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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Dynamical system with periodic orbit

We are given dynamical system $\phi $ in $R^2$, and know that it has periodic orbit (means $\phi(T,x_0)=x_0$ for some $T>0$ and $x_0 \in R$). We are asked to prove that the system has stationar ...
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43 views

Question about Kronecker factor

In his paper Ergodic methods in additive combinatorics, Bryna Kra said that the Kronecker factor $(Z_1, \mathcal{Z}_1, m, T)$ of $(X, \mathcal{X},\mu,T)$ is the sub-$\sigma$-algebra of $X$ spanned by ...
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Complex Symplectic Matrix

A symplectic $2n\times 2n$ real matrix is a constant matrix $M$ that satisfies $$MJM^T=J$$ where $J=\begin{bmatrix}0 &I_n \\-I_n & 0\end{bmatrix}$. We now that if such matrix is used as a ...
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80 views

A system of nonlinear differential equations

We have the following system in $\mathbb{R}^{2}$ $$\dot{y}_1=2-y_1y_2-y_2^2$$ $$\dot{y}_2=2-y_1^2-y_1y_2$$ i) Calculate the equilibrium points en determine their stability. ii) Draw the Phase ...
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122 views

Find values of the parameters in Predator prey model [closed]

$$r' = F_1(r, f) = r − cr^2 − drf$$ $$f' = F_2(r, f) = −f/4 + erf + gf^2$$ Consider the case where $g = 0$. For what values of the parameters, $c, d$ and $e$, which are all assumed to be positive, ...
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27 views

Is it possible to apply the Melnikov function to nonperiodic perturbations?

In the case of planar Hamiltonian system, the classical Melnikov function deals with the periodic perturbation. Is it possible to apply the Melnikov function to nonperiodic perturbations?
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39 views

Question about Singer's theorem

I recently studied Singer's theorem, but every proof I have read does not detail one important step that I still don't understand. This step can be written as the following: We have $h:I\subseteq ...
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1answer
73 views

Compact space, continuous dynamical system, stationary point

I'm having trouble proving that if $X$ is a compact metric space and every continuous function $f : X \rightarrow X$ has a fixed point, then every continuous dynamical system $ \varphi $ on $X$ has a ...
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2answers
79 views

Does any measure preserving system have an invertible extension?

Let $\mathsf{X} = \left\{ X,\mathcal{B},\mu,T \right\}$ be any measure preserving system. A sub-$\sigma$-algebra $\mathcal{A}\subseteq \mathcal{B}_X$ with $T^{-1}\mathcal{A}=\mathcal{A}$ modulo $\mu$ ...
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65 views

Differential Equation: Periodicity of a circle with zero radius in polar coordinates

I am given the following diff. equation in polar coordinates: $$\dfrac{dr}{dt} = r(1 + a~\cos \theta - r^2) \\ \dfrac{d \theta}{dt} = 1$$ where $a$ is a positive number and is less than $1$. I am ...
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76 views

Predator prey system question?

QUESTION 5. The following system describes a predator prey system in which the prey has an Allee effect. What is the threshold of the prey to persist when alone? Find the nullclines and the steady ...
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23 views

How do you get a measure space out of a dynamical system?

I'm reading a book on ergodic theory (by Cesar Silva), and also have read Stein and Shakarchi's third book on undergraduate analysis, where there is a section devoted to some ergodic theory. Both ...
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1answer
28 views

Order in taking limits?

So I am reading this physics paper that they define Kolmogorov entropy for dynamical systems as follows: $$K=\lim\limits_{\epsilon\to 0}\lim\limits_{T\to \infty}\frac{I(\epsilon,T)}{T}$$ They ...
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43 views

Reference about the Conley index thoery

I'm reading "Isolated invariant sets and the Morse index" by Charles Conley.But I'm lost in some of the concise description or definition.Could you recommend me some references or textbooks for the ...
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2answers
76 views

linear differential equation problem [closed]

Consider the following system of linear differential equations: $$\begin{split} \frac{dx}{dt}&=−3x+y\\ \frac{dy}{dt}&=x−3y \end{split}$$ Find the eigenvalues and eigenvectors associated ...
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62 views

Prove that a map is a homeomorphism and the inverse is bounded

I'm trying to unravel an obscure passage in a textbook, which states that if $\phi :\mathbb{R}^m\to\mathbb{R}^m$ is continuous, bounded and Lipschitz with constant $\varepsilon$ (which is still free ...
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1answer
61 views

Equilibria and stability

Find all equilibria for the following system and determine their stability: $$x'=y^2-4$$ $$y'=x^2-1$$
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87 views

Jordan canonical forms and diagonalizing.

