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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6 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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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 ...
3
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1answer
86 views

Period 3 window in chaos

Li and Yorke established the fact that period 3 implies chaos, which implies that if we have a period 3 orbit in a system then we have a chaotic system. I have seen that in bifurcation diagrams there ...
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0answers
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, ...
2
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1answer
33 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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0answers
24 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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1answer
55 views

Topological entropy of circle homeomorphism is zero. True or false?

may I know if it is true that $\ f: S^1 \to S^1$ a homeomorphism, then $h_{top}(f) = 0$, where $h_{top}$ stands for topological entropy. I believe this statement is true, but I cannot prove it.
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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 ...
3
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1answer
48 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 ...
2
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0answers
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
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?
2
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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
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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 ...
0
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1answer
28 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?
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0answers
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
398 views

Separatrix curve in dynamical systems

I have this question: What is a separatrix of a equilibrium point of a continuous dynamical system and why it is flow-invariant? Thanks Hello and thanks for the answer. I explain better. I'm ...
0
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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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0answers
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
36 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. ...
1
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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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0answers
10 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: $$\frac{dx}{dt}=b+\gamma H(x-1)$$ ...
6
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1answer
79 views

A problem about Cantor set and found when learning dynamical systems.

Consider the family of functions F(x)=$x^3 -\alpha$x, for $\alpha \gt 0$ Prove that if $\alpha$ is sufficiently large, then the set of points |$F^n(x)$| which do not tend to infinity is a Cantor ...
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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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1answer
199 views

Proving semi-conjugacy preserves chaotic behavior

http://www.math.upatras.gr/~bountis/files/def-eq.pdf In the above documentation it states "It is easy to check that a semiconjugacy also preserves chaotic behavior on intervals of finite length" on ...
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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. ( $ ...
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votes
2answers
76 views

What is “abstract” ergodic theory?

This is just a question about the usage of the term "abstract". What kind of questions in ergodic theory is considered "abstract" and what's a "regular" question? From some seminars it seems that ...
0
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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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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 ...
3
votes
2answers
514 views

What's the point of a Horseshoe map?

At the moment I'm doing a project about the Smale horseshoe map. This is a function which maps a square $D= \{(x,y)\in \mathbb{R}^2: 0\le x\le 1,\text{ } 0\le y \le 1 \} $ to a 'horseshoe'. It ...
3
votes
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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votes
0answers
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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0answers
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 ...
2
votes
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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votes
2answers
220 views

Feedback characteristics of nonlinear dynamical systems [closed]

$ \newcommand{\e}{\boldsymbol \eta} \newcommand{\h}{\boldsymbol h} \newcommand{\T}{T^{\mathsf{ref}}} \newcommand{\g}{\boldsymbol g} \newcommand{\e}{\boldsymbol \eta} \newcommand{\dt}{\partial_{t}} ...
3
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0answers
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 ...
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
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: $$Tr(J)<0$$ ...
4
votes
3answers
107 views

Is the product of a proximal system with itself proximal?

A topological dynamical system is a pair $(X,T)$ where $X$ is a compact metric space and $T$ is a continuous map from $X$ to itself. Two points $x,y\in X$ are said to to be proximal if for any ...
2
votes
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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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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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 ...
2
votes
1answer
43 views

How to find periodic points non-algebraically

For example, $f(x) = x - x^2$ by observation has a fixed point at $x = 0$, and an eventually periodic point at $x = 1$ (that goes to $0$). For the second iteration, to see if there's a periodic point ...
0
votes
2answers
469 views

Closed form solution of this second order linear difference equation?

$$ y(k + 2) - 3y(k + 1) +2y(k) = 2^k + k $$ Transform into a system of $n$ first order equations (Step 1) $$\begin{align} x_1(k) &= y(k)\\ x_2(k) &= y(k + 1) \end{align}$$ It follows that ...
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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 = ...
3
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0answers
85 views

How to choose $\epsilon$ and $\delta$ when proving stability/attractivity

I am having difficulty understanding how epsilon is chosen to prove that a dynamical system is attractive and/or stable. I have taken several analysis modules and was okay at proof writing, well now a ...
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 ...
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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 $n$ ...
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 ...
10
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1answer
610 views

Complicated exercise on ODE

I have this exercise extracted from a examination of qualitative theory of ODE (in which we study the existence and uniqueness of solutions, and stability using the function of Lyapunov) I don't know ...
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 ...