In dynamical systems, the motion of a particle in some geometric space, governed by some time dependent rules, is studied. The process can be discrete (where the particle jumps from point to point) or continuous (where the particle follows a trajectory). Dynamical systems is used in mathematical ...

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9 views

Dynamical Systems: Disease model, what happens to variable $m$?

In the diagram it shows that people can die from other causes at a rate $m$, however in the equations the $m$ and the variable $M_a$ disappear. Is there a mathematical reason for this to happen?
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1answer
28 views

How to handle the noise covariance matrices in a basic Kalman Filter setup?

I've recently been trying to learn about Kalman Filters; most explanations of the Kalman Filter confuse me in what is known / unknown. I'll assume the following setup: \begin{equation} \begin{split} ...
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1answer
118 views

How to model a system for tracking a person using kalman filter?

I need to model a system for human motion. The following link shows for to build a system for a plane. I am currently reading the documentation for a kalman filter library ...
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33 views

a dynamical systems view of the central limit theorem?

I have seen many heuristic discussions of the classical central limit theorem speak of the normal distribution (or any of the "stable distributions") as an "attractor" in the space of probability ...
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1answer
22 views

Non-atomic, ergodic measure which is left and right shift invariant.

Given a one-sided shift space, say $X = \prod\limits_{n=1}^\infty \mathbb Z_2$. Denote the left shift by $T$: $T(x_1 x_2 x_3\cdots) = x_2x_3 \cdots$. There are lots of examples of $T$-invariant ...
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1answer
10 views

Turing Instabilities

In the solution all partial derivatives are evaluated at the equilibrium point Why does the solution not talk about the fact that the determinant of the Jacobian Matrix=$f_ug_v-f_vg_u$ at the ...
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1answer
24 views

Volume Contraction

I need to determine if this system exhibits volume contraction: $\dot x =yz-x-x^3$ $\dot y =xz-y-y^3$ $\dot z =xy-z-z^3$ My approach is to just calculate the divergence of the vector field F: ...
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1answer
34 views

how to get the probability $p=1−(1−1/N)^{Tma_i}$ of extracting at least one ball in the urn

I am reading supplementary information of the paper Activity driven modeling of dynamic networks. It analogys the number of out degree of a activity node by Polya urns problem: it will equal to ...
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13 views

Periodic point of dynamical system

Hi please help me someone with the proof: We have a function $f:\mathbb{R}\longrightarrow\mathbb{R}$ continous and invertible, discrete dynamical system is given by $x_{n+1}=f(x_n)$ (a): prove that ...
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1answer
58 views

Notation question in linear estimators (Kalman filter)

I'm just learning about Kalman filters, and I'm trying to understand some notation. The book that I am reading through sets up a system with the state-space realization: $$\dot{x}(t) = A(t)x(t) + ...
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126 views
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1answer
389 views

Kalman filter and data extrapolation

Context of the situation: I have a system set up that can give me the position of a person in a room. I also have a light that shines on this position. However, the light are lagging behind by 0.300 ...
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26 views

What is the significance of studying the steady state behaviors of a system? What information do steady state models provide us?

For example, http://www.jameslovelock.org/page31.html In this 1983 paper by Lovelock and Watson modeling Daisyworld, in equations (10) through (14), the paper considers the non zero steady state ...
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2answers
113 views

Uniform unboundedness of linear operators

Question: Suppose that $(T_k)_{k=1}^{\infty}$ is a sequence of invertible linear operators on $\mathbb{R}^n$. Suppose that $\forall x \in \mathbb{R}^{n}\setminus \{0\}$, we have $$\lim_{k\to\infty} ...
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3answers
23 views

Reference Request: Parameter Dependent Center Manifold Theorem for ODEs

Suppose we have an $n$-dimensional first order ODE of the form $\frac{dx}{dt}= f_{\mu}(x)$ with $\mu \in \mathbb{R}^k$ a parameter and with an equilibrium at $x=0$ $(f_{\mu}(0) =0)$. For fixed $\mu$ ...
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15 views

Normal modes energy terms

I'm doing a normal modes problem and i'm not sure that i've got the right form for the the potential and the kinetic energies. The problem is that we have a beam of length $l$ and mass $m$ raised ...
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1answer
10 views

Definition of measure-preserving

In the definition of measure-preserving dynamical system, the crucial equation is $$ \mu (T^{-1} (A)) = \mu (A ) . $$ Why is it not the seemingly more natural $$ \mu (T (A)) = \mu (A ) ? $$
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1answer
31 views

Tent Transformation $x\mapsto3\min(x,1-x)$

Suppose I have a Tent Transformation which is defined by: \begin{align*}T(x)=\begin{cases}3x&\text{if $x\le\dfrac12$,}\\3(1-x)&\text{if $x\ge\dfrac12$.}\end{cases}\end{align*} After noticing ...
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1answer
24 views

