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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130 views

Is there a general formula for the angles of reflection in a rectangular billiards table?

While writing a program, I encountered a problem in which I needed to calculate the angles of reflection of in a rectangular billiards table. Let's say that we look at the table from above, and that ...
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354 views

Show two interval homeomorphisms are topologically conjugate .

My question is the following: suppose we have two homeomorphisms $f,g:[0,1]\to[0,1]$ such that $f(0)=g(0)=0$, $f(1)=g(1)=1$ and that neither $f$ nor $g$ have a fixed point in $(0,1)$. Can we show that ...
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149 views

Controllability properties of discrete vs. continuous systems

I'm not sure what differentiates discrete from continuous systems in trying to prove certain properties of controllability. I have a chain of proofs from class that prove each other for a continuous ...
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27 views

proving that the measure of the points that back infinitely many often is the same as the original set.

Let $X=(\Omega,M,P)$ be a probability space. Let $f:\Omega \to \Omega$ be a function such that for each $A\in M$ $P(A)=P(f^{-1}(A))$ Given an event $B\in M$ we define $B_0=\{x\in X; f^n(x)\in B ...
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73 views

For which $\alpha$ do the $\epsilon$-neighborhoods of $\{k\alpha \mod 1 \mid k = 1, \ldots , poly(1/\epsilon) \}$ cover $[0,1]$?

For which $\alpha$ do the $\epsilon$-neighborhoods of $\{k\alpha \mod 1 \mid k = 1, \ldots , poly(1/\epsilon) \}$ cover $[0,1]$? In this paper on quantum computing (last paragraph of page 25), Dorit ...
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1answer
64 views

compactness of the set of invariant measures

Suppose that we have a dynamical system on some compact space $X$ with discrete time space and transformation given by some $\phi : X \rightarrow X$. My question is when is the set $Prob(\phi)$ of all ...
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72 views

How to prove that a given set is strongly invariant?

Consider a dynamical system $\dot{x}=f(x)$, where $f:\mathbb{R}^n\to\mathbb{R}^n$ is of class $\mathcal{C}^1$. For any time $t$, denote a solution issuing from $x$ by $X(t,x)$. Consider the proper ...
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42 views

How do I determine whether the image of a function lies in an algebraic curve?

On p. 2 of the book Differential Equations and Dynamical Systems by Lawrence Perko we define the function $$x(t) = \left(c_1 e^{-t}, c_2 e^{2t}\right)$$ It is then stated that the curve defined by ...
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179 views

Linearization of Nonlinear System

I am trying to linearize the following nonlinear system to determine the type of equilibrium point present. The system is $x'=a-x-(4xy/(1+x^2))$ $y'=bx(1-(y/(1+x^2)))$ Do I do this by creating ...
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208 views

Stability for Nonlinear System

I am trying to assess the (Liapunov) stability of the equilibrium at $(0,0)$ of the system \begin{align*} x_1' &= -4x_2 + x_1^2 \\ x_2' &= 4x_1 + x_2^2. \end{align*} I plotted the phase ...
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91 views

Bifurcation and homoclinic orbits.

In two dimensions, if we have a dynamical system: $$\dot{x}=f_k(x,y)$$ $$\dot{y}=g_k(x,y)$$ with $f$ and $g$ smooth functions and $k$ is a paremeter. If $k=k^*$ is a bifurcation at which two ...
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75 views

Given any two points in an attractor, is one of them always (almost) reachable from the other?

Consider a continuous dynamical system with bounded orbits, described by some system of ODE's. Let the flow of the system be $\phi$; i.e. $x(t + T) = \phi(T, x(t))$ for all states $x$ and times $t, ...
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577 views

Proof that a set is forward invariant.

So I am trying to show that a set is forward invariant and I am not sure that this proof does the trick. For an ODE $\dot{x} = f(x)$, a set $S$ is called forward invariant if $x_0 \in S \rightarrow ...
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1answer
100 views

Prove that there are no fixed points in a non-autonomous system

Consider the non-autonomous dynamical system $$ \dot x = f(x,t) $$ with $x \in \mathbb{R}^n$. This may be converted to an autonomous system of dimension $n+1$ with $t = x_{n+1}$ and $\dot x_{n+1} = ...
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115 views

Perturbation of discrete dynamical system

Suppose we have a discrete dynamical system $$x_{k+1}=M(x_k)$$ where $M(x)$ is a diffeomorphism and $x\in \mathbb{R}^n$. We have a fixed point of the system $\rho$, ($M(\rho)=\rho$) that satisfies: ...
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324 views

State space representation of transfer function $ 1/s $

I have three questions here: 1) I'm looking for a method to compute state-space equations by hand when given a transfer function, so far I think the best I've found is here. 2) Here's an example ...
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4k views

Polar coordinates differential equation

I have the following ODE: $$\dot x=-y(x^2+y^2), \dot y=x(x^2+y^2)$$ I want to sketch the phase portrait (manually) and I want to find the flow $\phi_t$, the orbit $O(x_0)$ and the limit set ...
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1answer
194 views

phase portrait for two systems in $\mathbb R^2$

I have two systems in $\mathbb R^2$ (1) $\dot x=x^2-1$, $\dot y=y^2-1$ (2) $\dot x=y^2-1$, $\dot y=x^2-1$ How is it possible to sketch the phase portrait for each of the systems in $\mathbb R^2$ ...
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87 views

Calculating the Lyapunov index of elements of a matrix.

