Questions tagged [steady-state]

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

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35 views

Asymptotic behavior of $u_t= u_{xx}+au$

Consider the following one-dimensional reaction-diffusion equation: $$u_t= u_{xx}+au$$ on $\Omega=(0,1)$ with Dirichlet boundary conditions with $a>0$ and a nonnegative initial condition $u_0$. If $...
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1answer
69 views

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 ...
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1answer
41 views

How many steady state solutions does $u_t=d\Delta u+au-bu^2$ possess?

Consider the following evolution equation $$u_t=d\Delta u+au-bu^2$$ in a bounded and regular open subset $\Omega$ of $\mathbb{R}^N$, with smooth initial conditions $u_0\geq 0$ and homogeneous ...
2
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1answer
33 views

How to model room heat (with pump) problem

I'm trying to model a problem where a pump is being used to remove heat from a room at a rate of $R$. This pump uses an automated system that only activates when the outside temperature reaches some ...
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0answers
17 views

How to check that a steady-state solution to a wave scattering problem is the one we wanted?

Consider a 2D wave equation $$\partial_x^2 f(x,y,t)+\partial_y^2 f(x,y,t)=\partial_t^2 f(x,y,t)\tag1$$ on $\mathbb R^2$. The solution $f$ is to be bounded at $|\vec r|\to\infty$ (where $\vec r=(x\;y)...
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1answer
27 views

Equilibrium Solutions - Seeing Algebraically why starting at an equilibrium point leads to a constant solution

We define an equilibrium point as a point $\mathbf{a}\in\mathbb R^n$ such that for the dynamical system $ \frac{d\bf x}{dt}=\bf{F(x)}$ we have $\bf{F(a)}=\mathbf{0}$. Claim: A solution $\mathbf{x}(...
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0answers
30 views

Solving differential equation involving anti-symmetric part

I am looking for the steady state solution of a Fokker-Planck equation. The process involves a constant drift and position-dependent removal/insertion, thus leading to non-zero a steady state ...
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1answer
31 views

Finding stationary distribution given Generator matrix

From my Markov Chain, I have a generator matrix $G$= \begin{bmatrix} -20 & 20 & 0 \\ 12 & -32 & 20 \\ 0 & 12 & -12 \end{bmatrix} and I wish to find its ...
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1answer
35 views

Why am I getting non-sinusoidal output for sinusoidal input in mass-spring system?

A mass spring system is represented by following transfer function $$ H(s) = \frac{s^{2} + 0.1s + 10}{s^{4} + 0.2s^{3} + 20s^{2}} $$ but for sinusoidal input I am getting non sinusoidal output as ...
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0answers
28 views

conditions for asymptotic comparison to hold

I have the following simple dynamical system: \begin{align} x_1' &= a - f(x_2)x_1\\ x_2' &= bx_1 - cx_2, \end{align} where all parameters and initial conditions are positive. $f(x_2)$ is a ...
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1answer
56 views

Solve for steady state [closed]

I am trying to solve for the steady state (in the context of a DSGE economic model) and one of the equations is reffering to capital accumulation. Particularly: $$K_t=(1 - \Delta)K_{t-1} + \left( 1 - ...
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1answer
75 views

What would be the transition for ten coins?

There are ten coins and a move is made up of flipping any three adjacent coins: H H T T H T H T H T -> H T H H H T H T H T (flip: 2,3,4). How can this transition be represented? / EG: There are 102 ...
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1answer
26 views

Steady States and fractional Population

If I'm assuming that I have a population of size $N(t)$ that is growing, can my steady states be fractions? I'm quite confused because how can a population be a fraction? Note that the differential ...
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1answer
53 views

Why is the steady state error in this system incorrect?

(Note: I'm currently learning about this, but I'm having a hard time understanding why this system I am modelling is giving unexpected results when finding the steady state error) I have a system ...
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1answer
79 views

Given a steady state vector is it possible to calculate the corresponding transition (probability) matrix

Knowing that there is a probability matrix M (where all columns add to 1) which when applied to a given vector P produces the same vector P, what is the best solution to find M? I can get my head ...
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0answers
11 views

steady states stability of spruce budworm model

How do i find the stability of these steady states qualitatively and numerically (ludwig's spruce budworm model): $\frac{dS}{dt}=\gamma S(1−\frac{S\alpha}{\beta E})$ $\frac{dE}{dt}=\delta E(1−\...
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1answer
30 views

Finding the steady states of a quadratic ODE

How would I go about finding the steady states I know I need to set $\frac{dx}{dt}=0$ but then I'm struggling with the next step.
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1answer
47 views

Help using eigenvectors to solve Markov chain

I took Linear Algebra last semester and when learning about Markov Chains in my statistics class, I wanted to use eigenvectors/eigenvalues to find the steady-state vector rather than just using ...
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2answers
60 views

How do the steady states change as we change the parameter?