In my dynamical systems, we are asked to find the Jordan Canonical form of the Jacobian in order to analysis the linear stability at fixed points in a second order system. I believe that even for one ...
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251 views

calculus, predator-prey system

The following system describes a predator prey system in which the prey has an Allee effect. What is the threshold of the prey to persist when alone? Find the nullclines and the steady states of the ...
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68 views

Ergodic Rotation of the Torus

Consider the measure preserving dynamical system $(\mathbb{R}^2 / \mathbb{Z}^2, \mathcal{B} \otimes \mathcal{B}, \lambda \otimes \lambda, R_{(\alpha, \beta)})$. This is the torus with the borel ...
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47 views

Historical study of dynamical system

I am currently doing a historical study on my school project 'study of ODE' which slowly shift to the study of dynamical system as I am interested in pursuing my study of ode from linear system, phase ...
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74 views

How do I Linearize

How would I solve the following problem? Linearize around the fixed points $$\left\{\begin{align}\frac{\text{d}x}{\text{d}t}&=y-x^2\\\frac{\text{d}y}{\text{d}t}&=y-x\end{align}\right.$$ I ...
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133 views

Numerically calculate the boundary of a basin of attraction for a high dimensional dynamical system

I am looking for an efficient, non-exponential time algorithm to calculate the boundary of a basin of attraction for a stable fixed point in a high dimensional nonlinear dynamical system. The naive ...
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40 views

Simple proof of invariant sets

Let How to prove the unit circle is an invariant set? My way is that: At t1, x1(t1)^2 + x2(t2)^2 = 1, so the eqs become: Since both x1 and x2 are functions of 't', so solve it and obtain: ...
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30 views

Question on diffeomorphisms

Suppose that we are given an autonomous ode $\dot{x} = f(x)$ where $f : \mathbb{R}^{n} \to \mathbb{R}^{n}$. My (elementary question) is that is the time one map for the ode above a local ...
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51 views

Prove that $(A,B)$ is uncontrollable $\Longleftrightarrow$ $\exists P$ $\in$ $\mathbb{R}^{nxn}$, $P \neq 0$: $PA - AP = 0$, $PB=0$

In my course advanced system Theory I had the following question: Prove the following equivalence for the pair $(A,B)$ $\in$ $\mathbb{R}^{nxn}$ x $\mathbb{R}^{nxm}$: $(A,B)$ is uncontrollable ...
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35 views

Prove that A is marginally stable iff there exist a $P$ $\in$ $S^n$, $P \succ 0$ such that $A^T + PA \leq 0$

Asymptotic stability, which means that all eigenvalues of A are in the open left half plane is easily proven. See the scan in the attachment. However, in the book the proof for the second case where ...
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25 views

Reference Request: Simplex dynamics

I want to know if there's any general method to investigate linear system restrained on standard simplex. It's hard for me to start with such a system because if I directly regard it as general ...
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23 views

Converse of Noether's (first) theorem

Noether's (first) theorem states that if a Lagrangian $L$ admits a continuous symmetry, then the following quantity are conserved. $$\left(\frac{\partial L}{\partial \dot q}\cdot\dot ...
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1answer
38 views

Show that if $f\colon\mathbb{R}^2\to\mathbb{R}$ is $C^2$, then any nonempty $\omega$-limit for the equation $x'=\nabla f(x)$ is a critical point.

