $2n$ is even and the probability for $0<\sqrt{2}\theta<1$ is
$\frac{\sqrt{2}}{2}$, so $f^{\circ 2}(n)$ is odd with probability
$\frac{\sqrt{2}}{2}$.

By combining these four cases together, we deduce that the probability for
$f^{\circ 2}(n)$ to be odd is

$\frac{1}{2}\times\frac{1}{2}\times(\frac{1}{2}+\frac{\sqrt{2}}{2}+0+\frac{\sqrt{2}}{2})=\frac{2\sqrt{2}+1}{8}$

By continuing in the same way, we get that the probability for $f^{\circ
3}(n)$ to be odd is:

$\frac{1}{4}(\frac{1}{2}\frac{1}{2}+\frac{1}{2}\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\frac{\sqrt{2}}{2}+1\frac{1}{2}+\frac{\sqrt{2}}{2}\frac{\sqrt{2}}{2})=\frac{\sqrt{2}+7}{16}$

For $2\leq r\leq 24$, we computed the probability $p_{r}$ for $f^{\circ r}(n)$
to be odd (see Appendix). It seems (experimentally) that $p_{r}$ converges to
a number $\simeq 0.532288725\simeq\frac{8+3\sqrt{2}}{23}$ by Inverse Symbolic
Calculator . This leads to the following question/conjecture:

$\lim_{r\to\infty}p_{r}=\frac{8+3\sqrt{2}}{23}\ \ ?$

If so, consider the number $\alpha$ , then

$\alpha=1-\frac{8+3\sqrt{2}}{23}=\frac{15-3\sqrt{2}}{23}\simeq 0.467711,$

which matches with above computation (Analysis and discussion section). And
next, we would have:

$\delta=\frac{\sqrt{2}}{2^{\alpha}}=2^{\frac{1}{2}-\alpha}=2^{\frac{6\sqrt{2}-7}{46}}\simeq
1.022633$

##  5 Homoclinic orbits and chaos in the un-perturbed system using Collatz
problem

Although no universally accepted mathematical definition of chaos exists, a
commonly used definition originally formulated by Robert L. Devaney says that,
to classify a dynamical system as chaotic, it must have these properties:

  * •

1) it must be sensitive to initial conditions

  * •

2) it must be topologically mixing

  * •

3) it must have dense periodic orbits

Firstly we may consider the harmonically driven damped pendulum which is often
used as a simple example of a chaotic system, the equation is just

$\ddot{\phi}+\frac{1}{q}\dot{\phi}+\sin\phi=A\cos(\omega t)$ (3)

As long as $A$ and $\omega$ are small it behaves like a driven harmonic
oscillator, and asymptotically settles into regular oscillations with a fixed
period. However, as $A$ (or $\omega$) are increased, with the rest of
parameters fixed, the system undergoes a cascade of period doubling
bifurcations leading to chaotic behavior, which then gives way to regular
oscillations again when it is increased further. For example, when $q=2$ and
$\omega=2/3$ the first period doubling ("symmetry breaking") occurs at
$A\approx 1.07$ and the first chaos at $A\approx 1.08$. These rigorous results
seem to be obtained by numerical simulations. One can be actually interested
in situations where chaos does not occur [20]. Are there known rigorous
conditions on $A,\omega$ and $q$ that put the system below the first period
doubling? However this question does not belong to the aim of this paper but
it would be very interesting to conclude somethings about chaotics behaviors
of some dynamics and to discover new ways to supress chaos in the cubic-
Quintic Duffing Equation using some discrete iterated map which it is the aim
of our research in this paper. For the unperturbed system with fractional
order displacement, when $\varepsilon=0$, the differential equation (3) can be
reformulated as For $A=0$, to

$\ddot{x}-ax+bx^{3}+cx^{5}=0.~{}$ (4)

Let

$\Delta:=b^{2}+4ac.$ (5)

Equilibrium points for $\Delta>0$ are :

$\left(x,\dot{x}\right)=\left(\pm\sqrt{\frac{-b+\sqrt{b^{2}+4ac}}{2c}},0\right)\text{
: centers}$ (6)

