# Questions tagged [ds.dynamical-systems]

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

1,569 questions

**4**

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53 views

### Extension of Vector Field in the $\mathcal{C}^r$ topology

This question was previously posted on MSE.
Let $M\subset \mathbb{R}^n$ be a compact smooth manifold embedded in $\mathbb{R}^n$, we define $$\mathfrak{X}(M) := \{X: M \to \mathbb{R}^n;\ X\mbox{ is ...

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33 views

### Characterization of Time-homogeneous flows for conditional expectation

Let $X_t,Y_t$ be $\mathbb{R}^d$-valued processes. It is well known that for every $t\geq 0$, and every bounded function $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$, there exists a Borel function $f_t:\...

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176 views

+100

### Structural Stability on Compact $2$-Manifolds with Boundary

I'm studying the structural stability of vector fields and I'm interested in learning about this phenomenon on compact $2$-manifolds with boundary.
Let $M^2$ be a compact connected 2-manifold and $\...

**3**

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**1**answer

107 views

### Symmetries for Julia sets of perturbations of polynomial maps

This is a naive question. Consider the
Julia sets
of the map
$$ z \mapsto z^n + \lambda / z^k $$
with $z,\lambda \in \mathbb{C}$, and the exponents $n,k \in \mathbb{N}$.
For example, for $n=k=3$, ...

**0**

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51 views

### Runge-Kutta 4th order for predator-prey model [closed]

I'm trying to compute the numerical solution for a Predator-prey model with 3 equations. This is the model:
$$\frac{dx}{dt} =x(1-\frac{x}{k_1})-\frac{pxz}{1+ax+chy}\\
\frac{dy}{dt} =y(1-\frac{y}{k_2})...

**3**

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57 views

### Piecewise linear expanding maps

Let $(I_{n})$ be a countable infinite disjoint partition of $[0,1)$ into half-open intervals. Let $f:[0,1)\to [0,1)$ be the piecewise linear expanding map with $f(I_{n})=[0,1)$ for all $n$. I suppose ...

**1**

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**1**answer

100 views

### Trajectory leaving a set

Consider the differential equation $\dot{x}=f(x)$, where $f: \mathbb{R}^2 \to \mathbb{R}^2$ is smooth. Given a set $A \subset \mathbb{R}^2$, are there some results saying that whenever $x(0) \in A$, ...

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55 views

### Nearly eventually almost periodic functions

Call a function $f: [0, \infty) \to \mathbb R$ nearly eventually almost periodic with period $p > 0$ if for a.e. $x \in [0, p)$, the sequence ${f(x + np)}_{n \in \mathbb N}$ converges.
Suppose $f: ...

**2**

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55 views

### Does singularity confinement imply a fixed pattern of irreducible factors?

Consider a rational map
$f \colon (x_1,\ldots,x_n) \mapsto (P_1(x_1,\ldots,x_n),\ldots,P_n(x_1,\ldots,x_n))$, where the $P_i$ are rational functions. Via iteration this map defines a discrete ...

**2**

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**1**answer

78 views

### Discrete dynamical system and bound on norm

Let $z \in \mathbb R\backslash \left\{2 \right\}$ then I would like to understand the following:
Consider the dynamical system with $x_i \in \mathbb C^2:$
$$ x_{i} = \left(\begin{matrix} z &&...

**2**

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**1**answer

336 views

### Formal group law and Koenigs function conjecture?

Let $f(x,y)$ be a symmetric real function and a formal group law
$$G(x + y) = f(G(x),G(y)). \tag{1}$$
Consider the equation
$$ h(2x) = f(h(x),h(x)) = A(h(x)). \tag{2}$$
This equation has many ...

**1**

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**0**answers

42 views

### About strange invariant set of the Lozi mappings

Consider the Tent map: $f_{μ}(x)=μx$, if $x<0.5$ and $f_{μ}(x)=μ(1-x)$ if $x≥0.5$. In this page (https://en.wikipedia.org/wiki/Tent_map) it was stated that:
If $μ$ is greater than $2$ the map's ...

**8**

votes

**2**answers

244 views

### Dimension of orbit versus invariant functions

$\def\CC{\mathbb{C}}$Let $K = \CC(x_1, \ldots, x_n)$ and let $G$ be a countable group of automorphisms of $K$; in the cases I care about, $G \cong \mathbb{Z}$. Then the field of $G$-invariants, $K^G$, ...

**-1**

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86 views

### What functional equations are used to extend tetration from the whole numbers to the complex numbers?

In searching for information regarding extending tetration I have come across half a dozen published papers. But It seems that Abel’s functional equation is at the heart of the extension techniques I’...

**1**

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**1**answer

76 views

### A special oscillatory orbit in space

Edit: According to the comment of Prof. Eremenko I revise the question.
19 years ago, I have heard the following problem from a specialist of dynamical system. During these 19 years, I was in contact ...