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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13 views

Stability of dynamical system described in polar coordinates

Near a fixed point, a dynamical system $\dot{\bf{x}}=\bf{F}(\bf{x})$ can be approximated by $\dot{\bf{x}}=A\bf{x}$, where $A$ is the Jacobian matrix. From the trace and determinant of the Jacobian ...
5
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31 views

Limit points of the differential system $\dot {x}=y-x+x^3$, $\dot{y}=-x$

Consider the following system of differential equations: $$\dot {x}=y-x+x^3,\qquad \dot{y}=-x.$$ By linearization, it's easy to see that $(0,0)$ is a (nonlinear) sink. Show that there exists an ...
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23 views

Substitution in a system of ordinary differential equations when terms of the same order derivative for different variables occur in the same equation

Let's say I have a differential equation such as: y'' - 2ty' + y = 0, y(0) = 2.1, y'(0) = 1.0 I can solve this (among other ways) by substitution and conversion ...
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1answer
24 views

Coming Up With A Neutral Fixed Points Theorem

Question: If $f(x_0)=x_0,f'(x_0)=1$ and $f''(x_0)>0$, is $x_0$ weakly attracting, weakly repelling, or neither? (weakly attracting meaning $\exists\delta,\forall x\in ...
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2answers
33 views

Why is a linear autonomous system asymptotically stable iff for all eigenvalues $\lambda$ of $A$, $Re(\lambda) < 0$

I'm trying to understand asymptotic stability of linear antonymous systems. I'm not sure if for the system $x' = Ax$, $x(t) = 0$ is the only fixed point that can be stable. In any case, I can ...
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0answers
25 views

PDEs, infinite vector spaces, and functional analysis

Suppose I have a PDE for some quantity $q_{t}$:$= q(\boldsymbol{x},t)$ at some time $t$ on a 2D vector space $\boldsymbol{x} = (x_{1},x_{2})$, that looks, for the sake of the question, something like ...
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19 views

Entropy of isometric extension

A similar question to mine was asked before at the address below but it was not answered there so I am asking it again. Also there is a more specific case I am interested in. Topological entropy of ...
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23 views

Systems Theory - Complex Critical Point System

I have this system: $\dot{x}_{1}=x_{1}x_{2}+x_{2}$ $\dot{x}_{2}=x_{1}+x_{1}x_{2}^3$ Now i must find critical points. I have these solutions: ...
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16 views

Dynamical Systems - Jordan Form [closed]

Exercici (2.7.11) of Introduction to Dynamical Systems by D.K.Arrowsmith and C.M.Place Consider the differential equation $$\dot{u} = v, \quad \dot{v} = -v + \alpha u^{2} + \beta bu.$$ Make a linear ...
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1answer
52 views

Prove that $F(x) = 8x^{4} - 8x^{2} + 1$ is chaotic

I have to prove that the function $F(x) = 8x^{4} - 8x^{2} + 1$ is chaotic. I would like to use the definition of a chaotic function which is: Let $F$ be $F: V$ -> $V$. 1) Sensitive dependance on ...
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0answers
28 views

Calculate Derivative while Runge Kutta

I am thinking about writing a C++ code to solve an ODE using Runge Kutta method. As you know, RK method calculates the state space vector $X'$ in a few mid-points and uses these mid-points for ...
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1answer
44 views

Finding a strict Liapunov finction

I need to find a strict Liapunov function for this system at the equilibrium point $(0,0)$ $$x'= -2x-y^{2}$$ $$y'=-y-x^{2}$$ Also need to determine $\delta > 0$ as large as possible so that the ...
2
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0answers
29 views

Why are equilibria so important?

In studying nonlinear systems of differential equations, unlike linear systems, it turns out that we are more interested in equilibrium points rather than general solutions themselves. I mean, look ...
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0answers
12 views

How to prove a constructed set is a (n,ϵ)-spanning set for a [0,1] -> [0,1] homeomorphism

More specifically, I'm trying to figure out how to show that the following set is an $(n,ϵ)$-spanning set: $S = \{f^{-i}\big(\frac{j}{N}\big) \big| i = 0,1...n-1, j=0,1,...N\}$ where $N$ is selected ...
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25 views

Transversality of leaves to the spheres .