In Random Dynamical Systems - Arnold Ludwig, Example 3.3.9. The cocycle generated by a random matrix: $$A=\left(\begin{array}{cc} a & c\\ 0 & b \end{array}\right)$$ is given by: ...
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1answer
85 views

Flow, dynamical systems

We have $f:\mathbb{R}^N \to \mathbb{R}^N$ Lipschitz and $x(\cdot; x_0 )$ is the unique solution of $$x'(t)=f(x(t))$$ $$x(0)=x_0$$ Every solution of this ODE is global. Show, that $\Phi_t ...
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44 views

Generator of a transport semigroup on the torus

Consider on the Banach space $C( \mathbb T;\mathbb R)$ of continuous functions on the 1-dimensional torus $\mathbb T:=\mathbb R/\mathbb Z$ (equipped with uniform norm) the operator $L$ given by $$ L ...
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48 views

linear homogeneous periodic equation

I'm having trouble with the following problem: Consider a linear homogeneous equation in the plane: $x'(t)=A(t)x(t)$ (1) Assume the matrix $A(t)$ has period $T$, in other words $A(t+T)=A(t)$. Show ...
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156 views

Average number of predators and prey in Lotka–Volterra model?

Once again I wouldn't be surprised if this can be found maybe even on Wikipedia but I'm not a native English speaker and unfortunately couldn't find this myself. So assuming standard Lotka–Volterra ...
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185 views

Basic example of system controllability

Ok so I'm having a tough time understanding an example in my book, there is a great deal of hand-waving and I'm having a hard time following. I'd like to see the example worked out if possible, with ...
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38 views

How can I prove this theorem about differential inclusions?

Consider the following differential equations with initial conditions at time $t_0$ specified: $\dot{x}_1 = f_1(x_1,t) ; \,\,\,x_1: [t_0,T]\to\mathbb{R}^n, ...
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71 views

Separable non-linear ODE (with radicals)

I am trying to solve the equation $$ \frac{dy}{dt}=\sqrt{(\gamma-1+\frac{2\alpha\beta}{2\alpha-1})e^{-2\alpha y}-\frac{2\alpha\beta}{2\alpha-1}e^{-y}+1} $$ [1] $y(0) = 0$; $t_{0}=0$; $\alpha$, ...
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1answer
491 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 ...
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1answer
224 views

Random Dynamical Systems: Intuitive Understanding

I am having trouble understanding this definition, [Arnold: Random Dynamical Systems]: Definition: A measurable random dynamical system on the measurable space $(X,\mathcal{B})$ over a metric ...
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127 views

ODE system and show infinite number of positively invariant ellipsoids

The system of ODEs is: $$ \dot{x} = -2x+yz \\ \dot{y} = x-xz \\ \dot{z} = xy $$ I found two lines of equilibria etc. but I now need to find the parameters for this "energy" or Lyapunov function, so ...
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69 views

are all dynamical systems described by differential equations?

we defined in lecture a dynamical System as a one-parameter family of maps $\phi^t:M\rightarrow M$ such that $\phi^{t+s}=\phi^t\circ\phi^s$ and $\phi^0=Id$, where $M$ is some (smooth) manifold and ...
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1answer
47 views

A set of functions which is open in the space $C^1[0,1]$

Let $f:[0,1]\to [0,1]$ be a $C^1$ and increasing function such that $i)$ If $f(p)=p$ then $|f'(p)|\ne 1$ I want to prove that there exist an $\varepsilon>0$ such that if $g\in C^1$ and ...
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1answer
177 views

Kepler, cartesian coordinates and ellipses

I am trying to see if I am on the right track with this. The problem: A kepler orbit (an ellipse) in Cartesian coordinates is: $$(1−\epsilon^2)x^2 + 2\alpha \epsilon x + y^2 = \alpha^2.$$ The task ...
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1answer
86 views

If $F:\mathbb{R}\to\mathbb{R}$ is a lift of the circle homeomorphism $f$, show that $F^n$ is a lift of $f^n$ for every $n\in\mathbb{Z}$.

