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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56 views

a question on quasi-invariant measures (with respect to the irrational rotations) on the unit circle

Fix a $\sigma$-finite atom-less measure $\mu$ on the unit circle, which is quasi-invariant and ergodic under the rotation $T$ of the angle $2\pi\theta$, $\theta$ irrational. By a well-known result of ...
3
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0answers
15 views

Equality of measure sets of dynamical system

This is a homework question I have been crunching my brains on for a lot of time, but unfortunately I'm stuck. I would greatly appreciate any help! The problem is as follows: We have some continuous ...
1
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1answer
26 views

singularity and degeneracy of an ODE

I have trouble distinguishing the difference of singularity and degeneracy in the context of ODE theory. Could anyone give me a couple of examples in illustrating the difference of singular point and ...
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0answers
24 views

Stadium billiard reflection angles

Given a boundary and a massless particle with constant velocity with a certain direction, a billiard consists of an experiment where the particle collides with the walls conserving its velocity ...
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1answer
32 views

Lemma 1.7 in chapter 5 in Ergodic Theorems by Ulrich Krengel.

The lemma is on page 181. Here's the lemma and its proof: I don't understand how did they get: $$\lim | S^{n_k+1}f(\omega)|\leq \gamma (\alpha - \epsilon)+(1-\gamma)\alpha$$ The term that is ...
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1answer
31 views

Definition of Markov partition?

My teacher handed out an excerpt from a book by Robinson on chaotic dynamical systems. The excerpt is from a chapter on Markov partitions, and the following part has me confused: Let $$f(x)= ...
3
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1answer
48 views

periodic solution of $x''-\ (1-\ x^2-\ (x')^2)\ x'+x=0$

Assume differential equation $$x''-\ (1-\ x^2-\ (x')^2)\ x'+x=0$$ I want to discusse about non-constant periodic solution of it. Can someone give a hint that how to start to think. And does it have ...
3
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1answer
267 views

Estimating a dynamical system's behavior without using Liapunov theorem

Assume that we have the following dynamical system $$x'=(\epsilon x+2y)(1+z)$$ $$y'=(-x+\epsilon y)(1+z)$$ $$z'=-z^3$$ Then how can I show that any solution that started from the region $z>-1$ ...
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1answer
53 views

Hopf bifurcation phase portrait orientation

Say for a system $$\dot{x}=y$$$$\dot{y}=-x+\mu y -y^3$$ I have confirmed a hopf bifurcation occurs at the origin and that the branches are stable i.e., a stable limit cycle and the origin being stable ...
5
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1answer
122 views

Proof for the existence of a second fixed point in Poincaré's last geometric theorem

In "Geometry and Billiards" by S. Tabachnikov the author proves Poincaré's last geometric theorem: "An area-preserving transformation of an annulus that moves the boundary circles in opposite ...
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0answers
9 views

Locally Linear Systems-repeated $\lambda$

For a locally system whose corresponding linear system has repeated eigenvalues, the type of equilibrium point cannot be determined. I know that the locally Linear system equilibrium can possibly be a ...
3
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0answers
50 views

Teacher Challenge - multiple parts [on hold]

This was his challenge: "I would like you to consider the function $x^{r+\alpha}$, $r$ is an integer, $\alpha$ is a real number between $0$ and $1$. Differentiate it until you get a singularity ...
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1answer
23 views

Proving F is an Integral of the Linear Map L

In the question, I'm asked to show that \begin{align*} F\begin{pmatrix}x\\y\end{pmatrix}=x^2+y^2 \end{align*} is an integral for the linear map \begin{align*} L(\text{x})= \begin{pmatrix} ...
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2answers
340 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
26 views

“Evenly” dense orbit?

I want to prove the following: let $a$ be an irrational constant and $m$ an integer. Then $$\lim_{n \to\infty} \frac{1}{n}\sum_{k=0}^{n-1}e^{2\pi m i (x+ka)} = \begin{cases} 0, & m\not=0 \\ 1, ...
3
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0answers
31 views

Poincare map trouble

Consider $ X' = F(X)$, $F \in C^1(\mathbb{R}^2)$. Suppose that the system has an orbit $\mathcal{O}_p$ and $\Sigma$ an transversal section in $P$. Show that if $$\pi^{n+1}(\Sigma) \subset ...
10
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1answer
116 views

limit cycle of an ODE system

Consider a planar ODE system $z'=f(z)$ with $z=(x,y)$, $$ f(x,y)=(xy^2-x-y,y^3+x-y). $$ Using polar coordinates, one can get $$ r'=r(r^2\sin^2\theta-1),\quad \theta'=1. $$ With Mathematica one can ...
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1answer
20 views

About the Sharkovsky Forcing Theorem

(Sharkovsky Forcing Theorem ). If $m$ is a period for $f$ and $m⊲ l$ , then $l$ is also a period for $f$. I have the following question: Let $f$ be a such map having a period three, So $f$ is ...
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0answers
19 views

Orbit closure is uncountable, unless there is a periodic element.

