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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Finding discrete maps with prescribed cycle-structure (functional digraph-structure)

I apologize in advance for the naive nature of the following questions. I am also thankful to suggestions for improving the direction of the questions instead of direct answers. Let $f: \mathbb N \to ...
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18 views

Show a continuous bijection cannot have periodic points of prime period greater than 2

Suppose f:R↦R is a continuous bijection. Show that the system x_n+1=f(xn) cannot have periodic points of prime period greater than 2. Hint: Use Sharkovskii's Theorem to reduce the problem to the case ...
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17 views

Showing that the function $C(b)$ is a compact set for $|b| < 1$

I am reading "An Invitation to Dynamical Systems", and one of the challenge problems is to prove that $C(b)$ is a compact set where $C(b)$ is defined as the set of all numbers that can be expressed in ...
2
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56 views

modified ODE has same trajectories as original system and associated flow is defined for all $t \in \mathrm{R}$ [closed]

I really don't know where to start with this problem. Consider the differential equation $\dot{x} = f(x)$ with $f \in C^1(\mathrm{R}^n,\mathrm{R}^n)$. Consider the following modified differential ...
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36 views

$\cos(2\arccos(\frac{a}{a+1})x$

I have trying to prove that this cosine map: $$\frac{r}{4}((a+1)\cos\left(2\arccos\left(\frac{a}{a+1}\right)\ \left(X_n-\frac12\right)-a\right)$$ is a logistic map. What I have done so far: Using ...
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38 views

Holonomic constraints and degrees of freedom

Back in my undergrad I learned that in a dynamical system, if I add a holonomic constraint, I subtract one degree of freedom from the space of configurations. But one can think of situations in which ...
4
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1answer
57 views

A question on ergodic theory: topological mixing and invariant measures

This is a question on dynamical systems. Suppose I have a compact metric space $X$, with $([0,1], B, \mu)$ a probability space, with $B$ a (Borel) sigma algebra, and $\mu$ the probability measure. ...
2
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1answer
59 views

Existence of invariant states in a $C^*$-algebra

Let $\mathcal{A}$ be a C*-algebra and $\{\tau_t\}_{t\in\mathbb R}$ a weakly-continuous group of *-automorphisms. I've read the claim (without proof) that for any state $\eta$ (that is $\eta$ is a ...
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69 views

Understanding a proof concerning flows on tpological spaces

Let $M$ be a topological space and to each point $x\in M$ let there be an open interval $I(x)=(I_-(x),I_+(x))\subset\mathbb{R}$. To make it shorter, we set $E:=\bigcup_{x\in ...
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1answer
25 views

Function with many variables, diffrentiating. Hamiltonian and Langragian functions

If I have: $$H(q,p,t)=\dot{q}_ip_i-L(q,\dot{q},t)$$ How can I obtain the following relations: $$\mathrm{d}H=\dot{q}_i\mathrm{d}p_i-\dot{p}_i\mathrm{d}q_i-\frac{\partial L}{\partial t}\mathrm{d}t$$ ...
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FermiPasta-Ulam problem

Consider $H(q,p) = \frac{1}{2} \sum\limits_{j=1}^{n+1} {(p_j^2 + (q_{j}-q_{j-1})^2)}$ $H(q,p) $ is the Hamiltonian considered in the FermiPasta-Ulam problem. Consider canonical transformation $Q = ...
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139 views

Confluent Heun equation. Reduction to standard form.

I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions? $$(1-t^2)u_{tt}-tu_t+4\left[n\beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$ $C$ ...
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9 views

Stability of rest points of a time-dependent sequence

Consider the following sequence: $$x_{t+1} = (x_t)^t$$ A fixed point $a$ satisfies the following: $$a = a^t$$ for all $t$ sufficiently large. $$a(1 - a^{t-1}) = 0$$ Clearly, $a=0$ and $a = 1$ ...
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474 views

How to make a smart guess for this ODE

I am dealing with a strange problem currently, we have a differential equation $$y(x)^2 = \pm \sqrt{-A \cos(x) - B \cos^2(x)+y'(x)-C},$$ where $C, A$ and $B $ are parameters. (The case that either ...
0
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1answer
34 views

Level sets of a conserved quantity are trajectories of differential equation

If we have a differential equation $\mathbf{\dot{x}}=\mathbf{F}(\mathbf{x})$ and we have conserved quantity $E(\mathbf{x})$, which means $\dot{E}=0$, then I don't understand why level sets of $E$ are ...
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Notions of stability for differential equations

Consider a system of differential equations $$\dot{x} = f(x,u)$$ $$y = h(x,u)$$ where $x(t), u(t)$ are vectors in some $\mathbb{R}^n$. We define the infinity norm of a function in more-or-less in the ...
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45 views