Proving that $y(t)\to0$ given a dynamical system

Consider a nonlinear system of the form $$\dot{y}(t)=p(y(t)) + u(t)$$ where $$p(q) = a_kq^k+a_{k-1}q^{k-1}+\ldots+a_1q$$ $$u(t) = -\left(\alpha_ky(t)^k+\ldots+\alpha_1y(t)\right)-y(t)$$ with ...
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1answer
12 views

Proving global exponential stability of a perturbed system

Consider the system $$\dot{x} = \left(A + \frac{1}{2\varepsilon}BB^TP\right)x + Dg(t,y),\quad y=Cx,$$ where $g(t,y)$ is continuously differentiable and satisfies $$\Vert g(t,y)\Vert_2 \le k\Vert ...
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19 views

Eigenvalues and speed of convergence

Consider some dynamical system \begin{align} \dot x = f^1(x,y)\\ \dot y = f^2(x,y)\\ \end{align} There exists a fixed point at $E = (\tilde x, \tilde y)$, i.e. $f^1(\tilde x, \tilde y) = f^2(\tilde ...
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26 views

Periodic orbits of a dynamical system [on hold]

I have the system (time dependent) \begin{cases} u' - u(v-a) \\ v' = v(b-u) \end{cases} $a$ and $b$ being positive. I was wondering whether it is possible to apply the Bendixson-Dulac theorem to rule ...
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31 views

Why should $\phi'$ and $\phi''$ be $\mathcal O(1)$?

As Strogatz writes in his book Nonlinear Dynamics And Chaos (p. 64) There are often several ways to nondimensionalize an equation, and the best choice might not be clear at first. Therefore we ...
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1answer
34 views

Eigenvector for a non-linear system

Using the reversibility arguments alone, show that the system $\dot{x}=y$ $\dot{y}=x-x^{2}$ has a homoclinic orbit in the half-plane $x\leq 0$ This is a non-linear system. A ...
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17 views

Using Linear Kalman Filters with a Nonlinear System?

Can you answer these questions I have about using linear Kalman filters and extended Kalman filters with a nonlinear system? 1. Does using a linear Kalman filter mean that I must have a ...
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0answers
29 views

Classifying the Trajectories of Pendulum

The equation of the pendulum is: $$\ddot{\theta}+\frac{g}{l}\sin\theta$$ After some manipulation, we get $$H=\frac{\dot{\theta}^{2}}{2}-\frac{g}{l}\cos\theta=\mathrm{positive\ constant}$$ ...
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19 views

Poincare' map to find periodic solution.

Consider the equation $\dot{y} = (acos(t) + b)y - y^3$ $a > 0, b>0$. I know that I need to recast the equation as a first order system $\dot{y} = (acos(x) +b)y - y^3, \dot{x} = 1$. Also, we are ...
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0answers
42 views

Solve the dynamical system in polar coordinates

I have the system (it is time dependent, this is a simplified notation): \begin{cases} x' = x - y - x^3 \\ y' = x + y - y^3 \\ \end{cases} I can't seem to solve it for r, $\theta$. (The change of ...
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0answers
28 views

Alpha representation of probability distribution

Probability vector is an $n$-dimensional vector $p=(p_1,...,\ p_n)$ that the sum of whose components equals one, i.e. $p_1+...+p_n=1$. If we take the square root of each component of probability, we ...
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30 views

Dynamic Bayesian Networks without restrictions

Normally, when you create a Dynamic Bayesian Network, the restriction is that any random variable in time $t$ depends only on variables in time $t-1$. There are some other algorithms like AR-HMM ...
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46 views

Idea behind Poincaré Bendixson theorem

The Poincaré Bendixson theorem states: If R is a closed bounded subset of $\mathbb{R}^{2}$ containing no fixed points and $\Psi_{t}\left ( x_{0} \right ) \in R$ for all $t\geq 0$, then, the ...
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17 views

Index of a curve is independent of the curve c?