Let $a = (a_i)_{i \in \mathbb N}$ be a sequnece over some finite alphabet $\Sigma$. We may define on the space $X = \Sigma^{\mathbb N}$ a shift operation by $(Sx)_i = x_{i+1}$. Let $A$ be the orbit ...
1
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1answer
187 views

Maximal Positive Invariant Set — Some fine print

I would like to share something I noticed on the definition of Maximal Positively Invariant Sets. Definition 1. For a discrete-time system of the form $x_{k+1}=f(x_{k})$ (and $x_{k}\in ...
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0answers
35 views

Reducing a system of differential equations

Let $\mathbf F$ be a system of 1st order differential equations in $n>3$ variables $$\mathbf{F} : \mathbb{R}^n \to \mathbb{R}^n$$ $$\frac{d\mathbf{u}}{dt} = \mathbf{F}(\mathbf{u})$$ such that ...
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0answers
41 views

No periodic solution using Bendixson's criterion and Global analysis.

Theorem: Let $Z:U\subset \mathbb{R}^2\rightarrow \mathbb{R}^2$ a $\mathcal{C}^1$ field defined in a simply connected set $U$. If $\mathrm{div} Z(x)\neq0$ for all $x\in U$, then $Z$ does not have any ...
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1answer
17 views

A sufficient condition that domain of solution of differential equation became $\mathbb R$

If $ f:\mathbb R^n\to \mathbb R $ be bounded and continous then differential equation $$x'=f(x)$$ has a solution with domain $\mathbb R$. outlook of proof : if the maximal domain of solution is ...
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1answer
34 views

Dynamical System , Series : can't find the general terms

I have a dynamical system defined as follow : $$V_{n+3} - 6V_{n+2} +12V_{n+1} - 8V_n = 8, ~ \mbox{with}~ V_0=V_1=V_2=1$$ I have to find $V_n$ = ? So I began by solving this equation : $$x^3 -6x^2 ...
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1answer
18 views

Dynamical Systems, a question about first order DE asking for an example. [closed]

Construct an example of a differential equation depending on a parameter a for which some solutions do not depend continuously on a.
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0answers
51 views

If the limit matrix of a linear system has eigenvalues of negative real part, then the system is asymptotically stable

I appreciate if anyone can help me on this question: You are given the following linear system: $x'(t)=A(t)x(t)$. Suppose that $\lim_{t\to \infty}A(t)=A_{\infty}$ and that all eigenvalues of ...
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0answers
35 views

Homeomorphism between the group of $S(O)_{2}$ and the $S_1$.

During an exam I had to prove the following: "Let there be a dynamical system of $n=2$ dimensions and let the eigenvalues that correspond to it, to be imaginary with their real part equal to zero. ...
0
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0answers
11 views

Coupled Discrete Dynamical Systems in Complex plane.

Consider two dynamical systems $$Z_{n+1}=f(W_n, Z_{n-1})$$ and $$W_{n+1}=f(Z_n, W_{n-1})$$ where $z_0, w_0,z_{-1}, w_{-1}$ are given. The function $f, g$ are defined from $\mathbf{C}^2$ to ...
1
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1answer
41 views

Nature of Equilibrium Points

I would like to prove the following: "The nature of the equilibrium points (i.e. stability/instability) of a one-dimensional differential equation remains invariant under the effect of the ...
0
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1answer
25 views

Behavior of Non-Hyperbolic Equilibria?

So I'm working on a differential equation problem concerning epidemics - we're using the Kermack-McKendrick model. I've reached a point where I need to sketch phase portraits near my equilibria, ...
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0answers
333 views

How to numerically find Floquet multipliers (e.g., characteristic multipliers or Lyapunov exponents for periodic orbits from chaotic systems)?

I understand the theory (c.f., Perko or Nayfeh and Balachandran, Ch.3), but I do not understand how this is accomplished numerically: Given a (chaotic) dynamical system (for example, I am using the ...
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2answers
77 views

Every solution of the system is attracted to the center manifold

I am trying to solve the following problem. Determine a center manifold for the rest point at the origin of the system \begin{align} \dot x &=-xy \\ \dot y&= -y+x^2-2y^2 \end{align} a) ...
12
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4answers
373 views

Why solutions of $y''+(w^2+b(t))y=0$ behave like solutions of $y''+w^2y=0$ at infinity

Assume $w>0$ and $b(t)$ be continuous on $[0,+\infty)$ and $\int_0^\infty |b(t)| dt <\infty$ show that $y''+(w^2+b(t))y=0$ has solution $\phi(t)$ such that $$\lim_{t\to\infty} ...
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0answers
22 views

Cesaro bounded.

