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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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Bounded solutions of nonlinear third-order ODEs

I am interested in understanding the behavior of solutions to certain nonlinear third-order ODEs. Specifically, I am curious about conditions that guarantee all solutions remain bounded for $t \in [0, ...
Zhang Yuhan's user avatar
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Dynamical problem regarding radial and cross-radial acceleration

In a motion of two dimension, the radial and cross-radial components of acceleration are equal. Find the equation of the path. Radial acceleration: $\ddot{r}-r\dot{\theta}^2$ Cross-radial acceleration:...
Manjoy Das's user avatar
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Does this equation imply "non-linear" waves?

For one-dimensional bounded domain $x \in [0, L]$ consider $\partial_{t} v = -\frac{1}{\rho}\partial_{x}\rho - v $ $\partial_{t} \rho = -\partial_{x}(\rho v)$ with initial and boundary data as $v(0, ...
YoussefMabrouk's user avatar
1 vote
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28 views

Inversion of Nonlinear Dynamic using Koopman Operator

Consider a nonlinear dynamical system $X_{k+1}=F(X_k)$,to solve this system,we can transfer $X$ to a Koopman Invariant Subspace using a set of observation functions 'g' like $$g(X_{k+1})=K\mathcal g(...
UnnamedUser's user avatar
1 vote
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Proving that a dynamical system has at least one limit cycle, and that if there are several, they all have the same period $T(a, b)$.

This is exercise 7.3.7 of Nonlinear Dyanmics and Chaos by Strogatz. Consider $\dot{x}=y+a x\left(1-2 b-r^2\right), \dot{y}=-x+a y\left(1-r^2\right)$, where $a$ and $b$ are parameters $\left(0<a \...
Math_Day's user avatar
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Economic elasticity from a continuous-time dynamical systems perspective

I note that in system dynamics (Forrester (1961) Industrial Dynamics; Sterman (2000) Business Dynamics), which is applying differential-equation models to the social sciences, the concept of ...
gwr's user avatar
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Shift invariant measure and quasi-invariance

Let $X = \prod_{n\in \mathbb{N}}X_0$ for some finite set $X_0$ and let $T$ denote the shift transformation (namely, for any $x = (x_n)_{n\in \mathbb{N}}\in X$, $\big( Tx \big)_n = x_{n+1}$ for each $n\...
Sanae's user avatar
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2 votes
1 answer
42 views

Kalman Filter: Sequential Updates vs Simultaneous updates

There is this very thoughtful answer. It proves that if the observation noise is independent, then sequential update is the same as simultaneous update. Kalman filtering: Processing all measurements ...
Chen Chen's user avatar
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If a diffeomorphism has a dense orbit, are almost all of its orbits dense?

Let $M$ be a closed manifold. Suppose that a diffeomorphism $f:M\to M$ has a dense orbit. Is it true that almost every ofbit of $f$ is dense in $M$? Or, maybe, if the orbit of $x_0$ is dense, then all ...
Andrey Ryabichev's user avatar
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Show that the system $\dot{x}=-x-y+x\left(x^2+2 y^2\right), \dot{y}=x-y+y\left(x^2+2 y^2\right)$ has at least one periodic solution.

Show that the system $\dot{x}=-x-y+x\left(x^2+2 y^2\right), \dot{y}=x-y+y\left(x^2+2 y^2\right)$ has at least one periodic solution. My solution is as follows, \begin{align} r\dot{r} &= x\dot{...
Math_Day's user avatar
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1 answer
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A brick sliding in an horizontal plane after an initial push (under Coulomb's dry friction) - closed form solutions validation?

A brick sliding in an horizontal plane after an initial push (under Coulomb's dry friction) - closed form solutions validation? Posted later after comments: In summary, I am trying to understand what ...
Joako's user avatar
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Global uniqueness of the solution to a specific nonlinear ODE system with "global" initial condition

I get stuck trying to prove the uniqueness of the solution $y=\begin{pmatrix}y_1\\y_2\end{pmatrix}$ to the following system of nonlinear ODE's \begin{align*} y'_1&=\frac{y_1-y_2}{\xi(t)-y_1}\xi'(t)...
mariusz's user avatar
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Is Trajectories of a system same for different initial conditions on the trajectories?

