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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 models of diverse fields such as classical mechanics, economics, traffic modelling, population dynamics, and biological feedback.

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I am self-studying dynamical systems, and I came across a property that I am unsure if I correctly identified how it is found. That is, $$ \phi_t(x+\epsilon) - \phi_t(x) \approx \epsilon e^{t\lambda} $...
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Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be defined as; $$f(x) = \begin{cases} 3x & \text{if} \ x \ \leq \frac{1}{2} \\ 3-3x & \text{if} \ x \ > \frac{1}{2} \end{cases} $$ Proposition ...
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Consider the function $F:\mathbb{N}\to\mathbb{N}$ such that $F(n)=\tfrac{n^2-n}{\delta(n^2-n)}$, where $\delta$ returns the biggest prime factor of its input. I wonder if this function always ...
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This is a reference request; is this particular generalization of the $3\cdot n+1$ problem discussed in literature? What is known about it? Do any specific choices of $m$, $a_i$ lead to nontrivial yet ...
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Consider a system \begin{align} &\frac{dx}{dt}=y\\ &\frac{dy}{dt}=-y^2-\sin x \end{align} (0,0) is an equilibrium, and I want to know whether it is Lyapunov stable. If it is stable, then is it ...
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I have been investigating the number of limit cycles (loops) in the Generalized Collatz Problem defined by the map:$$T_k(n) = \begin{cases} (3n+k)/2 & \text{if } n \text{ is odd} \\ n/2 & \...
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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 ...
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I am interested in the following non-linear system of ODEs (all the parameters are positive): $$ dR_{1,t}=-\lambda_1 R_{1,t}\,dt + C\,(\beta_0-\beta_1 R_{1,t} + \beta_2 \sqrt{R_{2,t}})\, dt $$ $$ dR_{...
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Suppose we define a binary sequence $\varepsilon _{k}$ such that it satisfies the equation $$\varepsilon _{k} =\left\lceil \left(\frac{3}{2}\right)^{k}\left( 8-\frac{1}{3}\sum _{j=0}^{\infty }\left(\...
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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 ...
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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 ...
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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 ...
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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, ...
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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,...
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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 $M$, ...
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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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Body: Let $$ f(x) = \mu x(1 - x), \qquad \mu > 4. $$ Define $$ \Lambda = \{ x \in [0,1] \mid f^n(x) \in [0,1] \text{ for all } n \ge 0 \}, $$ the Cantor repeller (points that never leave ([0,1])). ...
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Let $X = \{1,...,d\}^{\mathbb{N}}$. Show that every infinite compact shift invariant subset of X has a non-periodic point. My Attempt: Suppose not. So $K\subset Per(\sigma)$, where $\sigma:X\...
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First of all one can check the reference in this link for notations and Theorems. My perturbed system is like equation (1), it's averaged form is (9) and I'm trying to apply Theorem 8. I'm trying to ...
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My question came from this paper by Lalley (p.g. 2114) where he showed that the hitting/harmonic measure from finite ranged random walk on free groups is a Gibbs state. Let $(\Lambda_+,\sigma)$ be a ...
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I'm studying the Wiener-Wintner (and related) ergodic theorems, and I've been running into a bit of confussion when passing the result from ergodic systems to non-ergodic ones. In most of the ...
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For a function $f : D^n \to \mathbb{R}$, $f \in C^1(D^n) \bigcap C(\overline{D^n})$, suppose $||f||_{L^\infty} \le 1$, we want to show $\inf | Df | \le c$, for all such $f$. A known result https://www....
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In mechanical engineering, there are many designs of a CVT(Continuously Variable Transmissions) that is able to change gear ratio continuously, but all of them are unsatisfactory for some reason. ...
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I am studying dynamical systems and wild spaces, and I'm interested in the title question to realise the Horned Sphere as an attractor for a homeomorphism. I am aware of a result by Morton Brown on ...
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In my textbook, generic sets are defined as sets that contain an open and dense set. In general, I am really confused on how to identify dense sets of matrices/linear maps. "Which of the ...
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Let $z_1 , \dotsc , z_k \in \mathbb{C}$. Is it true that there exists $n \in \mathbb{Z}^+$ such that $\operatorname{Re} ( z_j^n ) \geq 0$ for each $j$? Let $A = [ 0 , 1/4 ] \cup [ 3/4 , 1 ]$. By ...
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Setup: We have a real $k\times k$ matrix $A$ which has entries in $\{0,1\}$. Assume further that $A$ is irreducible and aperiodic. Consider the associated shift space \begin{equation}X := \left\lbrace ...
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Consider a discrete stochastic system with components $(x_k, y_k)$ updated as follows. If all components are strictly positive, i.e. $x_k > 0$, $y_k > 0$, then \begin{aligned} x_{k+1} &= x_k ...
