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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1answer
14 views
Range of stability for iterative map
Using linear stability analysis, I would like to compute the range of stability of the fixed points and the $2$-cycles of the following iterative map: $x_n = x_{n-1}^{2} - 3\mu$.
Setting $x = x^{2} - ...
1
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1answer
18 views
Strictly invariant sets of the rotation transformation on a discrete space.
Fix an integer $n>0$. Consider the space $X=\{a_0,a_1,...,a_{n-1}\}$ with transformation $T:X\to X$ defined by $T(a_i)=a_{i+1(\text{ mod n})}$.
What are the strictly invariant sets of this space? ...
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1answer
23 views
A good reference to learn the concept of partition function
I am looking for a good reference and easy to learn the concept of partition function in mathematics. Can anyone help me?
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2answers
53 views
Definition of metastability
I was reading Terence Tao's blog post on analogies between soft and hard analysis when I saw that the soft analysis statement "$x_n$ is convergent" corresponds to the hard analysis statement "$x_n$ is ...
3
votes
2answers
62 views
Show the map $f(x)=\frac12 (x+1/x)$ has an attractive fixed point in $(0,\infty)$
From numerical test, I know $x=1$ is an attractive fixed point of the function
$$
f(x)=\frac12 \left(x+\frac{1}{x}\right),
$$
on $(0,\infty)$.
Is there a way to prove it?
Since
$$
...
2
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1answer
48 views
Ergodic theory question about the support of a measure.
I am trying to work through some problems from " Ergodic theory with view towards number theory" but I am finding it a tad more difficult than expected. In particular, I have the following question:
...
2
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1answer
25 views
Examples of systems conforming the Lorentz Attractor
Might sound like a trivial question but would you please show me some examples of real systems conforming the Lorentz Attractor?
It can be any kind of system, just a little list. It can be a system ...
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1answer
43 views
How write these two motions into one system
f describe the horizontal motion
g describe the vertical motion
have error when put m(t) in x()
Maple code
...
1
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1answer
31 views
Conditions that Roots of a Polynomial be Less than Unity
Is is the case that Samuelson's result is a more specific result of Rouche's Theorem, or the Routh–Hurwitz stability criterion? Is it not the goal for a polynomial to be stable that all of its roots ...
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0answers
22 views
Strength of attraction of fixed points
Consider a smooth map $f: \mathbb{R} \rightarrow \mathbb{R}$ with an attracting fixed point $F$. Then, we have
if $f'(F) \ne 0$, $F$ is a "simple" attracting fixed point,
if $f'(F) = 0$, $F$ is a ...
1
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0answers
31 views
Normal form of a vector field.
Those are two analogous problems, the first one of which I have already accounted for.
Find the normal form of the vector fields:
a) Solved.
b) $$\dot x_1=x_2$$
$$\dot x_2=-x_1 $$
$$\dot x_3=\sqrt ...
3
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0answers
28 views
Definition and some elementary properties of the “vector turn map”
This is actually a follow up question. I have to apologise for the length of it. I didn't anticipate to be that long. I hope that it will be proved interesting nevertheless. I used the name "vector ...
13
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4answers
111 views
What are the important theorems in the theory of dynamical systems?
I recently stumbled over the section about dynamical systems in my physics textbook. I noticed that, although most of the rest of the book was very rigorous, this part contained nearly no firm ...
3
votes
3answers
118 views
Is this map a known one?
Let $A$ be a $2\times2$ real matrix, then define $f:S^1\to S^1$ by $f(\phi)=\arctan(A\cdot(\cos \phi,\sin \phi))$.
This can be viewed as a discrete dynamical system on $\mathbb{S}^1$ and I am trying ...
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1answer
37 views
Conjugating the flows of two dynamical system.
Consider the two one-dimensional linear odes
$$\dot x=\lambda_1x\qquad\dot x=\lambda_2x$$
Here $\lambda_1\not=\lambda_2$ and they have the same sign.
Now the solutions to those equations are ...
1
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1answer
41 views
Explicit form of Poincare's map for the following system of Ode’s
Problem: Write in explicit of Poincare's map for
$\ddot x+\delta\dot x+\omega_0^2x=\gamma\cos\omega t$.
Find the stationary points and examine their stability.
An attempt at a solution: Let ...
3
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0answers
64 views
Differential equation $y'(t) = 1-y(t) e^{y(t)-1}$
I am interested in finding a clean explicit solution (if possible) to the differential equation
$$
y'(t) = 1-y(t) e^{y(t)-1},
$$
where $0 \le t < 1$ and $0 \le y \le 1$.