Define

$x_{e}^{+}=\sqrt{\frac{-b+\sqrt{b^{2}+4ac}}{2c}}\text{ and
}x_{e}^{-}=-\sqrt{\frac{-b+\sqrt{b^{2}+4ac}}{2c}}$ (7)

The energy function for (4) is

$\frac{1}{2}\dot{x}(t)^{2}-\frac{1}{2}ax(t)^{2}+\frac{1}{4}bx(t)^{4}+\frac{1}{6}cx(t)^{6}=K$
(8)

where $K$ is the energy constant dependent on the initial amplitude
$x(0)=x_{0}$ and initial velocity $x^{\prime}(0)=\dot{x}_{0}$ :

$K=\frac{1}{2}\dot{x}_{0}^{2}-\frac{1}{2}ax_{0}^{2}+\frac{1}{4}bx_{0}^{4}+\frac{1}{6}cx_{0}^{6}.$
(9)

Dependently on $K$, the level sets are different. For all of them it is common
that they form closed periodic orbits which surround the fixed points
$(\mathrm{x},\dot{\mathrm{x}})=(x_{e}^{+},\ 0)$ or
$(\mathrm{x},\dot{\mathrm{x}})=(x_{e}^{-},\ 0)$ or all the three fixed points
$(x_{e}^{\pm},\ 0)$ and $(0,\ 0)$. The boundary between these two groups of
orbits corresponds to $K=0$, when

$\dot{x}_{0}=\pm
x_{0}\sqrt{\frac{1}{6}\left(6a-3bx_{0}^{2}-2cx_{0}^{4}\right)}.$ (10)

The level set

$\frac{\dot{x}^{2}}{2}-\frac{1}{2}ax^{2}+\frac{1}{4}bx^{4}+\frac{1}{6}cx^{6}=0\text{
}\quad$ (11)

is composed of two homoclinic orbits

$\Gamma_{+}^{0}(\mathrm{t})\equiv(x_{+}^{0}(\mathrm{t}),\dot{x}_{+}^{0}(\mathrm{t})),\text{
}\quad$ (12)
$\Gamma_{-}^{0}(\mathrm{t})\equiv(x_{-}^{0}(\mathrm{t}),\dot{x}_{-}^{0}(\mathrm{t})),\text{
}$ (13)

which connect the fixed hyperbolic saddle point $(0,\ 0)$ to itself and
contain the stable and unstable manifolds. The functions
$x_{\pm}^{0}(\mathrm{t})$ may be evaluated using these two formulas

$M(t_{0})=\int_{-\infty}^{+\infty}\dot{x}^{0}(t)[\gamma\cos\omega(t+t_{0})-\delta\dot{x}^{0}(t)]dt,\text{
}\quad$ (14)

with

$x^{0}(t)=\frac{A\text{sech}\left(\sqrt{k}t\right)}{\sqrt{1+\lambda\cdot\text{sech}^{2}\left(\sqrt{k}t\right)}}.$
(15)

. See Figure 5

The homoclinic orbit [21] separates the phase plane into two areas. Inside the
separatrix curve the orbits are around one of the centers, and outside the
separatrix curve the orbits surround both the centers and the saddle point.
Physically it means that for certain initial conditions the oscillations are
around one steady-state position [23], and for others around all the steady-
state solutions (two stable and an unstable).

Now ,in the second case which uses the iterated map (Collatze like problems
sequences ) as the RHS of equation (3) ,that is perturbed system, We have
noted transition to chaos(sensitivity to initial condition) ,Let us consider
the following IVP (intial value problem):

$\ddot{x}-ax+bx^{3}+cx^{5}=f,\dot{(}x(0))=0,x(0)=0$ (16)

such that :

$f:n\mapsto\left\\{\begin{array}[]{ll}\left\lfloor{n/\sqrt{2}}\right\rfloor&\text{
if }n\text{ even,}\\\ \left\lfloor{n\sqrt{2}}\right\rfloor&\text{ if }n\text{
odd.}\end{array}\right.$

Let $f^{\circ(r+1)}:=f\circ f^{\circ r}$, consider the orbit of $n=73$ for
iterations of $f$, i.e. the sequence $f^{\circ r}(73)$:

