\documentclass[11pt,reqno]{amsart} \usepackage[cp1251]{inputenc} \usepackage[english,russian]{babel} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xcolor} \usepackage[colorlinks]{hyperref} \sloppy \usepackage{titlesec} \titlespacing{\section}{0pt}{2.5ex}{1.5ex} \titlespacing{\subsection}{0pt}{1.5ex}{1ex} \titlespacing{\subsubsection}{0pt}{1.5ex}{1ex} \titleformat{\section}{\large\bfseries\centering}{\thesection}{1em}{} \titleformat{\subsection}[runin]{\bfseries}{\thesubsection.}{0.5em}{}[.\mbox{\ }] \titleformat{\subsubsection}[runin]{\bfseries}{\thesubsubsection.}{0.4em}{}[.\mbox{\ }] %%%%%---------- ORCiD \newcommand\orcidicon[1]{\href{https://orcid.org/#1}{\includegraphics[scale=0.02]{orcid.pdf}}} \def\udcs{???.??} %Here the author places classificators of the paper according to Russian classification system \def\mscs{??X??} %Here the author places classificators according to the AMS classification list. \setcounter{page}{144} \newtheorem{lemma}{Lemma} \newtheorem{theorem}{Theorem} \newtheorem{definition}{Definition} \newtheorem{corollary}{Corollary} \newtheorem{proposition}{Proposition} \hypersetup{linkcolor=blue,filecolor=black,urlcolor=blue, citecolor=blue} \def\logo{\textcolor{blue}{{\bf\Huge S\raisebox{0.2ex}{\hspace{0.55ex}\raisebox{0.05ex}e\hspace{-1.65ex}$\bigcirc$}MR}}} \def\semrtop { \vbox{ \noindent\logo \hspace{-3mm} \parbox{12cm}{ \begin{center} {\LARGE \textsf{СИБИРСКИЕ ЭЛЕКТРОННЫЕ}} \\[2mm] {\LARGE \textsf{МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ}} \\[2mm] {\Large \textsf{Siberian Electronic Mathematical Reports}} \\[1mm] {\Large\tt{\textcolor{blue}{http://semr.math.nsc.ru}}}\\[1mm] \raisebox{1ex}{ISSN 1813-3304 } \vspace*{-3mm} % {\small Èíñòèòóò ìàòåìàòèêè èì. Ñ.Ë.Ñîáîëåâà ÑÎ ÐÀÍ}\\[-1mm] % {\small Sobolev Institute of Mathematics SB RAS} \end{center} } \vspace{-1mm} \noindent \begin{tabular}{c} \hphantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa!} \\ \hline\hline \end{tabular} \vspace{3mm} {\flushleft\it Vol. 20, No. 2, pp. 144--145 (2023) \hspace{\fill}{\rm\small ÓÄÊ \udcs \hspace{-4mm}}} \newline %Volume, pages in Russian, filled by Editorial board {\rm\small \textcolor{blue}{https://doi.org/10.33048/semi.2023.20.???}}\hphantom{aaaaaaaaaaaaaaaaaaaaaaaa\;}{\rm\small MSC\ \ \mscs } }%\vbox } \begin{document} \renewcommand{\refname}{References} \renewcommand{\proofname}{Proof.} \renewcommand{\figurename}{Fig.} \thispagestyle{empty} \title[Stable solutions increasing on the initial time interval]{\large Construction and estimation of stable solutions of Cauchy problems increasing on the initial time interval} \author[E.A. Biberdorf, L. Wang]{ {\bf E.A. Biberdorf}\protect\orcidicon{0000-0002-2129-922X} {\bf AND L. Wang}\protect\orcidicon{0009-0000-3323-5840} } \address{Elina Arnoldovna Biberdorf \newline\hphantom{iii} Sobolev Institute of Mathematics, \newline\hphantom{iii} pr. Koptyuga, 4, \newline\hphantom{iii} 630090, Novosibirsk, Russia}% \email{\textcolor{blue}{ ermolova@math.nsc.ru}}% \address{Wang Li \newline\hphantom{iii} Novosibirsk State University, \newline\hphantom{iii} ul. Pirogova, 1 \newline\hphantom{iii} 630090, Novosibirsk, Russia}% \email{\textcolor{blue}{l.van3@g.nsu.ru}}% \thanks{\sc Biberdorf E.A., Wang L. Construction and estimation of stable solutions of Cauchy problems increasing on the initial time interval} \thanks{\copyright \ 2026 Biberdorf E.A., Wang L.} \thanks{\rm This work was accomplished within the state assignment of the Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, project no. FWNF-2026-0028.} \thanks{\it Received January, 1, 2023, Published December, 31, 2023}% %\semrtop \vspace{1cm} \maketitle {\small %\vspace{-12pt} %\centerline{{\it Dedicated to the memory of Sergey Godunov} %{\sc P.P. Petrov} %({\it filled by Editorial board}) %} \bigskip \bigskip \begin{quote} \noindent{\bf Abstract:} Two algorithms are presented that allow one to find initial data for asymptotically stable solutions to Cauchy problems for ODE systems that are increasing over the initial time interval. One algorithm uses eigenvectors, while the other is based on the matrix spectrum dichotomy method. Using these algorithms, locally increasing solutions were constructed for the flutter models and the Navier-Stokes system linearized in the neighborhood of plane-parallel Poiseuille flow. A two-sided estimate for the maximum norm of these solutions was obtained.