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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 ...

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22 views

Finding approximation to stable manifold of saddle point

I am stuck on the following exercise from Strogatz' book on dynamical systems (exercise 6.1.14). Consider the system $\dot{x} = x+e^{-y}, \dot{y} = -y$. This system has a single fixed point, $(-1,...
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18 views

Geometric significance of a bifurcation point with algebraic multiplicity $2$?

This is part of Strogatz exercise $3.2.3:$ This is the process by which I found the bifurcation point/points for $\dot x=x-rx(1-x)$: By the method of tangential intersection we have: $$x=rx(1-x)$$ $...
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37 views

Closed form of an improper integral to solve the period of a dynamical system

This improper integral comes from a problem of periodic orbit. The integral evaluates one half of the period. In a special case, the integral is $$I=\int_{r_1}^{r_2}\frac{dr}{r\sqrt{\Phi^2(r,r_1)-1}}$...
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9 views

Stable focus and Hopf bifurcations

If my system cross successively a subcritical Hopf and a supercritical Hopf bifurcations, should my inner fixed point necessarily go from stable focus to unstable focus ? Thanks
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30 views

Optimal control problem with a path constraint which involves controls at two distinct time points

I am faced with an optimal control problem in continuous time which includes a path constraint which involves controls at two distinct points in time. I do not know how to approach this problem. I do ...
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1answer
49 views

For what $n$ is $W_n$ finite?

Suppose, $W_n$ is the set of all words formed by letters '$a$' and '$b$', that do not contain $n$ same consecutive nonempty subwords (that means that for any nonempty word $u$, the word $u^n$ is not a ...
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1answer
37 views

Prove that $f$ has a periodic orbit of least period $n$ for each positive integer $n$.

$$f(x)=\begin{cases} 1/2+x&\text{ if }0 \leq x \leq 1/2,\text{ and }\\ 2-2x&\text{ otherwise. } \end{cases}$$ Moreover, let $I_0=[0,1/2]$, $I_1=[1/2,1]$. Prove that for each infinite ...
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24 views

Nondimensionalization of the logistic equation.

In the process of studying nonlinear dynamics by Strogatz, I saw how he did simplify the model for an insect outbreak with the use of nondimensionalization. So as an exercise I picked an equation and ...
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21 views
+50

Relation of parameter-dependent center manifold and parameter-independent center manifold

Considering a parameter-dependent ODE $\dot{x} = Ax + f_A(x,y,z,\mu) \\ \dot{y} = By + f_B(x,y,z,\mu) \\ \dot{z} = Cz + f_C(x,y,z,\mu) $ where A has only eigenvalues with zero real part, B only ...
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20 views

What terminology should I use when refrencing how close a sequence is to a loop for research?

I don't know what language I should use in order to ask what methods already exist that discuss how to take a sequence and assess it's likelihood of looping. For example, If I was interested in this ...
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1answer
13 views

Unit of time and normalization of time preference rates

Consider an infinite horizon cake eating differential game described by \begin{align} &\max_{u_1(t)} \int_0^\infty{e^{-r_1 t}\ln(u_1(t))dt}\\ &\max_{u_2(t)} \int_0^\infty{e^{-r_2 t}\ln(u_2(t))...
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14 views

Finding strong stable foliation explicitly [on hold]

Let $A’=(1+i\omega)A-(1+i\gamma)A\,\|A\|^2 $ for some $\omega$ not equal to $\gamma$ in $\mathbb{R}$. Can you please help me to find a strong stable foliation explicitly for this differential ...
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28 views

Classify the bifurcation that occurs at $\mu$ =0

$ dx/dt=\mu x+y+x^2+x^3 , dy/dt=-x+\mu y+x^2y$ What I have done so far is getting the matrix A with $A_{11}=\mu,A_{12}=1,A_{21}=-1,A_{22}=\mu$ at $(0,0)$.I can see the bifurcation is Hopf ...
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1answer
45 views

Exosystem for reference generation

Recently I have read a paper where an LTI system of the form $$ \begin{align} \dot{x}_p &= A_p x_p + B_p u \\ y &= C_p x_p \end{align}\tag{1} $$ for the control plant was considered. In ...
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11 views

Discretization of continuous model with white noise to use Kalman filter later

I have this system which describes dynamics of a car in 2D space. The dynamics are governed by Newton's law g(t) = ma(t). The final task is to use Kalman filter on discretized system to estimate it's ...
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2answers
46 views

Simple system of Linear ODEs

I have this simple system of ODEs for rates $r_1,r_2$ and $a(t), b(t)$: $$a' = r_2b - r_1a$$ $$b' = -r_2b + r_1a$$ I am trying to solve for $a(t), b(t)$, but I am not sure what I am doing wrong. I ...
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46 views

