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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Help on designing a dynamical system

I would like to build a four-dimensional dynamical system that has the following behavior: Here, $x_1, x_2, x_3$ and $x_4$ are the four dimensions, and each axis has a fixed point that should be a ...
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1answer
26 views

Stability for higher dimensional dynamical systems

I remember learning that in order for a steady state to be locally stable in a system of two equations, it is sufficient for the Jacobian evaluated at a steady state to have: $$Tr(J)<0$$ ...
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25 views

Plotting a 4D Dynamical System

Suppose I have a 4D dynamical system. Each axis has a fixed point, and there are orbits connecting the fixed points. It looks something like this: Each $Q_i$ is a fixed point on each axis of a ...
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1answer
76 views

Where did I make a mistake?

This is an excerpt from a dynamical systems paper: They provide a proof of this Lemma, and numerical simulations also show it should be true. It's clear the equilibrium point on each axis is ...
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1answer
28 views

Characterization of contraction mapping

Let $T$ be a mapping from $\mathbb{R}^n \to \mathbb{R}^n$. Fix $x^\star \in \mathbb{R}^n$, and suppose that the Jacobian matrix of $T(x) $ at $x = x^\star$is symmetric. Then, I know that if all the ...
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21 views

Asymptotical stability of a discrete dynamical system

There is a linear time invariant discrete system, \begin{align} x_{k+1}&=\tilde{A}x_k, \end{align} where $\tilde{A}$ is a block matrix represented by \begin{align} \tilde{A}= ...
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1answer
31 views

Poincaré-Bendixson theorem, periodic solutions/periodic orbits

According to my book (Hsu: ODE), a solution $\phi(t)$ to the system $x' = f(x)$ that is bounded for all $t \geq 0$ satisfies one of: 1) $\omega(\phi)$ contains an equilibrium, or 2) either $\phi(t)$ ...
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1answer
24 views

Finite automata as dynamical systems

In abstract (deterministic finite) automata theory the set of states of an automaton is an arbitrary set Q, and the transistion function is a specific set δ ⊆ Q × Σ × Q (with alphabet Σ, i.e. another ...
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2answers
39 views

What is the value as this sequence tends to infinity? [on hold]

Im curious to know whether the point $\frac{1}{4}$ has a closed form when it goes through the following sequence... $(\frac{1}{4}, (\frac{1}{4})^2+\frac{1}{4}, ...
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1answer
15 views

Stability of non-homogeneous and non-autonomous first-order difference equation

I am seeking to analyze the stability of steady points in a system of $n$ variables $x_1(t), ..., x_n(t)$. With discrete time $t$ the system is described by \begin{eqnarray*} x_i(t+1) = ...
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30 views

Building a dynamical system

Suppose I have a 4 dimensional system with 4 fixed points: $Q_1 = \left(p_1,0,0,0 \right)$, $Q_2 = \left(0,p_2,0,0 \right)$, $Q_3= \left(0,0,p_3,0 \right)$, and $Q_4 = \left(0,0,0,p_4 \right)$. ...
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3answers
58 views

Lyapunov stability at origin with identically zero test function

At the origin, determine stability of $$x' = y \\ y' = -\tan(x)$$ If we use the test function $V(x,y) = 0.5y^2 + \int_0^x tan(s)ds$, we get $\dot{V}=x'\tan x +y'y = y\tan x -y\tan x = 0$, so the ...
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0answers
14 views

Moser's Twist Theorem for maps with reflection

Suppose I have a simple two dimensional integrable twist map, such as $x_{1}=x_{0}+y_{0}, \quad y_{1}=y_{0}$. Suppose that I perturb it in such a way that Moser's Twist Theorem is satisfied. What ...
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2answers
34 views

LaSalle invariance, Lyapunov stability

Trying to understand the LaSalle invariance principle. Consider the system $x' = y \\ y' = -y-6x-3x^2$ a) Using the test function $V(x,y) = 0.5y^2+3x^2+x^3$, show that the origin is asymptotically ...
2
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0answers
24 views

finding an invariant measure

Are there any methods for finding an (infinite, absolutely continuous with respect to lebesgue measure) invariant measure $d\mu=f(u,v)dudv$ for something like the following? $$ T:Q\to Q, \ ...
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1answer
36 views

Complex Analysis Dense Set Problem

The Problem: Suppose $f(z) = e^{i\theta}z$. Show that if $\theta$ is not a rational multiple of $\pi$, then the orbit of $ z \in \mathbb{C}$ is dense in the circle with radius $|z|$ and at the center ...
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1answer
30 views

Example of polynomial in dynamics

I am looking for an example of a post-critically finite polynomial $P$ (i.e. all critical points have finite orbit), which has both the following: a critical point on its Julia set (such as the ...
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1answer
31 views

