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 ...

learn more… | top users | synonyms (1)

1
vote
1answer
14 views

continuously differentiable and local contraction

Let $F$ be a map from $\mathbb{R}^n$ to $\mathbb{R}^n$. Fix $x_0\in \mathbb{R}^n$. If $F$ is continuously differentiable near $x_0$ and the spectral radius of the Jacobian of $F$ at $x_0$ is less ...
2
votes
0answers
43 views

symplectic structure on $S^2$

i was looking for a symplectic structure on the $S^2 $. Originally i considered the Poisson-Structure of a rigid body, which was given by $\{F,G\}=\langle \Pi, \nabla F \times \nabla G \rangle$, for ...
1
vote
1answer
25 views

Empty omega limit set

I understand what is meant by a limit set but I don't understand what it would mean for this set to be empty. Could someone provide an example?
1
vote
0answers
31 views

Stable/Unstable Manifold heorem

Why does the stable/unstable manifold theorem imply that the power series expansion of the stable/unstable manifold is locally convergent? (local to the fixed point)
0
votes
1answer
13 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} ...
2
votes
0answers
35 views

Stability of an equilibrium

From a Center-Manifold reduction I get the following system: $$ \begin{pmatrix}\dot x \\\dot y\end{pmatrix}=\begin{pmatrix}-y(2x^2-2xy+y^2)\\x\end{pmatrix} $$ The aim is to analyze the stability of ...
1
vote
2answers
33 views

How to solve the following problem of S.H.M.? [on hold]

Problem A particle is moving in S. H.M of amplitude $a$ and period $T$ and when in a position of instantaneous rest is given a blow which imparts a velocity $u$ towards the mean center. ...
3
votes
0answers
74 views
+100

Solving ODE rigorously

I am given the ODE $$(f''(r)+\frac{f'(r)}{r})(1+f'(r)^2)-f'(r)^2f''(r)=0$$ and want to solve it rigorously for $r>0.$ So especially, I don't want to loose any solutions. $\textbf{Derivation of ...
1
vote
1answer
39 views

Does Homoclinic tangle violate deterministic law?

In a dynamical system, the stable and unstable manifold of a fixed point can intersect outside this point. I can understand the existence of homoclinic connections (orbits), which, as far as I'm ...
0
votes
0answers
9 views

Can you call it a saddle node bifurcation when using a step function?

I'm working on an assignment, and we are making an approximation in a dynamical system by replacing a Hill function with a step function. The system is then written: $$\frac{dx}{dt}=b+\gamma H(x-1)$$ ...
0
votes
0answers
26 views

$\int_0^t f(x) - x dx = f^{[t]}(0) - t - 1$

Let $t,x$ be nonnegative reals. Let $* ^{[k]}$ denote k th iteration. Find real-analytic $f(x)$ such that $\int_0^t f(x) - x dx = f^{[t]}(0) - t - 1$ Holds. We require analytic iterations. ( $ ...
0
votes
1answer
19 views

How do you go about finding the normal acceleration of a plane associated with polar co ordinates?

Question A plane, having just taken off, has a constant speed of v=94.3m/s. When $\theta=20^{o}$, the plane is climbing at an ever steepening rate of 0.17 rad/s. What is the normal acceleration of ...
3
votes
1answer
83 views

Can a fractal be a manifold?

Here it is said that it is not possible: Can a fractal be a manifold? if so: will its boundary (if exists) be strictly one dimension lower? But I am confused about this. What about the invariant ...
1
vote
2answers
36 views

Can someone explain the meaning of asymptotically stability in the following definition?

I am taking this from an online course note. Given a system $\dot x = Ax$, we say that: The origin is asymptotically stable if $x(t) \to 0$ as $t \to \infty \thinspace \forall x(0)$ I am ...
0
votes
0answers
29 views

Definition of omega limit set

We say that $p$ is an omega limit point of $x$ if there exists a sequence $\{t_n\}, t_n \rightarrow \infty$ such that the flow $\pi(t_n,x) \rightarrow p$. The set of all such points is called the ...
1
vote
0answers
20 views

Process Transition Algorithm

I have a process with 100 possible states and independent entities going through the process. All the Entities have been observed through a span of 5 years at the end of each month. When the ...
3
votes
0answers
37 views

Stability of origin of dynamical system

Usually you can note some nice structure in the problem which enables construction of a nice Lyapunov function. But this one is just a monster. Maybe there is a trick I've missed? Investigate the ...
0
votes
0answers
11 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 that ...
2
votes
1answer
36 views

