Questions tagged [dynamical-systems]

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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Assumption of continuously differentiable function in the Lyapunov Stability Criterion

According to the proof of Lyapnuov's theorem given in [1] the assumption of continuity of partial derivatives is necessary to prove asymptotic stability while for simple stability it is not. I wonder ...
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how the angle between vectors changes in a topological equivalence

I'm studying dynamical systems and i have the next problem: I have two dynamical systems in continuous time, let's say $\dot{x} =X(x)$ and $\dot{y} =Y(y)$ where there is a topological equivalence (or ...
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Question regarding contraction theory

I am following the paper (A Study of Synchronization and Group Cooperation Using Partial Contraction Theory) to understand how contraction theory used to analyze coupled oscillators. On page 211, the ...
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Complexification Averaging Method

I want to study about Complexification Averaging Method. What books do you suggest to start with? I want to completely understand this method and why should we use it. Thank you.
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Uniquene solution to minimisation of a Non Linear Objective Function

I am trying to estimate the path of a random described by the following SSM \begin{align} x_{t+1} = x_{t} + q_{t+1} \newline y_{t+1} = h(x_{t+1}) + r_{t+1} \end{align} where $h(x_{t+1}) =...
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Uniqueness in minimisation of Lp-Norm using Gradient Descent

I am trying to estimate the path of a random described by the following SSM ...
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A little trouble in proving that geodesic flow on Riemannian manifolds with bounded negative sectional curvature is Anosov (from Klingenberg's book)

I'm dealing with the proof from Klingenberg's "Riemannian Geometry" of the Anosov theorem on geodesic flows on complete Riemannian manifold with bounded negative sectional curvature, pages ...
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Set $T^{\mathbb N}x$ dense in $\mathbb S^1$ (Poincaré recurrence theorem)

Let $Ω =\mathbb S^1$ be the unit circle in $\mathbb R^2 = \mathbb C$, and let $T : Ω → Ω$ be multiplication by $e^{i\alpha}$. For $α \notin π\mathbb Q$ and every $x ∈ Ω$, is the set $T^{\mathbb N}x$ ...
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Understanding the Definition of the Koopman Operator

Consider a continuous time dynamical system $$\dot x(t) = F(x(t)),$$ where $x(t)$ is a coordinate vector of state and the right side of the equation $F$ is a non-linear smooth function. Let the state ...
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Estimate on derivative of ODE solution with respect to parameters

Consider the ODE $$ u'(t) = f(t,u,p), \qquad u(0) = v $$ where $p$ is a control parameter, and let $u(t;v,p)$ denote the solution to the problem above for fixed $v$ and $p$. It is apparently "...
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Ergodicity on a finite set [closed]

Let $\Omega$ be a finite set ($ \#\Omega = n$), how many dynamical systems on $\Omega$ are ergodic?
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Input for stopping in Dubin's path dynamics?

This question may seem pretty dumb. But I really want to know. I have the following linear dynamical system for Dubin's path, \begin{align*} \phi &= \begin{bmatrix} ...
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What is the relation between Poincaré sections and chaotic behaviour?

I've been studying Poincaré Sections. Here are some Poincaré Sections plots from the double pendulum. I've read that, intuitively, when plotting a chaotic orbit through a Poincaré Section, it will &...
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Lagrange multipliers: discrepancy between optimization and adjoint sensitivity results

Consider the constrained optimization problem: $min_x f(x)$ s.t. $g(x)=0$. For simplicity, let $f$ and $g$ be scalar functions. Under suitable conditions, the Lagrange multiplier theorem gives: $\...
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Definition of essentially T-invariant function

A function is $T$-invariant if $f(T(x))=f(x)$ for all $x\in X$. In text book: Introduction to Dynamical system by Brin, it defines essentially $T$-invariant: if $f(T(x))=f(x)$ almost every for $x\in X$...
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"Spanning" of solutions of ordinary differential equations

Suppose we have a switched ODE $$\dot{x} = A_{\sigma(t)}x,$$ where $A_{\sigma(t)}$ is a constant matrix given $\sigma(t)\in\mathcal{M}=\{1,2,\cdots,m\}$. If we fix the initial condition and can ...
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Necessity of the hypotheses of Lyapunov asymptotic stability theorem

In my ordinary differential equations course we saw Liapunov's theorem for asymptotic stability. I have a doubt about the necessity of the "negative definite" assumption. The statement we ...
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Every compact metric space contains some minimal set

A set $A$ is minimal if it is nonempty, closed, invariant (i.e. $f(A) \subseteq A$) and it does not contain any proper nonempty, closed, invariant subset. Let $X$ be a compact metric space and $f:X \...
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To determine the type of a singularity of a 2-D dynamic system.