Consider the following ODE $$ \frac{dx}{dt} = x \left(1-\frac{x}{m} - \frac{a}{1+x}\right), $$ where $a$ is a bifurcation parameter and $a\in(0,\infty)$ a positive constant. How do you find the ...
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1answer
28 views

Markov chain. Is steady state a scaled eigenvector of transition probability matrix

So suppose we have transition matrix P for a Markov chain and suppose it satisfies the relevant criteria so that $$ \lim_{n\rightarrow \infty} P^{(n)} = \pi $$ is well behaved and is some steady ...
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0answers
57 views

Finding a steady-state solution to a PDE

Problem I'm Facing: If it exists, find the steady-state temperature solution for the PDE $$u_t = u_{xx} + 1$$ with the boundary conditions $$u_x (0,t) = 1 \;\;\;\;\; u(L,t) = 1-L$$ This has ...
1
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1answer
54 views

Does $x_{t} = 1-x_{t-1}$ have a stable steady state solution?

At steady state, $x = x_{t} = x_{t-1}$. So I can solve for the steady state value of $x=0.5$. The general rule of determining the stability of the steady state is that the $|\text{slope}|<1$. But ...
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0answers
55 views

Solving a question with more unknowns than equations

I am not sure if I am thick in the head or not, but I am trying to replicate this paper: https://www.nber.org/papers/w19086.pdf. The equations characterising the equilibrium are given on Page $22$. ...
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0answers
69 views

When does any solution of an i.v.p. converges to some steady state?

Preliminaries Let $A = [0, 1]^N$. Consider a dynamical system $\dot{x} = f(x)$, where $x = x(t) : \mathbb{R} \to \mathbb{R}^N$ and $f : A \to \mathbb{R}^N$, where each $f_i$ is in the class $\...
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0answers
32 views

Properties of a mapping between two differential equations

Let $A\in\mathbb{R}^{n\times n}$ be a nonsingular matrix, $b\in\mathbb{R}^n$, and $f\colon\mathbb{R}\to\mathbb{R}$ be an arbitrary function. Consider the following differential equation $$\tag{$\ast$}\...
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1answer
71 views

On showing the existence of a Markov chain's steady state distribution

I want to show that the Markov chain with such transition matrices written below has a unique stationary distribution $\mu$. For a space dimension of 6 : $$\begin{equation} \Pi^{(6)} = \begin{...
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1answer
54 views

Limit of a solution to a differential equation is a steady state.

Suppose we have an initial value problem $$\dot{x}=f(x),$$ $$x(0)=x_0$$ where $f\in C^1(E)$ for some open $E$. Moreover, suppose we have a solution $x(t)$ such that $$\underset{t\rightarrow\infty}{\...
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0answers
23 views

Queuing theory for may task in univ

A gas station only has one pump for refueling the Pertamax type. The arrival of Pertamax-fueled cars to the gas station follows the process Poisson with an arrival rate of 15 cars / hour. However, ...
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0answers
83 views

Solution to the steady state diffusion equation

In the frame of reference moving in the zdirection , with a velocity, the steady state diffusion equation has te formL $\nabla^2u + \frac{2}{l}\frac{\partial u}{\partial z} = 0$ The solution to ...
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54 views

Lotka-Volterra equation (predator-prey): given any initial condition, how can one know the steady-state behavior?

I am trying to find out if it is possible to determine the steady state behavior of the predator-prey system defined by the nonlinear equations: $$\begin{eqnarray} \frac{dx}{dt}&=&ax-bxy \\ ...
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0answers
74 views

Finding relationship between input and output

I am just trying to figure out what key words I should look up to help me with the following problem. I have a control system to control a PWM motor and a sensor to detect the motors frequency for ...
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1answer
82 views

solve A=7A+2B+1C, B=2A+6B+2C, C=1A+2B+7C, A+B+C=1

The answer is A=B=C= 1/3 I can't seem to finish the logic below and I wonder if there is a faster way that anyone can suggest please. This is related to steady state probabilities and I know I could ...
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2answers
285 views

Heat partial differential equation with Neumann boundary conditions.

