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\setlength{\leftmargin}{0pt} \setlength{\parsep}{0pt} % turn on hanging indent if param 1 is 1 \ifodd #1 \setlength{\leftmargin}{\cslhangindent} \setlength{\itemindent}{-1\cslhangindent} \fi % set entry spacing \setlength{\itemsep}{#2\baselineskip}}} {\end{list}} \usepackage{calc} \newcommand{\CSLBlock}[1]{\hfill\break\parbox[t]{\linewidth}{\strut\ignorespaces#1\strut}} \newcommand{\CSLLeftMargin}[1]{\parbox[t]{\csllabelwidth}{\strut#1\strut}} \newcommand{\CSLRightInline}[1]{\parbox[t]{\linewidth - \csllabelwidth}{\strut#1\strut}} \newcommand{\CSLIndent}[1]{\hspace{\cslhangindent}#1} \setlength{\emergencystretch}{3em} % prevent overfull lines \providecommand{\tightlist}{% \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} \definecolor{quarto-callout-color}{HTML}{909090} \definecolor{quarto-callout-note-color}{HTML}{0758E5} \definecolor{quarto-callout-important-color}{HTML}{CC1914} \definecolor{quarto-callout-warning-color}{HTML}{EB9113} \definecolor{quarto-callout-tip-color}{HTML}{00A047} \definecolor{quarto-callout-caution-color}{HTML}{FC5300} \definecolor{quarto-callout-color-frame}{HTML}{acacac} \definecolor{quarto-callout-note-color-frame}{HTML}{4582ec} \definecolor{quarto-callout-important-color-frame}{HTML}{d9534f} \definecolor{quarto-callout-warning-color-frame}{HTML}{f0ad4e} \definecolor{quarto-callout-tip-color-frame}{HTML}{02b875} \definecolor{quarto-callout-caution-color-frame}{HTML}{fd7e14} \hypersetup{ pdftitle={Ordinary Differential Equations and Systems Dictionary}, pdfauthor={Enrico Bertolazzi}, colorlinks=true, linkcolor={blue}, filecolor={Maroon}, citecolor={Blue}, urlcolor={Blue}, pdfcreator={LaTeX via pandoc}} \usepackage{fontawesome5} % Per LaTeX moderno \definecolor{ocre}{RGB}{243, 102, 25} % Define the color used for highlighting throughout the book \chapterimage{orange1.jpg} % Chapter heading image \chapterspaceabove{6.5cm} % Default whitespace from the top of the page to the chapter title on chapter pages \chapterspacebelow{6.75cm} % Default amount of vertical whitespace from the top margin to the start of the text on chapter pages \usepackage{bookmark} % Importantissimo: dopo hyperref! \newtheorem{lemmaT}{Lemma}[chapter] % Theorems, the first argument is the theorem name, the second the theorem \newenvironment{lemma}{\begin{tBox}\begin{lemmaT}}{\end{lemmaT}\end{tBox}} \newtheorem{remarkT}{Remark}[chapter] % Theorems, the first argument is the theorem name, the second the theorem \newenvironment{refremark}{\begin{tBox}\begin{remarkT}}{\end{remarkT}\end{tBox}} \begin{document} %---------------------------------------------------------------------------------------- % TITLE PAGE %---------------------------------------------------------------------------------------- \titlepage % Output the title page {\includegraphics[width=\paperwidth]{background.pdf}} % Code to output the background image, which should be the same dimensions as the paper to fill the page entirely; leave empty for no background image { % Title(s) and author(s) \centering\sffamily % Font styling {\Huge\bfseries Ordinary Differential Equations and Systems Dictionary\par} % Book title \vspace{16pt} % Vertical whitespace {\LARGE Theory, Stability, and Boundary-Value Problems\par} % Subtitle \vspace{24pt} % Vertical whitespace {\huge\bfseries Enrico Bertolazzi\par} % Author name } %---------------------------------------------------------------------------------------- % COPYRIGHT PAGE %---------------------------------------------------------------------------------------- \thispagestyle{empty} % Suppress headers and footers on this page ~\vfill % Push the text down to the bottom of the page \noindent Copyright \copyright\ 2026 Enrico Bertolazzi\\ % Copyright notice \noindent \textsc{University of Trento}\\ % Publisher \noindent \textsc{\href{https://www.latextemplates.com/template/legrand-orange-book}{book-website.com}}\\ % URL \noindent Licensed under the Creative Commons Attribution-NonCommercial 4.0 License (the ``License''). You may not use this file except in compliance with the License. You may obtain a copy of the License at \url{https://creativecommons.org/licenses/by-nc-sa/4.0}. Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an \textsc{``as is'' basis, without warranties or conditions of any kind}, either express or implied. See the License for the specific language governing permissions and limitations under the License.\\ % License information, replace this with your own license (if any) \noindent \textit{English edition, 2026} % Printing/edition date %---------------------------------------------------------------------------------------- % TABLE OF CONTENTS %---------------------------------------------------------------------------------------- \pagestyle{empty} % Disable headers and footers for the following pages \tableofcontents % Output the table of contents %\listoffigures % Output the list of figures, comment or remove this command if not required %\listoftables % Output the list of tables, comment or remove this command if not required \pagestyle{fancy} % Enable default headers and footers again \cleardoublepage % Start the following content on a new page \bookmarksetup{startatroot} \chapter{Fundamental Forms and Classification}\label{fundamental-forms-and-classification} \index{fundamental forms and classification@Fundamental Forms and Classification} \begin{tcolorbox}[enhanced jigsaw, arc=.35mm, bottomrule=.15mm, bottomtitle=1mm, breakable, colback=white, colbacktitle=quarto-callout-note-color!10!white, colframe=quarto-callout-note-color-frame, coltitle=black, left=2mm, leftrule=.75mm, opacityback=0, opacitybacktitle=0.6, rightrule=.15mm, title={Purpose and Scope}, titlerule=0mm, toprule=.15mm, toptitle=1mm] This is not a set of lecture notes. It is a compact dictionary of definitions and results used in the accompanying courses. If you want to study any of these topics, you should follow a course or consult a textbook devoted to the subject. This dictionary is intended for readers who already know, or have previously studied, the material and need a concise reference whenever a reminder is useful. \end{tcolorbox} An ODE of order \(p\) can be written as \[F(t,y,y',\ldots,y^{(p)})=0, \qquad y^{(p)}=f(t,y,\ldots,y^{(p-1)}).\] Setting \(x=(y,y',\ldots,y^{(p-1)})^T\) gives a first-order system \[x'=f(t,x),\qquad x\in\mathbb{R}^n.\] The initial-value problem (IVP), or Cauchy problem, is \[\boxed{x'=f(t,x),\qquad x(t_0)=x_0.}\] An autonomous system has the form \(x'=f(x)\); a nonautonomous system has the form \(x'=f(t,x)\). A linear system is \[x'=A(t)x+b(t),\] and is homogeneous when \(b=0\) and has constant coefficients when \(A(t)=A\). \section{Important Scalar ODEs}\label{important-scalar-odes} \index{important scalar odes@Important Scalar ODEs} A separable equation satisfies \[y'=a(t)b(y),\qquad \int\frac{dy}{b(y)}=\int a(t)\,dt+C.\] Zeros of \(b\) may give constant solutions. For a linear equation, \[y'+a(t)y=b(t),\qquad \mu(t)=\exp\!\left(\int a(t)\,dt\right),\] the integrating-factor formula is \[y(t)=\mu(t)^{-1}\left[C+\int\mu(t)b(t)\,dt\right].\] A Bernoulli equation and its standard substitution are \[y'+a(t)y=b(t)y^\alpha,\quad \alpha\ne0,1, \qquad z=y^{1-\alpha}.\] A Riccati equation is \[y'=a(t)+b(t)y+c(t)y^2.\] Given a particular solution \(y_p\), the substitution \(y=y_p+1/z\) produces a linear equation. An exact equation satisfies \[M(t,y)\,dt+N(t,y)\,dy=0,\qquad M_y=N_t,\] and locally has the form \(\Phi(t,y)=C\), with \(\Phi_t=M\) and \(\Phi_y=N\). \bookmarksetup{startatroot} \chapter{Existence, Uniqueness, and Continuation}\label{existence-uniqueness-and-continuation} \index{existence uniqueness and continuation@Existence, Uniqueness, and Continuation} A classical solution \(x:I\to\mathbb{R}^n\) is continuously differentiable and satisfies the ODE pointwise. Its integral form is \[x(t)=x_0+\int_{t_0}^t f(s,x(s))\,ds.