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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50 views

Analyzing the singularity of ODE system

It is asked to analyze the singularities of the system $$\dot{x} = y e^y$$ $$\dot{y} = 1-x^2$$ I've found that the singularities are (1,0) and (-1,0) The linearization of the sysyem give the matrix ...
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95 views

Properties of join of open covers

I am studying the basics on Ergodic Theory from Fundamentos da Teoria Ergodica by Oliveira & Viana (you can actually find the pdf on their website), particularly, the properties of topological ...
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49 views

From multivariable system transfer function matrix to state space representation

I have the transfer function matrix $H(s) = \begin{bmatrix} {1\over s+1} & {2\over s+2} \\ {-2\over s^2+3s+2} & {2s\over s+1} \\ \end{bmatrix}$ And I want to ...
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40 views

Time period of oscillations of a point about the function's minimum value?

How am I to go about the following problem? Please do not explicitly solve it. Let $E_0$ be the value of the potential function at the minimum point $\xi$. Find the time period $T_0=\lim_{E\to ...
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26 views

Dynamics - (stable/unstable) focus - motion direction - CW/CCW?

How to determine the direction a stable focus (source) or unstable focus (sink) is rotating, given the eigenvalues $\lambda=\alpha\pm\beta i$ ? I know that if $\alpha > 0$ then it is source and if ...
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177 views

Has this difference equation :$x_{n+1}=Ax^2_{n}-Bx^2_{n-1}$ been studied before?

I posted this problem before in MO but only I would like to know if this difference equation :$$x_{n+1}=Ax^2_{n}-Bx^2_{n-1}$$ where $A(\theta)= \cos(\theta)$ and $B(\theta) =\sin(\theta)$ are ...
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26 views

What can be said about the space of vector fields for which a given, say $C^1$, function is a Lyapunov function?

I am learning Morse homology and I have been thinking about the following observation. One way of doing, say finite-dimensional, Morse theory is by fixing a Morse function $f\in C^{\infty}(M)$, where ...
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41 views

Fixed points that are NOT convergent points

Are there any fixed points that are NOT converget (aka attractig fixed points) in the sequence $x_n = 5\ln x_{n-1}$? How do you determine this?
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26 views

what if an orbit is contained in its $\omega$- limit set?

I guess it should be a periodic orbit, but I'm not sure whether there is an counterexample or not. can you give me a proof or an counterexample?
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26 views

about Poincare map

I saw that the Poincare map is defined by the flow of the periodic system with the least period $T$. that is, $$P(x):=\phi_T(x)$$ is a Poincare map with flow $\phi$ of time $T$. but I think if we ...
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1answer
30 views

Geodesic flow on a compact manifold is defined for all time

How can I prove that on a compact manifold, the geodesic flow is defined for all time? Is this as simple as citing the Hopf-Rinow theorem?
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29 views

Geodesic Flow is an Anosov Flow

I am trying to understand why geodesic flow on a compact surface of constant negative curvature is an Anosov flow. Klingenberg's book, Riemannian Geometry, says that in this case, the proof is very ...
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27 views

Can a countably infinite compact topological space have isolated point? Can it admit a minimal subsystem?

Examples I could think of are all sequences with their limit. But is every countably infinite compact space admit atleast one isolated point?
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55 views

How can I solve this variable-coefficient ODE system?

I originally have a linear, homogeneous, second-order variable coefficient ODE system of this form: $X''(x) = A(x)X(x)$, where $X(x) = $\begin{bmatrix} f(x) \\ g(x) \\ \end{bmatrix} ...
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30 views

Effect of weight perturbations on eigen values

I have a dynamical system of coupled differential equations: $\dot{X}$ = -X + S( AX + BI) where X = 7x1 vector, A = 7x7 and B = 7x1 vector. S(x) is a nonlinear sigmoid function. I want to find the ...
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38 views

Finding conditions of non existence of Periodic orbit

$$ x'=y \mbox{ and } y'=ax-by-x^2y-x^3 $$ I need non-existence of periodic orbits. Which conditions $a$ and $b$ in $\mathbb{R}$ must satisfy? First, one can see that if $a\leq 0$, then the system has ...
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18 views

“Damping factor” for a set of non-linear ODEs

I am not sure if this question is on-topic here I have a set of four non-linear ODEs representing a negative feedback. I have done parameter variation by random sampling to study the sensitivity of ...
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60 views

Meaning of the expression “orientation preserving” homeomorphism

The only time that I've heard the term "orientation-preserving map" was in Linear Algebra, but today I read the term orientation-preserving homeomorphism of the circle in the following context: If a ...
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28 views

Are these definition equivalent about a periodic point?