I have a system of differential equation $\frac{d\rm x}{dt}=f(\rm x, t)$, where $\text{x}\in A \subset \mathbb{R}^n$ and $t\geq 0$. The function $f$ is a smooth function. The set $A$ is the unit cube. ...
Madhan Kumar's user avatar
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Subsequence Process of Non-Markovian Stochastic Process

I have a problem that I haven't encountered before and would like to know if there is literature on the problem. Assume $X_t$ is a non-Markovian stochastic dynamical system and that $X_t \in S=\{1,2,...
E.S.'s user avatar
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Question on iterated fixed points of $x^2-2$

Let $P(x)=x^2-2$. Let $P_n(x)$ denote the $n^{th}$ iteration of P. I was asked to prove that the equation $P_n(x)=x$ has all distinct real roots. My attempt: I tried using induction, but I'm not sure ...
Dailin Li's user avatar
2 votes
0 answers
43 views

Excitability of the FitzHugh-Nagumo model

I have the following variant of the FitzHugh-Nagumo model: $$\dot{u} = u - u^3 - v \\ \dot{v} = \epsilon(u-a)$$ Where $\epsilon>0$ and $a$ is a constant. I need to give a value for $a$ such that ...
CauchyChaos's user avatar
3 votes
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220 views

A chaotic function related to the $3x+1$ problem? (Li-Yorke and the Collatz problem)

Let $ x $ be an infinite binary string. Define the function $ f(x) $ mapping $ x $ to the Cantor set of $ I = [0,1] $ as: $$ f(x) = \sum_{n=0}^{\infty} \frac{2 x_n}{3^{n+1}} $$ where $ x_n $ are the ...
mathoverflowUser's user avatar
0 votes
2 answers
66 views

Show that if $(A, B)$ is controllable then the D.T described by $x(k+1) = Ax(k-1) + Bu(k)$ with initial states $x(0), x(1)$ is also controllable

I show this statement on some lecture notes : It is obvious that if $(A, B)$ is controllable then the D.T with initial conditions $x(0), x(1)$ described by $$ x(k+1)= Ax(k-1) + Bu(k)$$ is also ...
tonythestark's user avatar
2 votes
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33 views

Coherence Resonance in Coupled Chaotic Oscillators [closed]

For a project I have to reproduce the results of this papers numerical analysis, specifically those power diagrams. I have written python code attempting to solve this numerical system but I have not ...
Iztok Kravos's user avatar
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1 answer
57 views

Orbit of vector field crosses transverse section in the same direction

Let $X\in\mathbf{C}^1(U,\mathbb{R}^2)$ a vector field on the open set $U\subset\mathbb{R}^2$. Let $D\subset\mathbb{R}$ open and $f:D\rightarrow U$ be a $\mathbf{C}^1$ map such that $\{f'(x),X_{f(x)}\...
Jack's user avatar
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84 views

Fundamental theorem of Markov chains for integrable functions

Let $(X_n)_{n\in \mathbb{N}}$ be an irreducible, positive recurrent and aperiodic Markov chain on a countable set $E$, with stationary distribution $\pi$. Let $f\in L^1(\pi)$. Is it true that $P^nf(x):...
No-one's user avatar
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Why does an invariant measure define a Schwartzman cycle?

If $M$ is a manifold with a flow $\phi_X:\mathbb{R}\times M\rightarrow M$ induced by a vector field $X\in \Gamma TM$. Any Borel measure $\mu$ defines a $1$-current $c_\mu$, i.e. an element in the dual ...
Nuke_Gunray's user avatar
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1 answer
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Best Coordinate system - Lagrangian problem

In $\Bbb R^3$ consider an heavy point $P$ whose mass $m$ on a circumference $\Gamma$ of radius $R$, centered in the origin. Now consider that $\Gamma$ lives in the plane $$\Pi = \{( x,y,z) \in \Bbb R^...
Turquoise Tilt's user avatar
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The reduction of the dimensionality of a system of PDE's.