\medskip \noindent{\bf Keywords:} Cauchy problem, stability, spectral criterion, matrix spectrum dichotomy. \end{quote} } \bigskip \section{Introduction} One of the most common mathematical methods of studying stability is the spectral approach. It is based on the linearization of the differential operator describing the process under study, with respect to small perturbations. Next, the presence or absence of points in the spectrum of the linearized operator, which are responsible for the growth of the perturbations, is determined in the corresponding instability region on the complex plane. Experimental data indicate that physical processes modeled by solutions of initial-boundary value problems can exhibit instability even when all eigenvalues of the linearized operator lie in the stability region. This phenomenon is observed, in particular, during subcritical laminar-turbulent transitions in viscous incompressible fluid flows (see references in \cite{wenzhang2}). It has been suggested that such <> instability is related to the fact that even in the case of spectral asymptotic stability, the linearized problem can have solutions that exhibit significant growth in the initial time interval. As an example of a paper investigating this mechanism of transient growth of disturbances in stable fluid flows, we can cite the article \cite{wenzhang2}. The main idea implemented in the article is an orthogonal decomposition of the solution space into two subspaces, with growing solutions represented as the sum of two nonzero terms belonging to one and the other subspace. In this paper, we conduct a comprehensive study of locally increasing solutions of the Cauchy problem $dy/dt=Ay,\;y|_{t=0}=y_{0}$ under the condition that the spectral criterion for asymptotic stability is satisfied. Below, we present two algorithms for finding such solutions and examples of their application. We also obtain estimates for the maximum of their norm (see section \ref{aprior_est}), among which the lower bound is of particular importance. The first algorithm (see section \ref{eig_vec_meth}) is based on analogies with the behavior of solutions of the Cauchy problem when the matrix has Jordan blocks. To calculate the initial data of the growing solution, this approach requires knowledge of the eigenvectors. The second algorithm (see section \ref{inv_subsp_meth}) is conceptually similar to the approach described in the aforementioned work \cite{wenzhang1}, but differs from it in its much more general nature and broader potential range of application. The new method also involves decomposing the solution space into subspaces, for which the initial data space is first decomposed. The matrix spectrum dichotomy method (see \cite{erfen3}-\cite{erfen15}) is used as a tool for this decomposition. The problem of dichotomy of the spectrum of a matrix pencil $A-\lambda B$ with respect to a curve $\gamma$ involves computing the matrix $H=H^*>0$, whose norm serves as a numerical criterion for the feasibility of separating the eigenvalues of the pencil $A-\lambda B$ by the curve $\gamma$. In particular, it can be used to estimate the distance from the spectrum to the curve. Projection matrices $P^2=P$ onto the reducing subspaces of the pencil corresponding to the parts of the spectrum lying on opposite sides of $\gamma$ are also computed, while the bases of the reducing subspaces are determined using singular value decompositions of the projectors. If $\gamma$ is a circle, then the algorithm for dichotomy of the spectrum with respect to it is an iterative process, with the main action at each iteration being the qr decomposition. The imaginary-axis dichotomy algorithm is the unit-circle dichotomy algorithm applied to a matrix exponential or a fractional-linear transformation. In Section \ref{aprior_est} of this paper, we formulate new properties of the line dichotomy and use them to obtain estimates for growing solutions of the Cauchy problem. \section{Eigenvector method} \label{eig_vec_meth} \subsection{Preliminary