This is a problem in my course homework. Suppose that $a>0$, $b>0$, considering the following 2-dimensional autunomous system: $$\begin{align}\frac{dx_1}{dt}&=F(x_1)=a-x_1-\frac{4x_1x_2}{1+...
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6 votes
3 answers
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Why does this cycle of 44 show up in the Collatz Conjecture?

Consider this function: $$f\left(x\right)=\frac{x-b^{\left(\operatorname{floor}\left(\log_{b}x\right)\right)}}{b^{\left(\operatorname{floor}\left(\log_{b}x\right)\ +\ 1\right)}-b^{\left(\operatorname{...
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Structural stability of Arnold's Cat Map

In Wikipedia it says that "hyperbolic automorphisms of the torus, such as the Arnold's cat map, are structurally stable", so I am trying to understand why this is true. What $C^1$-small ...
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Is the velocity vector tangent to a dynamic path?

Imagine a particle traveling along a curve (or surface). I cannot find anything on this topic that allows the curve (or surface) to change (or deform) as the particle moves. For a static curve, it is ...
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2 votes
1 answer
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Examples of hyperbolic sets of dynamical systems

I am studying the definition of hyperbolic set: a compact invariant set $\Lambda$ of a diffeomorphism $f$ such that the tangent space in each point of the set $\Lambda$ can be split in two invariant ...
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Upper bound on stepsize for guaranteeing convergence of forward euler method applied to nonlinear ODEs?

Consider the autonomous $d$-dimensional ODE given by $$ \dot{x}(t)=f(x(t)), \quad t\in [0,T], \quad x(0) = x_0 \in \mathbb{R}^d, $$ where $f$ is, in general, nonlinear. Consider now a discretized grid ...
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Is there a name for this function or a concept similar to it?

I'm wondering if anyone has heard or seen a function that looks or behaves like this one. $$f\left(x\right)=\frac{x-b^{\left(\operatorname{floor}\left(\log_{b}x\right)\right)}}{b^{\left(\operatorname{...
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2 votes
1 answer
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Examples of dynamical systems that have structural stability

I am looking for simple examples of structural stability, I read the definition of structural stability but couldn't figure out a concrete example of a system, its perturbated version and its ...
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Deduce stability from strict convexity of gradient systems

Given a twice differentiable function $f(x):\mathbb R^n\rightarrow \mathbb R$, and its corresponding gradient system $$\dot x=-\nabla f(x)$$ My question is: If $f(x)$ is strictly convex, can we ...
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6 votes
1 answer
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Why an LTI system with some zero eigenvalues still stable?

The textbook says an LTI system $\dot x=Ax$ is stable if and only if the eigenvalues of $A$ have the strictly negative real part. However, I found a counterexample. If $$A= \begin{bmatrix}-3 & -1 ...
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An inequality for fixed points of dynamical systems

Suppose $f:X\to X$ is a continuous map such that each iterate has finitely many fixed points. Define $h(f) = \limsup_{k\to \infty} \frac{\log \#Fix(f^k)}{k}$ where $\#Fix(f^k)$ is the number of fixed ...
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An exercise on counting fixed points from Milnor

I am working on the following exercise (Problem 6-b, pg. 6-22) from Milnor's Dynamical Systems notes: Problem 6-b Let $f$ be any self map such that each iterate has only finitely many fixed points. ...
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Central Limit Theorem for irrational rotation

If $ X=\mathbb{R}/\mathbb{Z}$ be the unit circle with $\mu$ Lebesgue measure, and suppose $\alpha\in X$ irrational, and $T:X\rightarrow X$ is defined by $T(x)=x+\alpha$. we want to find a function $f\...
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Why are the roots of a discriminant system called singularities in homotopy continuation?

In the article Introduction To Numerical Algebraic Geometry, the authors state on page six that The singularities along the solution paths are solutions of the system $\begin{cases}h(x, y, t) &= ...
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22 votes
2 answers
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Are these equations "properly" defined differential equations? (finite duration solutions to diff. eqs.)

Are these equations properly defined differential equations? Modifications were made to a deleted question to re-focus it. I am trying to find out if there exists any exact/accurate/non-approximated ...
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2 votes
2 answers
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Dynamics on the torus

It is well known that a $2-$torus foliated with lines of irrational slopes will produce dense curves on the torus. Likewise, rational slopes will lead to a periodic orbit. However, I am not seeing the ...
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The tent map system is transitive. Can we actually identify any of the infinitely many transitive points, however?

