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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16 views

Wronskian of a fundamental set of solutions

Consider the system of equations: $$\dot x_1=x_2$$ $$\dot x_2=-q(t)x_1-p(t)x_2$$ (Sorry I don't know how to do subscript notation for the 1's and 2's, an edit would be appreciated. Also the $x_1$ ...
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19 views

Wronskian, Linear Independence

Show that the following functions are linearly independent: $$e^t\begin{bmatrix}1\\1\\0\end{bmatrix}$$ $$e^{2t}\begin{bmatrix}0\\2\\1\end{bmatrix}$$ $$e^{-t}\begin{bmatrix}1\\0\\1\end{bmatrix}$$ ...
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16 views

Quantitative almost sure recurrence

I'm struggling to prove the following result, which is a special case of a quantitative recurrence result which is due to Michael Boshernitzan: Let $(X,d)$ be a compact metric space with finite upper ...
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1answer
31 views

Question that looks too easy to be true about the Gauss map

Using the itineraries of the Gauss map write the continued fraction expansion of the number $0 \leqslant \alpha \leqslant 1$ such that $$\displaystyle \alpha = \dfrac{1}{4+\dfrac{1}{3+\alpha}}$$ I ...
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1answer
17 views

Drawing toral automorphisms

How do we set about drawing a toral automorphism as in figure 5.1 in the picture above. How do we know where the points highlighted in yellow are? What happens if the eigenvalue (Im guessing some ...
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1answer
22 views

What does “tracktable trajectories” mean in this context?

In a dynamical system what does the following point to? "What range of trajectories seem to be trackable by XYZ method?"
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15 views

Is the conjugacy map between two distinct circle homeomorphisms unique?

Suppose $f,g,h_1,h_2$ are circle homeomorphisms with $f≠g$ and $fh_i = h_ig$ for $i=1,2$. Does it follow that $h_1 = h_2$? I restrict $f≠ g$ because I noticed that if $$f(x) := g(x) := R_\alpha(x) ...
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28 views

Convergence to a fixed point

When the following system is given: $x(k+1)=r-rx(k)$ where $r>=0 $ is a parameter Can someone explain why the fixed points are given by: $x(k+1)=x(k)=x^*$, so $x^*=\frac{r}{1+r}$? and how to ...
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2answers
33 views

$\lim\limits_{t\to\infty}t-x(t)=0\ ?$

Let $\displaystyle\cases{ x'=\frac{t-x}{1+t^2+x^2} & \cr x(1)=1 }$ be the Initial value problem, prove or disprove $\lim\limits_{t\to\infty}t-x(t)=0$ We've already proved that: for $t>1, ...
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34 views

How would one justify the claim that this differential cannot be solved analytically?

The Wikipedia article on the subject of free fall claims that: when the air density cannot be assumed to be constant, such as for objects or skydivers falling from high altitude, the equation of ...
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27 views

Algebraic approach to topological equivalence of dynamical systems

For continuous dynamical systems there is a notion called topological conjugacy or (somewhat weaker) topological equivalence. I gather that equivalence sends fixed points to fixed points and limit ...
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10 views

Question on stable manifolds

If $x\in M$ is a hyperbolic fixed point of a diffeomorphism $\phi:M\to M$, then the stable manifold $$ W^s=\{y\mid \lim_{n\to\infty}\phi^n(y)=x\} $$ is the image of an injective immersion $$ ...
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56 views
+50

About a result concerning Mersenne primes

I want to verify the proof of this result Theorem: If $p>2$ is a prime and $$H_{p}=\frac{(\sqrt3+2)^{2^{p-1}}+1}{(2^{p}-1)(\sqrt 3+2)^{2^{p-2}}}$$ is a natural number then $2^{p}-1$ is a prime ...
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1answer
26 views

Repairing solutions in ODE

Recently I encounter something interesting that I hope to hear from your opinions: Suppose we are given a ODE $\frac{dy}{dx}=y$, with no initial condition. Naively, we divide both sides by $y$ and ...
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49 views
+100

How to adapt the discrete-time to continuous, $(A) \Rightarrow (B)$?

in class was proved oseledets theorem for discrete time, following guidelines Ricardo Mañe book. Theorem discrete Oseledets (A) : Let $ M ^ n $ be a Riemannian manifold, $ f: M \rightarrow M $ be ...
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1answer
29 views

How I can find a similar expression for $x₀>1/2$

For the logistic map http://mathworld.wolfram.com/LogisticMapR=2.html the formula (4) in the link is valid only for $x₀<1/2$. How I can find a similar expression for $x₀>1/2$. The same question ...
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1answer
32 views

About quadratic map

let us consider the following quadratic map: $$s_{n}=s_{n-1}²+c$$ $$(*)$$ There is several papers disscuting the dynamics of (*). I want to know the behavior of this map for $c=-2$ and I am asking ...
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1answer
50 views

All fixed points of a function are globally stable or unstable.