The exercise is from Ulrich Krengel's book, Ergodic Theorems, on pages 173-174. First preliminary notions: a function $h$ with $T^*h=h$ is called harmonic, where $T$ is a contraction in $L_1$. $Y= ...
1
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3answers
69 views

Find the fixed points

We have $x_{n+1} = ax_n +b$ with $x_0$ given. We have to find the fixed points of this function, and decide for which values of $a$ they are stable. So I looked it up and found that a fixed point is ...
-1
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1answer
19 views

Undamped Forces

I want to make sure I am doing this problem correctly, especially when it comes to drawing the potential function V(x). Consider the system of differential equations: $$\dot {x}=y$$ $$\dot ...
2
votes
2answers
86 views

Closed orbits of vector fields under perturbation

Consider a vector field $V$ on an annulus $U$, say. Also, assume that the vector field $V$ has a closed orbit. I am looking for a reference that gives stability results of the following type: If the ...
1
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1answer
42 views

Closed orbits of dynamical systems

Consider the system $$\dot{x}=x-rx-ry+xy, \qquad \dot{y}=y-ry+rx-x^2,\qquad r=\sqrt{x^2+y^2},$$ which can be written in polar coordinates $(r,\theta)$ as $$\dot{r}=r(1-r), \qquad \dot{\theta}=r(1-\cos ...
0
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1answer
31 views

Stable and unstable manifolds of fixed points

I want to make sure I understand the definition of these terms. If someone could correct me or let me know if I am right I would appreciate it. The stable manifold of a fixed point is the set of ...
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0answers
23 views

Gronwall's inequality for LTI scalar systems with an input

Suppose we have a dynamical system such that $$\dot{x}(t)\le-\alpha x(t) + u(t),$$ with $\alpha>0$ for all $t\ge0$. Can we say that $$x(t) \le e^{-\alpha(t-t_0)}x(t_0) + \int_{t_0}^t ...
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0answers
49 views

Dichotomy for global existence or blow up for solutions of evolution problems.

Consider the problem (Nonlinear Schrödinger equation) \begin{equation} \left\{ \begin{array}{rl} iu_t + \Delta u\mp u|u|^{\alpha}=0\\ u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\ ...
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1answer
58 views

Question about having periodic solution.

Assume $a>0$ and $b>0$ and $g(x)=0$ when $|x|>1$ , $g(x)=k$ when $|x|\le1$ . Now show that in system of differential equation $$x'=y $$ $$y'=-[2b-g(x)]ay-ay^2$$ if ...
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0answers
48 views

The “muscle” behind the fact that ergodic measures are mutually singular

This is really motivated by the soft question at the end, but let me begin with something more circumscribed: Let $(X,\mathcal{B})$ be a measurable space and let $T:X\circlearrowleft$ be a self-map ...
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0answers
32 views

Hausdorff dimension of a ball

Let $\{f_1,\dots,f_m\}$ be an IFs and $E_n$ be the associated self similar set. It's known that $E_n$ is a union of disjoint balls $B(x_i,R\cdot r^n)$ (balls with same radius but not the same ...
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1answer
21 views

Omega limit set of omega limit set $\omega(\omega(a))$

Consider a dynamical system with a flow $\phi(t;a)$, and let $A\subset \mathbb{R}^n$. The omega limit set of $A$ is defined as the union of all $\omega(a)$ over all $a\in A$. Since for a given $a$, ...
1
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1answer
30 views

Proving that the Bernoulli self similar measure is doubling

Let $\mu_p$ a measure which is the push forward of the bernouli product measure $(p,1-p)^\mathbb N$. Let S=$\{f_1,\dots f_m\}$ an IFS, a system of functions with attractor $K$, means ...
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2answers
120 views

Are the values generated by non-linear equations truly random?

I was recently studying some literature on chaos theory and non-linear equations . where in various ciphers were created using non- linear equations like Lorenz equation . Are the data generated from ...
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0answers
39 views

Rational dynamical system with nonnegative paramaters and with nonnegative initial conditions

let $A$ be a rational system of the form :\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ ...
4
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1answer
44 views

Chaos in Newtons Method

Im trying to prove that Newtons method applied to ${\rm f}\left(\, x\,\right) =x^{2} + c$, is chaotic for $c > 0$. I know I need to prove: (a) The periodic points of ${\rm f}$ are dense in $X$, ...
0
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1answer
48 views

Dynamical Systems Periodic Orbits existing

Consider the nonlinear dynamical system $(1)$ : $x' = y(1 + x−y^2)$, $y' = x(1 + y−x^2)$, where $(x,y)\in\mathbb{R}^2$. (i) Determine the equilibrium points of $(1)$ (ii) Classify the ...