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Questions tagged [dynamical-systems]

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 models of diverse fields such as classical mechanics, economics, traffic modelling, population dynamics, and biological feedback.

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Why does chaotic behavior explode when $k > \pi$ for $f_{n + 1}(x) = \sin(kf_n(x))$?

Why does the recursion $f_{n + 1}(x) = \sin(kf_n(x))$ with initial conditions $f_0(x) = x$ quicly display global chaotic behavior when $k > \pi$? I have a very limited knowledge in nonlinear ...
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Show validity of Dulac criterion in not simply connected region

can anyone suggest me a not simply connected region where Dulac criterion is still valid? When I think on a not simply connected region the only thing that comes to my mind is an annulus, but in an ...
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Demostration of the minimun radius of curvature

I have a doubt about how to demostrate that the minimun radius of curvature is when the particle is at the maximum point. Using this formula:$\dfrac{\left[1+\left(\dfrac{\operatorname d \!y}{\...
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How to prove that the action of two dimensional torus is Hamiltonian?

Given the 2 dimensional torus: $T^{2}={(\varphi_{1} \mod 2\pi, \varphi_{2} \mod 2\pi)} $ that acts on $CP^{1}\times CP^{1}$ by: $ \left(\varphi_{1},\varphi_{2}):((z_{0}:z_{1}),(w_{0}:w)\right)) \to ...
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$\Sigma$-equivalence between $(y,z,1)$ and $(y+\mathcal{O}(3),z+\mathcal{O}(3),1)$

Consider the sets $\mathfrak{X}(\mathbb{R}^3) = \{X: \mathbb{R}^3 \to \mathbb{R}^3; X \mbox{ is smooth}\}$ and $\Sigma = \{0\}\times\mathbb{R}^2$. Let $X, Y$ be vector fields in $\mathfrak{X}(\mathbb{...
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1answer
28 views

Equivalent condition of a diffeomorphism having a dense orbit

Say $M$ is a manifold and $f: M \to M$ is a diffeomorphism. Assume also that, if we are given any nonempty open subsets $U$ and $V$, then there is $n \in \mathbb{Z}$ such that $f^n(U)$ intersects $V$....
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Discrete-time linear control with linear state/input constraints

Given a controllable discrete-time linear system $x(k+1) = A x(k) + B u(k)$ the input sequence leading from state $x_0$ to $x_f$ is given by $C^{-1} (x_f - A^n x_0)$ where $C$ is the controllability ...
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2answers
43 views

How do the steady states change as we change the parameter?

Consider the following ODE $$ \frac{dx}{dt} = x \left(1-\frac{x}{m} - \frac{a}{1+x}\right), $$ where $a$ is a bifurcation parameter and $a\in(0,\infty)$ a positive constant. How do you find the ...
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1answer
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Easy way to show the uniqueness of the solution of the time dependent linear system?

Consider the time dependent linear system in $\mathbb{R}^n$: $$ \dot{x} = Ax + b(t), $$ where $b: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is continuous. The unique solution satisfying the initial ...
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How two change the coordinate system of dynamical system with complex eigenvectors?

Let $A \in \mathbb{R}^{n \times n}$ be a real matrix. suppose $\lambda = \alpha + i \beta$ is a complex eigenvalue of $A$ with complex eigenvector $w = u+ i v$ in $\mathbb{R}^{n}$. Let $\bar{\lambda } ...
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Interpolation of Discrete Dynamical System [on hold]

Let $ A:\mathbb{N}\times \mathbb{R}^d\rightarrow \mathbb{R}^d, $ be a flow on $X$, such that there exists a function $f:\mathbb{R}^d\rightarrow \mathbb{R}^d$ satisfying $$ \lim\limits_{n\mapsto \infty}...
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1answer
37 views

How to linearize the system and then get the eigenvalue?

I have a system \begin{align} \dfrac{dx}{dt}&=-x^2 + 4 y^2, \\ \dfrac{dy}{dt}&=-8 - 4 y + 2 x y. \end{align} There two singular points $A_1(-2;-1), A_2(4,2)$. I need to know the type of these ...
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1answer
22 views

The structure of the limit set $w(x)$

Im currently studying differential equations and dynamical systems, specifically from the book of Gerald Teschl. I have a question about limit sets. Def: The $w_{+}(x)$ (resp $w_{-}$) of a point $x$...
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1answer
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How to show the uniqueness of the solution of the time dependent linear system in $\mathbb{R}^n$?