$73,103,145,205,289,408,288,203,287,405,572,404,285,403,569,804,568,401,\dots$

(see It seems that this sequence diverges to infinity exponentially, and in
particular, never reaches a cycle. Let illustrate that with the following
picture of $(f^{\circ r}(73))^{1/r}$, with $200<r<20000$ , See Figure 7

Figure 7:  expenential divergence for $200<r<20000$

And for initial sensitivity of IVP corresponding to the system (16) ,we have
obtained the following Figure up to $r=20000$ (small perturbation),see Figure
8:

Figure 8:  Sensitivity to initial condition for small perturbation up to
$r=20000$ for the Collatze sequence

We may now show a phase space plot of the trajectory up to $r=20000$,namely,
small perturbation ,See Figure9

Figure 9:  transition to chaos $r=20000$ for (16) using Collatze sequence for
73 iteration

This looks complicated, but in fact, most of the plot shows the initial period
of time during which the motion is approaching its final behavior which is
much simpler. The early behavior is called an "initial transient".

## Acknowledgments

This work is based on work of Sebastien Palcoux for his Arxiv research paper
entitled " unexpected behavior of some transcendental number like $\sqrt{2}$ "
[22]

##  6 declaration statement

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

##  7 Data Availability

The authors were unable to find a valid data repository for the data used in
this study

##  8 Appendix

Computation:

    sage: for i in range(3,26):
    ....:     print(sq2(i))
    ....:
    [1/4*sqrt(2) + 1/8, 0.478553390593274]
    [1/16*sqrt(2) + 7/16, 0.525888347648318]
    [3/32*sqrt(2) + 13/32, 0.538832521472478]
    [15/64*sqrt(2) + 13/64, 0.534581303681194]
    [5/128*sqrt(2) + 61/128, 0.531805217280199]
    [39/256*sqrt(2) + 81/256, 0.531852847392776]
    [93/512*sqrt(2) + 141/512, 0.532269260352925]
    [51/1024*sqrt(2) + 473/1024, 0.532348527032254]
    [377/2048*sqrt(2) + 557/2048, 0.532303961432938]
    [551/4096*sqrt(2) + 1401/4096, 0.532283123258685]
    [653/8192*sqrt(2) + 3437/8192, 0.532285334012406]
    [3083/16384*sqrt(2) + 4361/16384, 0.532288843554459]
    [3409/32768*sqrt(2) + 12621/32768, 0.532289246647030]
    [7407/65536*sqrt(2) + 24409/65536, 0.532288816169701]
    [22805/131072*sqrt(2) + 37517/131072, 0.532288667983386]
    [24307/262144*sqrt(2) + 105161/262144, 0.532288700334941]
    [72761/524288*sqrt(2) + 176173/524288, 0.532288728736551]
    [159959/1048576*sqrt(2) + 331929/1048576, 0.532288729880941]
    [202621/2097152*sqrt(2) + 829741/2097152, 0.532288725958633]
    [639131/4194304*sqrt(2) + 1328713/4194304, 0.532288724978704]
    [1114081/8388608*sqrt(2) + 2889613/8388608, 0.532288725350163]
    [1825983/16777216*sqrt(2) + 6347993/16777216, 0.532288725570602]
    [5183461/33554432*sqrt(2) + 10530125/33554432, 0.532288725561857]

Code:

    def sq2(n):
        c=0
        for i in range(2^n):
            l=list(Integer(i).digits(base=2,padto=n))
            if l[-1]==1:
                cc=1/4
                for j in range(n-2):
                    ll=[l[j],l[j+1],l[j+2]]
                    if ll==[0,0,0]:
                        cc*=1/2
                    if ll==[0,0,1]:
                        cc*=1/2
                    if ll==[0,1,0]:
                        cc*=(1-sqrt(2)/2)
                    if ll==[0,1,1]:
                        cc*=sqrt(2)/2
                    if ll==[1,0,0]:
                        cc*=1
                    if ll==[1,0,1]:
                        cc=0
                        break
                    if ll==[1,1,0]:
                        cc*=(1-sqrt(2)/2)
                    if ll==[1,1,1]:
                        cc*=sqrt(2)/2
                c+=cc
        return [c.expand(),c.n()]

