Template for Nonlinear Analysis: Modelling and Control. https://www.journals.vu.lt/nonlinear-analysis/
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% ************* Author's references ******************%
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@ARTICLE{article,
author = {F. Author and S. Authors and T. Author},
title = {Title of {A}rticle },
journal = {Journal},
year = {2027},
volume = {1},
number = {3},
pages = {101--120},
doi = {10.15388/namc.2023.30.42210},
}
@BOOK{book,
author = {A. Author},
title = {Title of Book},
publisher = {Publisher},
year = {2027},
volume = {3},
series = {B},
address = {Address},
edition = {5},
doi = {10.15388/namc.2024.30.42210},
}
@INPROCEEDINGS{proceedings,
AUTHOR = {F. Author and S. Author},
TITLE = {Title of proceedings},
BOOKTITLE = {BOOKTITLE},
YEAR = {2027},
editor = {Editor},
volume = {3},
series = {A},
pages = {1--6},
address = {address},
month = {month},
publisher = { publisher},
note = {note},
doi = {10.15388/namc.2022.30.42210},
}
@ARTICLE{LazerSIAM,
author = "A.C. Lazer and P.J. McKenna",
title = "Large-amplitude periodic oscillations in suspension bridges: {S}ome new connections with nonlinear analysis",
journal = "SIAM Rev.",
volume = "32",
number = "4",
pages = "537--578",
year = "1990",
url = "http://www.jstor.org/stable/2030894"
}
@ARTICLE{dancer,
author = "E.N. Dancer",
title = "On the {D}irichlet problem for weakly non-linear elliptic partial differential equations",
journal = "Proc. R. Soc. Edinb., Sect. A, Math.",
volume = "76",
number = "",
pages = "283--300",
year = "1977",
doi = "10.1017/S0308210500019648"
}
@BOOK{Fucik,
author = "S. Fu{\v c}{\'i}k",
title = "Solvability of Nonlinear Equations and Boundary Value Problems",
publisher = "Reidel",
address = "Dordrecht",
year = "1980",
doi = "10.1080/1023629033000062934"
}
@ARTICLE{GriSad,
author = "A. Gritsans and F. Sadyrbaev",
title = "Two-parametric nonlinear eigenvalue problems",
journal = "Electron. J. Qual. Theory Differ. Equ.",
volume = "10",
number = "",
pages = "1--14",
year = "2008",
doi = "10.14232/EJQTDE.2007.7.10"
}
@ARTICLE{Sad2,
author = "A. Gritsans and F. Sadyrbaev",
title = "Nonlinear Problems with Asymmetric Principal Part",
journal = "Math. Model. Anal.",
volume = "17",
number = "2",
pages = "217--226",
year = "2012",
doi = "10.3846/13926292.2012.661697"
}
@ARTICLE{Sad1,
author = "F. Sadyrbaev",
title = "Multiplicity in parameter-dependent problems for ordinary differential equations",
journal = "Math. Model. Anal.",
volume = "14",
number = "4",
pages = "503--514",
year = "2009",
doi = "10.3846/1392-6292.2009.14.503-514"
}
@ARTICLE{Sad3,
author = "A. Gritsans and F. Sadyrbaev",
title = "Two-parameter nonlinear oscillations: {T}he {N}eumann problem",
journal = "Math. Model. Anal.",
volume = "16",
number = "1",
pages = "23--38",
year = "2011",
doi = "10.3846/13926292.2011.559449"
}
@ARTICLE{sergejevaZILINA,
author = "N. Sergejeva",
title = "On nonlinear spectra",
journal = "Stud. Univ. {\v Z}ilina, Math. Ser.",
volume = "23",
number = "1",
pages = "101--110",
year = "2009",
doi = "10.3846/1392-6292.2009.13.87-97"
}
@ARTICLE{sergejevaLUMII,
author = "N. Sergejeva",
title = "On the {F}u{\v c}{\'i}k type problem with the integral nonlocal boundary conditions",
journal = "Electron. J. Qual. Theory Differ. Equ.",
volume = "21",
number = "1",
pages = "1--11",
year = "2016",
doi = "10.14232/ejqtde.2016.8.21"
}
@ARTICLE{2014SergejevaN,
author = "N. Sergejeva",
title = "The {F}u\v{c}\'{i}k spectrum for nonlocal {BVP} with {S}turm--{L}iouville boundary condition",
journal = "Nonlinear Anal. Model. Control",
year = 2014,
volume = 19,
number = 3,
pages = "503--513",
doi = "10.15388/NA.2014.3.13"
}
@INPROCEEDINGS{2009ProcDETAp,
author = "S. Pe{\v c}iulyt{\.e} and A. {\v S}tikonas",
title = "On the spectrum of the {S}turm--{L}iouville problem with one nonlocal boundary condition",
booktitle = "Proceedings of the International Conference on Differential Equations and Their Applications (DETA-2009),
Panev{\.e}{\v z}ys, Lithuania, September 10--12, 2009",
editor = "V. Kleiza and S. Rutkauskas and A. {\v S}tikonas",
publisher = "Technologija",
address = "Kaunas",
year = 2009,
pages = "117--122",
doi = "10.15242/2009.07.09"
}
\end{filecontents*}
% ************* end Author's references ******************%
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% ************* Author's definitions ******************%