Consider the time dependent linear system in $\mathbb{R}^n$: $$ \dot{x} = Ax + b(t), $$ where $b: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is continuous. Prove that $$ x(t) = e^{At}x_0 + \int_0^t e^{...
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1answer
40 views

Problem in Ergodic theory

Let $(X,T,\mu)$ be a classical dynamical system, where $(X,\mu)$ is a probability measure space and $T$ is a measure preserving invertible transformation. Let $U$ be the unitary on $L^{2}(X,\mu)$ ...
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1answer
58 views

Finding the critical value of bifurcation

I want to find the critical value/bifurcation value, $a_c$, I'm not sure if I'm approaching this the correct way. $$ \dot{x} = x(1-x) - \frac{1}{4}(a+1)^2\left(\frac{x}{x+a}\right)$$ A previous ...
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14 views

Horocylic flow and the velocity at which $c_t$ fills space

The horocyclic flow is the dynamics generated by the matrices $h^s_+,$ which are stable manifolds. Here I am considering the flow on the space of lattices with area $1.$ $$ h^s_+= \begin{pmatrix} 1&...
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1answer
36 views

Billiards in a holey square

Suppose you start a point-billiard (or light ray) in a square at a random location, shooting off at a random angle, reflecting with angle-of-incidence equals angle-of-reflection. In general, because ...
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1answer
50 views

Relation between the eigenvalues of matrices conjugated by a homeomorphism.

Let $A, B$ be $2\times 2 $ matrices satisfying: The eigenvalues $\lambda,\mu$ of $A$ satisfy $|\lambda|<1<|\mu|$. The eigenvalues $\lambda',\mu'$ of $B$ satisfy $|\lambda'|<1<|\mu'|$. ...
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Sub-basic pieces have local product structure.

I am reading the proof of the Spectral Descomposition Theorem for hyperbolic diffeomorphisms. I am going to (kind of informally) define, local product structure: Given a hyperbolic set $\Lambda$, ...
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Given commutative non-linear vector fields, the linearization commute

Given $f_1, f_2$ be $C^1$-vector fields on $\mathbb{R}^n$, we define the lie bracket $$ [f_1,f_2](x)=\frac{df_2}{dx}(x)f_1(x)-\frac{df_1}{dx}(x)f_2(x)$$ They are said to commute if the lie bracket ...
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Understanding the notation when finding action-angle coordinates

I'm trying to learn the basics of KAM theory and I wanted to first get to grips with Liouville integrability for Hamiltonian systems but I'm getting rather confused by the notation which seems to be ...
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1answer
55 views

How to show the solution of $\dot{x} = Ax $ is an invariant subspace?

Consider the linear dynamical system $\dot{x} = Ax $ in $V$ a finite dimensional vector space. The definition of an invariant subspace $U$ is as follows: For all $x_0 \in U$, (initial condition), ...
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Is the infinite product map $(∏_{i=1}^{∞}S_{i})×f$ topologically transitive

Let $f:X→X$ be a map. We say that $f$ is topologically mixing if for every open subsets $U,V$ of $X$, there exists $N$ such that for every $n≥N$ the set $f^{n}(U)∩V$ is non-empty. Let $S : X → X$ and ...
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15 views

Limit of Discrete Dynamical System

Let $\{A^n\}_{n \in \mathbb{N}}$, $\{v^n\}_{n \in \mathbb{N}}$ be sequences of $d\times d$ matrices and vectors in $\mathbb{R}^d$. Write the dynamical system defined by the discrete-time flow on $\...
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1answer
22 views

Is the Lorenz system well-posed in the Hadamard sense?

Apologies if this has already been discussed, but I searched the site and I couldn't find an answer. For the sake of simplicity, consider only ODEs, possibly depending on some vector of parameters $\...
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show that the system has a limit cycle using a theorem

I have the following system, and I'm trying to use a theorem to prove that it has a limit cycle. I proceeded finding the fixed points(Strogatz), $(0,0)$ in this case and then calculating the ...
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Limit Map of Discrete Dynamical System

Let $f:H\rightarrow H$ be a countinuous map from the separable hilbert space into itself, for every $x\in H$ define the discrete dynamical system $$ \xi_x^{n+1}\triangleq f(\xi^n_x);\qquad \xi^0_x\...
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2answers
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How to translate from a 2x2 state-space difference equation to a 2nd-order difference equation

I have a state-space evolution equation of the form $$\begin{bmatrix}u_k\cr v_k\end{bmatrix} = \begin{bmatrix}1-a & c \cr b(a-1) & 1-bc\end{bmatrix} \begin{bmatrix}u_{k-1}\cr v_{k-1}\end{...
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1answer
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Stable and unstable manifolds that are tangent to each other in a continuous dynamical system?

I am thinking of a scenario/ examples where the stable and unstable manifold of an equilibrium of a continuous dynamical system are tangent to each other? Any examples/ plots would be helpful?
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1answer
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How to get a basic knowledge on bifurcation analysis of dynamical systems and on the nature of delay equations?

My professor agreed to get me as her research assistant. She asked me to use DDE-biftool in Matlab to find computation of steady state and computation of stability etc. Some examples of the work can ...
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50 views

Energy Conservation and Speed of a Falling Satellite

I've been given the following problem (admittedly on a homework sheet), which I've solved, but I feel there has to be a neater method of solving it: A satellite falls freely towards the Earth ...
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Examples of prime ideals for Lie algebras

I am looking for examples of prime ideals for Lie algebras. In particular, I am interested in examples involving the Lie algebra given by the commutator of endomorphisms of a complex vector space and ...
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1answer
55 views

Unique ergodicity and first return time

I'm trying to solve the following problem: Let $T\colon X\to X$ be a continuous map on a compact metric space $X$, uniquely ergodic. Let $Y\neq \emptyset$ be an open set. Show that $t(x) = \min\left\{...
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1answer
32 views

Continuity of a function at $x^*$ implies that a property $P(f(x^*))$ holds for points near $x^*$?