Confusion regarding the $\omega$-limit of a set in a flow

In Salamon's Connected Simple Systems, p.8, the author writes that the $\omega$-limit of a set $Y$ inside a flow $\Gamma$ has the two equivalent descriptions $$ \omega(Y) = I(\overline{Y \cdot ...
0
votes
1answer
41 views

Finding angular acceleration

Given: $\mu_B=0.52$ $\theta=30^{\circ}$ Weight- $25$ lb $\omega=0$ $l=6$ ft $1/\kappa=3\sqrt 2$ radius of curvature. Find $\alpha$ My Equations of motion are the following: $\xleftarrow{+}\sum ...
2
votes
0answers
29 views

“Dictionary” of linearizations for nonlinear dynamical system

I have recently jumped on a control project that involves predicting output of a nonlinear system given some input. The team has used $N$ training input/output relationships to build a 'dictionary' ...
2
votes
2answers
52 views

Application of Poincaré-Bendixson theorem

Consider the system $$x' = 3xy^2-x^2y \\ y' = 5x^2y - xy^2$$ Show that the system has no periodic solutions. This is a tricky example. Linearization leads nowhere and I'm having a hard time ...
-1
votes
1answer
22 views

Find the fixed points of the following dynamical system

Find the fixed points of the following dynamical system\begin{align}\frac{dx}{dt}&= (a_1 -b_1x - c_1y)x \\ \frac{dy}{dt} &= (-a_2 +c_2x)y\end{align} Note that ALL the parameters are ...
1
vote
0answers
29 views

How to show that a seperatrix exists for the Fisher-KPP equation

We have the Fisher-KPP equation: $\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + ru(1-u)$ We can reduce this to a second order ODE: $cu_{\xi} = u_{\xi\xi}+u(1-u)$ where $\xi = ...
1
vote
1answer
27 views

Topological transitivity and surjectivity.

Let $X$ a compact metric space and $f: X \longrightarrow X$ a continuous map. The map $f$ es said to be transitive if for every pair of non-empty open sets $U, V \subset X$ there exists an integer $n$ ...
4
votes
0answers
59 views

Structure of first-coordinate-projection of set of solutions of “elliptic” diophantine equation $xy(6-(x+y))=6$

Say that a rational number $a$ is good iff there is a rational number $b$ with $ab(6-a-b)=6$, or equivalently iff $a^4 - 12a^3 + 36a^2 - 24a$ is the square of a rational number. Denote by $G$ the set ...
1
vote
1answer
80 views

When must proper closed invariant sets have strictly smaller Hausdorff dimension?

I'm quite new to dynamics, and trying to learn some of the basics with an application to my neck of the woods in mind. I have run across the property in the title a few times, often with little ...
2
votes
0answers
31 views

Use Gronwall's lemma and method of successive approximations to show that a unique continuous solution exists on

I have this problem from Perko(Page 85, Q-3) which says that : Consider the initial value problem: $$\dot{x}=f(t,x,\mu)$$ $$x(0)=x_0$$ Given that $E$ is open subset ...
0
votes
0answers
20 views

Diagonalization of A Dynamical System with Multiple Zero Eigenvalues

In Perko's book on Differential Equations and Dynamical Systems, 2.12 (Center Manifold Theory) is is stated that for dynamical system $\dot x=f(x)$ one can find the diagonal form of Jacobean at fixed ...
0
votes
0answers
20 views

Relation between ergodic terms and probabilistic terms

I am quite familiar with Ergodic Theory from the Math point of view, that is, when you have a dynamical system $f:M \to M$ with a measure which is preserved by $f$ and where $M$ is some measure space ...
7
votes
0answers
141 views

Eigenvalues of a quasi-circulant matrix

The following matrix cropped up in a model I am building of a dynamical system: $$A= \begin{bmatrix} 1 - \alpha & \alpha/2 & 0 & 0 &\cdots & 0 & 0 & \alpha/2\\ \alpha/2 ...
1
vote
1answer
36 views

Asymptotically stable vs Essentially Asymptotically Stable

I'm having a difficult time understanding these terms when dealing with dynamical systems (e.g. visualizing what these terms mean for a dynamical system). The dynamical systems I'm dealing with ...
1
vote
0answers
46 views

Question for experts in dynamical systems or symplectic geometry

I am looking for a theorem (read it once, but forgot about it and would now like to find a reference with proof): In symplectic geometry (at least for some particular subset of $\mathbb{R}^2$), there ...
0
votes
1answer
22 views

Notation regarding linearization near equilibrium point of dynamical system

Suppose we have $\frac{dx}{dt} = \dot{x} = f(x)$ with equilibrium point $x_e$ such that $f(x_e) = 0$. Then for the linearized approximation of the differential equation near $x_e$ we hope to use the ...
3
votes
1answer
38 views