Ergodic Theory, Bernoulli Measure, Cylinder Set

Let $N\geq2$ be an integer and consider the probability space ($\Sigma^+$,B, $\mu_p$) where $\mu_p$ is the Bernoulli measure with respect to probability vector $\ p = (p_1,...,p_N)$ . Show that for ...
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27 views

Making foliations C^1 smooth

I am not an expert in foliations, but I'm interested in the following conjecture of Cantwell and Conlon (from "An interesting class of $C^1$ foliations"): "If $F$ is a transversely orientable ...
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56 views

Free Vibrations - Simple Harmonic Motion

As an engineering enthusiast I have been practicing with numerous model assignments to see how well I could deal with dynamics if I were in an educational environment. Most problems seem simple to ...
3
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1answer
37 views

Commuting functions on the closed interval have the same value somewhere

We are given two commuting continuous functions $f,g:[0,1]\to[0,1]$. How can we prove that $f(x)=g(x)$ for some $x\in[0,1]$? A trivial observation is that if one of the two functions is a ...
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1answer
28 views

Stable (resp. Unstable) sets / subspaces of dynamical systems.

I read the book of Yuri. Kuznetsov, Elements of Applied Bifurcation Theory. He claims, $W^s(x_0) = \{x : \phi^t x\rightarrow x_0, t \rightarrow \infty\}$, defined therein as a stable set, same (as ...
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111 views

Properties of this Sturm-Liouville problem.

Given the ODE $$f''(x) + \left(\alpha_1 \cos(x) + \cos^2(x) - \lambda \right) f(x)= 0,$$ where $\theta \in [-\pi,\pi]$, $||f||_{L^2} < \infty$. I was wondering whether there is anything we ...
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Spectrum of the operator of differentiation along streamlines

Suppose $\mathbb T^2$ denotes the two-torus and suppose $\psi^0$ is a steady state smooth stream function on $\mathbb T^2$ and define $\mathbf u^0 = (-\partial_y \psi^0,\partial_x \psi^0 )$ be the ...
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1answer
42 views

Groups and conservation laws for discrete-time systems

I would like to know if the following concept has a name. I call it a "conservation law", but if it's defined in the literature it's probably called something else. I'm interested in any results based ...
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52 views

Hyperbolic Fixed Point

Let $f:M\rightarrow M$ be a $C^{1}$-class diffeomorphism . Let $x\in M$ be a fixed point. I've been looking for a while on Internet for a proof of the following fact, but i couldn't find : $\lbrace ...
3
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37 views

Ergodicity and Appropiate Partition of the Space

I'm trying to solve the following problem. Let $(X, \mathcal{B}, \mu)$ be a probability space and let $T \colon X \to X$ be a measure preserving function. Prove that if $T^n \colon X \to X$ is ...
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1answer
82 views

Energy Transfer in a Mechanical System - Standard Pulley Scenario

I understand that this is quite a basic question, I am new to dynamics and have trouble starting off questions, I found it quite difficult to find an example question alike to the one below thus I am ...
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1answer
38 views

bifurcation with more than parameter

Problem: Consider the scalar differential equation depending on the parameters $\alpha_1, \alpha_2$ ∈ $\Re$ $x˙ = \alpha_1 + \alpha_2 x − x^2$. Find a change of coordinates $y = \phi(x)$ such that ...
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Group action of $G<\mathbb Z^\infty_2$ over the Golden mean shift

I'm am looking for an action of an infinite subgroup of $\mathbb Z^\infty_2$ over the golden mean shift space $$X=\{x\in \{0,1\}^\mathbb N : x_i=1\Rightarrow x_{i+1}=0\}$$ such that any element of $G$ ...
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55 views

Physical interpretation of Ergodicity.

If $R_{\alpha}:[0,1] \to [0,1]$ is defined by $$R_{\alpha}(x)=x+\alpha $$ then $R_{\alpha} $is called a circle rotation, and it is known that $R_{\alpha}$ is ergodic iff $\alpha$ is irrational. I ...
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Sketching phase portraits [closed]

I am trying to answer this question: I would like to know how I go about drawing a phase portrait. All of the examples in my notes are simply the solution with no explanation, and this method of ...
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1answer
47 views

Marginal stability and centers of nonlinear dynamical systems

If I have a coupled nonlinear dynamical system, like $$\dot{x}=ax-bxy$$ $$\dot{y}=cxy-dy$$ by using jacobian matrix, I can find that the point ($\frac {d}{c}$,$\frac {a}{b}$) is a center. I think in ...
2
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34 views

toplogical entropy of general tent map

Measure theoretic entropy of General Tent maps The linked question made me wonder how to calculate the topological entropy of a general tent map. Let $I=[0,1]$ and $\alpha \in ( 0,1)$. Define $T: I ...
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1answer
32 views

