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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Trajectories that connect equilibrium points

Suppose I consider the autonomous system \begin{align*} x' &= F(x, y)\\ y' &= G(x, y) \end{align*} where $F$ and $G$ are nonlinear and my task is to draw the phase portrait of the above ...
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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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228 views

The Stable Manifold Theorem Applications

Definition: Let $\phi_t(x)$ be the flow of the nonlinear system $x'=f(x)$. The global stable manifold of $x'=f(x)$ at $0$ is defined by: $$W^s(0)=\bigcup_{t\leq 0}\phi_t(S)$$ Where $S$ is 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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1answer
2k views

Root Locus Diagrams - “Breakaway Point”

Say that we have a root locus diagram with n poles and m zeroes. And we determine that the root locus on the real axis lies between two of these poles and breaks away from the real axis and tends to ...
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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
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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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
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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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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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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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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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
62 views

Lyapunov equation for stability analysis - what's the point?

Straight from Wikipedia: In the following theorem $A, P, Q \in \mathbb{R}^{n \times n}$, and $P$ and $Q$ are symmetric. The notation $P>0$ means that the matrix $P$ is positive definite. ...
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1answer
29 views

What does a 3D periodic solution of a differential equation look like?

The Pointcare-Bendixson Theorem implies that if a solution stays in a bounded region with no equilibrium points then it is either a periodic solution or it approaches a periodic orbit as t goes to ...
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3answers
88 views

How to interpret complex eigenvectors of the Jacobian matrix of a (linear) dynamical system?

Consider a linear ODE system of the following form: $$ \frac {dx} {dt} = Ax $$ In case $A$ has real eigenvectors, I can interpret them as the directions in which the system will move, if the initial ...
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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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Periodic orbits of “even” perturbations of the differential system $x'=-y$, $y'=x$

Fix some even functions $f$ and $g$, differentiable, such that $f(0)=g(0)=0$ and $f'(0)=g'(0)=0$, and consider the autonomous differential system $$\left\{\ \begin{array}{lcr}x'&=&-y+f(x)\\ ...
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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
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
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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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614 views

Why is unique ergodicity important or interesting?

I have a very simple motivational question: why do we care if a measure-preserving transformation is uniquely ergodic or not? I can appreciate that being ergodic means that a system can't really be ...
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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 ...
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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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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
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
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 ...
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1answer
375 views

Separatrix curve in dynamical systems

I have this question: What is a separatrix of a equilibrium point of a continuous dynamical system and why it is flow-invariant? Thanks Hello and thanks for the answer. I explain better. I'm ...
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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
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
454 views

How to determine $\Omega(T)$?

Let $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and let $T\colon X\to X$ describe the following dynamics: 1 becomes a 2, 2 becomes a 0 and 0 becomes a 1 if at least one of its two neighbours is 1, ...
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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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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 ...
2
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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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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 ...
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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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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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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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0answers
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 ...
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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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1answer
107 views

Hopf Bifurcation of Reaction-Diffusion System

I'm considering the following reaction-diffusion system: $ \frac{\partial u}{\partial t} = f(u,v)+ D_1 \frac{d^2 u}{dx^2} $ $ \frac{\partial v}{\partial t} = g(u,v)+ D_2 \frac{d^2 v}{dx^2} $ where ...
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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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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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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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2answers
65 views

What is “abstract” ergodic theory?

This is just a question about the usage of the term "abstract". What kind of questions in ergodic theory is considered "abstract" and what's a "regular" question? From some seminars it seems that ...
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2answers
447 views

Closed form solution of this second order linear difference equation?

$$ y(k + 2) - 3y(k + 1) +2y(k) = 2^k + k $$ Transform into a system of $n$ first order equations (Step 1) $$\begin{align} x_1(k) &= y(k)\\ x_2(k) &= y(k + 1) \end{align}$$ It follows that ...
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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]& ...