Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [discrete-time]

For questions related to discrete time. Discrete time views values of variables as occurring at distinct, separate "points in time", or equivalently as being unchanged throughout each non-zero region of time ("time period")—that is, time is viewed as a discrete variable.

Discrete time views values of variables as occurring at distinct, separate "points in time", or equivalently as being unchanged throughout each non-zero region of time ("time period")—that is, time is viewed as a discrete variable.

Thus a non-time variable jumps from one value to another as time moves from one time period to the next. This view of time corresponds to a digital clock that gives a fixed reading of $10:37$ for a while, and then jumps to a new fixed reading of $10:38$, etc.

In this framework, each variable of interest is measured once at each time period. The number of measurements between any two time periods is finite. Measurements are typically made at sequential integer values of the variable "time".

A discrete signal or discrete-time signal is a time series consisting of a sequence of quantities.

For more, check out this link.

161 questions
12
votes
3 answers

What are the lines on a Bifurcation Diagram?

Here is the Bifurcation Diagram for the logistic map $x_{n+1}=rx_{n}(1-x_{n})$: And here is an enlargement of a particular section What are the faded lines that appear in the dark sections of the diagram, and why are they so pronounced?
Plato
  • 2,372
11
votes
4 answers

Determine $f(x)$ knowing $f(x)+f(x+\varepsilon)$

I encountered a problem at work that, in my opinion, has a fundamental mathematical reasoning to determine its solvability. Due to an unwitting software configuration, my associate recorded the audio of our meeting twice, resulting in the audio…
nickh
  • 485
5
votes
2 answers

Invariant curves induce invariant regions in discrete, 2D dynamical systems?

Consider a discrete dynamical system $x_{k+1} = f(x_k)$, where $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$, sufficiently smooth, and let $C \subseteq \mathbb{R}^2$ be an invariant, closed curve in the phase space. By Jordan's theorem, $C$ gives rise…
temo
  • 5,375
4
votes
0 answers

Discrete-time, continuous space heat equation inequality

For $u\left(x, t_{n+1}\right)$ that solves \begin{equation} \frac{u\left(x, t_{n+1}\right)-u\left(x, t_{n}\right)}{\Delta t}=\Delta u\left(x, t_{n+1}\right) \end{equation} with $u\left(x, t_{n+1}\right)=0$ on $\partial \Omega$. I'm trying to show…
usr342678
  • 65
4
votes
2 answers

Girsanov theorem for discrete-time stochastic processes

I am reading Buehler et al. (2022) "Learning to Trade II: Deep Hedging" and the slide on p. 44 states Fun fact: in discrete time, we can change also the volatility of a process by changing measure. I am familiar with continuous-time stochastic…
p.vitzliputzli
  • 141
  • 1
4
votes
1 answer

Discrete-time dynamical systems with variable state space dimensions (or output space dimensions)

I am trying to figure out how to formalize a dynamical system whose state vector can change dimensions from one step to the next. For example, I have a process (a discrete-time dynamical system, if you could call it) that at time step $t=k$ has a…
Brian S.
  • 113
4
votes
1 answer

Khalil-like book on discrete-time nonlinear systems

Is there an equivalent of the Khalil's Nonlinear Systems for discrete-time systems? I am particularly interested in matters of advanced stability analysis, perturbed systems, singular perturbation theory.
shnnnms
  • 343
3
votes
1 answer

Burning number and optimal burning sequences

I am reading about graph burning, which is a deterministic discrete-time process on graphs (Bonato, Anthony, A survey of graph burning, Contrib. Discrete Math. 16, No. 1, 185-197 (2021). ZBL1457.05068.) I have a problem with the definition of the…
Anirudh
  • 43
3
votes
1 answer

Is there always a sequential nature of time-stepping?

Pretend someone is solving a partial differential equation using something like the finite element method. Lets say they are calculating the propagation of a seismic wave through inhomogeneous terrain using a wave equation. If they wanted to start…
Dtron3030
  • 55
3
votes
0 answers

When does an optimal input sequence for a discrete-time system exist?

Suppose an LTI discrete-time system is given by the equations $$ x_{k+1} = Ax_k + Bu_k,\\ y_{k} = Cx_k + Du_k $$ with $x_k\in\mathbb{R}^{m}$, $y_k\in\mathbb{R}^{n}$ and $u_k\in\mathbb{R}^{p}$ and $\rho(A) < 1$ and $x_0 = 0$. Assume…
Benjamin Tennyson
  • 103
3
votes
0 answers

Determine the height $h(t)$ of the water level as a function of time $t$

Water flows out of a filled cylinder of base area $G$ through a small hole at the bottom. Let $h_0$ denote the initial height of the water at time $t_0 = 0$. Determine the height $h(t)$ of the water level as a function of time $t$. Determine the…
Euler007
  • 131
3
votes
1 answer

Converting between continuous and discrete-time stochastic processes

I'm reading through Dixit and Pindyck's Investment under Uncertainty, where I found the following passage. First, they introduce the Ornstein-Uhlenbeck process $$ dx = \eta (x - \bar{x})dt + \sigma dz\ , $$ and then claim that the above equation is…
Anthony
  • 833
  • 5
  • 17
3
votes
0 answers

Parseval's Identity Application

In this video, it is stated that Parseval's Identity "is how we go from discrete to continuous." However, I have not been able to find any material that expands on this use of Parseval's Identity. I have discovered that it is the inner product…
user10478
  • 2,118
3
votes
1 answer

Input in Differential Equations and Difference Equations

According to my understanding: In differential equations, the input is defined as the entire inhomogeneous part of the equation as it would be written on the right hand side. For example, the input of $y'' + 2y' - t + te^t + y = 0;\ y(0) = 1;\…
user10478
  • 2,118
3
votes
0 answers

What is the name of this interval operation?

I have encountered an basic operation on a set of intervals which I think should have a common name and be in textbooks. Unfortunately, my searches have turned up empty. Imagine you have a set of intervals $I$. From this set, you want to calculate a…
Hans van der Laan
  • 131
1
2 3
…
10 11