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\title{{\bf Lyapunov exponents for the Lorentz transformations}}

\author{Open Mathematics Collaboration\footnote{All {\it authors} with their {\it affiliations} appear at the end of this white paper.}\ \footnote{Corresponding author: \href{mailto:mplobo@uft.edu.br}{\texttt{mplobo@uft.edu.br}} | {\bf Open Mathematics Collaboration}}}

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\begin{document}
\maketitle

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\begin{abstract}

\noindent\texttt{We calculate the Lyapunov exponents of the complex stretch factor $f^-(z)=(1-z^2)^{-1/2}$ from the Lorentz transformations and of its reciprocal $f^+(z)=(1-z^2)^{+1/2}$ as well.}

\end{abstract}

\begin{center}
\noindent {\footnotesize\textsf{{\bf keywords}: Lyapunov exponent, chaos, discrete dynamical system, Lorentz transformation}}
\end{center}

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\begin{center}{\it\small The most updated version of this white paper is available at}\\\footnotesize{\url{https://osf.io/jsncm/download}\\\url{https://zenodo.org/record/4905180}}\end{center}

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\section*{Introduction}

\begin{enumerate}[leftmargin=*,resume]

\item Very little is known regarding the connection of mathematical maps and physical systems.

\item We proposed two conjectures for this end, one relating quantum superposition with the logistic map \cite{log-quant}, and the other giving an insight about the fundamental nature of time \cite{time-dynamical}.

\end{enumerate}

\newpage 

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\section*{Lorentz transformations}

\begin{enumerate}[leftmargin=*,resume]

\item The warp of spacetime in the realm of special relativity (inertial observers) is due to the following function, $$f(x)=\frac{1}{\sqrt{1-x^2}}=(1-x^2)^{-1/2},$$ in the natural system of units (NSU), where light has velocity $c=1$ \cite{taylor}.

\item In the NSU, $x$ is the fraction of the light speed, i.e., $0\leq x\leq1$.

\item $f(x)$ accounts for both time dilation and space contraction; let's call it the stretch function.

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Lyapunov exponent}

\begin{enumerate}[leftmargin=*,resume]

\item A chaotic system exhibits sensitive dependence on initial conditions and henceforth has a positive Lyapunov exponent.

\item We summarize the idea of the Lyapunov exponent in the following; more details can be found in \cite{strog}.

\item $x_0:=$ initial condition
\item $\delta_0:=$ a point nearby $x_0$ such that $|x_0-\delta_0|<<1$
\item $\delta_n=|x_n-x_0|$ is the separation of $x_n$ from $x_0$ after $n$ iterates.
\item $|\delta_n|\approx|\delta_0|e^{n\lambda}\rightarrow\lambda:=$ Lyapunov exponent
\item $f'(x)=\frac{df(x)}{dx}$

\item The analytical form of the Lyapunov exponent is thus given by $$\lambda=\lim_{n\to\infty}{\left(\frac{1}{n}\sum_{i=0}^{n-1}\mbox{ln}|f'(x_i)|\right)}.$$

\item Although $\lambda$ depends on $x_0$, {\it it is the same for all $x_0$ in the basin of attraction of a given attractor} \cite{strog}.

\item A positive Lyapunov exponent is a signature of sensitive dependence on initial conditions; it does not necessarily mean the system is in a chaotic regime, as we will see in the next section.

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Results}

\begin{enumerate}[leftmargin=*,resume]

\item We calculate numerically the Lyapunov exponents of $|f^-(z)|$ and $|f^+(z)|$ for the rational interval $0<z<1$ from the following definitions.

\item $$f^-:\mathbb C\rightarrow\mathbb C\mbox{~~~ such that ~~~}f^-(z)=(1-z^2)^{-1/2}$$

\item $$f^{-'}(z)=z(1-z^2)^{-3/2}$$

\item $$f^+:\mathbb C\rightarrow\mathbb C\mbox{~~~ such that ~~~}f^+(z)=(1-z^2)^{+1/2}$$

\item $$f^{+'}(z)=-z(1-z^2)^{-1/2}$$

\item Figs. 1$-$4 show the Lyapunov exponents of $|f^-(z)|$ and $|f^+(z)|$, respectively.

\item Figs. 5$-$12 show the iterations (time series) of $|f^-(z)|$ and $|f^+(z)|$, respectively.