Index of a curve C: $I_{C}$ is defined as the net number of counterclock wise revolutions made by the vector field as the vector field x moves once counterclockwise around the curve C. If C is a ...
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29 views

Regularity of the solutions of the infinite dimensional dynamical systems

Consider a densely defined unbounded operator $A:D(A)(\subset H)\to H$ which is infinitesimal generator of a strongly continuous semigroup $\mathbb{T_{t\ge0}}$ for the following dynamical system: ...
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1answer
17 views

Logarithmic Spiral- N-gon

In the mice problem, also called the beetle problem, $n$ mice start at the corners of a regular $n$-gon of unit side length, each heading towards its closest neighboring mouse in a counterclockwise ...
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0answers
24 views

Part of Picard–Lindelöf theorem proof

Say we have the sequence $$X_{n+1}(t) = X_0 + \int_{0}^{t} f(s, X_n(s)) ds \quad ,\ X_0(s) = X_0.$$ $f$ is continuous over I $\times$ U, I being an open interval in $\mathbb{R}$ and U an open set in ...
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21 views

Particle locating/collision prediction in bounded (two-dimensional) environments [on hold]

I believe that many physics engines, particle simulators, and even video games use discrete-event simulation to determine where a particle or object is at any moment, and the direction in which it is ...
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30 views

Question about n- expansive homeomorphism

Let $(X, d)$ be a compct metic space and $f$ be a homeomorphism on $X$ . Suppose $\Gamma_c(x)=\{y: d(f^{n}(x), f^{n}(y))<c \ , \forall n\in Z\}$ and for some $z\neq x$, $z\in \Gamma_c(x)$. ...
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0answers
21 views

Dynamical Systems problem

I have a problem that have been trying to solve but it's not going so good. I would like some guidelines on how to work myself around this problem: Two neighboring countries spy on each other and ...
1
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1answer
34 views

Finding a function within a dynamical system using a lyapunov function

Consider the system for $(x_1(t),x_2(t))$ \begin{align} \dot{x}_1 &= x_1^2+x_1^3+x_2\\ \dot{x}_2 &= x_1^2+u \end{align} Find a function $u=\phi(x_1(t),x_2(t))$ so that if $$V(x) = ...
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1answer
34 views

Michaelis-Menten steady state hypothesis

In part $ii) $the part underlined in green suggests that we substitute an equation we get from when $v'=0$ to garner a solution of $s'$ for all time from the time when $v'=0$. However $v'$ does ...
2
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2answers
68 views

Periodic solutions of $x'=x^2-1-\cos t$

Consider $x'=x^2-1-\cos t$. What can be said about the existence of periodic solutions for this equation? I'm not sure if periodic solutions exist, but if they do, they must have period equal to $ ...
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0answers
176 views

What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required. In random signal theory, this distribution is typically a ...
4
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1answer
470 views

Characterization of Asymptotic Stability via KL-class functions

Let us adopt the following definition of stability and asymptotic stability of a dynamical system of the form: $$ \dot{x}=f(x) $$ The trajectory of this system starting from the initial point ...
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33 views

Convergence of $\dot{x}(t) = -\alpha(t)x(t) + b\mathrm{e}^{-\lambda t}$

Let $x(t)\ge 0$ obey the following differential equation: $$ \dot{x}(t) = -\alpha(t)x(t) + b\mathrm{e}^{-\lambda t}, $$ where $b>0$, $\lambda>0$, $\alpha(t)\in\mathbb{R}$ is both lower- and ...
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23 views

To prove the properties of Denjoy's Maps

We need to show that the Denjoy homeomorphism constructed may actually be made $C_1$. a)For each integer $n$,let $$l_n=\frac{1}{(|n|+1)((|n|+2)}.$$Show that $$\sum_{n=-\infty}^{\infty} l_n ...
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1answer
17 views

Weak or strong Liapunov function

You are given the system $$\dot{x}=-x-xy^2; \dot{y}=2x^2y-x^2y^3$$ (a) What does the linearization about $x^*=(0,0)$ tell us about the local behavior. So $Df(x,y) = \begin{bmatrix} -1-y^2 ...
2
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1answer
87 views

Index of a function and a gradient flow

We know index of function $F:\mathbb{R}^n\to\mathbb{R}$ at critical point $x_0\in\mathbb{R}^n$ is the number of negative eigen values of Hessian matrix $DF^2(x_0)$. ...
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56 views

Construct a Lyapunov Function for the system

Construct a Lyapunov Function in order to show that the system \begin{align}\frac{dx}{dt}&=x(y^2 +1) +y \\ \frac{dy}{dt} &= x^2y +x\end{align} has no closed orbits (limit cycle) and hence ...
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1answer
54 views

Question about proof of rotation number of inverse map of circle homeomorphism

This question concerns a previous question, Rotation number of inverse maps on the circle. in which all the terminology and notation used below is defined. The question is given the rotation number ...
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1answer
27 views

How to find the straight line paths of saddle points for a nonlinear Hamiltonian system?

I have the system $$\dot{x}=y+2xy\\\dot{y}=-x+x^2-y^2$$ Which is Hamiltonian with $$H(x,y)=\frac{1}{2}x^2-\frac{1}{3}x^3+xy^2+\frac{1}{2}y^2$$ Now I want to plot the phase portrait for the system so ...