:::{.content-visible when-format="pdf"}\HeaderUnnumbered::::::{.hidden}{{< include mathematics.qmd >}}:::# Used Notation {#sec-used-notation .unnumbered}For easier orientation in the text, we include a basic list of used notation throughout the text:| Symbol | Meaning ||-------:|:--------||$\R, \R^+$ | set of all real numbers, $\set{x \in \R \divider x > 0}$ ||$\N, \N_0$ | set of all natural numbers, $\N \cup \set{0}$ ||$\norm{\cdot}_k$ | appropriate (vector, function, etc.) $L^k$-norm ||$x,\vi x,\rnd X, \vr X$ | scalar, vector, random scalar, random vector variable, respectively ||$\scal {\vi u} {\vi v}$ | scalar (dot) product between vectors $\vi u$ and $\vi v$ ||$f, \vi f$ | scalar-valued, vector-valued function ||$f(x)'$ | first derivative of a scalar-valued function $f$ with respect to scalar variable $x$ ||$\dot{x}(t), \dvi x(t)$ | derivative of a scalar, vector variable (dependent on time) with respect to time $t$||$f^{(m)}(x), \vi f^{(m)}(x)$ | $m$-th derivatives of a scalar-valued and vector-valued function with respect to a scalar variable $x$, respectively ||$\jacobi f, \jacobi \vi f, \jacobi_{\vi y} \vi f(\vi x, \vi y)$ | gradient of $f$, Jacobian matrix of $\vi f$, Jacobian matrix of $\vi f$ with respect to the 2nd vector argument $\vi y$ ||$\Jacobi$ | Jacobian matrix of a corresponding function $\vi f$ ||$\reOf{x}, \imOf{x}$ | real and imaginary parts of complex variable $x$, respectively ||$\oneToN{n}$ | $\set{i \in \N | i \leq n} = \set{1,2,\dots,n-1, n}$ ||$\stateSpace, \contStateSpace$ | state space of a dynamical system, often $\R^n$, state space of a semidynamical system, often $\contf{[-\tau, 0], \stateSpace}{}$ ||$\timeSet$ | algebraic structure capturing time for (semi)dynamical system, often $(\R, +)$ ||$\vi M$ | monodromy matrix of a cycle ||$\vi x_t$ | DDE history segment, $\vi x_t(s) \letDef \vi x(t + s)$, $-\tau \leq s \leq 0$ ||$\jacobi_{\tau_i} \vi f$ | Jacobian matrix of $\vi f$ with respect to the state variable corresponding to delay $\tau_i$ ||$\incrementFunc$ | increment function associated with a given numerical ODE solver ||$\globErr, \locErr, \locTrErr$ | global, local and local truncation error of a numerical integration method, respectively ||$\meshGrid, \indicesOf{\meshGrid}$ | discretization of a given section of 2-parameter plane to a grid, its 2D indices|$\manifold, \curve$ | manifold, typically of a cycle, curve ||$(a,b\rco$ | interval from $a$ to $b$ open on the left and open or closed on the right ||$\Phi$ | phase condition for a collocation problem ||$\solOp, \infGen$ | solution operator, infinitesimal generator of a linearized DDE ||$\evolOp$ | evolution operator of a (semi)dynamical system ||$\normalD{\mu, \sigma^2}$ | normal distribution with expected value $\mu$ and variance $\sigma^2$ ||$\rW(t)$ | Wiener process ||$\borelAlgebra$ | Borel $\sigma$-algebra |