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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Dynamical System transformation

How can the system $$\frac{dx}{dt}=-y+\epsilon x(x^2+y^2)$$$$\frac{dy}{ dt}=x+\epsilon y(x^2+y^2)$$ be transformed into $$\frac{dr}{dt}=\epsilon r^3$$ $$\frac{d\theta}{dt}=1$$ via polar coordinates? ...
2
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23 views

Repeatedly interpreting polar coordinates as Cartesian

Start with a Cartesian point $(x,y) \in \mathbb{R}^2$, convert it to polar coordinates $(r,\phi)$ ($\phi$ in radians), and then reinterpret $(r,\phi)$ as $(x,y)$, i.e., set $$(x,y) = (r,\phi) \;.$$ ...
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24 views

How to choose $\epsilon$ and $\delta$ when proving stability/attractivity

I am having difficulty understanding how epsilon is chosen to prove that a dynamical system is attractive and/or stable. I have taken several analysis modules and was okay at proof writing, well now a ...
2
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0answers
10 views

Extraction of quadratic terms with state-space representation

I am having trouble with transforming the dynamics of a 4DOF gyroscope to a neat state-space representation. The system has the following set of equations: $T_i + f_i(\omega, \alpha) = 0;\;i:1-4$ . ...
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29 views

Overview of nonlinear analysis, ODE and PDE, dynamical systems, and mathematical physics and their relationships

(Apologies in advance for my naive question.) The fields of (i) nonlinear analysis, (ii) ODE and PDE, (iii) dynamical systems, and (iv) mathematical physics are very huge, fertile, and, in a ...
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13 views

Topological Entropy and generator: Do we need that T is a homeomorphism?

There is the following statement in Walters concerning the computation of Topological Entropy in case of an expansive homeomorphism: Let $T\colon X\to X$ be an expansive homeomorphism on a compact ...
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1answer
12 views

What is the difference between a period $n$ point, and a point of least period $n$?

What is the difference between a period $n$ point, and a point of least period $n$? Simply what is the definition of the two of them, and how do they differ. I think I have a rough idea of one ...
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23 views

Why gradient-like dynamical systems are special case of Morse-Smale systems?

I'm studying Morse Theory and my question is exactly as stated in the above title. I can't see how a gradient-like dynamical system could be considered as a Morse-Smale system? Thanks in advance for ...
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17 views

Periodic critical points of a tent map [on hold]

Let $f_{c}$ be the tent map with slope equals to $c$. Does there exist a way to describe how many $c's$ have $1/2$ as a periodic point? Thank you
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25 views

Prove that if $X$ has $n-$ points then $T$ is topologically transitive iff there is a periodic point of period $n$.

Prove that if $X$ has $n-$ points then $T$ is topologically transitive iff there is a periodic point of period $n$. Scratch idea: Suppose $T$ is topologically transitive, then $T^n U \cap V$ is ...
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22 views

Inhomogeneous ODE system with singular Matrix

I'm trying to write a Mathematica script to solve this inhomogeneous ODE system of the form $\dot x(t)=M x(t) + g(t)$, where $M=\left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & ...
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1answer
25 views

Solution of $\begin{cases} y(t)=z''\\ z(t)=y'' \end{cases}$

Solve $$\begin{cases} y(t)=z''\\ z(t)=y'' \end{cases}$$ $y(0) = z(0) = 0$ $y(\pi/2) = z(\pi/2) = 1 $ My attempt: $$\begin{cases} y(t)=z''\\ z(t)=y'' \end{cases}\Rightarrow ...
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23 views

Limit of a sequence of eventually periodic points

This is a research-related question I've been thinking about for a while now -- it seems like a standard exercise in first-year analysis, but the solution eludes me. Let $f:\mathbb{T}\to\mathbb{T}$ ...
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3answers
28 views

Show that for a gradient system $\bf\dot x= f(x)$, $\frac{\partial f_i}{\partial x_j}-\frac{\partial f_j}{\partial x_i}=0$ for $1 \leq i, j \leq d$ [duplicate]

The dynamical system ${\bf \dot x} = {\bf f}({\bf x})$ is called a gradient system if there exists a function $V({\bf x})$ such that $$ {\bf f}({\bf x}) = - \nabla V({\bf x}) $$ Show that if ...
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16 views

Perturbation of Ordinary Differential Equation Example

This is from Arnold's book on ODE's. Does anyone know of a reference or example where I could see how a linear equation arose in the way he mentions? Earlier he claims that linear ODEs are useful ...
3
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167 views
+100

Prove that there exists an $n\in\mathbb{Z}\cup\left\{-\infty,+\infty\right\}$ such that… (Dynamics)

Let $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and on it the following dynamics described by $T\colon X\to X$ as follows: A 1 becomes a 2, a 2 becomes a 0 and a 0 becomes a 1 if at least one of its two ...
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32 views