This can obviously be ...
1
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0answers
37 views
Pull Back (change of variables)
Let be $h:\mathbb{R^2}\rightarrow\mathbb{R^2}$ a change of variables (diffeomorphism). Let be $X$ a vector fields in $\mathbb{R^2}$ and $f:\mathbb{R^2}\rightarrow\mathbb{R}$ a continuous application. ...
0
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0answers
20 views
Dynamic Linear Model - Joint distribution of observations and states
In my DLM, observations are denoted by $Y_t $ and the state vector by $ \theta_t$.
We assume we're in a closed system so that we can only learn about the future through past observations.
Our first ...
1
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0answers
28 views
Center Manifold Theorem
Let $f:M \rightarrow M$ be partially hyperbolic diffeomorphism of M with the usualy definition that at each p tangent space splits to Df invariant subspaces: $T_pM = E^s_p + E^c_p + E^u_p$ with the ...
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1answer
24 views
$Az=λz$ lead to $x(t) = c_1*e^{\lambda_1 t}z_1+c_2*e^{\lambda2 t} z_2+…+c_p*e^{\lambda_p t}z_p$ is a solution to $dx/dt=Ax$. Why?
I'm studying a course in dynamical systems. It's a pretty much linear algebra intensive course, and it's been a while since I did that sort of things.
In it, they say that if vector $z$ satisfies ...
2
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1answer
35 views
Dense orbits and invertibility
Let g be the logistic map $g(x) = 4x(1-x)$ and define $\phi (x) = \sin^2(\frac{\pi}{2}x)$, for $x \in [0; 1] $. Show that $\phi$ is invertible and $\phi \circ f = g \circ \phi$ , where $f$ ...
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0answers
47 views
Proving semi-conjugacy preserves chaotic behavior
http://www.math.upatras.gr/~bountis/files/def-eq.pdf
In the above documentation it states "It is easy to check that a semiconjugacy
also preserves chaotic behavior on intervals of finite length" on ...
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1answer
29 views
Logistic Equation: Equilibrium
Given a differential equation, for example, a logistic curve, how do I determine the equilibrium points, graphically?
Consider $$x'=ax\left(1-\frac{x}{b}\right)-\frac{x^2}{1+x^2}$$
It is clear that ...
3
votes
0answers
22 views
Chaos without period doubling
I have been studying the Duffing oscillator rather intensively lately, mainly based on the theory in of the book by Guckenheimer and Holmes. From all that I have gathered, it seems that most dynamical ...
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0answers
8 views
Caclulating limits of Hassell Model to show under/over compensation
For this question, you will need the following definition: For a discrete time model of intraspecific competition $N_{t+1} = F(N_t)$, if the limit $F_{\infty} = \lim_{N \to \infty} F(N)$ exists, ...
0
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1answer
32 views
Dynamical Systems. Bendixson's and Dulac-Bendixson's theorems.
I am looking for a place to read the proofs of Bendixsons and Dulac-Bendixsons theorems.
Namely let D be a simply connected set and the following system be defined in D.
$$\dot x=P(x,y)$$
$$\dot ...
2
votes
1answer
46 views
How important are the following undergrad courses when trying to pursue studies in chaos theory/dynamical systems?
I'm currently a physics major with a year left, and deciding whether to switch into mathematical physics, mathematics or applied mathematics. I'm definitely switching into one of them, as I can meet ...
1
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1answer
99 views
An issue with approximations of a recurrence sequence
By trying to give an approximation to a given recurrence sequence I encountered a problem.
To be more precise I have a method but it fails if the right condition is not met and I wonder how I should ...
2
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0answers
63 views
Show that this orbit has a zero Lyapunov exponent
I'm using J.Meiss -Differential dynamical systems, and have some trouble to understand a proof about Lyapunov exponents.
We have a dynamical system
$$ \dot{x} = f(x), $$
with the corresponding flow $ ...
7
votes
2answers
79 views
Recurrent point of continuous transformation in a compact metric space
Given a metric space $(X,d)$ and a transformation $T:X\rightarrow X$, a point $x\in X$ is said to be recurrent iff it belongs to the closure of its orbit $\{T(x), T^2(x),...\}$: more precisely, ...
1
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1answer
50 views
Why do manifolds with negative sectional curvature not have conjugate points?
I'm trying to understand why manifolds with negative sectional curvature not have conjugate points. In fact for me it is sufficient to understand it for surfaces, but of course i'd be interested in ...
