Questions tagged [steady-state]

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

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Exponential rate of convergence to steady state in the renewal equation

I'm currently working on the paper Invariants and exponential rate of convergence to steady state in the renewal equation and haven't made any progress for weeks and I'm beginning to despair. I know ...
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15 views

Steady-State Distribution

In according to the definition of Steady-State distribution on William Stewart's book: "A steady-state distribution must have all its value strictly positive, so >0 and sum to 1" But ...
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109 views

Stability of the dynamic system

I have a function ${\dot{\varphi } = \gamma - F(\varphi )}$ (where $\varphi$ - is 2${\pi }$-periodic function) and graph of function $F(\varphi)$. So it's needed to research this graph (to find the ...
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33 views

finding steady state continuous time markov chain using matlab

I have a Continuous Time Markov Chain with transition matrix $q$, I'm new to matlab and I want to find the steady state vector $P$ by solving these 2 equations : $$P\cdot Q=0$$ $$\sum Pi=1$$ lets say $...
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1answer
40 views

Find a state-space model for the plant

Consider the plant shown in the figure,plant. Find a state-space model for the plant. I know, I have to use the transfer function to get the state-state model. here I have a confusion about the ...
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2answers
28 views

Why is the Steady State Response described as steady state despite being multiplied to a negative exponential?

I'm evaluating Newton's Temperature Model $$dT/dt = k(T_e-T)$$ to find the response of the system.$T_e$ and $T$ are both functions of t. The response evaluates to $$T=e^{-kt}\int e^{ks} T_e(s)ds + ...
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1answer
37 views

Sequential dynamical system

I would like to understand the following example based on the following definition Definition: Alternatively, one can choose to update the states of the variables according to some fixed update order, ...
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1answer
79 views

Solving Differential Equations for Steady States

I have been looking at the following equations in an article* and wanted to know how the $E(∞)$ was derived. By substituting $\gamma E$ for $A$ and factorising it is easy to see how $E(∞)$ can equal 0 ...
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1answer
38 views

Equilibrium point stability given the derivative

Given a system described by the following equation: $$y'' + y'^4 + y'^2*u + y^3 = 0 $$ where $y(0) = 0$ and $ y'(0) \neq 0 $ , what is the stability of the equilibrium point? The eq. point is $0$ and $...
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41 views

Steady state error of Nonunity-feeback Systems

I am reading the book "Autotamic Control Systems" by Farid Golnaraghi and Benjamin C.Kuo, Tenth Edition. In the book: Figure 1 Consider the nonunity-feedback system above, where r(t) is ...
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52 views

Proving that non-absorbing Markov States have steady state probability of $0$

Suppose that I have a Markov chain that has absorbing states. Since there are absorbing states, lets group the Markov matrix into four blocks: the submatrix all states in the absorbing region(s) $A$, ...
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107 views

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 $$ \...
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20 views

What would be the steady state of x in following system X -k-> 2X, 2X -k*v(t)-> ∅ when v(t) = vo (const)?

I have a system of a mass action type. $\require{AMScd}$ \begin{CD} X @>{\text{$k$}}>> 2X, 2X @>{\text{$k\cdot v(t)$}}>> ∅ \end{CD} When $v(t) = v_0(const)$, what is the stable ...
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11 views

Log linearisation without linearising: formal correctness

For the Real Business Cycle model in macroeconomics we want to derive an equation for the deviation of an endogenous variable from steady state dependent on the changes of other variables from steady ...
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14 views

Interpreting the reciprocal of the reproductive ratio, $R_0$

$R_0$ is the average number of secondary cases arising from a single infectious individual in a fully susceptible population. In many of the compartmental models for epidemiology, the parameter $\...
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23 views

Stability of steady states of non linear difference equations.

I have a question about the conditions for stability of a steady state solution to a non linear differential equation. To a certain extent the argument given in the book I'm following is rigourous, ...
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6 views

DSGE parameters

I will be very thankful if you can help me with following problem: I want to estimate DSGE model based on existing literature. Now I have to use the real economic data to link the model with reality. ...
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1answer
35 views

What is the steady state probability?

