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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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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.