Mathematics Stack Exchange
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Questions tagged [steady-state]

For questions about steady states in systems theory, which are unchanging in time.

In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time.

In continuous time, this means that for those properties of the system, the partial derivative with respect to time is zero. In discrete time, it means that the first difference of each property is zero and remains so:

141 questions
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Creating a steady state vector

I'm confused on where the intuition came from to divide $w$ by the sum of its entries to find $q$. I don't really see the relation from the sum of its entries with "every solution being a multiple of the solution $w$".
stumped
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Steady state of diffusion-advection on the torus

Let $P$ be a positive scalar function and $\mathbf{v}(\mathbf{x})$ is an assigned smooth vector field. The quantity $P(t,\mathbf{x})$ evolves according to a transport equation of the kind $$ \partial_t P(\mathbf{x},t) = -\nabla \cdot […
Quillo
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$P_{\infty}=R$ in steady-state Kalman Filter when transition and observation matrix = $I$

Considering a simple Kalman Filter State update equation $x_t = x_{t-1} + w_t, w_t\sim N(0,Q)$ Observation equation $z_t = x_{t} + v_t, v_t\sim N(0,R)$ I'm curious, under what conditions, will we have steady-state $P_{\infty} = R$? In other words…
skf
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Why do we require positive recurrence for a Markov chain to have steady states?

Theorem 4.1 of the book Introduction to Probability Models (10th edition) by Sheldon Ross states that an "irreducible ergodic" Markov chain has limiting probabilities that exist. And ergodic further means that it must be positive recurrent and…
Rohit Pandey
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Find steady state of AIDS epidemic model

AIDS epidemic in a homosexual population The following diagram shows the AIDS epidemic in a homosexual population: Then the model can be described by $$ \begin{gathered} d X / d t=B-\mu X-\lambda c X \\ d Y / d t=\lambda c X-(v+\mu) Y \\ d A / d…
WhyMeasureTheory
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Steady states of $u_t= u_{xx}+\pi^2u$

I just put the following one-dimensional reaction-diffusion equation in Mathematica: $$u_t= u_{xx}+au$$ with $\Omega=(0,1)$ with Dirichlet boundary conditions. When $a<9$, no matter the initial condition I choose, the solution decays to $0$: (time…
David Lingard
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Alexandre Chorin steady state solution of Navier Stokes

I'm trying to learn fluid dynamics and numerical methods for solving the NS equations by reading research papers. But a lot of them are complicated so I kept searching for older references that look easier to understand. I found papers I like by…
betafractal
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Degenerate perturbation theory to nonlinear equation

I want to use perturbation theory to find the steady-state solution to the following nonlinear equation: $$ x_i\left(\sum_{j=1}^Nx_j^2\right)-a x_i + \epsilon \sum_{j\neq i}^N J_{ij}x_j=0, $$ where $i=1\cdots N$ and $\epsilon$ is a small parameters.…
Sean
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Steady State Temperature Distribution in a Rectangular Plate

We need to solve the following : $$ \nabla^2=0, 0\leq \leq, 0\leq y\leq b $$ satisfying the boundary conditions $$ (0,y)=0, 0\leq y\leq b \\ (,0)=(,)=0, 0\leq \leq \\ _x(a,)=T\sin^3(πy/a) $$ I proceeded as following: Page1 Page2 Page3 I'm pretty…
Ankit Kumar
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Stability of steady states using the Jacobian (linear approximation)

I'm studying the stability of steady states by means of the eigenvalues of $J$. So far the criteria is this: All eigenvalues $\gt 0 \implies$ unstable All eigenvalues $\lt 0 \implies$ stable. In 2D: one eigenvalue $\gt 0$, and another $\lt 0…
Cate
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When does a PDE have a steady-state solution?

I just started studying different types of PDEs and solving them with various boundary and initial conditions. Generally, when working on class assignments the professors will somewhat lead us to the answer by breaking a single question (solving a…
Mjoseph
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Natural-Forced and Transient-SteadyState pairs of solutions

We have the following circuit, where, $u(0)=V_{0}$. The ode that describes this circuit that has $V_{s}$ as input and the voltage $u(t)$ of the capacitor as output is the following: $\dot{u} + \tau u = \tau V_{s}$, where $\tau=\frac{1}{RC}$. If I…
Alex Kps Bdc
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How can a Markov chain have more than one but a finite amount of stationary distributions?

Here's my understanding of it: Assume we have an $n\times n$ stochastic matrix $P$ that represents our Markov chain such that $x$ and $y$ are stationary distributions for $P$. Then $P(x) = x$ $P(y) = y$ $P(ax+by) = P(ax) + P(by) = aP(x) + bP(y)$…
user665214
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Find steady state solution of heat equation when thermal conductivity depends on x

I am given the heat equation with the following boundary conditions: $$u,_x = (K_0(x)u,_x),_x$$ $$u(t,x=0) = 0$$ $$u(t,x=1) = 1$$ Where $$K_0 = \frac{e^x}{cos(x)}$$ In a steady state solution $u,_t$ goes to 0 and the boundary conditions become…
user2569905
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Mathematical biology problem involving discrete single-species models

The question is: Consider the nonlinear equation for population growth $$N_{t+1}=\frac{rN_t}{1+aN_t}$$ where $r>0$ is the basic reproduction rate, and $a>0$ is a constant. Part (a) Does this equation exhibit over,under or exact-compensation? My…
user395952
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