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% ************* end Author's definitions ******************% "author@somewhere.host" <author@somewhere.host>
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\begin{document}
\allowdisplaybreaks[2]
\begin{topmatter}
{\namctitle\footnote{This research was supported by grant No.\ xxxx.}}
{First Author{$^{\hh\mathrm{a}}$\footnote{The Author suported by }}\ID{1009-0009-6352-1616},
Second Author{$^{\hh\mathrm{b}}$\footnote{The Author suported by }}\ID{1000-0002-5872-5501},
Third Author{$^{\hh\mathrm{b}}$}\footnotemark[3]\COR{second.author@somewhere.host}\ID{1000-0002-0302-3449}
}
{\vv$^\mathrm{a\hh}$Institution a: Institute of Data Science and Digital Technologies, Faculty of Mathematics and Informatics, \\ \ \ Vilnius University, Lithuania\RORID{03nadee84}\\
\email{first.author@somewhere.host} \\
\vv$^\mathrm{b\hh}$Institute of Applied Mathematics, Vilnius University, Lithuania\RORID{03nadee84}\\
\email{second.author@somewhere.host};\ \email{third.author@somewhere.host}\\
}
\end{topmatter}
\baselineskip=\normalbaselineskip
\begin{abstract}
{
Abstract Text. A short abstract describing the research done, methodology, and achieved results is to be presented. The abstract should contain approximately 100 words. The volume of the article is up to 20 pages in NA journal style. The journal recognizes review articles as special ones. These articles (if any) will occupy the starting positions in the journal and may contain more than 20 pages. Text in article have to follow a few simple guidelines: complex mathematical expressions have to be justified (like in the excerpt below), algorithms have to be presented in the style of \texttt{alltt}, \textit{postscript specials} have to be absent in file format presenting images. The enumeration of references within any article has to be organized in alphabetic order (see the sample below). References have to be distinguished between journal articles, collective works, and books and presented in BibTeX\ \texttt{NAMCplain} style
}
\end{abstract}
\ArticleText %%======== write your article text======================================
\section{Introduction or the first section}\label{s:1}
This\footnote{Introduction text} is a sample demonstration of the requirements for articles submitted to the journal ``Nonlinear Analysis:\ Modelling and Control''. For visuality, we quote random portions of published articles.
\subsection{Examples of citation}\label{ss:1}
Citation of \cite[article]{article},\cite[book]{book}, \cite[proceedings]{proceedings}.
Bibliography entries are included at the beginning of the *.tex file.
When editing these entries, it is necessary to delete the `x.bib` file before compiling the LaTeX document (this step is not required when working in Overleaf).
\section{Examples of equations, algorithms, figures}\label{s:2}
Let us consider one-way coupled chaotic systems of the following general form (master-slave configurations or systems with a skew product structure):
%
\begin{equation}\label{eq:1}
\dot{X}=F(X),\qquad \dot{Y}=G(Y,X).
\end{equation}
%
Here $X\equiv\{x_1,x_2,\ldots,x_d\}$ is a $d$-dimensional state vector of the driving system, and $Y\equiv\{y_1,y_2,\ldots,y_r\}$ is an $r$-dimensional state vector of the response system. $F$ and $G$ define the vector fields of the driving and response systems.
\subsection{Complex mathematical expressions}\label{s:2.1}
%
\begin{align}
1
&\le\big|P(1)\big|=|a_d|\prod_{j=k+1}^d\alpha_j\prod_{j=k+1}^d\big(\alpha_j^{-1}-1\big)\prod_{j=1}^d(\alpha_j-1)\notag\\
&\le\frac{|a_da_0|}{M(P)}\bigg(\bigg(\frac{M(P)}{|a_0|}\bigg)^{1/(d-k)}-1\bigg)^{d-k}\bigg(\bigg(\frac{M(P)}{|a_d|}\bigg)^{1/k}-1\bigg)^k\notag\\
&= M(P)^{-1}\big(M(P)^{1/(d-k)}-|a_0|^{1/(d-k)}\big)^{d-k}\big(M(P)^{1/k}-|a_d|^{1/k}\big)^k\notag\\
&\le M(P)^{-1}\big(M(P)^{1/(d-k)}-1\big)^{d-k}\big(M(P)^{1/k}-1\big)^k,\label{eq:2}
\end{align}
%
\begin{equation}\label{eq:3}
\begin{split}
\varOmega
&=\big\{(r,z){:}\ \,0<r<R,\ 0<z<H\big\},\\
\varOmega_0
&=\big\{(r,z){:}\ \,r^2+z^2<R_0^2,\ z>0\big\},\\
\varGamma_1
&=\big\{(r,0){:}\ \,0\le r\le R\big\},\\
\varGamma
&=\big\{(0,z){:}\ \,0<z\le H\big\}\cup\big\{(R,z){:}\ \,0<z\le H\big\}\\
&\quad\cup\big\{(r,H){:}\ \,0<r<R\big\}\cup\varGamma_1,
\end{split}
\end{equation}
%
\begin{equation*}
u_{ij}=u(r_i,z_j,t),\qquad v_{ij}=v(r_i,z_j,t),\qquad d_{\alpha,ij}=d_{\alpha}(r_i,z_j), \\
\end{equation*}
%
$i=0,1,\ldots,N$, $j=0,1,\ldots,M_\alpha$, $\alpha=1,2$.