Given a dynamical system $(X,G)$, def1. A point $x\in X$ is called periodic, if there exist a syndetic set $S\subseteq G$, such that $Sx=\{x\}$. def2. A point $x\in X$ is called periodic, if $Gx$ is ...
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22 views

A question on recurrent points of group acitons

Given a dynamical system $(X,G)$, A point $x\in X$ is called recurrent, if for any neibourhood $U$ of $x$, there exist a $g\in G$, $g\neq e$ such that $gx\in U$. If $G$ is a topological group and ...
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20 views

Linear affine random dynamical systems - positive Lyapunov index proof check?

Consider the affine random dynamical system $$ X_n = \mathbf A_n X_{n-1} + R_n, $$ starting from an initial non-zero position $X_0$, where $\mathbf A_n\in\mathbb C^{d\times d}$ and $R_n\in\mathbb ...
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1answer
32 views

If $f(x) \le f(Tx)$ then $f(x)=f(Tx)$ almost everywhere ( $T$ is $\mu$-invariant )

Let $X$ be a probability space with probability $\mu$. Let $T:X\to X$ be a measurable and $\mu$-invariant transformation, i.e $\mu \left(T^{-1}A \right) =\mu A. $ for each measurable subset $A\subset ...
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16 views

Trajectories in orthogonal systems

Please forgive any awkward phrasing or misuse of terminology. My education isn't entirely formal. Question Am I right in guessing these orbits trace lissajous-ish figures on hyperspheres? ...
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25 views

Dynamics for a non-linear flow

I'm studying for an upcoming for an exam and I found a question I'm having trouble with in a past paper. a) Assume a non-linear flow $\dot x = F_\mu(x)$. At $\mu = 4$ the only stable dynamics is a ...
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25 views

how can I determine if a MIMO system is ''similar'' to another one?

I'm talking about closed loop regime. I have a complex model and a simplified one, I want to show that they behave similarly in closed loop. Both of them are MIMO. for SISO I could use the ...
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37 views

Show that the first quadrant of a dynamical system is invariant.

I have the following dynamical system $${dp \over dt} = p(1-p-q)$$ $${dq \over dt} = q(p-{1 \over 2}-q)$$ and I have to show that the first quadrant ( $p, q \ge 0$ ) is an invariant set. I know what ...
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1answer
21 views

Find a vector field on the plane such that $\omega$-limit set of a single point is two parallel lines.

I came across this puzzling question in "Chaos" by K. T. Alligood. Sketch a vector field in the plane for which the $\omega$-limit set of a single trajectory is two (unbounded) parallel lines. ...
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51 views

Linearisation of nonlinear system

I'm asked to write the linearisation of this nonlinear system around the equilibria: $$\begin{cases} x_{t+1}=-x_t+2x^2_t \\ y_{t+1} =-2x_t^2-y_t\end{cases}$$ The two equilibria are therefore: ...
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16 views

Linear affine random dynamical systems - positive lyapunov exponents?

Consider the affine random dynamical system $$ X_n = \mathbf A_n X_{n-1} + R_n, $$ where $\mathbf A_n\in\mathbb C^{d\times d}$ and $R_n\in\mathbb C^{d}$ with $R_n$ almost surely not equal to the zero ...
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12 views

Dimension of global attractor - How to find it?

I have a very general question. I have a dynamical system and found a global attractor, how can I determine the dimension of that global attractor? Do you have helpful hints, links etc.? I sit in ...
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63 views

About the existence of a regular solution in a chaotic system

Show that for some choice of $x_1\in\mathbb{R}$, the sequence given by: $$ x_{n+1} = n-3^{x_n} $$ satisfies: $$ \lim_{n\to +\infty} x_n= +\infty. $$ I was able to prove the statement by showing ...
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232 views

Checking the stability of an equilibrium point

I have the linearization of a non-linear system about an equilibrium point as follows $$ \dot x = (-A+M)x, $$ where $x\in\mathbb{R}^3$, $A$ is a positive definite matrix and $M$ has its eigenvalues ...
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37 views

Sketching bifurcation diagrams.