remarks} As an example, let us consider the system $y' = A_{\delta} y$, the matrix of which at $\delta=0$ is a Jordan cell of size $5 \times 5$, in which all eigenvalues are located in the left half-plane: $$ A_{\delta}=\begin{bmatrix} -0.5 &15 & & & \\ & -0.5& 15 & & \\ & &-0.5 & 15 & \\ & & & -0.5 & 15\\ \delta & & & & -0.5 \end{bmatrix}. $$ With a small perturbation of $\delta=5\cdot10^{-6}$, the Jordan structure is destroyed, and the matrix $A_{\delta}$ also acquires an eigenvalue in the right half-plane. This means that a small perturbation transforms asymptotic stability into instability. %\small \begin{figure}[t] \centering \includegraphics[width=0.95\textwidth]{dop2.jpg}%{yuedangjuzhen3.png} \caption{Solution of the Cauchy problem corresponding to the matrices $A_{0}$ and $A_{\delta}$, $\delta=5\cdot10^{-6}$}\label{fig1} \end{figure} %\normalsize Fig. 1 shows an obvious situation where the norm of the exponent of the stable matrix $A_{0}$ tends to zero, while the exponent of the perturbed matrix $A_{\delta}$ grows infinitely (here and below, unless otherwise stated, the spectral norm is used for matrices, and the Euclidean norm is used for vectors). However, in addition to this, on the initial interval of the variable $t$, the matrix exponent $e^{A_{0}t}$ also exhibits rapid growth. In this case, this is easily explained by the fact that the matrix exponent of a Jordan cell such as $A_0$ has elements of the form $t^ke^{\lambda t}$. The largest contribution to the growth of the norm comes from the element in the last column of the first row, which contains the maximum power of the parameter $t$. Let us turn to the question of choosing the initial data for the solution of the Cauchy problem, growing over the initial time interval. It is easy to establish that by choosing the last adjoint vector in the Jordan chain $y_0=[0,0,0,0,1]^T$, we obtain maximum growth, and the same amplification mechanism will be activated as for the matrix exponential. The graphs of the norms of this solution and the matrix exponential coincide (Fig. \ref{fig1}). Generally speaking, if the system matrix contains a Jordan block, then a growing solution can be obtained by taking the last adjoint vector as the initial data. However, due to the density of the set of diagonalizable matrices in the matrix space, when using a computer for calculations, it is possible to obtain only a few very close eigenvectors for the matrix $A_0$: $$v_1=[1,0,0,0,0]^T,\;\;v_2=[-1,10^{-18},0,0,0]^T,\;\;v_3=[1,-10^{-18},10^{-35},0,0]^T,$$ $$v_4=[-1,10^{-18},-10^{-35},10^{-52},0]^T,\;\;v_5=[1,-10^{-18},10^{-35},-10^{-52},10^{-69}]^T.$$ Nevertheless, the previously selected vector $y_0=[0,0,0,0,1]^T$ yields the same growing solution. Note that in this case, the vector $y_0$ differs in that it is orthogonal to four of the five eigenvectors and orthogonal to the fifth with high accuracy. Based on this observation, as the initial data, we will choose a vector that forms, in some sense, the maximum angle with all eigenvectors. \subsection{Angles between vectors} Let $v_1,\dots, v_k$ be some group of normalized linearly independent vectors from an $n$-dimensional space, $k\le n$, $\|v_j\|=1$, and let the normalized vector $y_0$ be their linear combination: $y_0=Va$, where $V=[v_1, \dotsm v_k]$, $a=[a_1, \dots, a_k]^T$, $\|y_0\|=1$ (see the diagram in Fig. \ref{fig2}, left). The following definition is quite natural. {\bf Definition 1.} If $\alpha_j$ is the angle between vectors $y_0$ and $v_j$, then the angle between $y_0$ and the set of vectors $v_1,\dots, v_k$ will be called $\alpha=\min_j \alpha_j$. Obviously, this angle satisfies an equality that completely defines it. \begin{equation} \label{alpha1} |\cos\alpha|= \max_j|\cos \alpha_j|=\max_j |(v_j, y_0)|=\|V^*y_0\|_{\infty}, \end{equation} where $(\cdot, \cdot)$ is the scalar product. Based on the representation (\ref{alpha1}), the angle between $y_0$ and the set $\{v_j\}$ can be determined in a different way, using not the minimum, but, in a certain sense, the average value of the angles $\alpha_j$. {\bf Definition 2.