Consider a form in the complex plane such that its linear part is $\omega_0=\lambda_1xdy-\lambda_2ydx$ in the Poincare domain: $\lambda_1\lambda_2 \ne 0$ and $\lambda_1/\lambda_2 \notin \mathbb{R}^-$. ...
2
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1answer
23 views

Finding external angles for Misiurewicz points in the Mandelbrot set

In the Mandelbrot set for the quadratic polynomial $z \to z^2 + c$, rational external angles with even denominator are pre-periodic and have corresponding external rays which land at Misiurewicz ...
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3answers
27 views

The general solution of $y(n+1) = ay(n)^2$

I would like to find the general solution of the difference equation $y(n+1) = \alpha y(n)^2 $. I know that the general solution to $y(n+1) = y(n)^2$ is $y(n) = \exp({c\cdot 2^{n}})$. However, I've ...
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1answer
26 views

Understanding what a Diffeomorphism is.

I am self-studying Rob Devaney's "An introduction to Chaotical Dynamical Systems". "Decide whether each of the following functions are 1-1, onto, homemorphisms or diffeomorphisms on their domains of ...
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0answers
32 views

Stability proof of the difference equation $y(n+2)-y(n) = 0$

I'd like to be able to prove that the solutions of the following equation $y(n+2)-y(n) = 0$ are stable, but I'm having trouble defining a correct $\delta(\epsilon)$ such that the stability condition ...
2
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3answers
97 views

Basin of attraction of the fixed map $f(x) = x-x^3$

Prove that the interval $(-\sqrt 2 ,\sqrt 2 )$ is the basin of attraction of the fixed point $0$ of the map $f(x)=x-x^3$, for $x \in \mathbb{R}$. How one would prove this? In the examples I've seen ...
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0answers
82 views

Show that any sequence $(u_{n})$ must tends to infinity as $n→∞$

The motiation to this question can be found in About the solution of a difference equation My question is: Show that any sequence $(u_{n})$ verifying the equation in the above question must tends to ...
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0answers
12 views

Index of a limit cycle

How do we show that the index of a limit cycle is 1. I can see why (the vector tangential to any simple closed curve must rotate 2pi before returning to its original angle of inclination) but I am ...
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4answers
67 views

Can one generate a sequence of natural numbers whose density has a given distribution?

Suppose $\{ p_{k} \}$ is a collection of real numbers with the following properties: 1) $p_k \in (0,1)$ $~~~~$(i.e. $0$ and $1$ are not allowed values) 2) $\sum_{k=1}^{\infty} p_k =1$ An ...
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2answers
45 views

Lattice points of forward orbit of $z+z^{-1}$ are finite.

Let $f(z) = z+\frac{1}{z}$. Show that for any non-zero rational number $x$, the set $$\{f^n(x)\}_{n\geq 0} \cap \mathbb{Z}$$ is finite. For which $x$ is this set largest and what is its cardinality? ...
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0answers
15 views

Are preperiodic points subgroup?

Suppose that $G$ is a group and $f$ a group endomomorphism of $G$. Let $H = \{g \in G \mid f^n(g) = f^m(g) \textrm{ for some positive integers } n,m \textrm{ with } n \neq m\}$ be the set of ...
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1answer
19 views

Sarkovskii ordering is not a well-ordering?

The Wikipedia article on Sarkovskii's theorem claims that the Sarkovskii ordering of the natural numbers is not a well-ordering, stating: Note that this ordering is not a well-ordering, since the ...
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1answer
18 views

Eventual Image $Y\equiv\bigcap_{n\geq 1}T^n(X)$

Consider $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and $T\colon X\to X$ is defined 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 neighbors is 1. ...
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1answer
33 views

Gronwall's Lemma type problem

I have a function $X(t)\geq 0$, with initial condition $X(0)=X_0\geq 0$ and constants $\alpha < 0$, $\beta > 0$ and $\gamma <0$ such that $$\frac{d}{dt} X(t)^2 \leq \alpha X(t)^2 + \beta ...
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1answer
50 views

How to prove that $F(x) = \lim_{n \to \infty} F^{n}( f ' (0) \cdot F^{-n}(x)) $?

