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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23 views

Determine whether a set is Invariant, Positively invariant or negatively invariant

I have just started a dynamical systems course and I am a bit confused as to how to determine if something is positively or negatively invariant, or just invariant. I know the defintions for ...
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24 views

Is there a fiber bundle approach to nonlinear oscillations?

I've recently been learning about nonlinear oscillations, and I noticed a seemingly strong connection between how the equations of motion are solved/approximated, and fiber bundles (or vector bundles ...
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23 views

How to determine if these r.v are independently Gaussian?

I have a time series vector ${(X_i)}_{i=1}^n$ which is the output of a non-linear dynamical system in $R^d, d=1$ of unknown distribution. Using Takens delay emebedding theorum, if I embed the 1 ...
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33 views

Lotka-Volterra coordinates transformation

I would like to ask the following: Given a Lotka-Volterra predator-prey system, \begin{align} & \frac{dx}{dt}={\alpha}x-{\beta}xy \\ & \frac{dy}{dt}=-{\gamma}y+{\delta}xy \end{align} , with ...
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2answers
73 views

Lotka-Volterra First Integral and Fixed Point

I have the following problem that I am dealing with, quite a long time, I must say. Let us assume that we have a predator-prey, Lotka-Volterra system given to us by: \begin{align} & ...
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69 views

Lyapunov exponents of a linear upper block triangular system

I seem to be stuck at formally showing something that intuitively seems to be true. I have a linear non-autonomous system of the form $$ \dot{x} = A(t)x $$ where $A(t):\mathbb{R}\to ...
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51 views

Types of solutions of ODEs and periodic orbits.

I am currently studying specific types of solutions of ordinary differential equations. If a vector field can be autonomous or non-autonomous, in which of these cases are there periodic orbits if ...
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37 views

Homoclinic orbits

Consider an autonomous vector field on the plane having a hyperbolic fixed point with a homoclinic orbit connecting the hyperbolic fixed point. Can a trajectory starting at any point on the ...
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1answer
45 views

Row Reducing with Imaginary Numbers

So I have the system $$ X' =\begin{pmatrix} 0 & 1 \\ -k & -b \end{pmatrix} X $$ where I assume $$ 0 \le b < 2 \sqrt{k} $$ which results in one complex ...
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2answers
83 views

Logistic family and chaos

It is a well known fact that the map $f(x)=4x(1-x)$ is chaotic on $[0,1]$. By chaotic I mean the usual definition, i.e.: a) the periodic points of $f$ are dense in $[0,1]$, b) $f$ is topologically ...
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58 views

Writing ODE system with a complex variable

I'm looking at the system of ODEs: $$\begin{cases}\dot{x} = -y + kx + xy^2\\ \dot{y} = x + ky - x^2\end{cases}$$ I'm trying to introduce a complex variable $z = x+iy$ to write this as a single first ...
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103 views

Measure of the set of periodic points of a measure preserving map

Given a continuous (Lebesgue) measure preserving map $T$ from a compact convex region to itself that has an aperiodic point (i.e. a point $p$ such that $p \ne T^n(p)$ for any $n$), does the set of ...
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2answers
125 views

Bump Functions in Dynamical Systems.

Define $B(x) = e^{-1/x^2}$ for $x > 0, B(x) = 0 $ otherwise. sketch the graph of B(x); prove that B'(0) = 0. When $x = 0$, $B(x) = 0$. It follows that the rate of change of a constant ...
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0answers
28 views

Uniformly convergent iterates for a function analytic on the complex unit disk

I have asked this question to several leading mathematicians in dynamical systems, and they all told me to ask someone else, until I was directed back to asking my advisor, with whom I first posed the ...
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1answer
36 views

Help figuring out output signal of LTI system.

Would greatly appreciate any help in figuring out the output signal of my discrete time LTI system. My input signal is cos(ωn) and my frequency response is H(e^jω)=(1+e^−jω)/2.
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1answer
61 views

Strongly mixing uniquely ergodic dynamical system

I'm looking for an example of a dynamical system which is both (measure-theoretically) strongly mixing and uniquely ergodic. I've looked around and found lots of discussion of systems which are ...
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69 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 ...
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1answer
134 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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26 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
31 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
42 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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33 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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27 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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26 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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74 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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36 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
50 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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33 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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28 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 $\lambda_1/\lambda_2 \notin \mathbb{R}^-$. ...
3
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1answer
65 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
29 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
32 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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37 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 ...
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3answers
127 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
91 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
19 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
81 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 ...
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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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
24 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
40 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
46 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 ...
3
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1answer
52 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 ...
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0answers
19 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
93 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
16 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) ...
2
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38 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
18 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
42 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
35 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 ...