GS guarantees that the asymptotic dynamics of the response system is independent of its initial conditions and is completely determined by the driving system. Geometrically, this implies a collapse of the overall evolution onto a stable synchronization manifold $M=\{(X,Y){:}\ \,\varPhi(X)=Y\}$ in the full phase space of the two systems $X\oplus Y$. It is easy to show that the linear stability of the identity manifold $Y'=Y$ in the extended phase space $X\oplus Y\oplus Y'$ is equivalent to the linear stability of the manifold $M=\{(X,Y){:}\ \,\varPhi(X)=Y\}$ in the original $X \oplus Y$ phase space.
\subsection{The following excerpt demonstrates presentation of an algorithm}\label{s:2.2}
%
{\small
\begin{alltt}
Input: A list of segmentation parameters and the
corresponding \(C1\), \(C2\) and \(C3\) values.
Output: A selection of the optimum segmentation and its
parameters.
BEGIN
Search the input list for local minima in \(C2\),
and list the local minima into \(L\);
Search \(L\) and locate the minimum of \(C1 + 10 C3\sp{2}\),
present this element of \(L\) as the result;
END
\end{alltt}}
\subsection{Just another excerpt presenting image}\label{s:2.3}
In the works of S.\ Fu\v{c}\'ik, this spectrum was studied first as an object related to ``slightly'' nonlinear problems.\ The interest in these type of problems grew also in connection with the theory of suspension bridges (see, e.g., \cite{LazerSIAM}).\ From the mathematical point of view, the Fu\v{c}\'ik equation became a source of numerous investigations generalizing and refining the results by Fu\v{c}\'ik.
The Fu\v{c}\'ik equation with Sturm--Liouville conditions has similar structure of spectrum, and the description of it can be found in \cite{dancer} or \cite{Fucik}.
Completely different spectrum was obtained for the problem composed of equation
%
\begin{equation} \label{eq:4}
x''=-\mu x^{+}+\lambda x^{-}
\end{equation}
%
with nonlocal conditions
%
\begin{equation} \label{eq:5}
x(0)=0, \qquad \int\limits_0^1 x(s)\,\mathrm{d}s=0.
\end{equation}
%
A recent paper \cite{sergejevaLUMII} deals with the problem composed of equation \eqref{eq:4} with conditions
%
\begin{equation} \label{eq:6}
x'(0)=0, \qquad \int\limits_0^1x(s)\,\mathrm{d}s=0.
\end{equation}
Some branches of the spectra for these two problems are depicted in Fig. \ref{fig:1}.
\begin{figure}[ht]
\centering
\includegraphics[width=5.8cm]{Fig1a.pdf}\quad
\includegraphics[width=5.8cm]{Fig1b.pdf}
\caption{The spectrum for problem \eqref{eq:4}, \eqref{eq:5} and for problem \eqref{eq:4}, \eqref{eq:6}.}\label{fig:1}
\end{figure}
\section{Conclusion}\label{s:5}
\paragraph{Acknowledgment.}
We would like to thank dr. N. Sergejeva for the excerpts of article \cite{2014SergejevaN}.
%%======== end of article text======================================
\vspace{-6pt}
% ----------------------------------------------------------------
\paragraph{Author contributions.}
% ----------------------------------------------------------------
All authors (F.A., S.A. and T.A.) have contributed as follows: Methodology, S.A.; formal
analysis, F.A., S.A and T.A.; software, F.A. and T.A.; validation, F.A. and T.A.; writing--original
draft preparation, F.A.; writing--review and editing, S.A.. and T.A.; visualization, F.A. and T.A. All authors
have read and agreed to the published version of the manuscript.
\vspace{-6pt}
% ----------------------------------------------------------------
\paragraph{Conflicts of interest.}
% ----------------------------------------------------------------
The authors declare no conflicts of interest.
% ----------------------------------------------------------------
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