To transform an $ n $th-order linear ordinary differential equation (ODE) into a system of $ n $ first-order ODEs, you use a change of variables by introducing new variables to represent the higher-...
ayman lakehal's user avatar
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1 answer
65 views

I want a formal mathematical definition of the configuration space of a system (mathematically defined examples appreciated)

The Wikipedia article on configuration space offers this "formal definition": In classical mechanics, the configuration of a system refers to the position of all constituent point particles ...
Nate's user avatar
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1 answer
34 views

The identity of the variational equation (ODE)

Consider the nonautonomous system: $\dot{x}=f(t,x)$, $s(t, t_0,x_0)$ is the solution trajectory starting from $s(t_0,t_0,x_0)=x_0$. I wonder why $\frac{\partial}{\partial t} \Vert s(\tau, t, x)\Vert ^...
Rui Tachibana's user avatar
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0 answers
6 views

sign of singularity on legendrian arc in surface

I am learning contact geometry and the following lemma is stated in the notes of Prof. Etnyre on Convex Surfaces. Lemma Let $L$ be a Legendrian arc in a surface $\Sigma\subset (M,\xi)$ and $x$ a point ...
Csaba Daniel Farkaš's user avatar
1 vote
0 answers
38 views

Relation between Arnold tongues and Virasoro algebra

My knowledge in many mathematics topics is far from completeness, so please consider my question as so-called soft question. I know that Viroso group is a central extension of group of circle ...
Artem Alexandrov's user avatar
1 vote
1 answer
29 views

For the stable 3-cycle in the logistic map, where are the other cycles?

According to Sharkovsky's theorem, the existence of a 3-cycle means there will also exist cycles of all periods. In the logistic map, with the stable 3-cycle that starts at 1 + sqrt(8) with ...
Brian Whetten's user avatar
2 votes
1 answer
36 views

Stability of Hamiltonian system on degenerate critical point.

I'm trying to find information on the stability of the following ODE: $$ x'' = x^4-x^2.$$ We know that it has a Hamiltonian $H(x,y) = \dfrac{y^2}{2} - (\dfrac{x^5}{5} - \dfrac{x^3}{3})$. The orbits ...
Guybrush's user avatar
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0 answers
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How to prove the existence of global attractors?

In conclusion, based on compactness and continuity of the semigroup, there are some methods to prove the existence of global attractors for the autonomous systems. In detail, there are five methods to ...
Liangjia Guo's user avatar
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0 answers
43 views

Stability in Dynamic Systems

I am really having a hard time figuring out the solution of this problem : Consider a continuous flux on X. Let M ⊂ X be non-empty, compact, and invariant. Prove or disprove that M is stable if and ...
Omar Blake's user avatar
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0 answers
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Reference for proof of stability of almost linear systems

In Ordinary Differential Equations and Stability Theory by David Sanchez, he gives criteria for classifying spiral points and nodes and saddle points of a nonlinear system in terms of a linear system. ...
cgb5436's user avatar
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2 votes
2 answers
107 views

More on rationally independent subsets of $\mathbb{R}$.

Suppose that $\lambda_{1}, \lambda_{2}, \lambda_{3}\in\mathbb{C}\setminus\{0\}$ and that $\frac{\lambda_2}{\lambda_1}, \frac{\lambda_3}{\lambda_1}\in\mathbb{R}^{+}\setminus\mathbb{Q}$ such that the ...
user 987's user avatar
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0 votes
2 answers
48 views

Is the translation on torus $\mathbb{T}^2$ ergodic with the haar measure?

Let $f: \mathbb{S}^1 \times \mathbb{S}^1 \to \mathbb{S}^1 \times \mathbb{S}^1$ $f(x,y)=(cx,y)$ such that $c$ is a complex number and $|c|=1$. Consider the haar measure. Is it an ergodic system? Why? I ...
Gabriel Corrêa's user avatar
1 vote
0 answers
45 views

Numerically solving Pontryagin's Maximum Principle

I am trying to understand why when solving this boundary problem in matlab is significantly impacted by the $A_0$ matrix. The optimal control problem I am trying to solve reads: $$ \begin{aligned} ...
zzgsam's user avatar
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0 answers
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In how far is the Lyapunov spectrum characteristic for a dynamic system?