$\newcommand{\O}{\mathcal{O}}\newcommand{\G}{\mathcal{G}}\newcommand{\T}{\mathcal{T}}$TLDR; skip to the end of the preamble - we know that the tent map system is topologically transitive. However, do ...
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Show that $x_0=1$ is a Detractor Point when $f(x)=x^3$

Let $f(x)=x^3$. The fixed points are $\lbrace0,1,-1\rbrace$. I have shown that $0$ is an attractor point, becaues it is Lyapunov stable and is contained in $(-1,1)$. First, I supposed that $x\in(0,1)$,...
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What is integrating a variational equation?

This continues from How to understand the largest Lyapunov exponent? It is said that we can differentiate the equation, $$\tau\frac{dh_i}{dt} = F_i = -h_i + \sum_{j=1}^N J_{ij} \phi(h_j),$$ against $...
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3 votes
1 answer
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How to understand the largest Lyapunov exponent?

I've posted the question in the physics site too. It is said that ..the largest Lyapunov exponent, which measures the average exponential rate of divergence or convergence of nearby network states. ...
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2 votes
1 answer
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Tent map is topologically transitive

Let $T:[0,1] \to [0,1]$ be the function $Tx= 2x$, if $ x \in [0, \frac{1}{2}]$ and $Tx = 2-2x$, if $ x \in ( \frac{1}{2} , 1] $. We say that a map is topologically transitive if, for any pair $U, V$ ...
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4 votes
1 answer
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Best betting strategy for an unfair random walk with a skewed payoff

Say you start with bankroll $B$ and i.i.d. random variables $U_i$ with distribution $p=P(U_i=r)>.5$ and $q=1-p=P(U_i=-1)$. Your earnings from bet $i$ is $W_iU_i$, where $W_i$ is your wager at step ...
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2 votes
0 answers
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Ruling out limit cycles in 2 dimensions

Let $f:[0,1]^2 \rightarrow \mathbb{R}^2,$ where $f_1(x,y) = g(y)-x$ and $f_2(x,y) = g(x)-y.$ Here $g(\cdot)$ is a strictly decreasing polynomial function such that $g(0)=1$ and $g(1)=0.$ I am ...
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Can one obtain a Poincaré Section without having the explicit solution of the system of differential equations?

I'm studying Poincaré Sections. Every example I see, it calculates the solution $x(t)$ and then it uses it to calculate the Poincaré map. I don't see the point of the Poincaré map then. What ...
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Help with intuition behind invariance of ODE solution on planar curve.

Intuitively, I think I understand this, but more formally if we have a smooth vector field $$ F(x,y) = \begin{bmatrix} f(x,y) \\ g(x,y) \end{bmatrix} $$ such that for a smooth plane curve $C: x \...
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1 vote
1 answer
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a question on visualizing a state of DAE and a question from a continous time nonlinear dynamics

Could someone explain me what is the fundamental difference between the dynamical system of the kind $\dot x = f(x)$ and $E \dot x= f(x)$ where $E$ is a singular matrix with real entries. For the ...
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-1 votes
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Find real-valued closed formulas for the trajectory x(t+1)=Ax(t), where

Find real-valued closed formulas for the trajectory $x(t+1)=Ax(t)$, where $$ A=\pmatrix{−4 & 3 \\ -3 & −4}\quad \text{and}\quad x(0)=\pmatrix{1 \\ 0} $$ (image link)
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3 votes
1 answer
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Help Proving that Set Invariant for a Dynamical System.

I'm practicing for an upcoming test, and this one has been giving me some problems. Suppose we have $$ \begin{cases} \dot x = x^2 + 2y - 4 \\ \dot y = -2xy \end{cases}. $$ Let $R = \{(x,y) : |x| \leq ...
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1 vote
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Questions about Ratio Set of a Dynamic System

Given a dynamic system $(X, \Omega, \mu)$ ($\Omega$ the $\sigma$-algebra and $\mu$ a measure), we assume there is a group $G$ acting on the system in the sense that, for each group element $g$, $g$ ...
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2 votes
1 answer
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globally asymptotic stable gradient system has unstable point

Given a gradient system $$\frac{d\theta_1}{dt}=-\sin(\theta_1-\theta_2)$$ $$\frac{d\theta_2}{dt}=-\sin(\theta_2-\theta_1)$$ The system is a gradient system since $$\frac{d\vec \theta}{dt}=-\nabla V(\...
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3 votes
1 answer
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Plotting the bifurcation diagram for Ikeda map

I'm trying to plot the bifurcation diagram for Ikeda map. I wrote a code in Python to get the points of this diagram, but it seems that for $u > 1$ the points diverge and my code doesn't work ...
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2 votes
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center manifold and bifurcation: 2D Bifurcation system reduction

I have this system to study $$ \left\{ \begin{aligned} \frac{dx}{dt} &= y-x - x^2 \\[5pt] \frac{dy}{dt} &= \mu x - y - y^2 \end{aligned} \right. $$ I have derived the Jacobian around fixed ...
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