} The angle between $y_0$ and $\{v_j\}$ in the sense of the mean value is the value $\alpha$ for which the relation is satisfied \begin{equation} \label{alpha2} |\cos\alpha|= \frac{1}{k}\sqrt{\sum_{j=1}^k|\cos \alpha_j|^2}= \frac{1}{k}\sqrt{\sum_{j=1}^k|(v_j, y_0)|^2}=\frac{1}{k}\|V^*y_0\|. \end{equation} In this case, the sign $\|\cdot\|$ denotes the Euclidean norm. Now let us turn to the problem of choosing $y_0$ that forms the maximum angle with the set $\{v_j\}$, that is, minimizes the value $\|V^*y_0\|$. Consider the spectral (also known as singular) decomposition $V^*V=W\Lambda W^*$, where the diagonal elements of the matrix $\lambda$ are positive, nonzero, and arranged in descending order $\lambda_1\ge\lambda_2\ge \dots\ge \lambda_k$. Let $w_j$ be the columns of the matrix $W$. Using this notation, we formulate and prove the lemma, which is the theoretical basis of the first algorithm. \begin{lemma} \label{lemma1} The vector $y_0=V\cdot w_{k}$ forms the maximum possible angle in the sense of Definition 2 with the vectors $v_1, \dots, v_k$. \end{lemma} \begin{bf} Proof. \end{bf} Let $y$ be an arbitrary normalized vector belonging to the linear span of vectors $\{v_j\}$: $y=V c$, where $c=[c_1,...,c_k]^{T}.$ Note that since $y$ and $v_j$ are normalized, it follows that $\|c\|=1$. Consider the norm of the product $$ \|V^*y\|=\|W\Lambda W^*c\|=\|\Lambda W^*c\|=\|\Lambda \tilde c\|, $$ where $c=W\tilde c$. It is obvious that $\min_{\tilde c}\|\Lambda \tilde c\|=\lambda_k$ and is achieved when $\tilde c=[0,\dots, 0,1]^T$, that is, when $y=y_0=V \cdot w_k$, which is what was required to be proven. \vspace*{3mm} This result leads to the following algorithm. \noindent\textbf{Algorithm 1}\\ {\large Given:} matrix $A$.\\ {\large Step 1.} Calculate the matrix $V$ consisting of the eigenvectors of matrix $A$.\\ {\large Step 2.} Find a factorization for matrix $S=V^{*}V$ such that $S=W\Lambda W^{*}$, where $W=\begin{bmatrix} w_{1} & w_{2}&\cdots&w_{k} \end{bmatrix}$ is an orthogonal matrix, $\Lambda=\mbox{diag}(\lambda_{1}, \dots, \lambda_{k}) $, $\lambda_{j} > 0$ are arranged in descending order.\\ {\large Step 3.} $y_{0}=V\cdot w_{k}$ \begin{figure}[t] \centering \includegraphics[width=0.49\textwidth]{algorithm1.png} \includegraphics[width=0.49\textwidth]{algorithm2.1.png} \caption{Scheme for algorithm 1 (left), for algorithm 2 (right)}\label{fig2} \end{figure} \subsection{Example: Wing flutter model} Let's consider one of the mathematical models of wing flutter described in \cite{fulada2}. It is a system of ordinary differential equations, where the system matrix is expressed as follows \begin{small} \begin{equation} \label{fld1} \begin{array}{cc} A=\begin{bmatrix} -vD &-(G+v^2F) \\ I & 0 \end{bmatrix},\;\;& G=\begin{bmatrix} 37.7 & & & \\ & 169& & \\ & & 899& \\ & & & 1792 \end{bmatrix}, \\ D = 0.73\cdot 10^{-2}I,& F=10^{-3}\begin{bmatrix} 0 &-1.97 &0 &0 \\ 0.12& 0 &-4.19 &0.171 \\ 0&0.176 &0 &0 \\ 0&-0.154 & 0 & 0 \end{bmatrix}. \end{array} \end{equation} \end{small} Here $v$ is the flow velocity, $G$ is the matrix of natural oscillation frequencies, $vD$ is the friction matrix, $v^{2}F$ is the matrix of interactions between the wing parts, and $y$ is the displacement vector. For the numerical experiment, we use the value $v=425$, which is below the critical value, i.e., all eigenvalues of the matrix (\ref{fld1}) lie in the left half-plane: $$ \lambda_{1,2}= -3.0922 \pm i10.9845, \;\; \lambda_{3,4}= -0.0103 \pm i10.9845, $$ $$ \lambda_{5,6}= -1.5513 \pm i29.3672, \;\; \lambda_{7,8}= -1.5512 \pm i 42.2974. $$ To select the initial data for the Cauchy problem, we use Algorithm 1. To do this, we first apply it to the group of eigenvectors $v_1,v_2,v_3,v_4$ corresponding to the eigenvalues $\lambda_1, \dots, \lambda_4$, and then to all eigenvectors $v_1, \dots, v_8$. The norms of the resulting solutions are shown in Fig. \ref{fig7}. Both solutions exhibit growth in the initial period of time. \begin{figure}[h] \centering \includegraphics[width=0.95\textwidth]{lgsuanfa1(fulada).png} \caption{Solution of the Cauchy problem for the flutter model system at $v=425$.