I'm kind of struggling with an exercise I found in a book about Poincaré-Bendixon theory and I would like some help. The exercise is precisely what I wrote on the title: I have to show that if ...
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138 views

Logistic map bifurcation

Ok I am trying to do this on matlab, but I need to understand how to find the bifurcation values for logistic map by hand first. So here is the logistic map: $$ x_{i+1} = f(x_i) \qquad \text{where} ...
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book recommendations for further learning about dynamical system and bifurcation

I have read the book "Introduction to Applied Nonlinear Dynamical Systems and Chaos"by Stephen Wiggins.Could someone recommend books on dynamical systems and bifurcation theory for further learning?
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232 views

Proving Nonhomogeneous ODE is Bounded

I am trying to prove the following: Given a solution $x(t)$ of the IVP $\dot x=A(t)x+h(t)$, where $A(t), h(t)$ are continuous on $0<t<\infty$, prove that x(t) is bounded for $t\ge1$ if both ...
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Lyapunov function with an added constant

I have the system: $x=f(x)$, where $f(x)=\left[\begin{array}{c} x_1^2-x_2^2-1 \\ 2x_2\end{array}\right]$, and I'm stumped because I can't find a way to get around the "$-1$" when finding the Lyapunov ...
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35 views

Periodic points of topologically conjugated functions in dynamical systems?

I'm working on a homework problem which seems obvious, but I am having a hard time proving/completing. The problem can be stated as follows: Let $f,g:$ $\mathbb R$ $\rightarrow$ $\mathbb R$ be ...
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16 views

Looking for methods/results for explicitly bounding iterations of rational functions

This is a cross-post of http://mathoverflow.net/questions/155775/looking-for-methods-results-for-explicitly-bounding-iterations-of-rational-funct But I received no answer there to the actual ...
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56 views

Topological conjugation between two flows

This is problem 1.53 from Ordinary Differential Equations by Chicone (2nd Ed). Prove that there are open intervals $U, V\subset \mathbb{R}$ both containing the origin and a differentiable map $H: U ...
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Flow of a particle in dynamical system

Suppose I have the linear system $\dot{x}=Ax$, with $A=\left[ \begin{array}{cc} -1 & 0 \\ 0 & 2\\ \end{array}\right]$. I know that the phase portrait of the linear system has a saddle in ...
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87 views

Normal form of differential equation $y'=y-y^2$

I've been told that I can transform any equation of form $y'=ay+g(y)$, where $g(y)$ is polynomial, to form $y'=y$ (Poincaré-Dulac theorem if I'm not mistaken). So I want to find the normal form of ...
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329 views

Why computing Fatou coordinate is so hard?

I'm trying to make images of Fatou coordinate for some polynomial maps. If I'm not wrong there is no explicit general formula/method for computing Fatou coordinate near parabolic fixed point. Is ...
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1answer
67 views

Why was this change of variables made in this system of differential equations?

I'm trying to understand an example from my notes. I'm given a system of linear differential equations as follows $$x'=2x-y$$ $$y'=2x-2$$ The notes solve these by making the change of variables ...
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69 views

Topology on the set of infinite sequences

Help, who can give me an example to show the following definition of $\lim$ and closed set let $\Sigma$ be a set of states, let $\Sigma^\omega$ denote the set of all infinite sequences of elements in ...
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41 views

Simple problems related to Hilbert's 16th problem

I am currently taking a course in dynamical systems and am planning to do a project on Hilbert's 16th problem/counting limit cycles. Unfortunately, every major related theorem I've found is too ...
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Why was holonomic system named like that?

I want to know how to literally translate it. so what is the literal meaning of it?
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92 views

Reference request: Nonlinear dynamics graduate reference

There are already a number of requests for textbooks detailing nonlinear stability theory, chaos theory etc. but many of them are more introductory (e.g. Strogatz - Nonlinear Dynamics and Chaos) I've ...
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29 views

Give an example of a homeomorphism of a compact metric space that has a dense orbit but no dense semiorbit.

I am at a loss in trying to find such a function. I know it must only be bijective and continuous to be a homeomorphism since it is a compact metric space. However, every function I've tried to ...
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89 views

Introductory text on perturbation theory for dynamical systems

I am working on my thesis which is about oscillations and as far as I realise I need to know about perturbation theory and methods in solving differential equations, specifically dynamical systems. A ...
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What is the method for solving this class of problem?

$\frac{dx}{dt} = y$ $\frac{dy}{dt} = -\partial y - \mu x - x^2$ Find the fixed points and discuss stability. I'd at least like to know what I should be googling, any help is appreciated.
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39 views

Kinematics Question Help

I'm having trouble with this question: A particle moves so that its position vector with respect to the origin $O$ of a reference frame $Oxyz$ is $$ \mathbf{r}(t)=bcos(wt) ...
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1answer
41 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 ...