Questions tagged [dynamical-systems]

Question 1

Consider the function $F:\mathbb{N}\to\mathbb{N}$ such that $F(n)=\tfrac{n^2-n}{\delta(n^2-n)}$, where $\delta$ returns the biggest prime factor of its input. I wonder if this function always ...

Question 2

This is a reference request; is this particular generalization of the $3\cdot n+1$ problem discussed in literature? What is known about it? Do any specific choices of $m$, $a_i$ lead to nontrivial yet ...

Question 3

Consider a system \begin{align} &\frac{dx}{dt}=y\\ &\frac{dy}{dt}=-y^2-\sin x \end{align} (0,0) is an equilibrium, and I want to know whether it is Lyapunov stable. If it is stable, then is it ...

Question 4

I have been investigating the number of limit cycles (loops) in the Generalized Collatz Problem defined by the map:$$T_k(n) = \begin{cases} (3n+k)/2 & \text{if } n \text{ is odd} \\ n/2 & \...

Question 5

Question: Let $\varphi_t: M \to M$ be a continuous flow with no fixed points on a compact metric space $M$. If $\varphi_t$ has a periodic orbit, is there a positive lower bound on the period of all ...

Question 6

I am interested in the following non-linear system of ODEs (all the parameters are positive): $$ dR_{1,t}=-\lambda_1 R_{1,t}\,dt + C\,(\beta_0-\beta_1 R_{1,t} + \beta_2 \sqrt{R_{2,t}})\, dt $$ $$ dR_{...

Question 7

Suppose we define a binary sequence $\varepsilon _{k}$ such that it satisfies the equation $$\varepsilon _{k} =\left\lceil \left(\frac{3}{2}\right)^{k}\left( 8-\frac{1}{3}\sum _{j=0}^{\infty }\left(\...

Question 8

Suppose that $X$ is a compact metric space with metric $d$ and $T:X\to X$ is continuous. Assume that $(X,T)$ is uniquely ergodic with the Borel probabilistic measure $\mu$ and $(X,\mathcal{B}(X),\mu,...

Question 9

Consider the dynamical map that terminates on all natural numbers: $f_o:x\mapsto (x+1)/2$ if $x$ odd $f_e:x\mapsto x/2$ if $x$ even This is easily proven to terminate for all natural numbers. Now ...

Question 10

Question The dynamical system: Let $f(x)=\begin{cases}(x+1)/2&&x\textrm{ odd}\\x/2&&x\textrm{ even}\end{cases}$ is the the bit shift map with binary strings reversed. It terminates ...

Question 11

I have the following system: \begin{cases} \dot{x}=x-y,\\ \dot{y}=x^2-4 \end{cases} The point (2,2) is an unstable spiral and (-2,-2) is an saddle point. I know they are connected ...

Question 12

My intuition for the answer is NO, here is my thought: Let $T(x,\lambda)$ be the map depending on one parameter $\lambda$, assume at $(x_0,\lambda_0)$ a tangent bifurcation occurs for the $T^2$ map, ...

Question 13

Let $X$ and $Y$ be Banach spaces. Let $U$ be an open subset of $X$ and $f:A\rightarrow Y$ be a function with $U\subseteq A\subseteq\overline{U}$. Then we have two possible definitions of $f\in C^{1}(A,...

Question 14

Let $M$ be a smooth connected manifold of dimension $\geq 2$, and let $\phi: \mathbb{R} \times M \to M$ be a complete flow on $M$. Suppose there exists a point $x_0 \in M$ whose orbit is dense in $M$, ...

Question 15

Fix a discrete-time system with input sequence $(x_t)_{t\ge 0}\subset\mathbb{R}^d$, output $(y_t)_{t\ge 0}\subset\mathbb{R}^p$, and $L$ internal levels. For each level $\ell\in\{1,\dots,L\}$ choose a ...

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Questions tagged [dynamical-systems]

Integer sequence with largest prime factor

Reference request: a particular generalization of the Collatz problem

Determining stability of the equilibrium point for a nonlinear system

Why does the Generalized Collatz map ($3n+k$) with $k=3^x+2^x$ produce 1,023 cycles at $x=15$, but collapse to 1 cycle for $x \ge 21$?

Periodic Orbits of Arbitrarily Small Period for a Flow Without Fixed Points

Study of non linear dynamical system

Can we determine whether the binary sequence we obtain through this equation has ever more ones than twice the zeros?

Prove that $(X,T^k)$ is uniquely ergodic [closed]

What is the name of the property of $x\mapsto 3f(x)$ that its orbits are wellordered by $f$? [closed]

Is the reverse bit shift map times $3^n$ guaranteed to have sequences with no further multiples of $3$?

Prove that exists an heteroclinic orbit

Can even periodic points of a map undergo tangent bifurcation?

Definition of $C^{1}$ on closure of open set

Does a dense orbit imply topological transitivity for flows on manifolds?

Does multi-rate update depth bound the Volterra degree?

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