Logistic map (discrete dynamical system) vs logistic differential equation

I have to roughly illustrate the logistic discrete dynamical system (as a model for population growth) to some non mathematics students. I'm not an analyst or an expert of dynamical systems. Looking ...
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0answers
35 views

Sharkovsky's Theorem and Triangular Functions

I'm trying to prove that Sharkovsky's Theorem Let $\vartriangleleft$ denote the Sharkovsky ordering given (informally) by $\underbrace{1\vartriangleleft 2 \vartriangleleft 4\vartriangleleft 8\...
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1answer
27 views

A state-space representation of an integro-differential equation implies a false statement

I would like to convert the equation $\ddot{y}+\int_0^t y(\tau)d\tau=0$ to state-space representation. Below, I present my attempt, which seems to be contradicting, and then ask my question at the end....
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57 views

Homeomorphism in compact two dimensional manifold, periodic points, and Euler Characteristic.

I want to prove that if a homeomorphism (a continuous bijection with continuous inverse) in a two dimensional manifold doesn't have a periodic point, then the Euler Characteristc of the manifold is ...
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28 views

XPP: how do I plot with something that is not a variable

I am investigating the motion of a robot with regards to four variables. Here is the code for my system: $$p'=\dot{p}$$ $$\dot{p}'= \frac{2 Nwa (Npa_{1}-\dot{p})}{1+exp(kd q^2)} + \frac{2 Nwb (...
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1answer
56 views

Conditions when solutions of $\dot{x} = f(x)$ exist for all time

I am reading the following textbook: Introduction to Applied Nonlinear Dynamical Systems and Chaos by Wiggins p.92 (top) Consider $$\dot{x} = f(x)$$ where $f(x)$ ...
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27 views

Intuition on Strongly Stable Sets in a dynamical system

I am trying to follow the beginnings of the following paper, I will post the relevant definitions after. https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/conleytype-...
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26 views

Balanced Word to Balanced (Sturmian?) Sequence

Let $E \in \{0,1\}^{n}, n\in \mathbb{N}$, be a balanced finite word: for every two subwords $U,V$ of the same length, the number of $1$'s in $U$ differs from the number of $1$'s in $V$ by at most one. ...
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1answer
65 views

Convergence of a sequence on the unit sphere of Bahach or Hilbert space

Let $X$ be a Banach or Hilbert space and $A$ be a bounded linear operator on $X$, and fix an element $x \in X$. Then I want to know that are there any good ways or theories to deal with the ...
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1answer
61 views

Find the Floquet multipliersfor the Markus & Yamabe system

I need some help with the following excersice.Find the minimum period and the Floquet multipliers $\bf\lambda_{1},\lambda_{2}$ of the following matrix. $A(t)=\begin{bmatrix}-1+\frac{3}{2}cos^2(t) &...
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1answer
32 views

Equipotential curves of a Julia set

In 'Dynamics in One Complex Variable' is states that a polynomial $f$ of degree $n$ maps the equipotential $G^{-1}(c) = \{z; G(z)=c\}$ to $G^{-1}(nc)$. I have been thinking about this and I can not ...
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20 views

Find circle map parameters $K, \Omega$, for certain predetermined orbits.

Find $K, \Omega \in \mathbb{R}$ such that the circle map $\theta_{n-1} = [\theta_n + \Omega + \dfrac{K}{2\pi}\sin(2\pi \theta_n)]\quad(\mod 1)$ has a fixed point and a period $3$ orbit. I was reading ...
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74 views

Regularity of a hyper-surface defined through a flow

Let $X:\mathbb{R}^n \to \mathbb{R}^n$ be a smooth and bounded vector field, such that $$ X_n \ge c|(X_1, \dots, X_{n-1})| \ge \epsilon > 0\,. $$ Under these assumptions one can prove that integral ...
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27 views

Space of orientation-preserving diffeomorphisms on the circle. [duplicate]

Diff$_+(\mathbb{T})$ is the space of orientation-preserving diffeomorphisms on the circle. Then is it true that it is connected? I have looked through the similar questions on here, but seem to find ...
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1answer
41 views

SIR model with disease fatality rate: disease always eventually disappears

This is a question in Edelstein-Keshet's book Mathematical Models in Biology: "show that in a SIR model with disease fatality at rate $\eta$ the disease will always eventually disappear". I took the ...
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1answer
47 views

Gradient dynamical systems have no nonconstant recurrent solutions

This is given as something we should intuitively understand, but I don't see how this is trivial. We were given that a solution is recurrent if $X(t_n) \to X(0)$ for some sequence from $t_n$ to ...
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88 views

Minimal dynamical systems in $2^{\mathbb N}$

If we have $\Delta$ a finite set (For simplicity we can just assume it's $2$) and we are looking at $\Delta^\Bbb N$, we can look at this set as a dynamical system with respect to the action: $T((a_n))...
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31 views