}\label{fig7} \end{figure} \section{Method of invariant subspaces} \label{inv_subsp_meth} \subsection{Brief description of the method} Recall that the asymmetric spectral problem in the classical formulation can be very sensitive to errors when calculating both eigenvalues and eigenvectors. In contrast, the matrix spectrum dichotomy problem offers several advantages. One of these is the algorithm's ability to <>, meaning that the computational robustness of the obtained result is determined simultaneously during its execution. For this reason, this method is preferable for a large number of problems. The basic idea of the method for choosing initial values for solutions of the Cauchy problem growing over a finite interval remains the same: we must determine $y_0$ that forms the largest possible angle with the eigenvectors. However, in this case, we generalize the concept of eigenvectors to invariant subspaces of a matrix. Therefore, to obtain $y_0$, we must partition the initial data space into invariant subspaces. The simplest case is a partition into two subspaces, which is accomplished using the dichotomy method. We choose a line that divides the set of eigenvalues into two parts, one of which contains the critical value (i.e., the one closest to the imaginary axis). We apply the matrix pencil dichotomy method with respect to this line, resulting in bases of invariant subspaces corresponding to the separated groups of eigenvalues. Next, we choose the initial vector $y_{0}$ such that the angle between it and the bases of both subspaces is maximal (see the diagram in Fig. \ref{fig2} on the right), using Definition 2 and Lemma \ref{lemma1}. \vspace*{3mm} \noindent\textbf{Algorithm 2}\\ { Given:} matrix $A$, line $\gamma$\\ {\large Step 1.} Using dichotomy, calculate the projectors $P_{1}, P_{2}$ onto the invariant subspaces of matrix $A$ corresponding to the eigenvalues lying on opposite sides of $\gamma$.\\ { Step 2.} Calculate the singular value decompositions of the projectors $P_{j}=U_{j}S_{j}V_{j}^{*}$\\ {Step 3.} From the first $k_{1}$ and $k_{2}$ columns of matrices $U_{1}$ and $U_{2}$, form a matrix $$V=\left[ U_{1}(:,1:k_{1}), U_{2}(:,1:k_{2}) \right ], \mbox{ where } k_{j} = \mbox{tr}P_{j}.$$ {Step 4.} Find a factorization for the matrix $S=V^{*}V$ such that $S=W\Lambda W^{*}$, where $W=\begin{bmatrix} w_{1} & w_{2}&\cdots&w_{n} \end{bmatrix}$ is an orthogonal matrix, $\Lambda=\mbox{diag}(\lambda_{1}, \dots, \lambda_{n}) $, $\lambda_{j} > 0$ are arranged in descending order.\\ {\large Step 5.} $y_{0}=V\cdot w_{n}$ \subsection{Examples} \subsubsection{Wing flutter model} Let us apply Algorithm 2 to the flutter model (\ref{fld1}) considered earlier. We use the dichotomy of the spectrum with respect to the following lines: the real axis (highlighted in red and denoted by the number 1 in Fig.\ref{fig9}), the line $Im \;\omega=12$ (blue, number 2), the line $Re \; \omega =-1$ (green, number 3), the line $Im \; \omega =3 Re\, \omega+15$ (black, 4). As a result, four variants of the initial conditions for the Cauchy problem are obtained, presented in Table \ref{tab1}. \begin{figure}[t] \centering \includegraphics[width=0.7\textwidth]{v=425(xianhetezhengzhi).png} \caption{The spectrum of the matrix (\ref{fld1}) at $v=425$ and the lines used in the example dividing it into parts}\label{fig9} \end{figure} \begin{table}[h] \caption{Initial values for the flutter model (\ref{fld1})}\label{tab1} \centering \begin{tabular}{|l|l|l|l|l|} \hline $10^{-4}\cdot y_0$ & line 1 & line 2 & line 3 &line 4 \\ \hline $y_0^1$ & 0 & -62-3i & -19 &123-96i\\ \hline $y_0^2$ & -i & 161+8i & -87 &563-196i \\ \hline $y_0^3$ & -i & 203-7i & -46 &551-169i \\ \hline $y_0^4$ & 1-9i & -3+i & 1 &-10+3i \\ \hline $y_0^5$ & 4-30i & 1-10i & -1690 &-5-1545i \\ \hline $y_0^6$ & -15+100i & 24-470i & -7095 &-2318-6003i \\ \hline $y_0^7$ & -40+265i & 522-9920i & -6840 &-565-7425i \\ \hline $y_0^8$ & 1476-9886i & -53+1016i & 131 &6+200i \\ \hline \end{tabular} \end{table} These initial data correspond to solutions of the Cauchy problem, the Euclidean norm of which for each $t$ is shown in Fig.\ref{fig10}. \begin{figure}[t] \centering \includegraphics[width=0.99\textwidth]{lgall.png} \caption{Solution of the Cauchy problem with matrix (\ref{fld1}), lines are used for separation see fig.