Notation question in linear estimators (Kalman filter)

I'm just learning about Kalman filters, and I'm trying to understand some notation. The book that I am reading through sets up a system with the state-space realization: $$\dot{x}(t) = A(t)x(t) + ...
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1answer
55 views

Numerically solving a system of partial integro-differential equations in Matlab [closed]

Given the following system of partial integro-differential equations - $\frac{dS(t)}{dt}=\Lambda-\mu S(t)-\beta S(t)F(t),\\ \frac{\partial I(t,\omega)}{\partial t}+\frac{\partial ...
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1answer
61 views

Construction of Rauzy Fractals with substitutions without a fixed point

The formal definition of a Rauzy fractal can be found at the beginning of this paper Using Sage-math-cloud, I can generate Rauzy fractals of substitutions that I choose. Should I choose the ...
3
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1answer
67 views

Period of Fibonacci mod $b$?

It is not too difficult to show that the Fibonacci numbers mod $b$ form a periodic sequence. I would like to say something interesting about the period. There is a small shortcut to the brute-force ...
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0answers
28 views

What are the general steps to turn a PDE into a dynamical system $\dot x(t)= Ax(t) + Bu(t)$

It is said that every boundary value PDE such as the heat equation can be turned into dynamic system of the type $\dot x(t)= Ax(t) + Bu(t)$ with appropriate I.C. Can someone elaborate as to how to ...
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0answers
7 views

Determine points with bounded orbits of discrete dynamical system defined by quadratic polynomial and chaos

Consider the discrete dynamical system definded by the function $f(x) = ax^2+bx+c$ for real parameters $a,b,c$ with $a \neq 0$, $(b-1)^2 \geq 4ac$. How does the set $\Lambda$ of all $x \in \mathbb R$ ...
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1answer
58 views

Averaging for nonlinear systems

I am trying to figure out how the following result has been obtained. Consider a function $J:\mathbb{R} \longrightarrow \mathbb{R}$ and a dynamical system: $$ \dot{ \hat{x} }(t) = k a \sin ( \omega ...
3
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1answer
58 views

A Hamiltonian vector field on $\mathbb{R}^{4}$ which has closed orbit but does not have critical point

Is there a polynomial function $H:\mathbb{R}^{4} \to \mathbb{R}$ without critical points but the corresponding hamiltonian vector field possess at least one closed orbit?
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36 views

An equivalent definition of the rotation number of a circle homeomorphism

Let $f : \mathbb S^1 \to \mathbb S^1$ be an orientation-preserving homeomorphism. The classical definition of the rotation number is the following: we lift $f$ to a homeomorphism $F : \mathbb R \to ...
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19 views

Finding a Ljapunov function for discrete dynamical system with 3 variables.

Consider the discrete dynamical system given by $(x_{k+1},y_{k+1},z_{k+1}) = f(x_k,y_k,z_k)$, where $f(x,y,z) = (x(1-ay),y(1-b+ax),z+by)$ with $a,b \in (0,1)$ are parameter and we are only interested ...
0
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1answer
30 views

Nonconstant solutions of discrete predator and prey model and Perron-Frobenius

Consider the discrete dynamical system given by $x_{n+1} = A x_n$, where $A = \begin {pmatrix} a & -b\\c &d\end {pmatrix}$ and $x_n = \begin {pmatrix} u_n\\v_n\end {pmatrix}$. Are there ...
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0answers
13 views

A 2-variable stochastic difference equation exhibiting 2 stable orbits with switching?

I have some social science data to which I would like to fit a stochastic difference equation in two variables. I will describe the dynamics of the system that I have observed. I am hoping someone ...
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17 views

Inverse evolution of a dynamical system

Background Considering a one-dimensional chaotic map $T$. The domain of the map (the closed interval $[0,\,1]=I_1 \cup I_1$) is partitioned into two regions (i.e. $I_0\cap I_1=\emptyset$): $[0,\,A] = ...
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1answer
34 views

A simple question on forward and backward invariant sets

A subset $A\subset X$ is forward invariant if $f^{t}(A)\subset A$ for all $t\ge 0$ and backward invariant if $f^{-t}(A)\subset A$ for all $t\ge 0$. I want to show that the complement of a forward ...
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19 views

Topological Semi conjugacy between Henon map and Logistic Map

I am currently teaching myself dynamical systems and have come across a problem I am not quite able to figure out. More specifically, I am unable to find a conjugator function to establish a semi ...
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28 views

Definition of Hamiltonian system through integral invariant

I've read that Poincare's integral invariance can be used as a definition of a Hamiltonian system. That is to say, if $g^t$ is a phase flow satisfying $$\oint_{\gamma} \omega = \oint_{g^t \gamma} ...
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21 views

