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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36 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 ...
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14 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 ...
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12 views
+50

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 ...
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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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1answer
27 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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0answers
67 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 ...
9
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2answers
367 views
+150

Understanding this ODE

(I did not change anything, I just rewrote the ODE in a simpler form): I started with an ODE (first ODE) : $-(1-x^2)y''(x) +x y'(x) - \left( \alpha x + \gamma x^2 \right) y(x) = \lambda y(x),$ ...
5
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2answers
60 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
102 views
+200

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 ...
3
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0answers
52 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 $\alpha, ~\beta$ ...
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1answer
34 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 ...
2
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0answers
33 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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0answers
17 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 ...
2
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1answer
31 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 $T^{-1}$ ...
1
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1answer
38 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
20 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 ...
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2answers
48 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
39 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 ...
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1answer
22 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 ...
2
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0answers
41 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 ...
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1answer
100 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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1answer
38 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
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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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 $n=2$.
3
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1answer
40 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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1answer
95 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 ...
2
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1answer
15 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 ...
1
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1answer
58 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 ...
2
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0answers
15 views

If $p$ is a regular point $X$ such that $p \in \omega(p)$ then $\omega(p)$ is periodic orbit. [closed]

Let $X$ be a field in $\mathbb{R}^3$, $C^1$ class. If $p$ is a regular point $X$ such that $p \in \omega(p)$ then $\omega(p)$ is periodic orbit.
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16 views

Maximum intervals of a solution and singularities [closed]

Let $X$ be a vector field of $C^1$ calsse in $\Delta \subseteq \mathbb{R}^n$. Prove that if $\varphi(t)$ is a trajectory of $X$ defined maximum range $(\omega_-,\omega_+)$ with: $$\lim_{t \rightarrow ...
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 ...
1
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1answer
67 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 ...
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0answers
11 views

problems based on connected bodies in dynamics

Two particles of masses 2m & 3m lie together on a smooth horizontal table.A string which joins them hangs over the edge and supports a pulley carrying a mass of 4m .Show that the latter mass ...
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26 views

About infinite places

In terms of dynamical systems, what it means an Infinite or finite Place? (This appears in Section 2 of this paper https://ueaeprints.uea.ac.uk/18601/1/sintdynsys.pdf) Greetings !
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1answer
40 views

Prove the formula for the Lie derivative of a differential form

If $X$ is a vector field then by $\mathcal F^t_X$ I will denote it's flow. If $\alpha \in \Lambda^k$ then by definition $$ \mathcal L_X \alpha = \frac{d}{dt}(\mathcal F^t_X)^*\alpha \, ...
2
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1answer
47 views

Problem with a pushforward of vector field formula (Michael Taylor, “Partial Differential Equations”)

Let $X$ denote a vector field and let $\mathcal F^t_X$ denote its flow. If $X$ and $Y$ are two vector fields we denote by $\mathcal F^t_{X\#}Y$ the vector field satisfying $$ \mathcal ...
1
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1answer
81 views

Why does an automorphism of the disk that is an involution not have fixed points in its boundary?

I'm reading Milnor's Dynamical Systems in One Complex Variable and I'm stuck with a detail in one of his proofs and would appreciate some help! Let $F$ be an automorphism of $\overline{\mathbb{D}}$ ...
0
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0answers
27 views

Parameter estimation of Lorenz system (nonlinear dynamical system)

My problem is as follows. I have to estimate parameters of Lorenz system using given data. Lorenz system is described by following system of ODEs: $$ \frac{dx}{dt} = \sigma(x-y) \\ \frac{dy}{dt} = ...
-1
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1answer
35 views

Invariant Line of a System of Equations [closed]

I am unable to follow the part of the solution I've highlighted in green.
-1
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1answer
25 views

Level Sets of Lyapounov Functions

I understand the solution to part a) it's the second part i'm having trouble with. I understand that $\frac{dv}{dt}=0$ implies that the distance from the the fixed point remains constant, however I ...
0
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1answer
32 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?
1
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1answer
55 views

Compact $\omega$-limit set $\Rightarrow$ connected

Consider the flow $\varphi: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ and $L_{\omega}(x)$ the $\omega$-limit set of a point $x \in \mathbb{R}^n$. How can I show that if $L_{\omega}(x)$ is ...
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1answer
31 views

Unstable manifold real Analytic

I don't follow the part of the solution underlined in green. How do we know that the unstable manifold is analytic?
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12 views

Topological semi-conjugacy between two systems

I have a few questions regarding the clarification of the difference between semi-conjugacy and conjugacy. I am trying to figure out a explicit conjugator function to semi-conjugate (a-x^2, x) and ...
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0answers
23 views

Stable/Unstable Manifold theorem

I don't understand how the stable/unstable manifold theorem implies that the power series of of the unstable manifold converges
1
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1answer
47 views

“Reparametrizing” a differential system of the first order (Vinograd theorem?)

Consider a continuous function $f:\Omega\subset\mathbb R^n\longrightarrow \mathbb R^n$ such that for every $x\in\Omega$ the Cauchy problem: $$(\ast)\left\{\begin{array} {ll} y'=f(y)\\ y(0)=x ...
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0answers
24 views

A interesting question from mapping dynamical systems

We have known that in a (discrete) topological dynamical system (TDS) (X,T), a TDS is called initial sensitive, if $\exists \epsilon > 0$, s.t. $\forall x\in X, \forall \delta > 0, \exists y ...
3
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0answers
49 views

Rotation number of inverse maps on the circle.

I'm still a bit lost in my studies of rotation numbers. Any help is much appreciated! Let's say we have a homeomorphism $F: \mathbb{R} \rightarrow \mathbb{R}$ which is a lift of a homeomorphism ...
1
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1answer
40 views

Finite family of subtori in the torus $(S^{1})^{n}$

Working on a problem on matroids, I've already ask a question about some subtori. Here's the link to a previous problem: Topological subspace in $(S^{1})^{n}$ Anyway, here's another problem related ...