If two dynamic systems have the same Lyapunov spectrum on their respective attractor (of equal dimension, of course), which results relate to the properties of these two dynamic systems on those ...
algebruh's user avatar
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2 votes
1 answer
102 views

Linearization of ODE flows on manifolds about arbitrary solutions

On Euclidean space, for an ODE $$\dot x_t = F(x_t)$$ there is a natural linearization about $x_t$ given by $$\dot y_t = y_t \cdot \nabla F(x_t).$$ On a manifold $M$, if we have a vector field $V$, we ...
Physical Mathematics's user avatar
1 vote
0 answers
70 views

Recommended dynamical systems book [closed]

I'm searching for strong books on dynamical systems and discrete dynamical system. My level is a master's degree on Mathematics but i want to study in depth the theory of ODE and dynamical systems.
bernardo valente's user avatar
0 votes
0 answers
58 views

Checking 'No-resonance' condition for the eigenvalues of a discrete Laplacian matrix with Dirichlet boundary condition

1-D discrete Laplacian matrix (finite difference scheme) has eigenvalues as (page-2 in ref.): $$\lambda_j = sin^2(\frac{j\pi}{2(N+1)});\ j\in \{1, 2, ..., N\}$$ Where $N$ is the number of ...
Manish Kumar's user avatar
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0 answers
67 views

Prove existence of solution for IVP

I am trying to solve this ODE problem. I am studying for an exam. Let $f \colon \mathbb{R}^n \rightarrow \mathbb{R}^n$ be $C^1$ such that $|f(x)| \leq 1 + |x|^{\alpha}$. Consider the IVP \begin{...
user123456's user avatar
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0 answers
25 views

Space of functions, Banach spaces, reference books to find basic properties of Bochner integral, Laplace and Fourier transforms.

I'm looking for references where I can find definitions and basic properties of Bochner Integral in Banach Spaces and its basic properties, such as: Every continuous function is integrable, ...
Silvinha's user avatar
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1 vote
0 answers
29 views

Finding and classifying Hénon map bifurcations and periodic points

I am stumped on how to answer the following question: Consider the Hénon map given by $$\textbf{H}(x, y) = (a-x^2+by, x)$$ Assume $0<b<1$. Classify the bifurcations that occur at $a = -\frac{1}{...
JOlv's user avatar
  • 99
1 vote
1 answer
107 views

Stable Kalman Filter estimator with given covariance matrices

I asked this question a while back. Essentially considering the follow basic Kalman Filter, following the Wikipedia convention. \begin{equation} \begin{split} x_k &= F_kx_{k-1} + B_k u_k +w_k\\ ...
Taylor Fang's user avatar
3 votes
0 answers
27 views

Vector field with almost non-periodic orbits

Let $M$ be an n-dimensional smooth manifold (Open or compact). I want to know if it is possible to construct an smooth vector field with exactly one singularity, such that the set of periodic integral ...
Pablo Cid's user avatar
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0 answers
27 views

How are measurement functions expanded in terms of Koopman eigenfunctions in the Koopman mode decomposition?

I am trying to understand the Koopman mode decomposition as presented in Modern Koopman Theory for Dynamical Systems - Brunton et al. Specifically, in page 16-17, the Koopman mode decomposition where ...
Nikos H.'s user avatar
1 vote
1 answer
62 views

About density in $\mathbb{R}^{3}$

Suppose that $\frac{\lambda_2}{\lambda_1}, \frac{\lambda_3}{\lambda_1}\in\mathbb{R}^{+}\setminus\mathbb{Q}$. Does anyone have any suggestions to prove the following $$\overline{\mathbb{R}\left(1,\frac{...
user 987's user avatar
  • 645
1 vote
0 answers
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Could this vector field exists on a manifold?

I want to "generalize" the smooth flow $F(t,x)=e^{t}x$ on a Riemannian manifold in the following sense: Let (M,p) be a pointed positive dimensional smooth complete Riemannian manifold (that ...
Pablo Cid's user avatar
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0 answers
14 views

Indexes of nonhyperbolic equilibrium points in planar vector fields

There is a well-known theorem in dynamical systems stating that if $\gamma$ is a "sufficiently nice" closed curve (continuous, piecewise smooth, nonconstant function from $[a,b]$, say $[0,1]$...
Boris Dimitrov's user avatar
2 votes
0 answers
60 views

structural identifiability of ordinary differential equations is preserved when adding terms independent of parameters

Assume I have an ordinary differential equation of the form: \begin{equation*} \frac{dx}{dt} = f(x,t,\Theta), \ x(0) = x_0 \end{equation*} with $f:\mathbb{R}^n \times \mathbb{R} \times \mathbb{R}^l \...
Paul Joh's user avatar
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