\] \section{Peano's Theorem}\label{peanos-theorem} \index{peano� s theorem@Peano’s Theorem} Continuity of \(f\) implies local existence, but not necessarily uniqueness. \section{Picard--Lindelöf Theorem}\label{picardlindeluxf6f-theorem} \index{picard-lindelof theorem@Picard–Lindelöf Theorem} If \(f\) is continuous in \(t\) and locally Lipschitz in \(x\), the IVP has a unique local solution. The Lipschitz condition is \[\|f(t,x)-f(t,y)\|\le L\|x-y\|.\] If \(f\) is \(C^1\) with respect to \(x\), it is locally Lipschitz in \(x\). Picard iteration is \[x^{(k+1)}(t)=x_0+\int_{t_0}^t f(s,x^{(k)}(s))\,ds.\] \section{Carathéodory Solutions}\label{carathuxe9odory-solutions} \index{caratheodory solutions@Carathéodory Solutions} Carathéodory theory permits measurable dependence on \(t\) and continuous dependence on \(x\), together with suitable bounds. Solutions are absolutely continuous and satisfy the ODE almost everywhere. \section{Maximal Interval}\label{maximal-interval} \index{maximal interval@Maximal Interval} The unique solution extends to a maximal interval \((\alpha,\beta)\). It can be continued as long as its graph remains in a compact subset of the domain of \(f\). At-most-linear growth, \[\|f(t,x)\|\le a(t)+b(t)\|x\|,\] provides standard conditions for global existence. \bookmarksetup{startatroot} \chapter{Dependence on Data and Grönwall's Inequality}\label{dependence-on-data-and-gruxf6nwalls-inequality} \index{dependence on data and gronwall� s inequality@Dependence on Data and Grönwall’s Inequality} If \(f\) is Lipschitz in \(x\), then \[\|x(t)-y(t)\|\le \mathrm{e}^{L|t-t_0|}\|x(t_0)-y(t_0)\|.\] Grönwall's inequality states that \[u(t)\le a+\int_{t_0}^t b(s)u(s)\,ds \quad\Longrightarrow\quad u(t)\le a\exp\!\left(\int_{t_0}^t b(s)\,ds\right).\] For \(x'=f(t,x,\mu)\), sufficient regularity gives continuous or differentiable dependence of the solution on initial data and parameters. \bookmarksetup{startatroot} \chapter{Flows, Variational Equations, and Sensitivities}\label{flows-variational-equations-and-sensitivities} \index{flows variational equations and sensitivities@Flows, Variational Equations, and Sensitivities} For \(x'=f(x)\), the flow satisfies \[\varphi_0=I,\qquad \varphi_{t+s}(x)=\varphi_t(\varphi_s(x)).\] For nonautonomous systems one uses \(\Phi(t,s,x)\) with the corresponding composition property. Linearization along \(x(t)\) gives \[\delta x'=\nabla_x f(t,x(t))\,\delta x.\] The sensitivity with respect to the initial condition satisfies \[S=\frac{\partial x(t;t_0,x_0)}{\partial x_0},\qquad S'=\nabla_x f\,S,\quad S(t_0)=I.\] Parameter sensitivity satisfies \[S_\mu'=\nabla_xf\,S_\mu+f_\mu.\] \bookmarksetup{startatroot} \chapter{Nonautonomous Linear Systems}\label{nonautonomous-linear-systems} \index{nonautonomous linear systems@Nonautonomous Linear Systems} For \[x'=A(t)x+b(t),\] a fundamental matrix satisfies \[\Phi'=A(t)\Phi,\qquad \det\Phi\ne0.\] The state-transition matrix obeys \[\Phi(s,s)=I,\qquad \Phi(t,r)\Phi(r,s)=\Phi(t,s),\qquad \Phi(t,s)^{-1}=\Phi(s,t).\] Variation of constants gives \[\boxed{x(t)=\Phi(t,t_0)x_0+ \int_{t_0}^t\Phi(t,s)b(s)\,ds.}\] The Wronskian is \[W(t)=\det[x_1(t)\ \cdots\ x_n(t)],\] and Liouville's formula is \[\det\Phi(t)=\det\Phi(t_0) \exp\!\left(\int_{t_0}^t\operatorname{tr}A(s)\,ds\right).\] \bookmarksetup{startatroot} \chapter{Constant-Coefficient Linear Systems}\label{constant-coefficient-linear-systems} \index{constant-coefficient linear systems@Constant-Coefficient Linear Systems} For \(x'=Ax\), \[x(t)=\mathrm{e}^{A(t-t_0)}x_0.\] The matrix exponential satisfies \[\mathrm{e}^{At}=\sum_{k=0}^\infty\frac{t^kA^k}{k!},\qquad \frac{d}{dt}\mathrm{e}^{At}=A\mathrm{e}^{At},\qquad (\mathrm{e}^{At})^{-1}=\mathrm{e}^{-At},\] \[\det(\mathrm{e}^{At})=\mathrm{e}^{t\operatorname{tr}A}.\] If \(AB=BA\), then \(\mathrm{e}^{A+B}=\mathrm{e}^A\mathrm{e}^B\). If \(A=V\Lambda V^{-1}\), then \[\mathrm{e}^{At}=V\mathrm{e}^{\Lambda t}V^{-1}.