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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31 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
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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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 ...
3
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2answers
60 views

Do Anosov flows exist on two dimensional compact manifolds?

Question: Let $M$ denote a two-dimensional, smooth, compact Riemannian manifold (without boundary). If $\phi:\mathbb{R}\times M \rightarrow M$ is a smooth flow, then prove that $\phi$ is not an ...
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0answers
24 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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0answers
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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0answers
21 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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0answers
15 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 ...
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0answers
88 views

reworking attractor equations

The equations/algorithms of these attractors (the link below) for given parameters and a number of iterations output a set of positions, the size of the set is the same as the number of iterations: ...
2
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0answers
28 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 ...
2
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1answer
22 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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0answers
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 $\lambda_1/\lambda_2 \notin \mathbb{R}^-$. ...
1
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1answer
41 views

A question on Bernoulli measures and mixing

this is a question on ergodic theory. Suppose I have an integer $N \geq2$ and a probability space $(\sum^{+} , B, \mu_{p})$, where $\mu_{p}$ is the Bernouilli measure with respect to probability ...
4
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1answer
272 views

Estimating a dynamical system's behavior without using Liapunov theorem

Assume that we have the following dynamical system $$x'=(\epsilon x+2y)(1+z)$$ $$y'=(-x+\epsilon y)(1+z)$$ $$z'=-z^3$$ Then how can I show that any solution that started from the region $z>-1$ ...
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 ...
4
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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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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
81 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 ...
2
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4answers
66 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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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 ...
1
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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 ...
1
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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
16 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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2answers
367 views

Closed form solution of this second order linear difference equation?

$$ y(k + 2) - 3y(k + 1) +2y(k) = 2^k + k $$ Transform into a system of $n$ first order equations (Step 1) $$\begin{align} x_1(k) &= y(k)\\ x_2(k) &= y(k + 1) \end{align}$$ It follows that ...
2
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0answers
63 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 ...
2
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2answers
275 views

Ergodicity of the First Return Map

I was looking for some results on Infinite Ergodic Theory and I found this proposition. Do you guys know how to prove the last item (iii)? I managed to prove (i) and (ii) but I can't do (iii). Let ...
2
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1answer
44 views

What is the density of the SRB measure conditioned to unstable manifolds?

I have a question regarding the SRB measure. As Lai-Sang Young puts it, the SRB-measure is the invariant ergodic invariant measure most compatible with volume. (see [ ...
2
votes
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 ...
3
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1answer
49 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 ...
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 ...
1
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2answers
68 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 ...
4
votes
2answers
181 views

Dynamical systems and differential equations reviews/surveys?

I would be very glad if someone could point me to modern reviews/surveys on these topics. To be concrete, I'll provide some examples: S. Smale, Differentiable dynamical systems D. V. Anosov, On the ...
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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 $f$ ...
6
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2answers
349 views

Polygonal billiards and uniform distribution

According to this article in Wikipedia: A billiard is a dynamical system in which a particle alternates between motion in a straight line and specular reflections from a boundary. When the particle ...
2
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0answers
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 $V(x,y)=x^2+y^2$ , ...
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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) ...
0
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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 ...
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 ...
1
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1answer
386 views

Lotka Volterra predator prey system

I am doing a project work mainly saying the relation between jacobian matrix and lotka volterra predator prey method , and I had a doubt,when I find eigenvalues of the system,I got purely imaginary ...
1
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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 ...
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1answer
37 views

Showing that the linear twist map is sensitive dependent

Choose $\Delta=\frac{1}{2}$ (I believe this value should work). let $\delta > 0$ and let $\textbf{x}_1=(x_1,y_1) \in X$. I assuming that $d$ is the Euclidean distance. Somehow I think we ...
2
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1answer
103 views

Solutions of a periodic non-autonomous system

I must find solutions for the system $$ \left( \begin{matrix} \dot{x_1}\\ \dot{x_2} \end{matrix} \right) = \left( \begin{matrix} \cos(t) & -\sin(t)\\ \sin(t) & \cos(t) \end{matrix} ...
0
votes
2answers
2k views

Solving lyapunov equation, Matlab has different solution, why?

I need to solve the lyapunov equation i.e. $A^TP + PA = -Q$. With $A = \begin{bmatrix} -2 & 1 \\ -1 & 0 \end{bmatrix}$ and $Q = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$. Hence... ...
1
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0answers
8 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 ...