Topological entropy and degree of smooth mappings

Where can I find the literature "Topological entropy and degree of smooth mappings" by Misiurewicz. Thanks for any help.
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15 views

Approximating non-continuous dynamic system by smooth functions

I have a dynamic system $\dot{y}=f(y)$, with $y\in \mathbb{R}^4$ and $f=(f_1(y), f_2(y), f_3(y), f_4(y)$. Here, $$ f_1(y)=\left\{\begin{array}{cc} -1 & \text{ if } y_2>0 \\ [-1,1]& ...
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36 views

Krylov-Bogoliubov theorem in discrete and continuous time

Consider the following two similar frameworks (the first one in discrete time, the second one in continuous time) in which I am trying to apply the Krylov-Bogoliubov theorem. 1) (Discrete time.) ...
2
votes
1answer
22 views

Convergence of a sequence of subspaces

Let $E_n\subset \mathbb R^n$ be a sequence of subspaces. What does it mean $E_n$ convergence to a subspace $E\subset \mathbb R^n$? I saw this when reading about hyperbolic sets. Where can I read ...
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33 views

Periodic phase curves

I'm currently reading Arnolds "Mathematical Methods of Classical Mechanics" and I'm having a hard time solving some of the problems in Chapter 2. I think that the following problem is fairly simple ...
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2answers
44 views

stability of equilibria for $n$-dimensional nonlinear systems of differential equations: examples

I'm currently self-studying dynamical systems. I'm trying to summarize what can be said about the stability of equilibrium points for an $n$-dimensional non-linear system of differential equations: ...
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1answer
19 views

System of separable diff. eqns, explicit solution and curves, Lotka-Volterra model

In the book on p.68 is a system of differential equations for a Predator-Prey model (Lotka-Volterra) given as: $$ \dot x=x(\alpha-c\gamma) \\ \dot y=y(\gamma x -\delta) $$ On the next page, it is ...
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1answer
28 views

Infinite closed shift-invariant set of binary sequences with zero entropy?

Does there exist an infinite closed shift-invariant $X \subset \{0,1\}^\Bbb Z$ with zero topological entropy? How to think of an example? Will periodic points of shift$|_X$ have zero entropy?
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65 views

How do i find formula for the recurrence relation :$x_{n+1}= x^2_{n}-x^2_{n-1}$ with :$x_{-1}=0,x_{0}=\frac{3}{4}$? [on hold]

How do i find formula for the recurrence relation according to the below initial conditions: $x_{n+1}= x^2_{n}-x^2_{n-1}$ with :=$x_{-1}=0,x_{0}=\frac{3}{4}$ ? Note:$n=0,1,2\cdots$ Thank you for ...
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35 views

A continuous function with a dense set of periodic points but without sensitive dependence on initial conditions

The Question: Give an example of a continuous function, $f$, on the interval, $I$, such that the set of periodic points of $f$ is dense in $I$, but f does not have sensitive dependence on initial ...
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1answer
21 views

Difficulty understanding the concept of writing an L-System?

I've recently tried my hand at L-Systems, but I'm having some difficulty wrapping my head around it. I watched this video on the subject which is pretty good, but I had a question around the 1:43 ...
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0answers
21 views

Does an exponential bound on a Lyapunov candidate imply asymptotic stability?

If I have a Lyapunov candidate $V:[0,\infty)\rightarrow \mathbb{R}$ and I'm able to show that $$ V(t)\le k e^{-\eta t} V(0),\qquad \forall t\in[0,\infty) $$ can I conclude something about ...
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1answer
16 views

Local center manifold theorem.

Local center manifold theorem, under certain assumptions, state that for the \begin{cases} \dot x = Cx+F(x,y) \\ \dot y = Py+G(x,y)\\ \end{cases} there exist a function $h(x)$ such that ...
2
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39 views

Stability for a infinite dimensional dynamical system

Suppose I have a infinite dimensional dynamical system as $\dot{x_n}=Ax$ where $A$ is an infinite-dimensional matrix and $\{x_n\}$ is a sequence of states. I was wondering if you can help me find ...
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1answer
33 views

About expansive homeomorphim

We say $(X,f)$ is expansive if there is $c(f)>0$ such that if $d(f^{n}(x), f^{n}(y))< c(f)$ for every $n\in Z$ then $y=x$. Let $(X,f)$ is expansive with constant $c(f)$ and for infinite set ...
2
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26 views

Non-Conservative System

I'm having a bit of trouble understanding the concept of a conservative system mathematically. A problem in a textbook (Arnold's Mathematical Methods for Classical Mechanics) is asking me to give an ...
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37 views

Trajectories of predator prey equation

I am studying the predator prey equation recently, and here is an example: Let $x'=x(1-0.5y)$ and $y'=y(-0.75+0.25x)$. This is a predator prey equations. Then ...