Application of Sharkovskii's theorem

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ continuous, $n \geq 3$ and $x_1< \dots < x_n$ so that $f(x_i)=x_{i+1}$ for all $i=1,\dots n-1$ and $f(x_n)=x_1$. In order to apply the theorem I have ...
3
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0answers
33 views

Fixed points of dynamical system

I am given a system $$ \dot{\theta_1} = C - \sin{\theta_1} + D\sin{(\theta_2-\theta_1)}, $$ $$ \dot{\theta_2} = C + \sin{\theta_2} + D\sin{(\theta_1-\theta_2)}, $$ $$ C,D \geq 0 $$ and asked to ...
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2answers
53 views

Need a textbook for math course

The undergrad course is called intro the applied math, and it covers: "The unit introduces some of the principal mathematical techniques such as difference equations, differential equations and ...
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1answer
20 views

Linearize system around trajectory

I know how to linearize a nonlinear system around equilibrium point (with Taylor series). There are lots of example on the internet about it. However, I didn't find a simple explication about ...
2
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1answer
25 views

How to find periodic points non-algebraically

For example, $f(x) = x - x^2$ by observation has a fixed point at $x = 0$, and an eventually periodic point at $x = 1$ (that goes to $0$). For the second iteration, to see if there's a periodic point ...
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15 views

Suspension flow and topological equivalence(s)

Let $M$ be a compact smooth manifold, $\tau:M\to\mathbb{R}_{\geq 0}$. Let $f:M\to M$ is a surjective piecewise-smooth map. There is a standard construction of suspension allowing to extend $f$ to a ...
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51 views

is there a diffeomorphism with only finite orbits but of infinite order?

Note: after not receiving any answer for some time, I asked this in mathoverflow, and got an answer there. The Question: Is it possible for a diffeomorphism $\phi$ (of a smooth manifold $M$) to have ...
3
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1answer
41 views

What are the fixed points of $f_ c = c · \sin$ for $c > 1$?

I’m doing an exercise for a lecture on dynamical systems. We are asked to classify all bifurcations of the dynamical system $f_c = c·\sin$ for real $c > 0$. We are given that bifurcations of ...
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1answer
31 views

How to describe behavior of population system, given by system of ODEs?

So I have a system of equations:$$x'(t)=x(t)(4-2x(t)-y(t))\\y'(t)=y(t)(3-x(t)-y(t)) $$ What I understand so far is: if we have x being the population of prey and y is the population of predators. x ...
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1answer
12 views

smoothness of invariant manifolds

Suppose M be compact manifold and f be a diffeomorphism on M.and A be hyperbolic set respect to f.How can we proof that the global stable and unstable manifolds of A are embedded manifolds?
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39 views

Lotka-Volterra Problem From Arnold's Ordinary Differential Equations

Problem 1 of section 2.7 of Arnold's Ordinary Differential Equations book asks to prove that the period of the oscillations in the Lotka-Volterra model tends to infinity as the initial condition ...
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12 views

Computing the index around a curve with respect to a field, invariance?

If I understood the course book Nonlinear Dynamics and Chaos right, The index can be found by $$\newcommand{\dd}{\mathrm{d}} \newcommand{\id}{\mathrm{d\,}} I_{C}=\frac{1}{2\pi}\oint_C ...
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1answer
32 views

How can this function be considered to have a saddle node bifurcation?

Say I have the function $f(x,\mu) = (1 + \mu)x − x^2 − 0.1$. By definition a Saddle Node bifurcation occurs if: $f_{\mu_0}(0) = 0$ $f'_{\mu_0}(0) = 1$ $f''_{\mu_0}(0) \neq 0$ $\frac{\delta ...
0
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1answer
12 views

Velocity field arrows along null clines as well as outside null clines

For Question 8 (as well as in general), I don't understand how to sketch velocity field arrows along the null clines as well as outside the null clines. For this question the f1 null cline would be ...
0
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1answer
48 views

Prove that $T$ has an orbit of period 3

Suppose that $T$ is continuous map from an interval $I$ to itself. Moreover, suppose that there exists $x_1 < x_2 < x_3 < x_4 $ such that $$T(x_1) = x_2, T(x_2) = x_3, T(x_3) = x_4\ \ ...
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35 views

Idea behind Lyapunov functions? [closed]

Despite my best efforts I am having some trouble understanding the concept of Lyapunov functions. Suppose we have a 2D dynamical system $$\dot{x} = f(x,y) \\ \dot{y}=g(x,y)$$ with a fixed point at ...
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1answer
44 views

Show that f: $\mathbb{R}$/$\mathbb{Z}$ $\to$ $\mathbb{R}$/$\mathbb{Z}$ orientation reversing. Then f(x) = x has exactly 2 solutions.

