# Questions tagged [ds.dynamical-systems]

Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.

1,636
questions

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### What is the current status on methods to find limit cycles?

What are the current best methods to show analytically the existence of a limit cycle in a $n$-dimensional system of the form:
$$
\frac{\mathrm{d}}{\mathrm{d} t} \vec{x}(t)=\vec{f}(\vec{x})
$$
Where $...

**6**

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**0**answers

65 views

### On the first sequence without collinear triple

Let $u_n$ be the sequence lexicographically first among the sequences of nonnegative integers with graphs without collinear three points (as for $a_n=n^2$ or $b_n=2^n$). It is a variation of that one.
...

**19**

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424 views

### On the first sequence without triple in arithmetic progression

In this Numberphile video (from 3:36 to 7:41), Neil Sloane explains an amazing sequence:
It is the lexicographically first among the sequences of positive integers without triple in arithmetic ...

**27**

votes

**13**answers

3k views

### Unconventional examples of mathematical modelling

Disclaimer. The is admittedly a soft question. If it does not meet the criteria for being an acceptable MO question, I apologize in advance.
I'll soon be teaching a (basic) course on mathematical ...

**1**

vote

**0**answers

38 views

### Trapping lightrays under nonstandard reflections and/or paths

Almost every version of trapping lightrays with mirrors is either resolved---usually negatively---or open:
"It is unknown whether one can construct a polygonal trap for a parallel beam of light": ...

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101 views

### Does anyone have proof of this Lemma or know another reference where I can find [on hold]

[12] is this article
Hirsch, MW. (1970). Stable manifolds and hyperbolic sets. In Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968). UC Berkeley.
I could not find it

**4**

votes

**0**answers

95 views

### Can every dynamical system be interpreted in terms of (unitary) conjugation in an operator algebra

Let $H$ be a Hilbert space and $X$ be a compact Hausdorff space with a homeomorphism $\alpha: X \to X$. Assume that $C(X)$ is a commutative sub algebra of $B(H)$, namely $C(X)$ is embedded in $B(H)$...

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41 views

### The number of minimal components of a dynamical system via certain invariants of corresponding cross product $C^*$ algebra, some precise examples

Let $X$ be a compact Hausdorff space and $\alpha$ be a homeomorphism of $X$.
So we have a natural action of $\mathbb{Z}$ on $C(X)$ which generates the cross product algebra $C^*(X,\alpha)$...

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132 views

### About a lemma of the Bowen Book [closed]

I believe the passage I highlighted within the red square is wrong, therefore invalidating the proof of this motto. If it really is wrong. Does anyone know another demo and in which book can I find it?...

**4**

votes

**1**answer

88 views

### A non-geodesible foliation of $S^3$ or $S^2\times S^1$

Is there a $1$-dimensional foliation of $S^3$ which is not a geodesible foliation? Is there a $1$-dimensional foliation of $S^2\times S^1$ which is not a geodesible foliation?
If the answer is ...

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45 views

### Existence of the eigenvalue of the dual operator of the transfer operator

In the passage that I marked in green apparently the author uses a relationship between fixed point and eigenvalues. The result that I know of to ensure the existence of this eigenvalue requires that ...

**8**

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161 views

### The condition on $\alpha$ that $\alpha^n$ is convergent modulo 1

We consider numbers $\alpha\in \mathbb{R}$ with $|\alpha|>1$. Is there any result about a characterization of those $\alpha$ so that $\{\alpha^n\}_{n\in \mathbb{N}}$ is convergent modulo 1?
I ...

**2**

votes

**1**answer

120 views

### Orbit-based metric

Let $(X,d)$ be a metric space and $f:X\rightarrow X$ be continuous. Then, is there any meaning/research done on the metric
$$
D(x,y)\triangleq \sum_{n \in \mathbb{N}} \frac1{2^n} d(f^n(x),f^n(y));
$$
...

**3**

votes

**1**answer

196 views

### Attractors in random dynamics

Let $\Delta$ be the interval $[-1,1]$, then we can consider the probability space $(\Delta , \mathcal{B}(\Delta),\nu)$, where $\mathcal{B}(\Delta)$ is the Borel $\sigma$-algebra and $\nu$ is equal ...

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51 views

### On invariant cones of the Katok map

I am studying the Katok map and similarly constructed examples of nonuniformly hyperbolic surface diffeomorphisms. An important part of the analysis of these diffeomorphisms is the invariance of a ...