By a lift of a circle homeomorphism $f$, I mean that $F:\mathbb{R}\to\mathbb{R}$ satisfies $\pi\circ F = f\circ \pi $ where $\pi$ is the natural projection $\pi:\mathbb{R}\to\mathbb{R}/\mathbb{Z}$. ...
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236 views

Show that Bellman-Gronwall's inequality

I'm trying to prove the theorem general of the Bellman-Gronwall's inequality: Assume that $u(t)$ be real valued non - negative continuous function, and such that $$u(t)\le u(\tau ...
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149 views

An ergodic theorem on the circle

Let $S^1$ be a circle (i.e. a closed $1$-dim. manifold) and let $F$ be a non-vanishing smooth vector field on $S^1$. Denote by $(t,x) \mapsto \Phi_t^x$ the flow generated by $F$. I want to show ...
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31 views

Is there a 2D 3-colorstate mobile automaton that grows like $ln^{0,5}(t)$?

Define an integer function $f(t)$ for an integer $t>25$ such that $|f(f(t)) - ln(t)| < \sqrt {ln(t)}+2$. Define $L(X(t))$ as the number of nonwhite states at iteration $t$ of mobile automaton ...
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111 views

Is there a generalization of the ODE Comparison Theorem to n dimensional systems such as this one?

Is the following theorem true? If so, under what conditions? If not, why not? For any finite set of points $S$, let $conv(S)$ denote the convex hull of $S$. Let $f:\mathbb{R}^{n+1} \to \mathbb{R}^n ...
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1answer
85 views

Showing the limit of a flow must be an equilibrium point under certain restrictions.

I'm stumped on how to approach this one: Consider the autonomous ODE $\dot{x} = f(x)$, $x \in \mathbb{R}^n$ with initial condition $x(0) = x_0$ and $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$ (at ...
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98 views

Generic invertibility of a Gramian-like matrix

Motivation: Consider the cascade arising from connecting the system $\dot{x}_1 = A_1 x_1 + B_1 u_1$ with the system $\dot{x}_2 = A_2 x_2$, $y_2 = C_2 x_2$ according to $u_1 = y_2$, namely: $\dot{x} = ...
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1answer
149 views

Maps with every point being periodic

Does there exists a characterization of continuous maps $f:[0,1]\rightarrow [0,1]$ with every point $x\in [0,1]$ being periodic (i.e. if for every $x\in [0,1]$ there exists $n\in\mathbb{N}$ such that ...
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2answers
147 views

Prove that $x \equiv 0$ of $\dot{x}(t)=a(t)x$ is Uniformly Asymptotically Stable

I have a problem: Consider the scalar equation: $$\dot{x}(t)=a(t)x \tag{I}$$ where $a(t) \in C(\mathbb{R}^+)$. Prove that $x \equiv 0$ of $(I)$ is Uniformly Asymptotically Stable iff ...
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1answer
51 views

Properties for internal stability of a discrete-time system

These are two parts of a larger proof I'm working on, can't figure how i) implies ii) though. Dynamic system: $x_{(k+1)} = Ax_{k}, x(0)=x_0$ Where $A \in \mathbb{R}^{n\times n} $ is a real ...
4
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2answers
218 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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2answers
97 views

A continuous function that when iterated, becomes eventually constant

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function, and let $c$ be a number. Suppose that for all $x \in \mathbb{R}$, there exists $N_x > 0$ such that $f^n(x) = c$ for all $n \geq ...
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1answer
156 views

There is non-trivial function satisfy the given condition?

Let $f:[0,1]\to\Bbb{R}$ to be a function satisfying that $$ f(x)=\begin{cases} \frac{f(2x)}{2} &\text{if }x<1/2 \\ \frac{f(2x-1)}{2}+\frac{1}{2} & \text{if } x\ge1/2\end{cases} \qquad ...
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2answers
60 views

Long run behavior of a dynamical system

I have to deal with a dynamical system that looks as follows, with $H$ being the initial state (parameters next to arrows denote transition probabilities between the states $H$, $L_1$ and $L_2$): I ...
2
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1answer
103 views

Parametrization of level sets of a smooth function

Let $H:\mathbb{R}^2\rightarrow\mathbb{R}$ be given by $H(q,p)=p^2/2+3q^2/2$ (single-well potential). This function has a critical point at $(0,0)$. Define $T:\mathbb{R}^+\rightarrow \mathbb{R}$ by, ...
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2answers
285 views

Show that System $(I)$ is stable iff $X(t)$ is bounded.

I have a theorem: For a linear homogeneous system: $$\dfrac{dx}{dt}=A(t)x \tag{I}$$ Where $A(t)=(a_{ij}(t))_{n \times n} \in C(\mathbb{R}^+,\mathbb{R}^{n \times n})$ Suppose that $X(t)$ be the ...
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52 views

How is open set defined in linear map space

I got this statement in my homework: Prove that the invertible linear contractions are an open set in $Mat(2\times 2;\mathbb{R})$ I know what "invertible linear contraction" , "open set" and ...
2
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249 views

“is topologically mixing” vs. “is topologically transitive” in the defition of chaos

Wikipedia gives essentially "is topologically mixing and has dense periodic periodic orbits" as the definition of chaos, and this paper shows that its (the paper's) definition of chaos is equivalent ...