I am analyzing the iterated sequence of the function $\lambda \sin( \pi x)$ for $x, \lambda \in [0,1]$, where $x_n=f(x_{n-1})$ for a paper I am writing. I know that all fixed points of this function ...
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60 views
+50

Quadratic Irrationality of the Periodic points of the Gauss map

If $G:[0,1] \rightarrow [0,1]$ is the Gauss map which is defined as $$G(x) = \left\{\frac{1}{x}\right\} = \frac{1}{x} - \left\lfloor\frac{1}{x}\right\rfloor,$$ show that if $x$ is periodic of order ...
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2answers
69 views

Fun with Newton's Method - Infinitely many cycles

I'd like to preface this problem by saying that I have absolutely no clue if it is solvable or not. This is just the result of some musings, and I'm looking for either some guidance, or to be pointed ...
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1answer
35 views

Problem with itinerary of a coding problem with infinite 1's

If $f(x)=2x \ mod \ 1$ on $[0,1)$. Then if we code $x \in [0,1)$ with its itinerary w.r.t. the partition $P_0=[0,1/2)$ and $P_1=[1/2,1)$. Can you show that there is no point $x$ whose itinerary has ...
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0answers
23 views

Conjugacy of linear systems with one zero eigenvalue

I have a question from Hirsch, Smale, and Devaney's "Differential Equations, Dynamical Systems and an Introduction to Chaos." Consider all linear systems with exactly one eigenvalue equal to 0. ...
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32 views

Sharkovskii's theorem in two dimensions?

Question A weak form of Sharkovskii's theorem in $1$D dynamical systems states that, if a continuous function $f:I\to I$ does not include a periodic point of least period $2$ on $I$, then ...
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1answer
36 views

1 dimensional flows and phase portraits

I have a flow defined by $\dot{x} = x-x^4+1 :=f(x)$. I need to sketch its phase portrait. Firstly, I have to find its fixed points, these occur at $f(x)=x$. So, $x^4=1 \Rightarrow x= \pm1$. Next, I ...
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0answers
28 views

Are stable manifold for gradient flows embedded submanifold?

Generally, the stable manifolds $W^s(p)$ of a diffeomorphism $\phi:M\to M$ is no embedded submanifold. The injective immersion $$ E^s:T_p^sM\to M $$ does not need to be a homeomorphism onto its image ...
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2answers
39 views

Derivative of a projective transformation

Assume $A$ is a matrix from $R^{n\times n}$, $A:R^n\rightarrow R^n$. Then $A$ induces a projective transformation $f:RP^{n-1}\rightarrow RP^{n-1}$. For example, $\\$ $$\begin{pmatrix} 4 & 0 ...
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1answer
23 views

Find for which r this system converges to a fixed point

Given the following (discrete time) system $x(k+1)=r-rx(k)$ where $ r>=0 $ is a parameter Find for which $r>=0$ all solutions of this system converge to a fixed point Verify if there exist ...
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0answers
34 views

A question of integral from Krengel's book in Ergodic Theorems.

As the picture depicts, I don't understand how did he get the RHS of: $$\int_0^{2X(\omega)} t^{-1} \psi(dt) \leq m(\log^{+} 2X(\omega))^{m-1} \int_{0}^{ 2X(\omega)} t^{-1} dt$$ Presumably it ...
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9 views

Deriving the $F_3$ type generating function in Hamiltonian formulation

I'm working on some practice questions and I am a bit confused with this one: Generating functions of the type $F_1(q,Q)$ satisfy the condition: $$pdq-PdQ = dF_1$$ Starting from this condition ...
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12 views

Basin of attraction

Let $$g(x)=\frac{2}{5}x^3-\frac{7}{5}x$$. The fixed points are 0 and $$\sqrt6$$. There is a period-2 orbit of 1 and -1. The critical points are $$\sqrt\frac{7}{6}$$ a. calculate the Schwarzian ...
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31 views

Dynamical System Problem

If (X, f) and (g, Y ) are dynamical systems (with semigroup |N_0 lets say) and π : Y → X is a semiconjugacy, then periodic points for g are periodic for f. Give an example that the opposite is not ...
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23 views

Stability of a fixed point

Determine the stability of all the fixed points of the following functions: So basically I understand how to find fixed points and whether they are attracting/repelling...but I am confused on how to ...
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1answer
23 views

Clarifying understanding of Poisson Brackets in Hamiltonian Dynamics

I'm just reading through my textbook and would like to clarify my understanding of 'Canonically related variables'. In my textbook, it says that if $Q_i$, $P_i$ are related to $q_i$, $p_i$ by a ...
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1answer
21 views