\ref{fig9}}\label{fig10} \end{figure} In all four cases, despite the significantly different initial conditions, the solution has a maximum. Moreover, when using straight lines 3 and 4, the solution is real-valued in one case and complex-valued in the other, with both solutions increasing by almost two orders of magnitude. \subsubsection{Infinite strip panel flutter model} We will examine the model previously discussed in works \cite{erfen9},\cite{alr1}. The dimensionless equation for the perturbed state $u(t,x,y)$ of an infinite plate has the form $$M\Delta^2u+\gamma \vec{v}\nabla u+\gamma \frac{\partial u}{\partial t}+\rho h \frac{\partial^2 u}{\partial t^2}=0.$$ Here $\vec{v}$ is the free-stream velocity, $\Delta^2$ is the biharmonic operator, $\gamma$ is the gas specific heat index, $E$ is Young's modulus, $h$ is the constant plate thickness, $\nu$ is Poisson's ratio, $\rho$ is the material density, and $M=Eh^{3}/(12(1-\nu^2))$ is a constant. Different methods of fastening the strip edges at $y=0$ and $y=1$ are described by the corresponding boundary conditions. For hinged and rigid edge fastening, they are: $$ u=0, \frac{\partial^2u}{\partial y^2}=0 \;\;\mbox{and}\;\; u=0, \frac{\partial u} {\partial y}=0 $$ respectively. Substituting a solution of the form $u(t,x,y)=\varphi(t,y)e^{-i\alpha x}$ into the equation and boundary conditions leads to an initial-boundary value problem with one spatial variable for each fixed $\alpha$. For finite-dimensional approximation of the boundary value and spectral problems, it is necessary to discretize the differential operators taking into account the boundary conditions. For this purpose, matrices of collocation derivatives are used, the choice of which is described in more detail in the paper \cite{erfen9}. We only note that for the case of hinged fastening, the matrices described in the paper \cite{cem1} are suitable, and for rigid fastening, those in the paper \cite{tre1}. As a result, we obtain the Cauchy problem for a system of second-order ordinary differential equations $$\rho h\frac{d^2 \varphi}{d t^2} + \gamma \frac{d \varphi}{d t}+ C\varphi=0, $$ where $$C=M[D^{4}_{N}-2\alpha^2 D^{2}_{N}+\alpha^4 I_{N}]+ \gamma v_{y} D_{N}- i \alpha \gamma v_{x} I_{N},$$ $D_N$ is the collocation matrix derivative, $I_N$ is the identity matrix. Using the standard transition to a new unknown vector function, we obtain a first-order ODE system, but twice as large: \begin{equation} \label{flat_Cosh} B \frac{d}{d t} \psi=A \psi, \mbox{ where } A=\begin{bmatrix} C &\gamma I \\ 0 & I \end{bmatrix},\;\; B= \begin{bmatrix} 0 & \rho h I \\ I & 0 \end{bmatrix}, \;\;\psi= \left( \begin{array}{r} \varphi\\\frac{d}{dt} \varphi \end{array} \right) , \end{equation} We fix the value of the velocity vector components $v_x=0.1606$, $v_y=0$ for hinged fastening, $v_x=0.0906$, $v_y=0$ for rigid fastening and calculate the initial data for the Cauchy problem using Algorithm 2 with the following constant values: $$ \alpha=1,\; \gamma=1.4,\; E=7 \cdot 10^5,\; h=3 \cdot 10^{-3},\;\nu=0.33,\;\rho=2.7 \cdot 10^3,\; N=30. $$ Note that the values of $\alpha$ and $\vec{v}$ are such that the entire spectrum of the pencil $A-\omega B$ is located strictly in the left half-plane. \begin{figure}[t] \centering \includegraphics[width=0.49\textwidth]{eiglinejiao.png} \includegraphics[width=0.49\textwidth]{eiglinegang.png} \includegraphics[width=0.98\textwidth]{jiejiaofulada.png} \includegraphics[width=0.98\textwidth]{jiegangfulada.png} \caption{Top: eigenvalues of the pencil $A-\omega B$ and the lines with respect to which the spectrum is divided (on the left, hinged fastening, on the right, rigid fastening); the norm of the solution of the Cauchy problem for the system (\ref{flat_Cosh}) (in the center, hinged fastening, below, rigid fastening).} \label{fig15} \end{figure} In the case of hinged edge fixation, the first step of Algorithm 2 performs a spectrum dichotomy with respect to the following lines:\\ 1. line $Re \; \omega = -0.08$ (see Fig. \ref{fig15}, top left, marked in red);\\ 2. line $Re \; \omega = -0.02$ (Fig. \ref{fig15}, top left, black);\\ 3. line $Im \;\omega =-1578Re \; \omega -134$ (Fig. \ref{fig15}, top left, blue).