Finding normal coordinates of a system

We have coupled oscillators with equations of motion: $$\ddot{x} = -10x+18y$$ $$\ddot{y}=-3x+5y$$ At $t= 0$ we have $x=a$ and $\dot{x}=\dot{y}=y=0$. I found the solution to be $$\begin{pmatrix} x(t) ...
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1answer
47 views

Bifurcation Diagram stability

I'm wondering how you know which branches are stable. For example taking $\alpha$ as the bifurcation parameter $$y'=\frac{y(1500-y)}{3200}-\alpha.$$ So I plot $\alpha=\frac{y(1500-y)}{3200}$ and flip ...
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1answer
49 views

near identity change of coordinates

Problem: Consider the scalar differential equation $$x' = \frac{4x – 24x^2 – 16x^3}{1 – 12x – 12x^2}.$$ which has a fixed point at $x^* = 0 $. For $x$ close to $x^* = 0 $ find a near identity ...
3
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46 views

Showing iterates of a complex function on the upper half plane converges uniformly on compact sets

The following is an irksome problem that my complex analysis class is having trouble solving: Let $*$ be an operator that takes a function $F:\mathcal{H}\to\mathcal{H}$ to a function ...
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26 views

measurability restriction operator

Let $M\subset \mathbb{R}^k$ compact. For every $x\in M$ we define $L(x): \mathbb{R}^m \rightarrow \mathbb{R}^m $ a linear isomorphism Let $G_n (\mathbb{R}^m)=\{ W: W\ \mbox{is subspace of} \ ...
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1answer
50 views

Recommend resources on dynamical systems and singularities

I'm looking for resources on bifurcation theory and systems of non-linear differential equations, but am very particular about the way it is taught/explained. I would like the approach to be based on ...
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25 views

Subshifts of finite type; No fixed or period 2 points

I'm working out of Devaney's Introduction to Chaotic Systems, and one of the problems I'm working on is to construct a subshift of finite type in $\Sigma_3$ with no fixed or period two points, but ...
2
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2answers
81 views

the global stable and unstable manifolds

Show that $x^* = (1, 2)$ is a fixed point of the system $x_1' = 2 + 3x_1 − 2x_2 − x_1^2 + 2x_1x_2 − x_2^2$ $x_2' = 3 + 4x_1 − 3x_2 − x_1^2 + 2x_1x_2 − x_2^2$ Determine $W^s(x)$ and $W^u(x)$, the ...
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1answer
32 views

Is a cascaded chaotic system is still chaotic?

I am curious whether a new system which cascades two individual chaotic systems is always chaotic. My feeling says if two chaotic systems $S_1$ and $S_2$ satisfying the constraints that $${\rm ...
2
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34 views

Relationship between Reproductive Ratio and Jacobian in Population Model

In class we defined the Reproductive Ratio, $R_0$ of a population modelled by SIR, SEIR,... as the average number of secondary infections caused by an average infected individual in an average ...
2
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1answer
38 views

Lyapunov Exponents for independent-nonidentically distributed matrices?

My question is highlighted in bold at the end. $\mathrm{\underline{Background}}$ Consider a product of i.i.d. $d\times d$ random matrices $A_{i}$ (with $\mathbb{E}\log\left\Vert A_{i}\right\Vert ...
3
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1answer
57 views

Systems of Linear Differential Equations - population models

I have to solve the following first-order linear system, $x(t)$ represents one population and the $y(t)$ represents another population that lives in the same ecosystem: (Note: $'$ denotes prime) ...
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1answer
34 views

What are the Routh Hurwtiz Criteria for 3$\times$3 Matrices?

The Criteria I know (for dynamical systems) is... The eigenvalues of a matrix are guaranteed to be negative if Tr($J$)<0 and det($J$)>0, where $J$ is the Jacobian of some 2 dimensional dynamical ...
2
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1answer
89 views

Show that system is Transcritical bifurcation

In what ways can you show that transcritical bifurcation occurs? For example take the system $$\dfrac{dx}{dt}=xr+2x^2 $$
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1answer
100 views

Demonstrating that the Mandelbrot Set is connected

I know that demonstrating the Mandelbrot Set is connected requires a non-trivial proof, and that Mandelbrot himself was fooled at first. But can it be demonstrated visually that the set is connected? ...
0
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1answer
48 views

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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1answer
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 ...