Consider the following PDE problem for $u = u(t, x), \ 0 \leq x \leq 4, \ t > 0$ \begin{align} u_{t} &= u_{xx} \\ u_{x}(t,0) &= u_{x}(t,4) \\ &= -2 \\ u(0,x) &= \begin{cases} 0 &...
1
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1answer
43 views

Conditions on the heat source so that a steady state solutions exists.

Consider the heat equation, $$\frac{\partial u}{\partial t} = \frac{\partial }{\partial x} \left(K_0(x) \frac{\partial u}{\partial x}\right)+Q(t)$$ $$u(t,x=1)=0$$ $$u(t,x=2)=1$$ where $K_{0}=x^2$ ...
3
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1answer
42 views

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 $u,_t$ ...
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0answers
54 views

Long-term ball distribution in urns

Suppose, we have $k$ urns each starting with $n$ balls (all balls are of the same color). At each step, we draw balls from each urn, where each ball could be drawn independently at random with ...
0
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1answer
67 views

How does state transition matrix indicate time-varying system, but $A$ matrix is constant?

When I determine whether a linear dynamical system is time-varying or not by looking at the state transition matrix, I get confused by the following: Normally, I look at the $A$ matrix to determine ...
1
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1answer
68 views

Differential equation Steady state solution

Hi I am looking for the steady state solution (stationnary solution independent of t when t grow to infinity) of the following equation: $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\...
2
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1answer
369 views

Finding steady-state vector for stochastic matrix P

Let $P$ be a stochastic matrix and $E$ the $n\times n$ identity matrix. Assume that $P^q$ = $E$ for some integer $q \geq 2$, but $P \neq E$. Find a steady-state vector for $P$. Give an example for ...
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0answers
32 views

Smallest number of states to escape

There are $100$ people, each of whom own a number between $1$ and $100$ inclusive. Known that one number appear for more than $50$ times, they'll go to a switch in a random order. The last person ...
2
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1answer
227 views

l'Hopital's rule for 2 variables to compute Jacobian matrix

I have a system of three ODEs and I have computed the Jacobian matrix. One of the steady states is (0,0,0) and I am trying to linearize the system around this ...
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2answers
330 views

Find the steady-state solution from two given differential equations

Would anyone be able to show me how to solve the question in the link? Or give me guidance on where to begin? I'm not sure what is meant by finding the steady-state solution, or how I would go about ...
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2answers
32 views

Which steady state will be attained by these system of equations?

I have these two coupled ODEs... $\dfrac{dc_1}{dt} =- k_1 c_1 c_2 + k_2c_2$ $\dfrac{dc_2}{dt} = Y k_1 c_1 c_2 - k_2c_2$ $c_1$ has two steady states, $\dfrac{k_2}{k_1}$ and $\dfrac{k_2}{Yk_1}$. ...
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2answers
738 views

Designing a Markov chain given its steady state probabilities

As explained in https://en.wikipedia.org/wiki/Markov_chain, for a three state Markov Chain with the transition matrix given as $P = \begin{bmatrix} 0.9 & 0.075 & 0.025 \\ 0.15 & 0.8 & ...
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0answers
37 views

Physical Interpretation of Steady State or Equilibrium Temperature.

In my PDE book, in order to solve nonhomogeneous pde's we start off (with a nonhomogeneous heat equation) by obtaining an equilibrium temperature $u_E(x)$. It's time independent, and must satisfy $$...
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1answer
87 views

Mathematical Biology: Continuous Single-Species Models

Consider the modification to the Malthusian equation $$\frac{dN}{dt}= rS(N)N ,$$ where $r > 0 $ is the per capita growth rate, and $S(N)$ is a survival fraction. For some organisms, finding a mate ...
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0answers
54 views

Interpreting the Steady State Solution of a BVP

I have a question about a Robin Boundary Condition on a BVP. The BVP is given by: $$\begin{cases} u_t=u_{xx} & x\in(0,1), t>0\\ u(0,t)-u_x(0,t) = 20 & u(1,t)=10\\ u(x,0)=e^x & x\in(...
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0answers
65 views

Solving a symbolic nonlinear system

I have a system of 5 non linear ODEs and I am trying to find the steady states of the system by solving the following. The variables are a,b,c,d,e. ...
0
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1answer
254 views

how to prove that in a finite markov chain, a left eigenvector of eigenvalue 1 is a steady-state distribution?

The pic linked below is a part of the notes in MIT discrete-time stochastic process open course, I can't understand what was the intuitive idea that the theorem is trying to prove, nor how it ...
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0answers
27 views

Does a steady state exist for a looped flow network which allows negative flow?

I am modelling something with a flow network. There are a number of layers to the network, each layer has a different number of nodes. Each node receives some flow from all the nodes in the previous ...