\\ When the strip edges are rigidly fixed, a spectrum division with respect to the lines is used:\\ 1. line $Re \; \omega = -0.08$ (Fig. \ref{fig15}, top right, red);\\ 2. line $Re \; \omega = -0.02$ (Fig. \ref{fig15} top right, black);\\ 3. line $Im \;\omega =10$ (Fig. \ref{fig15} top right, blue).\\ The dependence of the norm of solutions to the Cauchy problem (\ref{flat_Cosh}) on $t$ with initial data obtained using Algorithm 2 can be seen in the central and lower panels of Fig.\ref{fig15}. For both the hinged and rigid fixations, using lines 1 and 2 (red) leads to a solution that exhibits a slight increase and an extremely slow decay. The latter is likely due to the critical eigenvalue being very close to the imaginary axis. The use of line 3 (blue color) for dichotomy in both cases allows us to obtain solutions that sharply increase by two orders of magnitude, while the rate of subsequent decrease is just as small as in the cases of lines 1, 2. \subsubsection{Navier-Stokes system} By analogy with the derivation of the Orr-Sommerfeld equation, the linearization of the Navier-Stokes system of equations for a viscous incompressible fluid $$ \left\{\begin{array}{l} \dfrac{\partial U}{\partial t}+(U,\nabla)U=-\nabla P+\dfrac{1}{Re}\Delta U,\\ \mbox{div}\, U =0\end{array}\right. $$ in the neighborhood of a plane-parallel Poiseuille flow $U=(1-y^2,0,0)^T$, $P=-2x/Re$ relatively small perturbations of the form $[u(t,y),v(t,y),w(t,y)]^Te^{i(\alpha x+\beta z)}$, $p(t,y)e^{i(\alpha x+\beta z)}$ and successive elimination of variables leads to a one-dimensional initial-boundary value problem for the function $v$ : \begin{equation} \label{N-S} %\left\{ \begin{array}{c} i Re \left(\dfrac{\partial^2}{\partial y^2}-k^2\right)^2\dfrac{\partial}{\partial t} v=\\ = \left[\left(i\alpha Re(1-y^2)-1\right)\left(\dfrac{\partial^2}{\partial y^2}-k^2\right)^2+2i\alpha Re\right] v, \\ v|_{y=\pm 1}=v_y|_{y=\pm 1}=0, \;\;v|_{t=0}=v_0 \end{array} %\right. \end{equation} Here $-1\le y\le 1$, $Re$ is the Reynolds number, $\kappa^2=\alpha^2+\beta^2$. We use the collocation matrix derivatives $D_N$ (see \cite{tre1}) to discretize the derivatives with respect to the variable $y$ ($y_j$ are the collocation points), resulting in the Cauchy problem for the system of ordinary differential equations where \begin{equation} \label{OZpencil}\begin{array}{l} B\dfrac{d}{dt}v=A v, \mbox{ where }\\ A=\left(i\alpha Re\,\mbox{diag}(1-y_j^2)-I\right)\left(D_N^2-k^2I\right)^2+2i\alpha Re\,I, \\ B=i Re \left(D_N^2-k^2I\right)^2. \end{array} \end{equation} \begin{figure}[t] \centering \includegraphics[width=0.45\textwidth]{Eigenvalues oula2.png} \includegraphics[width=0.45\textwidth]{oula2gejie.png} \caption{Eigenvalues of the matrix pencil $A+i\omega B$ (\ref{OZpencil}) and lines dividing the spectrum into parts (left), norms of solutions of the system (\ref{OZpencil}) (right).}\label{fig11} \end{figure} For the numerical experiment, we use the parameter values $Re = 5000$, $\alpha=1.02$, $\beta=0$, for which the zero solution is asymptotically stable, and the number of collocation points $N=100$. We obtain the initial vector $v_0$ using Algorithm 2. The dichotomy method is applied with respect to the following lines: \begin{enumerate} \item line $Im \; \omega = -0.05$ (Fig. \ref{fig11}, red); \item line $Re \; \omega = 0.4$ (Fig. \ref{fig11}, black). \end{enumerate} In Fig. 11, the left shows how these lines divide the spectrum of the matrix pencil $A+i\omega B$, and the right shows the norms of the solutions to the problem (\ref{N-S}) for these two cases. It can be noted that both obtained solutions grow over the initial time interval, with the growth amounting to approximately two orders of magnitude. \section{A priori estimates of solutions of the Cauchy problem} \label{aprior_est} For the maximum norm of the solution of the Cauchy problem for the system $y'=Ay$ there is a well-known estimate obtained by M.G. Krein \cite{God1}, \cite{Dem1}: $$ \|y(t)\|\le\sqrt{\mbox{cond}\,(H)}e^{-t/(2\|H\|)}\|y_0\|, $$ where $\mbox{cond}$ is the condition number, and $H$ is the solution of the Lyapunov equation $$ HA+A^*H=-I, $$ which can be represented in integral form $$ H=\frac{1}{2\pi}\int_{-\infty}^{+\infty}(A^*+i\xi I)^{-1}C(A-i\xi I)^{-1}d\xi, \;\;C=I, $$ and coincides with the matrix criterion for the dichotomy of the spectrum of the