Used Notation

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, {xR|x>0}
N,N0 set of all natural numbers, N{0}
k appropriate (vector, function, etc.) Lk-norm
x,x,X,X scalar, vector, random scalar, random vector variable, respectively
u,v scalar (dot) product between vectors u and v
f,f scalar-valued, vector-valued function
f(x) first derivative of a scalar-valued function f with respect to scalar variable x
x˙(t),x˙(t) derivative of a scalar, vector variable (dependent on time) with respect to time t
f(m)(x),f(m)(x) m-th derivatives of a scalar-valued and vector-valued function with respect to a scalar variable x, respectively
f,f,yf(x,y) gradient of f, Jacobian matrix of f, Jacobian matrix of f with respect to the 2nd vector argument y
J Jacobian matrix of a corresponding function f
Rex,Imx real and imaginary parts of complex variable x, respectively
[n] {iN|in}={1,2,,n1,n}
X,XC state space of a dynamical system, often Rn, state space of a semidynamical system, often C([τ,0],X)
T algebraic structure capturing time for (semi)dynamical system, often (R,+)
M monodromy matrix of a cycle
xt DDE history segment, xt(s):=x(t+s), τs0
τif Jacobian matrix of f with respect to the state variable corresponding to delay τi
Inc increment function associated with a given numerical ODE solver
err,le,lte global, local and local truncation error of a numerical integration method, respectively
M,I[M] discretization of a given section of 2-parameter plane to a grid, its 2D indices
M,C manifold, typically of a cycle, curve
(a,b interval from a to b open on the left and open or closed on the right
Φ phase condition for a collocation problem
S,A solution operator, infinitesimal generator of a linearized DDE
φ evolution operator of a (semi)dynamical system
N(μ,σ2) normal distribution with expected value μ and variance σ2
W(t) Wiener process
B Borel σ-algebra