Consider the system: $$\dot{x}=x(\mu - x + y^2), \ \dot{y} = y(1-x+y^2)$$ I've been asked to consider bifurcations of the fixed points $(\mu, 0)$ and $(0,1)$ at $\mu = 1$ and $\mu = -1$ ...
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51 views

Bottlenecks in Dynamical Systems

Consider the equation $\dot{x} = r+x^2$. When $0 < r \ll 1$, this system experiences a bottleneck effect. Then the time $T$ spent in this bottleneck can be approximated by: $$T_{bn} = ...
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23 views

$A$ Attractor $\implies\exists$ open neighborhood $V$ of $A$ such that $\omega(x)\subset A$ for all $x\in V$

Let $X$ be a compact metrizable topological space and $f\colon X\to X$ continuous. Let $A\subset X$ be an attractor. Show that there exists an open neighborhood of $A$ such that for the ...
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24 views

Relation between hitting and return time.

Given a dynamical system $(X,\mathcal{B},\mu,T) $ where $X$ is a space, $B$ is borel $\sigma$-algebra, $\mu$ is a probability measure and $T$ is a $\mu$ invariant transformation i.e. ...
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1answer
33 views

Recurrent points and rotation number

I need to show that if $f:S^{1} \rightarrow S^{1}$ is a preserving-order diffeomorphism and $f$ has irrational rotation number, then $f$ has at least one recurrent point. How can I prove that? ...
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23 views

Bifurcations of dynamical systems, different parameters [closed]

I've found the following excercise and I've broken my head about it, but I don't know how to answer it. So we have a system $$\frac{dx}{dt} = f(x,a_1,a_2,...,a_n)$$ And say we first take $a_1$ as ...
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25 views

Periodic Shifts of finite type

I am learning about periodic shifts and want to look at finite types of periodic shifts. So I have this theorem: A shift space $X$ is an $M$-step shift of finite type iff whenever $uv, \; vw\; \in ...
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41 views

Existence of a recurrent point [duplicate]

I need to show that if $f:S^{1} \rightarrow S^{1}$ is a preserving-order diffeomorphism and $f$ has irrational rotation number, then $f$ has at least one recurrent point. How can I prove that? ...
2
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1answer
29 views

Continuous Time Dynamical Systems with 2-cycles

I know that discrete time dynamical systems such as $x_{n+1} = rx_n(1-x_n)$ exhibit 2-cycles for some parameter values of r. I'm curious if there exist continuous time dynamical systems that exhibit ...
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101 views

How to interpret complex eigenvectors of the Jacobian matrix of a (linear) dynamical system?

Consider a linear ODE system of the following form: $$ \frac {dx} {dt} = Ax $$ In case $A$ has real eigenvectors, I can interpret them as the directions in which the system will move, if the initial ...
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18 views

colvolution function

I am trying to understand the equaliti denoted in the attache picture. Any help? Thank you!
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1answer
31 views

writing a state of a dynamical system

Is it un/common to write the state of a dynamical system in the following manner: $$ \begin{pmatrix}x_{t+1} \\ v_{t+1} \end{pmatrix} = \begin{pmatrix}A & B \\ C & D ...
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1answer
34 views

modelling the behavior of a particle

I want to study the evolution of a particle as a function of time, then in dynamical systems, the usual thing to do would be to define the state of the particle. Usually we are able to do this by ...
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1answer
48 views

Proving conjugacy to the Logistic Map

I have a map which I have to show is a conjugate to the Logistic Map ( $x_{n+1} = rx_n(1-x_n)$ ). The map in question is as follows. $x_n = \sin^2(\pi\theta_n)$ $\theta_{n+1} = N^n\theta_0$ mod $1$ ...
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47 views

What is corresponding Lie group for Lie algebra of vector fields in dynamical systems?

According to Ado's theorem, for every finite dimensional abstract Lie algebra there is a Lie group. Finite dimensional analytic (or meromorphic) Vector fields (in dynamical systems) over the filed of ...
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1answer
38 views

A generalization of Poincare-Birkhoff theorem

What could be the statment of a possible generalization of Poincare Birkhoff theorem for $M\times [0,\; 1]$ where $M$ is a compact orientable manifold?
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74 views

Solution for a differential equation

I am stuck in getting the solution for the following non-linear differential equation: \begin{equation*} x^2 + B\frac{dx}{dt} = A\sin(wt) \end{equation*} Is there any method to solve this kind of ...
4
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1answer
105 views

Does every homeomorphism of a compact metric space lift to the Cantor set?

This is a follow-up to this question. It is well-known that any compact metrizable space can be expressed as a quotient of the Cantor set. But can every homeomorphism of such a space be lifted to a ...