\begin{figure}[htpb]
\begin{center}
\resizebox{1\columnwidth}{!}{
\includegraphics{f-0-1.png}}
\caption{The Lyapunov exponents of $f^-(z)=(1-z^2)^{-1/2}$ plotted for $z_0\in\{0.01,0.02,0.03,...,0.99\}$.}
\label{fig:f-0-1}
\end{center}
\end{figure}

\begin{figure}[htpb]
\begin{center}
\resizebox{1\columnwidth}{!}{
\includegraphics{f-1-100.png}}
\caption{The Lyapunov exponents of $f^-(z)=(1-z^2)^{-1/2}$ plotted for $z_0\in\{1.01,1.02,1.03,...,99.99\}$.}
\label{fig:f-1-100}
\end{center}
\end{figure}

\begin{figure}[htpb]
\begin{center}
\resizebox{1\columnwidth}{!}{
\includegraphics{f+0-1.png}}
\caption{The Lyapunov exponents of $f^+(z)=(1-z^2)^{+1/2}$ plotted for $z_0\in\{0.01,0.02,0.03,...,0.99\}$.}
\label{fig:f+0-1}
\end{center}
\end{figure}

\begin{figure}[htpb]
\begin{center}
\resizebox{1\columnwidth}{!}{
\includegraphics{f+1-100.png}}
\caption{The Lyapunov exponents of $f^+(z)=(1-z^2)^{+1/2}$ plotted for $z_0\in\{1.01,1.02,1.03,...,99.99\}$.}
\label{fig:f+1-100}
\end{center}
\end{figure}

\newpage

\begin{figure}[htpb]
\begin{center}
\resizebox{1\columnwidth}{!}{
\includegraphics{itf-.png}}
\caption{Iterations (time series) of $|f^-(z)|=|(1-z^2)^{-1/2}|$ for the initial conditions $Z_0=\{0.15,0.30,0.45,0.60,0.75,0.90\}$. Each color represents one different initial condition $z_0\in Z_0$. Note that all conditions led to three attractors.}
\label{fig:itf-}
\end{center}
\end{figure}

\item The attractors $a_i$ of the system in Fig. \ref{fig:itf-} are the following.
\begin{center}
\begin{tabular}{c|c|c|c}
 $z_0$ & $a_1$ & $a_2$ & $a_3$ \\ \hline
 0.15 & 6.59 & 0.15 & 1.01 \\  
 0.30 & 3.18 & 0.30 & 1.05 \\
 0.45 & 1.98 & 0.45 & 1.12 \\
 0.60 & 1.33 & 0.60 & 1.25 \\
 0.75 & 0.88 & 0.75 & 1.51 \\
 0.90 & 0.48 & 0.90 & 2.29
\end{tabular}
\end{center}

\newpage

\begin{figure}[htpb]
\begin{center}
\resizebox{1\columnwidth}{!}{
\includegraphics{f-x0=12.png}}
\caption{Iterations (time series) of $|f^-(z)|=|(1-z^2)^{-1/2}|$ for the initial condition $z_0=12$. The attractors are 11.999999999999925, 0.0836242010007096, and 0.9965217285917831.}
\label{fig:f-x0=12}
\end{center}
\end{figure}

\begin{figure}[htpb]
\begin{center}
\resizebox{1\columnwidth}{!}{
\includegraphics{f-x0=34.png}}
\caption{Iterations (time series) of $|f^-(z)|=|(1-z^2)^{-1/2}|$ for the initial condition $z_0=34$. The attractors are 0.9995673804686134, 33.99999999999647, and 0.029424494316828042.}
\label{fig:f-x0=34}
\end{center}
\end{figure}

\begin{figure}[htpb]
\begin{center}
\resizebox{1\columnwidth}{!}{
\includegraphics{f-x0=1234.png}}
\caption{Iterations (time series) of $|f^-(z)|=|(1-z^2)^{-1/2}|$ for the initial condition $z_0=1234$. The attractors are 0.9999996716479318, 1234.0000001380304, and 0.0008103730374718962.}
\label{fig:f-x0=1234}
\end{center}
\end{figure}

\newpage

\begin{figure}[htpb]
\begin{center}
\resizebox{1\columnwidth}{!}{
\includegraphics{itf+.png}}
\caption{Iterations (time series) of $|f^+(z)|=|(1-z^2)^{+1/2}|$ for the initial conditions $Z_0=\{0.15,0.30,0.45,0.60,0.75,0.90\}$. Each color represents one different initial condition $z_0\in Z_0$. Note that all conditions led to two attractors.}
\label{fig:itf+}
\end{center}
\end{figure}

\item The attractors $a_i$ of the system in Fig. \ref{fig:itf+} are the following.
\begin{center}
\begin{tabular}{c|c|c}
 $z_0$ & $a_1$ & $a_2$ \\ \hline
 0.15 & 0.15 & 0.99 \\  
 0.30 & 0.30 & 0.95 \\
 0.45 & 0.45 & 0.89 \\
 0.60 & 0.60 & 0.80 \\
 0.75 & 0.75 & 0.66 \\
 0.90 & 0.90 & 0.44
\end{tabular}
\end{center}