I'm working through Numerical Optimization by Nocedal and Wright, and I'm having trouble with some of its proofs that seem too handwavy to me. Take the first theorem for example: Theorem: If $x^∗$ ...
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Name of a particular kind of identity block matrices

Do matrices that have block structure made of identity matrices with negative identity matrices along the main diagonal have a name? More specifically, do matrices such as: $$ {{A}_{n}}=\left[ \begin{...
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How to process Adjacency Matrix to Jacobian Matrix to ODE SysteM?

Hy all, if I have adjacency matrix, let say $ 3\times 3$ \begin{bmatrix} 1 &0& 0 \\ 0& 2& 2\\ 1& 3& 1 \end{bmatrix} How to process this to Jacobian matrix and then to ODE ...
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1answer
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Does anyone knows how to average function in three dimensions

$$ \begin{split} \frac{dr}{dt} &= -\epsilon \sin^2(\theta)z \\ \frac{d \theta}{dt} &= -1-\epsilon \cos\theta \sin\theta z\\ \frac{dz}{dt} &= \epsilon(r^2-T) \end{split} $$ ...
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Can a family of functions $F_{\lambda}$ experience a saddle-node bifurcation and also experience a period -doubling bifurcation?

Suppose $F_{C}=(x-C)^{2}$ is a family of functions. I found that at $C=-\frac{1}{4}$ there is a saddle-node bifurcation. So for $C<-\frac{1}{4}$ there are $0$ fixed points, $C=-\frac{1}{4}$ there ...
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1answer
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For what values does my dynamical system produce periodic orbits?

For what values of $a$ does the function, $f$, contain periodic orbits, where $f$ is given by: $$f(x)=a+x \mod 1.$$ It seems for any rational number $a$ you get periodic orbits although I don't know ...
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1answer
47 views

Exchanging limits when functions may not converge uniformly

Let $X$ be a compact metric space and for each $i \geq 1$, let $f_i \colon X \to \mathbb{N}$ be continuous functions satisfying: $f_{i+1}(x) \geq f_i(x)$ and; for each $n \in \mathbb{N}$ ...
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2answers
41 views

Difference Equation Initial Value Problem

Solve the IVP: $$y^2_{k+2}-4y^2_{k+1}+m\cdot y^2_k=k, m \in \mathbb{R},$$ $y_0=1, y_1=2, y_2=\sqrt{13}$ I started by taking $k=0$ $y^2_2-4y^2_1+my^2_0=k$ $\Rightarrow13-16+m=k$ $\Rightarrow m=k+3$ ...
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32 views

Family of functions

I took a dynamical systems coursre and I have a hard time understanding the definition of a family of functions: Let $y_{k+n}+a_1y_{k+n-1}+\dots+a_ny_{k}=R_{k}$ be a difference equation Then we call ...
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About topological conjugacy

The Smale horseshoe map $f$ is desribed in this page: What's the point of a Horseshoe map? A striking feature of this system is the stability of its dynamics: given any diffeomorphism $g$ ...
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1answer
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Prove that $s(v-u)=uvh$ in Distance, Time and Speed Word Problems

Question: A train is timmed to run from Howrah to Delhi at an average speed of $u$ kilometers per hour. Due to some engine trouble the starting of the train was delayed by $h$ hours. At Delhi the ...
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1answer
81 views

If $X,Y$ are two equivalent vector fields in two open sets $A$ and $B$, such that $A\cup B = M$. Are $X$ and $Y$ equivalent?

Let $X$ and $Y$ be smooth vector fields on $\mathbb{T}^2$. Definition 1: Let $A$ be an open subset of $\mathbb{T}^2$, we say that $X$ and $Y$ are equivalent in $A$, if there exists a homeomorphism $...
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1answer
29 views

Why does the limit set contain forward orbit?

I just started reading on dynamical systems and the author made a claim that I don't quite understand. If someone could elaborate it would be highly appreciated! So here is the problem: Let $X$ be a ...
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1answer
46 views

Understanding $T(x)=3x\pmod 1$ [closed]

Let $T:[0,1]\to[0,1]$ be such that $T(x)=3x\pmod{1}$ which is measurable with respect to the $\sigma$-algebra of Borel on $[0,1]$, which we denote by $\mathscr{B}_{[0,1]}$. Prove that Lebesgue ...
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0answers
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How to demonstrate the efficiency of a dynamic factor in comparison with some constant coefficient with a chart?

I proposed a measure which estimates the coefficient of each sample dynamically, So the coefficient of two sample s1 and s2 can be different. To show the effectiveness of this measure, I want to ...
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22 views

What is the steady-state solution of stochastic differential equations?

I'm starting to study stochastic differential equations by my own and would like to know what is the meaning of steady-state solutions of a stochastic differential equation and how to compute it. For ...