Dynamics $\delta x(t)=\delta x(0) e^{\lambda t}$ of Henon Attractor

Recall the question I asked before: Linearized perturbation dynamics of Henon Attractor So, I have the following separation dynamics for Henon attractor ($a = 1.4. b = 0.3$) $$\delta x(t)=\delta x(0)...
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1answer
83 views

Describing mappings using dynamics of time-dependent ODE-flows

Let $f\colon\mathbb{R}^n\to\mathbb{R}^n$. When is it possible to find some $g\in C^1([0,1]\times\mathbb{R}^n, \mathbb{R}^n)$, uniformly Lipschitz continuous w.r.t the second argument, such that if $u_{...
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1answer
85 views

Stability of Mathieu equation: $x''(t)+\cos t \,x(t)=0$

The equation $$ x''(t)+\cos t \,x(t)=0 \quad (1) $$ can be transformed to the system: $$\vec{x}'= \begin{pmatrix} 0 & 1\\ -\cos t & 0 \end{pmatrix} \vec{x}=A(t) \cdot x(t) $$ with minimum ...
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18 views

Brachistochrone curve

Determine the necessary conditions of optimum to bring the boat from point 1 $(0,0)$ to point 2 $(x,y)$ for a minimum time, $y$ isn't specified. Equation for fluid: $$ s(x)=-\frac{2}{625} x^2 + \frac{...
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0answers
24 views

ordering of variables for computing the Jacobian and eigenvalues

I'm a engineering student (i.e. no solid foundations on "true" mathematics), sorry if my question is silly. When I was computing the Jacobian to study the stability of equilibria points on power ...
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0answers
45 views

Show that this system has at least one unbounded solution as $t \to \infty$

Assume the system $$x'(t)=\begin{pmatrix} \frac12-\cos t & 2 \\ 1 & \frac32+\sin t \end{pmatrix}x(t)=A(t)\cdot x(t)$$ with minimum period: $T=2\pi$. Let $\mu_1,\mu_2$ be its characteristic ...
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1answer
48 views

Eventually periodic point or $\omega(x)=\emptyset$

Let $X$ be a complete metric space and $T:X\rightarrow X$ continous. if exists a point $x$ with closed orbit, then $x$ is eventually periodic or $\omega(x)=\emptyset$ I've tried so far: If $\omega(x)...
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1answer
45 views

Analizing the stability of the equilibrium points of the system $\ddot{x}=(x-a)(x^2-a)$

$\require{amsmath}$ $\DeclareMathOperator{\Tr}{Tr}$ $\DeclareMathOperator{\Det}{Det}$ Investigate the stability of the equilibrium points of the system $\ddot{x}=(x-a)(x^2-a)$ for all real values ...
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61 views

Poincaré-Bendixson for 3D systems?

The Poincaré-Bendixson theorem completely characterizes the $\omega$-limit sets of planar systems. I would like to know whether extensions exist to 3D systems which tend to 2D systems in the ...
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26 views

Linearization of nonlinear dynamic system

I am reading the following paper: ergodic theory of chaos and strange attractors, by J.-P. Eckmann (can be easily downloaded) My question is around equation (1.7)...
2
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1answer
50 views

LaSalle for time varying systems

I am looking for an explanation, why LaSalles theorem is in general not applicable to time varying systems. Can someone provide an example system with $$ \dot{x}(t) = A(t)x(t) \tag{1} $$ I.e., why ...
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2answers
37 views

Find one domain of attraction for this system

Assume the system: \begin{align} \begin{pmatrix} x \\ y \\ \end{pmatrix}' &= \begin{pmatrix} -(1-y)x \\ -(1-x)y \\ \end{pmatrix}...
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1answer
45 views

Linearized perturbation dynamics of Henon Attractor

I am reading the following paper: ergodic theory of chaos and strange attractors, by J.-P. Eckmann (can be easily downloaded) My question is around equation (2....
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0answers
27 views

Understanding ODE and phase portraits on Manifolds

I'm studying on my own Dynamicals Systems and having difficulties to undersrtand ODE defined on a manifold $M$. Firstly, let $X: \Omega \subset \mathbb{R}^{n} \rightarrow \mathbb{R}^{n} $ a vector ...
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27 views

Symbolic dynamics in continue dynamical system

Do you know a good and studied example of symbolic dynamics applied to a continue dynamical system? I mean, if there is an example of a continue dynamical system for which there is a Poincaré map for ...
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47 views

Poincaré map under small pertubations

Let $\gamma$ be a periodic orbit of a vector field $X$. Through a point $x_{0} \in \gamma$ we consider a section $\Sigma$ transversal to the field $X$ and V a small neighbourhood which contains $x_{0}$...
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2answers
47 views

Dynamic system $f(x) = 2x$ mod $1$

I am reading the following paper: ergodic theory of chaos and strange attractors, by J.-P. Eckmann (can be easily downloaded) My question is from an example on p....