\newpage

\begin{figure}[htpb]
\begin{center}
\resizebox{1\columnwidth}{!}{
\includegraphics{f+x0=12.png}}
\caption{Iterations (time series) of $|f^+(z)|=|(1-z^2)^{-1/2}|$ for the initial condition $z_0=12$. The attractors are 12.0 and 11.958260743101398.}
\label{fig:f+x0=12}
\end{center}
\end{figure}

\begin{figure}[htpb]
\begin{center}
\resizebox{1\columnwidth}{!}{
\includegraphics{f+x0=34.png}}
\caption{Iterations (time series) of $|f^+(z)|=|(1-z^2)^{-1/2}|$ for the initial condition $z_0=34$. The attractors are 34.0 and 33.98529093593286.}
\label{fig:f+x0=34}
\end{center}
\end{figure}

\begin{figure}[htpb]
\begin{center}
\resizebox{1\columnwidth}{!}{
\includegraphics{f+x0=1234.png}}
\caption{Iterations (time series) of $|f^+(z)|=|(1-z^2)^{-1/2}|$ for the initial condition $z_0=1234$. The attractors are 1234.0 and 1233.9995948135477.}
\label{fig:f+x0=1234}
\end{center}
\end{figure}

\end{enumerate}

\newpage

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\section*{Python scripts}

\begin{enumerate}[leftmargin=*,resume]

\item \url{https://osf.io/5cx4r}
\item \url{https://osf.io/rwg73}
\item \url{https://osf.io/rychk}

\end{enumerate}

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\section*{Final Remarks}

\begin{enumerate}[leftmargin=*,resume]

\item We showed that the complex version of the Lorentz factor has both negative and positive Lyapunov exponents while preserving its periodic behavior under the dynamics of a certain subset of initial conditions.

\item The stretch function comprised in the Lorentz transformations might shed some light on the quantum nature of spacetime in case it follows such a discrete dynamical system in its innermost fundamental blocks \cite{log-quant,time-dynamical}.

\end{enumerate}

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\section*{Open Invitation}

\noindent {\it Review}, {\it add} content, and \texttt{co-author} this white paper \cite{microarticles,ojmp-guidelines}.\\ {\it Join} the {\bf Open Mathematics Collaboration}.\\ Send your contribution to \href{mailto:mplobo@uft.edu.br}{\texttt{mplobo@uft.edu.br}}.

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\section*{Open Science}

The {\bf latex file} for this {\it white paper} together with other {\it supplementary files} are available in \cite{ojmp-project,zen}.

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\section*{How to cite this paper?}

\noindent\url{https://doi.org/10.31219/osf.io/jsncm}\\
\noindent\url{https://zenodo.org/record/4905180}

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\section*{Acknowledgements}

\begin{itemize}[leftmargin=*]

\item[$+$] {\bf Center for Open Science}\\\url{https://cos.io}

\item[$+$] {\bf Open Science Framework}\\\url{https://osf.io}

\item[$+$] {\bf Zenodo}\\\url{https://zenodo.org}

\end{itemize}

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\section*{Agreement}

\noindent All authors agree with \cite{ojmp-guidelines}.

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\section*{License}

\noindent {\it CC-By Attribution 4.0 International} \cite{cc}

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\begin{thebibliography}{1}

\bibitem{log-quant} Lobo, Matheus P. “The Logistics of Quantum Spacetime.” {\it OSF Preprints}, 10 May 2021.\\\url{https://doi.org/10.31219/osf.io/s2dnt}

\bibitem{time-dynamical} Lobo, Matheus P. “Time Is a Discrete Dynamical System.” {\it OSF Preprints}, 21 May 2021.\\\url{https://doi.org/10.31219/osf.io/8f4yg}

\bibitem{taylor} Taylor, Edwin F., and John Archibald Wheeler. {\it Spacetime Physics}. Macmillan, 1992. \url{https://eftaylor.com/spacetimephysics}

\bibitem{strog} Strogatz, Steven H. {\it Nonlinear dynamics and chaos with student solutions manual: With applications to physics, biology, chemistry, and engineering}. CRC press, 2018.

\bibitem{microarticles} Lobo, Matheus P. “Microarticles.” {\it OSF Preprints}, 28 Oct. 2019.\\\url{https://doi.org/10.31219/osf.io/ejrct}

\bibitem{ojmp-guidelines} Lobo, Matheus P. “Simple Guidelines for Authors: Open Journal of Mathematics and Physics.” {\it OSF Preprints}, 15 Nov. 2019.\\\url{https://doi.org/10.31219/osf.io/fk836}

\bibitem{ojmp-project} Lobo, Matheus P. “Open Journal of Mathematics and Physics (OJMP).” {\it OSF}, 21 Apr. 2020. \url{https://osf.io/6hzyp/files}

\bibitem{zen} \url{https://zenodo.org/record/4905180}

\bibitem{cc} CC. Creative Commons. {\it CC-By Attribution 4.0 International}.\\\url{https://creativecommons.org/licenses/by/4.0}

\end{thebibliography}

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\section*{The Open Mathematics Collaboration}

\noindent {\bf Matheus Pereira Lobo} (\texttt{lead author,} \href{mailto:mplobo@uft.edu.br}{\texttt{mplobo@uft.edu.br}})\textsuperscript{1,2}\\ \url{https://orcid.org/0000-0003-4554-1372}\\

\noindent\textsuperscript{1}Federal University of Tocantins (Brazil)\\ \textsuperscript{2}Universidade Aberta (UAb, Portugal)

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\end{document}