matrix $A$ with respect to the imaginary axis. In practice, this estimate is usually quite high, but for a large value of $\mbox{cond}\,(H)$ it is a warning that the solution may grow significantly over the initial time interval. The following theorem also describes the conditions under which a solution to the Cauchy problem with a local maximum can exist. Note that the theorem provides not only an upper bound but also a lower bound. \begin{theorem} \label{maximum} Let the matrix $A$ have its entire spectrum in the left half-plane. Let the angle between two fixed eigenvectors be $\alpha$, and let the corresponding eigenvalues be $\lambda_j=\alpha_j+i\beta_j$, $j=1,2$, $\alpha_1\le\alpha_2<0$. Then there exists a solution to the Cauchy problem $y'=Ay$, $y(0)=y_0$, $\|y_0\|=1$, with a local maximum of the norm $M$ such that \end{theorem} \begin{equation} \label{max-est} \begin{array}{l l} \text{ctg}\,\alpha \dfrac{\alpha_{2}-\alpha_{1}}{|\alpha_{1}|}e^{-1}0$. This implies the inequality $$\mbox{ctg}\alpha \geq \frac{\|P\|^2-1}{2\|P\|}.$$ Thus, the left-hand side of the inequality is proven. The proof of the right-hand side of the inequality is as follows. Fix vectors $u=\textstyle\sum_{i=1}^{l}a_{i}U_{i}$ and $v=\textstyle\sum_{j=1}^{m}b_{j}V_{j}$ such that the angle between them is minimal. Among the vectors orthogonal to $v$, choose a vector $x$, $\|x\|=1$, such that the projection of $x$ onto the subspace spanned by the columns of $U \; Px=cu$ is a vector collinear to $u$. Vectors $u,cv$ and $x$ form a right triangle such that one leg is vector $x$, the other leg is directed along vector $v$, and the hypotenuse is directed along vector $u$. By the definition of sine, we obtain the equality $$\sin\alpha=\frac{\|x\|}{\|Px\|}\geq \frac{1}{\|P\|}.$$ Next, we easily obtain $$\mbox{ctg}\;\alpha=\frac{\sqrt{1-\sin^2\alpha}}{\sin\alpha}\leq \sqrt{\|P\|^2-1}$$ Thus, the right-hand side of the inequality is also proven. \vspace*{3mm} {\bf Proof of Theorem \ref{theorem3}.} Consider the circle $C_{\rho_1}$. Its equation is $\xi \bar \xi=\rho_1^2$. It is easy to establish that the equation of its preimage $C_-$ is $$ \lambda\bar\lambda-\left(1-\frac{2}{1-\rho_1^2}\right)\lambda-\left(1-\frac{2}{1-\rho_1^2}\right)\bar\lambda+1=0. $$ It follows that the center of this circle is at the real point $O_-=-(1+\rho_1^2)/(1\rho_1^2)$, and the radius is $R_-=2\rho_1/(1-\rho_1^2)$. This means that the distance from the circle $C_-$ to the imaginary axis is $S_-=(1-\rho_1)/(1+\rho_1)$. Similarly, we obtain $R_+=2\rho_2/(1-\rho_2^2)$ and $S_+=(1-\rho_2)/(1+\rho_2)$. Next, we use the estimate $\rho_{1,2}<\sqrt{\frac{\|H\|-1}{\|H\|+1}}$ \cite{God1}, which results in the inequalities (\ref{C_pm}). The theorem is proved. %*************************************************************************************** \section{Conclusion} In this paper, for initial-value problems satisfying the spectral stability criterion, a mechanism for the formation of solutions that grow over the initial time interval is revealed. Theorem \ref{maximum} shows that the initial data of such solutions are special linear combinations of non-orthogonal eigenvectors, with the local maximum of the solution proportional to the cotangent of the angle between them. Theorem \ref{maximum} also substantiates a two-sided estimate for the maximum value of the solution norm. While the upper bound is, in a sense, a refinement of M.G. Krein's well-known estimate, the lower bound represents a new tool for analyzing ODE systems. The presence of this estimate indicates that the system has a solution that, given normalized initial data, grows beyond a given value. Theorems \ref{ctg-est-th} and \ref{theorem3} allow us to obtain estimates without calculating eigenvalues and eigenvectors, but by applying the matrix spectrum dichotomy method and estimating the positions of the eigenvalues. The angle between the eigenvectors is estimated using the angle between the subspaces in which they lie. This fact suggests that the left-hand side of the inequality (\ref{max-est}) can be strengthened in this way only for $2\times 2$ matrices, as shown in the example (\ref{example}). 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