$M(t_{0})=\int_{-\infty}^{+\infty}\dot{x}^{0}(t)[\gamma\cos\omega(t+t_{0})-\delta\dot{x}^{0}(t)]dt,\text{
}\quad$ (17)

Let

$x^{0}(t)=\frac{A\text{sech}\left(\sqrt{k}t\right)}{\sqrt{1+\lambda\cdot\text{sech}^{2}\left(\sqrt{k}t\right)}}.$
(18)

## References

  * [1] J.J. O Connor and E.F. Robertson, Lothar Collatz, St Andrews University School of Mathematics and Statistics, Scotland; (2006)
  * [2] H. Niederreiter and I. E. Shparlinski. Dynamical systems generated by rational functions. In Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (Toulouse, 2003), volume 2643 of Lecture Notes in Comput. Sci., pages 6–17. Springer, Berlin, 2003.
  * [3] Livio Colussi, The convergence classes of Collatz function, Theoretical Computer Science; vol.412 issue 39 pp. 5409–5419 (2011).
  * [4] Jeffrey C. Lagarias and Daniel J. Bernstein, The 3x+1 conjugacy map, Canadian journal of mathematics; vol.48 issue 6 pp. 1154-116 (1996).
  * [5] Alex V. Kontorovich, Yakov G.Sinai, Structure theorem for (d,g,h)-Maps, https://arXiv.org/abs/math/0601622,2006
  * [6] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford university; (2008).
  * [7] Poonen, B.: Characterizing integers among rational numbers with a universal-existential formula. Am. J. Math. 131(3), 675–682 (2009)
  * [8] van der Poorten, A.J.: Solution de la conjecture de pisot sur le quotient de Hadamard de deux fractions rationnelles. CR Acad. Sci. Paris 306(97), 102 (1988)
  * [9] Rabin, M.O.: Probabilistic algorithm for testing primality. J. Number Theory 12(1), 128–138 (1980)
  * [10] Shalev, A.: Some results and problems in the theory of word maps. In: Erdos Centennial, ˝ pp. 611–649. Springer (2013)
  * [11] Smith, A.: The congruent numbers have positive natural density. Preprint (2016). arXiv:1603.08479
  * [12] Tao, Z.: On the representation of large odd integer as a sum of three almost equal primes. Acta Mathematica Sinica 7(3), 259–272 (1991)
  * [13] Collatz, L.: On the origin of the (3n+1) problem. Journal of Qufu Normal University, Natural Science Edition 12(3), 9–11 (1986)
  * [14] Lagarias, J.C.: The 3x+1 problem: An annotated bibliography (1963–2000). ArXiv math (NT0608208) (2006)
  * [15] Wirsching, G.J.: The Dynamical System Generated by the 3n+1 Function. Lecture Notes in Mathematics, vol. 1681. Springer, Berlin (1981)
  * [16] Hata, M. "Rational Approximations to pi and Some Other Numbers." Acta Arith. 63, 335-349, 1993.
  * [17] Amdeberhan, T. and Zeilberger, D. "q-Apéry Irrationality Proofs by q-WZ Pairs." Adv. Appl. Math. 20, 275-283, 1998.
  * [18] Oliveira e Silva, T. "Computational Verification of the 3x+1 Conjecture." Sep. 19, 2008. http://www.ieeta.pt/~tos/3x+1.html
  * [19] John William Strutt, 3rd Baron Rayleigh (1894). The Theory of Sound. Vol. 1 (Second ed.). Macmillan. p. 123.
  * [20] H.W. Haslach H.W. Haslach . “Post-buckling behavior of columns with non-linear constitutive equations,” In ternational Journal of Non-Linear Mechanics 20
  * [21] P. Holmes& J. Marsden P. Holmes& J. Marsden .‘A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam,”Archives for Rational,Holmes, Philip and Marsden, Jerrold E. ,292 ,419-448.
  * [22] Sebastien Palcoux.Unexpected behavior involving $\sqrt{2}$ and parity,2020 https://mathoverflow.net/q/353493/51189
  * [23] Pyragas,1996 Pyragas,, "Continuous control of chaos by self controlling feedback", Academic Press, San Diego, , pp. 118-123.