Let $F(x)$ be a real-analytic function near $0$ ,with $0$ as one of its fixpoints and $f ' (0) > 1$. $$F(x) = F \circ F \circ F^{-1} = \lim_{n \to \infty} F^{n} \circ F \circ F^{-n} = \lim_{n \to ...
2
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0answers
64 views

homogeneity of non-algebraic function [closed]

Suppose one has a system $$ \dot x_1 = f(x_1) $$ where merely a few properties are known. Properties are $f$ is nonlinear, Lipschitz continous, bounded, $f(0) = 0$ and $f \geq 0$. Is it possible to ...
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0answers
15 views

Possible Relations between Properties of a Polynomial and its Periodic Points

Question: Let $f(x)$ be a polynomial in $\mathbb{Z[x]}$. Is there a relation between the property $P_i$ of $f$ and the number of its periodic points with period $p$ (x is a $p$-periodic point of $f$ ...
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2answers
69 views

Suggestion for a Lyapunov function

Consider the differential system $$ x'=x+y $$ $$y'=x-y+xy$$ What would be a Lyapunov function for this system at $(0,0)$? I have considered functions $V(x,y)=ax^{2n}+by^{2m}$ but none of ...
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0answers
10 views

A M-system is a E-system?

(X,T)is a transitive system, if the minimal point is dense in X,then we call (X,T) is a M-system. if there exist a full measure m(i.e. supp(m)=x )and m is a T-invariant measure,then we call (X,T) ...
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31 views

suggestion for lyapunov function

Consider differential equation \begin{align}x'&=-t(x+y)\\ y'&=-y+x-y(y^2-6).\end{align} Can some one suggest a lyapunov function for it. I have examined $V(x,y)=x^2+y^2$ , ...
2
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0answers
14 views

How to find the value of a parameter such that the map has a period-doubling bifurcation?

For example: $f(x)=x_{n+1}=\mu+x_n^2$. Is it when $|f'(x^*)|=1$, where $x^*$ is a fixed point of the system? In this case, $\mu=1/4$? Also how to determine whether it is supercritical or ...
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1answer
31 views

To show solutions of a linear system lie on parabolas in phase space.

Given a linear system $\dot{x}=x$ $\dot{y}=2y$ To show solutions of a linear system lie on parabolas in phase space. Which solutions (if any) do not lie on parabolas? It is the second question ...
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1answer
32 views

Prove that if $\lambda_j$ are the eigen values of $Df(\bar x)$, and if $\lambda_j<1$, then $\bar x$ is assymptotically stable.

We study the discrete dynamical system in $\mathbb{R^n}$ with differentiable function $f(x)$: $$x_{n+1}=f(x_n)$$ $1.$ Assume that $\bar x$ is a fixed point and consider small perturbations around ...
2
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1answer
43 views

if $f$ is weakly mixing then $f^n$ is ergodic?

if $f$ is weakly mixing then $f^n$ is ergodic?I think this is false but I cant find a counter example because I dont know transformations weakly mixing but not mixing.can you prove or give a ...
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0answers
9 views

Expressing function as superposition of suitable functions

I came across the following paragragh in the paper entitled "simulation of Power-Law Relaxations by Analog circuits: Fractal Distribution of Relaxation Times and Non-integer Exponents" In linear ...
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1answer
36 views

Bi-infinite sequences of $0$'s and $1$'s

Consider the set $\Sigma=\{0,1\}^\mathbb{Z}$, i.e. the space of bi-infinite sequences of 0's and 1's, and the left-shift $\sigma:\Sigma\to\Sigma$. Define a distance in $\Sigma$ as follows: ...
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1answer
29 views

