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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Mathematical Fundamentals of Physical Simulation

Simulation in general refers to all sorts of modelling done on computers, but what I'm looking for is general mathematical body of theory for doing simulation. That is, theories that explain e.g. how ...
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6 views

About the solenoidal sets

In the book Dynamic Reported - The definition of the solenoidal sets is: Let $I_{0} \supset I_{1}\supset I_{2}\dots$ be periodic intervals with periods $m_{0}$, $m_{1}$,$\dots$. If $m_{i} \to \infty$ ...
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1answer
10 views

Orbit of transformation on point in measure space returns to subset

Let $X$ be a measure space and $T:X\to X$ a measure preserving transformation. The Poincare recurrence theorem states that for any $T$ and any $A\subset X$ with $\mu(A) > 0$ (we take $\mu(X)=1$) ...
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8 views

Bakers map has dense orbit.

Let $X=[0,1]^2$, $L$-the Lebesgue measure on the $\sigma$-field of Borel sets. Define the map $T:X \to X$: $$T(x,y)= \begin{cases} (2x,y) \textrm{ for } x \in [0,\frac{1}{2}), y \in [0,1] \\ (2x-1, ...
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1answer
36 views

Ordering periodic orbits

I want to prove the proposition: Proposition- Let $f:I \to I$ be continuos, and let f have a (2n+1)- periodic orbit {$x_{k}=f^{k}(x_{0})$, $k=0,1,\dots,2n$}, but no (2m+1)-periodic orbit for ...
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26 views

References for methods for convergence analysis of discrete time dynamical systems with non lipschitz nonlinearity

Let a nonlinear dynamical system be described by the difference equations $$x(n+1)=f(n,x(n)),\ n\ge 0$$ with the function $f$ being nonlinear and non-lipschitz. Assume that $f$ is linear and bounded ...
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24 views

Non-hyperbolic zeros of vector field

I'm wondering the following: Let $V$ be a vector field on a (compact Riemannian) smooth manifold $M$ with non-degenerate zeros. Let $p$ be a non-hyperbolic zero of $V$. Can we perturb $V$ slightly so ...
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1answer
49 views

Definition of topological entropy

What the meaning of the limit that appears in the definition of topological entropy? Let $X$ a compact metric space and $f\colon X\to X$ a continuous function and a subset $K \subset X$. The ...
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1answer
20 views

What is measure theoretical entropy in multidimensional symbolic dynamical systems?

Can any one describe the term entropy used in dynamical systems, and what is roll of entropy in symbolic dynamical systems and please give the brief introduction on measure theoretical entropy?
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1answer
19 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 ...
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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 ...
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1answer
39 views
+50

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?
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34 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)
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1answer
16 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
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38 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 ...
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2answers
37 views

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

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. ...
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2answers
130 views

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 ...
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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 ...
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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: $$\frac{dx}{dt}=b+\gamma H(x-1)$$ ...
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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. ( $ ...
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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 ...
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1answer
88 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 ...
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2answers
37 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 ...
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30 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 ...
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21 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 ...
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39 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 ...
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0answers
18 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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1answer
37 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 ...
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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
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0answers
30 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' ...
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2answers
53 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 ...
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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 ...
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0answers
32 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 = ...
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1answer
28 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 $n$ ...
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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 ...
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1answer
81 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 ...
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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 ...
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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 ...
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21 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 ...
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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 ...
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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 ...
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0answers
47 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 ...
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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 ...
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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 ...
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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
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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: $$Tr(J)<0$$ ...
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32 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
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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 ...
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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 ...
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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}= ...