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The bellow theorem and its proof are presented exactly as in James C. Robinson's book, Infinite-Dimensional Dynamical Systems. My doubts are right after it. Theorem 14.5. If $ f \in H $ then the ...
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I am stuck in solving the following point of this exercise. We consider maps $f_\alpha:[−1,+1]\setminus\{0\}→[−1,+1]$, where $\alpha\in (1,2]$, given by $$f_\alpha(x)=\alpha x−\text{sign}(x)$$ where $\...
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I was playing around on Desmos a few months ago, with a basic Mandelbrot Set visualizer I made, experimenting with what new shapes I could make by changing the function iterated on $\Bbb{R}^2$. This ...
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I am trying to solve this exercise, Exercise 4.14 ((*) Like the Lorenz return maps). (Compare Sparrow [209].) We consider maps $f_{\alpha} : [-1, +1] \setminus \{0\} \to [-1, +1]$, where $\alpha \in (...
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In a 2d dynamical system with only 1 parameter, there are different bifurcation types in x and y directions, e.g. pitchfork along x and saddle node along y. What is the proper term for this ...
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I have been studying the Koopman operator, which is an operator that transforms any dynamical system, even non-linear systems, into a linear system in a potentially infinite dimensional space. Suppose ...
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We Considered the following setting in class: Let $\rho_0$ and $\rho_1$ be two $C^{\infty}$ functions that are strictly positive and decay fast enough. Let $u : \mathbb{R}^n \to \mathbb{R}$ be a ...
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I am investigating the mathematical properties of a nonlinear, coupled system intended to unify structures from several deep areas of mathematics (e.g., Birch and Swinnerton-Dyer, Hodge, Navier–Stokes,...
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$G\subset {\rm SL}(n,\mathbb R)$ is a simple, connected Lie group and let $\{\alpha(t)\}_{t\in\mathbb R }$ be a non-trivial one parameter subgroup of diagonalizable matrices. I wonder how to prove the ...
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Definition (The Rule): Let '$n$' be a positive integer with digits $$n = d_1 d_2 d_3 \dots d_k$$ Cube each digit. Assign alternating signs: $ d_1^3-d_2^3+d_3^3-d_4^3...d_k$ Conjecture: For every ...
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I come across the notion of asymptotically periodic source which has a positive lyapunov exponent but seemingly the orbit will land on the source. I am not sure whether I have misunderstood the ...
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Im reading the book Intoduction to Dynamical Systems by Brin and Stuck. Currently im studying the ergodic theory section and to be more precise the result about Anosov Diffeomorphisms. There is this ...
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I am looking for an explicit example of a $C^1$ diffeomorphism $f:\mathbb{R}\to\mathbb{R}$ with the following properties: It has a fixed point $p$ such that $f(p)=p$. This fixed point is non-...
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Given two flows $\phi$ and $\psi$, they are conjugated iff there exist a bijection $h$ such that $h(\phi(t,x))= \psi(t,h(x))$ $\forall x $ in the domain. Is conjugation unique? If not, when can be ...
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Given a linear time-invariant system, $$\dot x = Ax$$ We know that all of its equilibrium points are contained in $\text{ker}(A)$. Suppose $\ker(A)$ contains more than one point and I pick two ...
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I am studying a 3D system of ODEs derived from a biological model (tumor growth). All parameters are defined to be positive, as negative values would lack biological meaning (e.g., negative loss rates ...
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Good day guys. Recently, I've been working on a dynamics problem defined as follows. Given $$X(1) = 1,\quad Y(1) = 1,$$ for $t > 1$, let $(X(t), Y(t))$ evolve according to $$Z(t) = \frac{Y(t) + X(t)...
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Let $E$ be a Banach space. Fix $T \in E$ and $\alpha \in [0,1]$. Let $R:E \to E$ be a Lipschitz map with constant $L$. Define the iteration $$ x_{n+1} = (1-\alpha)T + \alpha R(x_n). $$ Fixed points ...
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I am trying to do an exercise on Peano's Theorem. I can't solve this one because I don't know if it works in a closed set. Prove that the Cauchy problem: \begin{cases} \dot{x}(t)=\dfrac{-t+\sqrt{...
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I am currently looking a bit into non-autonomous ODEs. Their analysis requires a new notion of hyperbolicity. One that is commonly used is a defintion via an exponential dichotomy of the variational ...
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Consider a continuous unit-speed curve $X:[0,L]\to\mathcal{S}$ in the unit square $\mathcal{S}=[0,1]^2$ with specular reflection at the boundary (we can also unfold to the flat torus $\mathbb T^2$). ...
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Let $V$ be a vector field on $\mathbb{R}^2$ such that the set $$ C = C_x \cup C_y \cup C_x' \cup C_y' = \text{positive $x$-axis} \cup \text{positive $y$-axis} \cup \{(x, x^2) \mid x \geq 0\} \cup \{(y^...
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