Trajectories that connect equilibrium points

Suppose I consider the autonomous system \begin{align*} x' &= F(x, y)\\ y' &= G(x, y) \end{align*} where $F$ and $G$ are nonlinear and my task is to draw the phase portrait of the above ...
4
votes
0answers
40 views

Help on designing a dynamical system

I would like to build a four-dimensional dynamical system that has the following behavior: Here, $x_1, x_2, x_3$ and $x_4$ are the four dimensions, and each axis has a fixed point that should be a ...
2
votes
1answer
41 views

Stability for higher dimensional dynamical systems

I remember learning that in order for a steady state to be locally stable in a system of two equations, it is sufficient for the Jacobian evaluated at a steady state to have: $$Tr(J)<0$$ ...
3
votes
0answers
31 views

Plotting a 4D Dynamical System

Suppose I have a 4D dynamical system. Each axis has a fixed point, and there are orbits connecting the fixed points. It looks something like this: Each $Q_i$ is a fixed point on each axis of a ...
3
votes
1answer
81 views

Where did I make a mistake?

This is an excerpt from a dynamical systems paper: They provide a proof of this Lemma, and numerical simulations also show it should be true. It's clear the equilibrium point on each axis is ...
0
votes
1answer
43 views

Characterization of contraction mapping

Let $T$ be a mapping from $\mathbb{R}^n \to \mathbb{R}^n$. Fix $x^\star \in \mathbb{R}^n$, and suppose that the Jacobian matrix of $T(x) $ at $x = x^\star$is symmetric. Then, I know that if all the ...
0
votes
0answers
28 views

Asymptotical stability of a discrete dynamical system

There is a linear time invariant discrete system, \begin{align} x_{k+1}&=\tilde{A}x_k, \end{align} where $\tilde{A}$ is a block matrix represented by \begin{align} \tilde{A}= ...
3
votes
1answer
37 views

Poincaré-Bendixson theorem, periodic solutions/periodic orbits

According to my book (Hsu: ODE), a solution $\phi(t)$ to the system $x' = f(x)$ that is bounded for all $t \geq 0$ satisfies one of: 1) $\omega(\phi)$ contains an equilibrium, or 2) either $\phi(t)$ ...
2
votes
2answers
49 views

Finite automata as dynamical systems

In abstract (deterministic finite) automata theory the set of states of an automaton is an arbitrary set Q, and the transistion function is a specific set δ ⊆ Q × Σ × Q (with alphabet Σ, i.e. another ...
0
votes
1answer
18 views

Stability of non-homogeneous and non-autonomous first-order difference equation

I am seeking to analyze the stability of steady points in a system of $n$ variables $x_1(t), ..., x_n(t)$. With discrete time $t$ the system is described by \begin{eqnarray*} x_i(t+1) = ...
1
vote
0answers
32 views

Building a dynamical system

Suppose I have a 4 dimensional system with 4 fixed points: $Q_1 = \left(p_1,0,0,0 \right)$, $Q_2 = \left(0,p_2,0,0 \right)$, $Q_3= \left(0,0,p_3,0 \right)$, and $Q_4 = \left(0,0,0,p_4 \right)$. ...
1
vote
3answers
65 views

Lyapunov stability at origin with identically zero test function

At the origin, determine stability of $$x' = y \\ y' = -\tan(x)$$ If we use the test function $V(x,y) = 0.5y^2 + \int_0^x tan(s)ds$, we get $\dot{V}=x'\tan x +y'y = y\tan x -y\tan x = 0$, so the ...
2
votes
0answers
17 views

Moser's Twist Theorem for maps with reflection

Suppose I have a simple two dimensional integrable twist map, such as $x_{1}=x_{0}+y_{0}, \quad y_{1}=y_{0}$. Suppose that I perturb it in such a way that Moser's Twist Theorem is satisfied. What ...
3
votes
2answers
37 views

LaSalle invariance, Lyapunov stability

Trying to understand the LaSalle invariance principle. Consider the system $x' = y \\ y' = -y-6x-3x^2$ a) Using the test function $V(x,y) = 0.5y^2+3x^2+x^3$, show that the origin is asymptotically ...
2
votes
0answers
28 views

finding an invariant measure

Are there any methods for finding an (infinite, absolutely continuous with respect to lebesgue measure) invariant measure $d\mu=f(u,v)dudv$ for something like the following? $$ T:Q\to Q, \ ...
0
votes
1answer
39 views

Complex Analysis Dense Set Problem

The Problem: Suppose $f(z) = e^{i\theta}z$. Show that if $\theta$ is not a rational multiple of $\pi$, then the orbit of $ z \in \mathbb{C}$ is dense in the circle with radius $|z|$ and at the center ...