Differential system in 4 nonlinear equations

I have some problems in solving the following differential system. To simplify notation, I write $x$ for $x(t)$ and $x'$ for $\frac{\text{d}}{\text{d} t} x(t)$ $$ \begin{cases} w'=w^{a+1} & \\ ...
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11 views

Estimates for iterations of unimodal maps

I've been trying to read Jakobson chapter on Ergodic Theory of One-Dimensional mappings (Encyclopaedia of Mathematical Sciences Volume 2, 1989, pp 179-199) and a I have a small question about one of ...
2
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2answers
64 views

Proving a function is chaotic on an interval

I'm self studying from the book 'First course in chaotic dynamical systems' and am having a hard time grasping how to prove that a function is chaotic. For the function we have $T(x) = 2x$ for $x ...
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1answer
18 views

Is $T:(x,y)\mapsto(x+\alpha, y+x)$ mod $1$, expansive on $\mathbb{R}^2 / \mathbb{Z}^2$?

Let $\mathbb{T}^2=\mathbb{R}^2 / \mathbb{Z}^2$. Let $T: \mathbb{T}^2 \rightarrow \mathbb{T}^2$ be the transformation. Let $\alpha \in \mathbb{R}$. $$T(x,y)=\left(x+\alpha, x+y\right) \mod 1 $$ One ...
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1answer
38 views

Show that $S= \{ \left(\frac{i}{k},\frac{j}{nk} \right) : 0 \leq i < k, 0 \leq j < nk \} $ is an $(n,\epsilon)$-spanning set

Let $\mathbb{T}^2=\mathbb{R}^2 / \mathbb{Z}^2$. Let $T: \mathbb{T}^2 \rightarrow \mathbb{T}^2$ be the transformation. Let $\alpha \in \mathbb{R}$. $$T(x,y)=\left(x+\alpha \mod 1, x+y \mod 1 \right) ...
2
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1answer
15 views

Show that $T^n(x,y)=\left(x+n\alpha \mod 1, y+nx+\frac{n(n-1)}{2}\alpha \mod 1 \right)$

Let $\mathbb{T}^2=\mathbb{R}^2 / \mathbb{Z}^2$. Let $T: \mathbb{T}^2 \rightarrow \mathbb{T}^2$ be the transformation. Let $\alpha \in \mathbb{R}$. $$T(x,y)=\left(x+\alpha \mod 1, x+y \mod 1 \right) ...
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1answer
45 views

Show that $\lim_{n \rightarrow \infty} \left(\prod_{i=1}^{n} (a_i+1) \right)^{1/n} $ using Birkhoff Ergodic Theorem

Show that for Lebesgue-almost every $x \in [0,1)$, the geometric mean $$\lim_{n \rightarrow \infty} \left(\prod_{i=1}^{n} (a_i+1) \right)^{1/n} $$ exists and has common value. What is this? (no ...
1
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1answer
37 views

If $x=[a_0,a_1,\dots]$ show that $\mu$-almost every $x \in (0,1/N]$ is infinitely recurrent

Let $G$ be the Gauss map, $$G(x)= \begin{cases} 0 & \text{if} \ x=0 \\ \{\frac{1}{x} \}=\frac{1}{x} \ \mathrm{mod} \ 1 & \text{if $0<x\leq 1$}\end{cases}$$ and $\mu$ be the ...
1
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0answers
25 views

Conditions to guarantee unique limits of trajectories.

For a real function $f$ on $\mathbb R^n$, such that no trajectories of the gradient escape to infinity, what are necessary and/or sufficient conditions so that each trajectory limits to a unique point ...
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2answers
34 views

Show that $ \lim_{n \rightarrow \infty} \frac{|f(T^n(x))|}{n}=0 $

Let $T: (X, \mathcal{A},\mu) \rightarrow (X, \mathcal{A},\mu)$ be ergodic wrt a measure $\mu$ on $(X,\mathcal{A})$. Show that for any $f \in L^1(X,\mathcal{A})$ and $\mu$-almost every $x \in X$ we ...