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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21 views

Limit of a Discrete Dynamical System

For the system defined below, the point by point evolution remains bounded for all $t$ so I could see that some sort of limit exists. However, the question is what sort of limit is it -- a single ...
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32 views

Baker's transformation: continuity, orbits of irrational and rational points

I've reading the Pugh's Analysis book and I have problems with one exercise. This says: The baker's transformation: a rectangle of dough is stretched to twice its length and folded back on itself. ...
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1answer
25 views

The largest real eigenvalue of a matrix is bigger than 1

I have a problem which is interesting: given a real matrix $A_{n\times n}$, when this matrix has a largest real eigenvalue which is strictly bigger than 1. If possible, can you give some conditions ...
2
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2answers
74 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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1answer
34 views

Definition of strong mixing and definition of measure-preserving

I have a few questions regarding measure-preserving dynamical systems $(X,\mathcal{A},\mu,T)$. 1) The definition of measure preserving is always stated as $$\mu(T^{-1}B)=\mu(B),$$ for all $B$. I ...
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1answer
72 views

Solving for Center Manifold with Parameter

I have a system of ODEs given by $$\frac{dX}{d\tau}=\beta X\left(1 - \frac{X+Y}{N}\right)$$ $$\frac{dY}{d\tau}= Y\left(1 - \frac{X+Y}{N}\right)$$ where $\beta $ is a parameter. How should I ...
5
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2answers
73 views

System of non-linear ODE's

do you have any suggestions to solve analytically the Non-linear ODE system $\dot x=18 x^2 y-3p x^2+6p xy$ $\dot y=18 x^2 y-6p xy $ where $p$ is a real constant. Thank you very much cheers
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1answer
35 views

Prerequisite of Dynamical system and applied PDE

For the further research interest, I want to focus on the application of Dynamical systems and PDE in the field of robotics and neuroscience, particularly from a mathematical points of view. ...
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17 views

dynamics in a group [on hold]

can we find the dynamics of elements in a group and a ring.is it suggestable to take the topic for research in mathematics.I applied dynamics to elemetns of a group and found some properties of the.I ...
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1answer
31 views

Existence of ergodic joining

Let $\underline{X}=(X,\mathcal{B},\mu,T)$ and $\underline{Y}=(Y,\mathcal{B},\mu,S)$ be ergodic measure preserving systems on Borel probability spaces. A joining of $\underline{X}$ and $\underline{Y}$ ...
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28 views

What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?

I'm trying to solve the system $$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = ...
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1answer
32 views

periodicity of an interval exchange transformation(IET)

Let $T$ be an IET. That is, $T:[0,1] \rightarrow [0,1]$ is a piecewise orientation-preserving isometry. Let $D$ be the set of points whose entire forward iterates are well-defined. I have the ...
8
votes
3answers
207 views

Determining the maximum value for the solution of this delay differential equation?

I am working on the following delay differential equation $$\frac{df}{dt}=f-f^3-\alpha f(t-\delta)\tag{1},$$ where $\frac{1}{2}\leq\alpha\leq 1$ and $\delta\geq 1$. I know that there are three ...
9
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2answers
421 views

Understanding this ODE

(I did not change anything, I just rewrote the ODE in a simpler form): I started with an ODE (first ODE) : $-(1-x^2)y''(x) +x y'(x) - \left( \alpha x + \gamma x^2 \right) y(x) = \lambda y(x),$ ...
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24 views

Careers in Mathematical Modeling of Non-Linear Dynamical Systems

What are some industrial uses of nonlinear dynamical systems and the subsequent mathematical models and simulations? Further, what particular industries and specific companies make use of such ...
4
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1answer
241 views

The Denjoy Theorem

I'm currently studying Denjoy's theorem, which says the following: Theorem If $f$ is a diffeomorphism of $S^1$ with a irrational number rotation $ \rho$ and the variation of $f^{'}$ (denoted by ...
3
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1answer
353 views

How do I find transfer function of a discrete-time system when its state-space form is given?

