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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54 views

Difficulty in understanding the Dyadic map and its application

The Dyadic map also called as the Bernoulli Shift map is expressed as $$x(k+1) = 2x(k) \bmod 1$$. Consider a discrete map $F : X \rightarrow X$ in the interval. Let this map be the Tent Map. In Link1: ...
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47 views

Could all iterates of $s(n)=2n+1$ be composite for some starting $n$?

Let $s(n)=2n+1$ and $\sigma(n)=\{n,s(n),s^2(n),s^3(n),\ldots\}$, where $s^3$ denotes functions composition, $s^3(n)=s(s(s(n)))$. For example $\sigma(11)=\{11,23,47,95,\ldots\}$. As another example ...
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51 views

Gentlest ascent in dynamical system

I have a question about the following excerpt from the paper("An Iterative Minimization Formulation for Saddle-Point Search") by Gao,Leng, Zhou on gentlest ascent in dynamical systems. I am having ...
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61 views

Slow fast systems

I have some questions concerning fast slow system like the van der pol equation say we have $\epsilon x′_1=-\frac13 x_1^3+x_1 − x_2$ and $x′_2= x_1$ Does $\epsilon x'_1$ means that $x_1$ is faster ...
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62 views

Topological Conjugacy between tent and skewed tent map

Consider the family of skew tent maps $\mathcal{S}$ on $[0,1]$, such that: $S(0)=S(1)=0$; The peak (maximum) of the tent occurs at $S(a)=b$; $\max(a,1-a)<1$ which implies the map to be locally ...
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36 views

Connection between possibility of non-monotonic solutions to first-order delay differential equations and 1-d discrete dynamical systems?

Is there a connection between the possibility of non-monotonic solutions, including periodic or other oscillatory solutions, arising in first-order autonomous delay differential equations such as the ...
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1answer
21 views

Approximating monotonically increasing differential equation

I am trying to make sense of the Appendix of the paper (Cooper, 1986). The following model is presented: $$\dot{(BX)}=\gamma_1BX \\ \dot{(BXB)}=\gamma_2(BX)B \\ \dot{B}=\gamma_3(BXB)$$ Without ...
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1answer
69 views

Is the topological entropy of the “caterpillar waves” 0?

Please let me first describe the general background. The state of the system at time $t$ will be described by a scalar or phase $u=u^t$. Both $t$ and $u$ are discrete. $u$ take values from ...
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42 views

Function not equal a.e. to continous function on real line and on circle

I am looking for a proof of the following fact: Suppose that $H: \mathbb{R} \rightarrow \mathbb{R}$ is a periodic function with period $1$. Suppose further that there is no continuous function ...
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1answer
34 views

Why is T continuous?

Let $T\colon X\to X$, with $X=\left\{0,1,2\right\}^{\mathbb{Z}}$, desribe the following dynamics: 1 becomes 2 2 becomes 0 0 becomes 1 if at least one of its two neighbours is 1, otherwise it remains ...
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29 views

controlling position of pendulum with motor

There is a pendulum with a motor mounted at its point of rotation. The motor can generate a rotational force at any time, thus changing the dynamics to $θ" = −a*sinθ − b*θ" + u$ where $u$ is the ...
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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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1answer
56 views

Lyapunov function

My lecturer gave us the definition of a Strong Lyapunov function. She then said that if V is positive definite but $dV/dt$ is also positive definite (instead of negative definite) in a region ...
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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
43 views

Second Order Non-Linear Ordinary Differential Equation

I have the equation $$x_{tt}+cx_t+x=x^2$$ where $c$ is constant and $x=x(t)$. If the $x^2$ wasn't on the right hand side of the equation then I could solve this easily by the method of ...
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1answer
35 views

Torus translation is ergodic if and only if the components of the translation vector are rationally independent.

I'm reading Ergodic Theory and Differential Dynamics by Ricardo Mane. There is a theorem in the book that states the following: If x $\in$ $R^n$, the translation L $_{\pi(x)}$: $T^n \rightarrow T^n$ ...
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2answers
46 views

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? ...
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28 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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37 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 ...
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25 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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Overview of nonlinear analysis, differential equations (ODE and PDE), dynamical systems, and mathematical physics, and their relationships [closed]

The fields of (i) nonlinear analysis, (ii) ODE and PDE, (iii) dynamical systems, and (iv) mathematical physics (e.g., electromagnetism, general relativity, gravitation, etc,...) are very huge, ...
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24 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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40 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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29 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
28 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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28 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
39 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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20 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 ...
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226 views

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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35 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 ...
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35 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
70 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
39 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 ...
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1answer
32 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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24 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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79 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 ...
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1answer
43 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
40 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
14 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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45 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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38 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 ...
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1answer
17 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 ...
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1answer
54 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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1answer
51 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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57 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 ...
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29 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
34 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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32 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 $x^3 - 3x - 1 = 0$. ...
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1answer
44 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. ...