A lot of board games involve rolling dice and moving around a cyclical board. Monolopy is the most common example. On the 16 position board below, the player’s piece was on the bottom row as depicted ...
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1answer
29 views

Board game: steady state probability of being in starting location (GO)

Considering a board game which has 20 positions only. There are 2 fair dice that can be rolled and the player moves in a clockwise position over the board positions. Considering that the dice are fair ...
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42 views

Application of PDE for solving steady state heat problem.

A thin rectangular homogeneous thermally conducting plate occupies the region $0 \leq x \leq a$, $0 \leq y \leq b$. The edge $y = 0$ is held at temperature $Tx(x − a)$, where T is a constant and the ...
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24 views

I could use some direction or help to see if I am on the right path for this problem.

The problem is to solve the PDE: $$ \frac{\partial^2 u}{\partial t^{2}} - \frac{\partial}{\partial x}\left(K(x)\frac{\partial u}{\partial x}\right) $$ subject to the following boundary conditions:...
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10 views

Need some intuition on terms used in solving dynamic systems: explosive, bounded, stable, stochastic equilibrium, …

E.g. What is an explosive equilibrium? In what sense it it an equilibrium, as it is, I guess, not stable? What is a bounded equilibrium? Is it the same as a stable equilibrium? What is a non-bounded (...
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12 views

What's the difference between indeterminacy and explosiveness in the context of dynamic systems?

One important question to ask is if the model has as unique stable (asymptotically stationary) solution (determinacy) or multiple solutions (indeterminacy). But what's the difference between ...
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5 views

What are predetermined variables in the context of dynamic systems with expectations feedback?

The techniques to solve the determinacy of a dynamic system (= DSGE model in economics), does this branch of maths have a name? E.g. for non-rational expectations, this technique is named by Evans, G....
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11 views

Control problem steady state

How can i solve this problem ? Find frequency $w_0$ such that $A_y <= 0.1$ for $w>w_0$ and $y_{ss}$ is the steady state solution. The function $G(s)$ is the transfer function and the input ...
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73 views

How do you find the steady-state proportions of a matrix

I need to find the steady-state proportions of a given 3x3 matrix that explicitly does not have a steady state. I would normally solve for the steady-state by using 1 as an eigenvalue and solving for ...
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9 views

Some insight on Linear Stability Analysis.

so I'm trying to analyse a model of differential equations using linear stability analysis, and I'm slightly confused about some of the parameters used here. For example, let's say I use the ...
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1answer
35 views

Time to reach equilibrium in convection-diffusion

I have a convection-diffusion PDE in the form of: $$\frac{{\partial x}}{{\partial t}} = w\frac{{{\partial ^2}x}}{{\partial {z^2}}} - \frac{{\partial x}}{{\partial z}}$$ Assuming I know the initial ...
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13 views

Finding the half-life of the process by which $x_{t} = ax_{t-1} + c - d$ approaches its limit

We are given the dynamic system $x_{t} = ax_{t-1} + c - d$, where $0 \leq a \leq 1$, $c > 0$, and $d > 0$, which has a steady state where $x_{t}=\frac{c - d}{1-a}$, defined when $a \neq 1$. ...
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2answers
434 views

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) ...
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1answer
40 views

Steady state of a discrete difference equation

I've encountered this Theorem in my mathematical biology notes: $$\bar{𝑥}\text{ is a stable steady state of }f(x_n) = x_{n+1}\text{ iff }|f^{'}(\bar{𝑥})|<1.$$ The definition of a steady state ...
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1answer
39 views

Phase diagram contradicts analytical results

Consider the following dynamic system \begin{align} f(x,y) = \left(\frac{1}{4}\left(\frac{1}{2y} - x\right), \frac{10y^2 - 12xy + 3}{24y - 16x}\right). \end{align} There exists a steady state at $(x_\...
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1answer
42 views

Steady state distributions

Suppose we have a discrete-state discrete-time Markov Chain with n states. We know that this Markov chain has a unique steady-state distribution. If you additionally know that the transition matrix $P ...
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1answer
174 views

Steady-state probability for a Markov process

Friends, I got stuck with formulating a Markov chain that I just came up with. What I want to do is to obtain a steady-state expression for $b_0$ as a function of $b_k$, in order to plug in the ...
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1answer
49 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
87 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
62 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 ...
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1answer
41 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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28 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
45 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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32 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
289 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
72 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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32 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
158 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
77 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
37 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
194 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
141 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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1answer
35 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.