I read this and this Wikipedia pages, but both of them are explaining continuous-time systems. My question is about discrete-time case. For example, given the state-space equations of the second ...
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1answer
37 views

Fixed Matrices over finite field by a map

Consider a set $M_n$ of all possible square matrices of dimension $n$ over a finite field $F_q$. Clearly the cardinality of the set $M_n$ is $q^{n^2}$. Let us consider a map $f:M_n$ $\longrightarrow$ ...
3
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0answers
84 views

Intersections of two exponential curves in a plane

I am struggling to show that two exponential curves in $\mathbb{R}^n$ do not overlap except finite distinct points. Could anyone help me? Let $A\in\mathbb{R}^{n\times n}$ and ...
15
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1answer
465 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 ...
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1answer
35 views

Partial derivation of a population kinetic's equation

In reviewing my biophysics' course on population kinetics I am stuck in finding which equation was actually used to derive from. It uses an example to "explain" the analytical method, in order to ...
3
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0answers
53 views

Boundedness of solutions of Difference equation

Consider a second order difference equation in complex plane, \begin{equation} z_{n+1}=\frac{\alpha + \beta z_{n}}{1+z_{n-1}},\qquad n=0,1,\ldots \end{equation} where the parameters $\alpha, ~\beta$ ...
2
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1answer
32 views

Poincaré Recurrence Theorem (measure theory version)

I had a look on the proof of the following Recurrence Theorem of Poincaré: Let $(\Omega,\Sigma,T,m)$ be a conservative dynamical system in measure theory for which the function $T^{-1}$ ...
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0answers
36 views

Stability properties of discretization of ODE

I am trying to find some conditions which guarantee that a continuous time dynamical system and it's discretization have the same behavior with regard to equillibrium points. Specifically that if the ...
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1answer
73 views

Delayed System Help

It is well-known that a small delay may or may not cause stable equilibrium to become unstable. Can anyone help that if for $\tau=0$ the equilibrium solution is unstable and if $\tau>0$ is there a ...
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0answers
18 views

How to generate a Poincare section for discrete particle trajectory?

I'm a novice when it comes to generating Poincare sections, and I can't seem to get it right. I have a particle moving in a 3D periodic field, and I wish to generate a Poincare section of its ...
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2answers
49 views

Extension of Poincaré-Bendixson Theorem to $\mathbb{R}^3$

Hartman mentioned in his ODE book (chapter 7) that Poincaré-Bendixson Theorem is limited to $\mathbb{R}^2$ or $2$-manifold because of Jordan Curve Theorem. Since there is generalization for ...
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1answer
40 views

Differential Equation Examples for different type of critical point

For a linear system $X'=AX$, there are only limited types of critical points according to the eigen values of $A$. When I want to considering non-linear dynamical system in $\mathbb{R}^2$ and ...
1
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1answer
21 views

A property of Sturmian sequence

I need the following simple property of Sturmian sequence: Let $\omega\in \{0,1\}^{\mathbb{N}}$ be a Sturmian sequence, we define the orbits space ...
2
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0answers
42 views

perturbation theory solution of forced Duffing's equation

Question: Find the leading order of the asymptotic expansion for large t: $\frac{d^2x}{dt}+\varepsilon\beta\frac{dx}{dt}+x+\varepsilon x^3=Fcos(\frac{1}{3}\big(1+\varepsilon\omega)t\big)$ I have ...
0
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1answer
25 views

Henon Map Parameter

In case of Hennon map two parameters $a$ and $b$ to be set.The Hénon map takes a point $(x_n, y_n)$ in the plane and maps it to a new point $x_{n+1} = 1-a x_n^2 + y_n$, $y_{n+1} = b x_n$. The map ...
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0answers
26 views

The future of the orbit of a point is a closed set [duplicate]

$X$ is a metric space and $f: X \rightarrow X$ is a dynamical system. Prove: $w(x_{0})$ is closed. Here the set $w(x_{0})$ is the future of the orbit of $x_0$, defined as $$\omega(x_0) = \{y \mid ...
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1answer
33 views

Limit sets and Poincare -Bendixson

Does a) imply that the limit set is the entirety of the points $1/3<y^2+z^2<1/2$. If so what does the periodic solution in b) look like?
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1answer
40 views

Recommendation for dynamical system with complex behaviors

I want to learn the behaviors of dynamical systems, especially the in form of $X'=f(X)$ and $X'=f(t,X)$ in $\mathbb{R}^3$. I know Lorentz system is such a system(typically ...
1
vote
1answer
101 views

How an empty set is collapsed to a point?