Show that the system is controllable (i.e. prove P has full rank)

Given the matrix: $$A = \begin{pmatrix}m&1&0&0&0\\ 0&m&1&0&0&...\\ 0&0&m&1&0&...\\ 0&0&0&m&1&...&\\ ...
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1answer
62 views

Mathematica Question regarding NSolve

I'm trying to solve for 2-cycles for the equation $$x=xe^{r[1-x]}$$ using Mathematica. I've tried using NSolve, FindRoot, and Solve. When I use NSolve I input it as $$\textrm{NSolve}[xe^{r[1 - xe^{r[1 ...
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0answers
20 views

Why this operator is not fredholm?

Define $f:S^{2}\to \mathbb{R}$ by $f(x,y,z)=z$. Let $D:=D_{\nabla f}$. As I learned from the following post this operator is not counted as a fredholm operator.( I did not underestand, ...
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4answers
99 views

On finding the equilibrium solutions to a system of differential equations

I am asked to find all equilibrium solutions to this system of differential equations: $$\begin{cases} x ' = x^2 + y^2 - 1 \\ y'= x^2 - y^2 \end{cases} $$ and to determine if they are stable, ...
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1answer
40 views

Newton Map in Dynamical Systems

Let $$p(x)=(x^2-1)(x^2-4)=x^4-5x^2+4$$. Let $$N(x)=Np(x)$$. Notice that N(x) goes to +-infinity at $$a1=-\sqrt{2.5}$$ $$a2=0$$ $$a3=\sqrt{2.5}$$ a. Sketch the graph of N(x). So the Newton map would ...
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1answer
38 views

Dynamical Systems: Transition Graphs

Let f be a continuous function defined on the interval [1,4]with f(1)=4, f(2)=3, f(3)=1, and f(4)=2. Assume that the function is linear between these integers. a. Sketch the graph of f b. Label ...
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105 views

Hopf bifurcation for a delayed DE

Consider the system of delay differential equations given by $$ \begin{array}{lcl} x^{\prime}(t) &=& (1+\alpha_1)y(t-\tau)-\alpha_2\alpha_3x(t)-\alpha_2(1-\alpha_3)z(t),\\ y^{\prime}(t) ...
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1answer
21 views

Ergodic behaviour for bounded random dynamical system

Considering an iterated system described by $$ X_n =\gamma_nX_{n-1} , $$ where $\gamma_i$ are non-negative i.i.d. variables. It is easy to show that the expectation will grow unbounded for $\mathbb ...
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17 views

Rotational invariant curve definition

I have a question about the definition of rotational invariant curve. The definition I have is the following "By an invariant curve for a twist map $F$ we mean a simple closed curve that is invariant ...
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26 views

How can I prove these two fields are locally topologically conjugated?

The problem is to prove that the fields $x'=x$ and $x'=x^3$ are locally topologically conjugated in the origin. I found that the corresponding flux for the first equation is $\phi(x_0,t)=x_0e^t$.The ...
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52 views

Connection between the Cantor set and the Tent map $T_3$

I would like to prove part c) using the ternary expansion as shown in the second half of the image. I understand how to compute up to what is shown in the image. I am struggling to understand the ...
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1answer
40 views

How to find a transfer function the transfer function from angular velocity to the current used by the motor? [closed]

I need to find the transfer function of a control system with the input being an angular velocity (from a sensor measuring the rotation of a wheel of a vehicle) and the output being a current used to ...
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0answers
59 views

Tubular neighborhoods in the proof of the Morse homology theorem

I have a question regarding the proof of the Morse homology theorem given by D. Salamon in "Morse theory, the Conley index and Floer homology". The full text can be found here: ...
4
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2answers
46 views

Iterating a periodic function

I'm curious about what happens if you iterate a function that is periodic. What happens to the period? For example, consider iterating a function like $\sin(x)$ or $\tan(x)$ several times. It should ...
4
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1answer
113 views

Is chaos theory really a theory? Why not just call it non-linear dynamics?

This may just be semantics, but it's always confused me. What is the thesis of Chaos Theory? I have read an entire book about it, and as far as I can tell, its just a bunch of analytical techniques, ...
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18 views

Solutions of unstable switched systems

Consider $n$ affine systems, $\dot{x} = A_i x + a_i$, where $\sigma(A_i)\subset \mathbb{C}^{+}$ for each $i\in \{1,\dots,n\}$. Here $A\in\mathbb{R}^{k \times k}$, $a\in\mathbb{R}^k$, ...
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1answer
23 views

Difference between slow and inertial manifolds?

Could anyone provide a clear definition of the basic difference between two of the invariant manifold types - the inertial and slow manifolds?