Im having some problem with the following question. Show that if $f: \mathbb{R}/\mathbb{Z} \to \mathbb{R}/\mathbb{Z}$ orientation reversing, then $f(x) = x$ has exactly $2$ solutions. ($f$ has $2$ ...
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82 views

Help in understanding the prove whether a map is one-to-one and disconnected for homeomorphism from the book

I need to prove that amp is a homeomorphism. I am following the basics from the book For the proof I have taken the help of the book "An introduction to dynamical system" Download link ...
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1answer
25 views

Locally evaluate nonlinear dynamic system's stability using eigenvalues

I don't have a large mathematical background, but I'm working with Computational Neuroscience. I have a large Synaptic Matrix (x axis: presynaptic NeuronID, y axis: postsynaptic NeuronID) in a Modular ...
0
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1answer
18 views

Lagrangian of bead on a rotating hoop

I'm trying to find the Lagrangian for a bead on a rotating circular loop (constant angular velocity $\omega$, radius $a$) in two different ways and I'm unsure why these are giving different answers. ...
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1answer
27 views

Numerical phase plane?

In my Differential Dynamical Systems text book, I came across the following question: Sketch the local behavior you obtained in the phase plane and compare with a numerical phase plane plotter that ...
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30 views

3-cycles of the iterated quadratic maps and the Galois Theory of $x^3 = a(x-1)$

Taken from these notes [1] on Galois Theory, I would like to show that iterating the map $$p: x \mapsto x^2 - x - 2 $$ has a cycle of order 3 when you start with the root of $x^3 - 3x - 1 = 0$. ...
0
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1answer
23 views

Periodic cycles of the Poincare map

For a dynamical system $\dot{x} = f(x)$, I understand the Poincare map is defined by successive intersections of an (n-1) dimensional surface $\Sigma$ with trajectories in n dimensional phase space. ...
4
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1answer
39 views

Recommended second textbook for dynamical systems?

I recently finished a course on dynamical systems supplemented by Strogatz's textbook. There are a few parts of the book that we didn't cover (in particular, the material on fractals), but the ...
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1answer
33 views

Angle-doubling map is mixing

Let $$ T:\mathbb{S}^1\to\mathbb{S}^1\\ x\mapsto 2x $$ be the angle-doubling map on the circle. We know that this transformation is ergodic. We want to prove that is mixing. I have to show that $$ ...
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1answer
21 views

What is the difference between a trajectory and an orbit in dynamical systems?

In dynamical systems, the integral curves for a differential equation that governs a system are referred to as trajectories or orbits, then what is the difference bewteen they?
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1answer
240 views

Equality by iteratively applying $(a,b)\rightarrow [(a+1,2b)\text{ or }(2a,b+1)]$?

I play a game starting with $2$ positive integers $a$ and $b$. At each step of the game I can double one of the integers and add $1$ to the other integer. Is there always a procedure for any ...
6
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1answer
110 views

Properties of the space of $ T $-invariant probability measures over a compact topological space.

Let $T: X \to X$ be a continuos map defined on the compact space (Maybe Haussdorff) $X$. Denote by $M_T$ the space of $T$-invariant probability measures defined over the Sigma Algebra of Borel sets. ...
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53 views

What's the difference between $Df$ and $Tf$?

I'm reading Michael Shub's Global Stability of Dynamical Systems. In chapter 4, he defined hyperbolic set and said the splitting $E^s$ and $E^u$ are $Tf$ invariant. So I assume this $Tf$ is the ...
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1answer
40 views

A differentiable manifold of class $\mathcal{C}^{r}$ tangent to $E^{\pm}$ and representable as graph

The stable manifold theorem tell us: A local stable manifold $W^{s}_{loc}(x^{*})$, which is a differentiable manifold of class $\mathcal{C}^{r}$ and dimension $n_{-}, $ tangent to the ...
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11 views

How to understand this verbal description of the dynamics?

Let $T$ map $\left\{0,r,l\right\}^{\mathbb{Z}}$ to itself by having the r's move right, the l's move left and an r and an l annhihilate each other when they meet or cross. How would you understand ...
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33 views

Terminology for functions such that $f(x)\ge x$ for all $x$

Is there a common terminology for a real function $f$ such that $$f(x)\ge x$$ for all $x$? same question for the conditions $\forall x,f(x)>x$; $\forall x,f(x)\le x$; $\forall x,f(x)<x$. (I'm ...
2
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0answers
42 views

Question related to the ballistic motion

A point mass will move in the gravitational field of the Earth according to the equation $$\ddot R =-\frac{GM_eR}{|R|^3},$$ where $R$ is the position vector of the point mass measured from the ...
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1answer
42 views

Conditions for Deriving $R_0$ for SIR Model Using Survival Function Method

I'm taking a look at the SIR model given by the system of differential equations \begin{align} \frac{dS}{dt} & = - \beta S I \\ \frac{dI}{dt} & = \beta S I - \gamma I \\ \frac{dR}{dt}& ...