\] For \(J=\lambda I+N\) with \(N^p=0\), \[\mathrm{e}^{Jt}=\mathrm{e}^{\lambda t} \sum_{k=0}^{p-1}\frac{t^kN^k}{k!}.\] The forced system has the solution \[x(t)=\mathrm{e}^{A(t-t_0)}x_0+ \int_{t_0}^t\mathrm{e}^{A(t-s)}b(s)\,ds.\] If \(b\) is constant and \(A\) is invertible, \[x_\star=-A^{-1}b,\qquad x(t)=x_\star+\mathrm{e}^{A(t-t_0)}(x_0-x_\star).\] \bookmarksetup{startatroot} \chapter{Higher-Order Scalar Linear ODEs}\label{higher-order-scalar-linear-odes} \index{higher-order scalar linear odes@Higher-Order Scalar Linear ODEs} Consider \[a_p(t)y^{(p)}+\cdots+a_0(t)y=g(t).\] For constant coefficients, the characteristic polynomial is \[p(\lambda)=a_p\lambda^p+\cdots+a_0.\] A real root \(\lambda\) of multiplicity \(m\) gives \[\mathrm{e}^{\lambda t},\quad t\mathrm{e}^{\lambda t},\quad\ldots, \quad t^{m-1}\mathrm{e}^{\lambda t}.\] A conjugate pair \(\alpha\pm i\beta\) gives \[\mathrm{e}^{\alpha t}\cos\beta t,\qquad \mathrm{e}^{\alpha t}\sin\beta t.\] The superposition principle is \[L[y]=g,\qquad y=y_h+y_p.\] \bookmarksetup{startatroot} \chapter{Autonomous Systems and Phase Space}\label{autonomous-systems-and-phase-space} \index{autonomous systems and phase space@Autonomous Systems and Phase Space} For \(x'=f(x)\), an equilibrium \(x_\star\) satisfies \[f(x_\star)=0.\] A set \(S\) is positively invariant if \(x_0\in S\) implies \(\varphi_t(x_0)\in S\) for every \(t\ge0\). A first integral satisfies \[\frac{d}{dt}I(x(t))=\nabla I(x)^Tf(x)=0.\] \bookmarksetup{startatroot} \chapter{Stability of Linear Systems}\label{stability-of-linear-systems} \index{stability of linear systems@Stability of Linear Systems} For \(x'=Ax\), \[\text{asymptotically/exponentially stable} \quad\Longleftrightarrow\quad \operatorname{Re}\lambda<0 \quad\text{for every }\lambda\in\operatorname{spec}(A).\] If some eigenvalue has positive real part, the system is unstable. If all eigenvalues have nonpositive real part, stability requires the eigenvalues on the imaginary axis to be semisimple. \bookmarksetup{startatroot} \chapter{Gradient, Conservative, and Hamiltonian Systems}\label{gradient-conservative-and-hamiltonian-systems} \index{gradient conservative and hamiltonian systems@Gradient, Conservative, and Hamiltonian Systems} For a gradient flow, \[x'=-\nabla V(x),\qquad \frac{dV}{dt}=-\|\nabla V(x)\|^2\le0.\] A Hamiltonian system has the form \[z'=J\nabla H(z),\qquad J=\begin{pmatrix}0&I\\-I&0\end{pmatrix}.\] If \(H\) is autonomous, then \(dH/dt=0\). The flow is symplectic: \[(D\varphi_t)^TJ(D\varphi_t)=J.\] \bookmarksetup{startatroot} \chapter{Boundary-Value Problems}\label{boundary-value-problems} \index{boundary-value problems@Boundary-Value Problems} A boundary-value problem (BVP) is \[x'=f(t,x),\qquad B(x(a),x(b))=0.\] The linear case is \[x'=A(t)x+b(t),\qquad B_ax(a)+B_bx(b)=c.\] Shooting parameterizes the missing initial data and solves a residual equation \(F(s)=0\). Multiple shooting uses several IVPs linked by continuity conditions. Collocation enforces the ODE at selected points of a finite-dimensional approximation. \bookmarksetup{startatroot} \chapter{Final Concept Map}\label{final-concept-map} \index{final concept map@Final Concept Map} \[\boxed{x'=f(t,x)\Longrightarrow \{\text{existence, uniqueness, flow, sensitivity, stability, numerics}\}.}\] \[\boxed{x'=A(t)x+b(t)\Longrightarrow \{\Phi(t,s),\text{ Liouville, variation of constants}\}.}\] \[\boxed{x'=Ax\Longrightarrow \{\mathrm{e}^{At},\operatorname{spec}(A),\text{ Jordan form, linear stability}\}.}\] \[\boxed{x'=f(x),\ f(x_\star)=0\Longrightarrow \{\text{equilibrium, invariants, autonomous dynamics}\}.}\] %---------------------------------------------------------------------------------------- % ANALYTICAL INDEX %---------------------------------------------------------------------------------------- \cleardoublepage \renewcommand{\indexname}{Analytical Index} \begingroup \raggedright \printindex \endgroup \end{document}