In the original book of Conley Index Theory: Isolated Invariant Sets and the Morse Index chp3.3, p6, Charles Conley mentioned that ...
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2answers
283 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 ...
2
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1answer
27 views

Question on Gauss map - application of Birkhoff's ergodic theorem

Take a Gauss map $G: [0,1] \longrightarrow [0,1]$ which is $$G(x) = \frac{1}{x} \mod 1, 0<x<1$$ and $0$ if $x=0$. Let $\mu$ be the Gauss measure. For $x \in [0,1]$ let $[a_{1}(x), ...
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2answers
42 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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1answer
39 views

Index of a function and a gradient flow

We know index of function $F:\mathbb{R}^n\to\mathbb{R}$ at critical point $x_0\in\mathbb{R}^n$ is the number of negative eigen values of Hessian matrix $DF^2(x_0)$. ...
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1answer
507 views

Prove that the tent map has exactly nine 6-cycles.

Prove that the tent map T(x)= {2x if 0<=x<=1/2 and 2-2x if 1/2
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0answers
20 views

For all topological conjugation $$h: \Delta_1 \rightarrow \Delta_2$$ we have to $h(\omega(p))=\omega(h(p))$, for all $p \in \Delta_1$

Let $X_1$ and $X_2$ fields in $\Delta_1,\Delta_2$ subset open in $\mathbb{R}^n$. Then, for all topological conjugation $$h: \Delta_1 \rightarrow \Delta_2$$ we have to $h(\omega(p))=\omega(h(p))$, for ...
3
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1answer
41 views

Does a pseudo-Anosov homeomorphism of a punctured surface possess infinitely many periodic points?

In A Primer on Mapping Class Groups by Farb and Margalit theorem 14.19 implies that every pseudo-Anosov homeomorphism $f:S \rightarrow S$ on a compact surface $S$ possesses infinitely many periodic ...
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33 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 ...
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18 views

if $X$ is a vector fild in $\mathbb{R}^3$ and $h$ is a periodic orbit, then $X$ have a singularity? [duplicate]

if $X$ is a vector fild in $\mathbb{R}^3$ and $h$ is a periodic orbit, then $X$ have a singularity? and in dimension $n$? I know there is singularity when $n=2$.
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1answer
96 views

Irrational Rotation

Let $\sigma$ be a homeomorphism of $S^1$. Then the following statements are equivalent; (1) O(z) is dense in S for some z in S, (2) O(z) is dense in $S^1$ for every z in $S^1$, (3) $\sigma$ is ...
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vote
1answer
69 views

a system of finite difference equations

Let $a,b>0$ such that $ab<1$ consider the system$$x_{t+1}=x_ty_t+ay_t$$ $$y_{t+1}=x_ty_t+bx_t$$ I would like you to help me answer the following: find values $a$ and $b$ ​​for which the ...
2
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1answer
16 views

Varying definition of Controllability Gramian

I have always used the following definition of controllability gramian of $(A,B)$ over the time window $[t_0,t_f]$ $$W_c[t_0,t_f] = \int_{t_0}^{t_f}e^{A(t_f - t)}BB'e^{A'(t_f - t)}$$ I have used ' as ...
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1answer
61 views

A phase diagram outlining

I'm trying to solve this differential equation $$x^{ \prime}=f(x)-nx-y$$ $$y^{\prime}=\frac{(f^{\prime}(x)-r)y}{\alpha}$$ where $f:[0,+\infty[\rightarrow \mathbb{R}_{+}$ is an increasing and concave ...
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0answers
436 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 ...
3
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133 views

Proving that $f$ has points with prime period 3

Exercise: Let $f:\mathbb{R}\longrightarrow\mathbb{R} $ be continuous, $n>3$, and $x_1,x_2,\cdots,x_n$ be points such that $x_1<x_2<\cdots<x_n$. Show that if $f(x_i)=x_{i+1}$ for ...