2
votes
0answers
29 views
Are geodesic flows on surfaces with negative curvature Anosov?
I'm just going through the original book by Anosov, where he tries to proof this result.
I don't quite understand it.
So let $\phi_t:TM\rightarrow TM$ be a geodesic flow on a compact surface $M$ of ...
3
votes
1answer
34 views
Similarity between Henon attractor and logistic map?
I noticed that layering Henon attractor images with avalue=1 and bvalues from -0.2 to 0.3 looks like a distorted version of the logistic map.
In the image below you can see the layered images (left).
...
1
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2answers
78 views
Classifying local behavior of fixed points using eigenvalues from linear stability analysis of 3D system
I've learned about classification of fixed points of 2D systems using linear stability analysis and I am wondering how if at all I can apply the same process to analyzing local behavior of fixed ...
0
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1answer
50 views
Calculating the angular velocity
I have an inverted pendulum with a accelerometer mounted on the top that at rest gives me a vector up opposite to gravity, which is used to calculate the angle of the pendulum. Is it possible to ...
2
votes
1answer
42 views
Linear stability analysis on a constrained three-dimensional system of ODE
Let $\begin{cases} \dot x = f({\bf u}) \\ \dot y = g({\bf u}) \\ \dot z = h({\bf u})\end{cases}$ be a well-defined nonlinear system with ${\bf u} = (x,y,z)$ and restricted to domain $x,y,z \geq 0$. ...
0
votes
1answer
35 views
Unfolding a Billiard Trajectory
The following image is from page 1019 of http://www-fourier.ujf-grenoble.fr/~lanneau/references/masur_tabachnikov_chap13.pdf
From what I understand about unfolding billiards we are representing the ...
4
votes
3answers
74 views
Repeated nested roots
Quite some years ago, I remember being asked the following question:
Suppose $\alpha = \sqrt{2+\sqrt{2+\sqrt{2+\ldots}}}$, what is $\alpha$.
The solution was given by squaring $\alpha$ and solving ...
1
vote
1answer
28 views
Are there solutions when the boundary conditions are particle positions at 2 different times instead of positions and speeds at an initial time?
Is it possible to find solutions for a dynamic system when the boundary conditions are particle positions at 2 different times instead of positions and speeds at an initial time?
The question is ...
0
votes
0answers
39 views
The infinite-time Shadowing Theorem.
Let $S$ be a hyperbolic set in $\mathbb{R}^n$. $f$ is a diffeomorphism in $C^1$.
$y= \{ y_k\}_{k=- \infty}^{+ \infty} \in l^{\infty}$ is $\delta-$ pseudo orbit of $f$ in $S$.
$G: l^{\infty} ...
4
votes
1answer
68 views
The Starry Rebound
An (infinitely small) ball starting out in the middle of a 5 pointed star table (outer 5 points 10m radius, inner 5 points 5m radius) has a starting angle of a random value from 0 to 360 degrees. The ...
0
votes
1answer
60 views
Results for $y^{\prime\prime}(x) = a(x)y(x)$, where $a(x) > 0$.
I'm looking for references to any known results regarding solutions to the following 2nd order ODE $y''(x)=a(x)y(x)$, where $a(x)>0$ and $x \in \mathbb{R}$. Any help would be appreciated.
2
votes
1answer
19 views
Doubling Map and Measure
First off! This is a homework question, so I DO NOT want an answer to the question I'm writing, I really just want an explanation of the final bit (which I'll make clear).
So if we have $T:[0,1)\to ...
2
votes
1answer
58 views
How to classify equilibrium points
I have the two differential equations:
$$\frac{dN_1}{dt} = N_1(2 - N_1 - 2N_2)$$
$$\frac{dN_2}{dt} = N_2(3 - N_2 - 3N_1).$$
I worked out the equilibrium points to be at $N_1 = 0, \frac{4}{5}$ and ...
1
vote
2answers
123 views
What is the shape of parabolic critical orbit?
The parabolic critical orbits of discrete dynamical system form n-th arm stars :
which shapes are conjugated with "regular" n-th arm stars
Here are 2 images of parabolic critical orbits for 2 ...
2
votes
0answers
48 views
Bernoulli shift on $S^\mathbb{Z}$
Why is the Bernoulli shift ergodic? I know the proof for $S^\mathbb{N}$. The best would be if you could argue from this fact to the integer case, so that way I can see how one usually goes from ...
0
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0answers
27 views
Dynamical Systems and Combinatorics
Suppose we are interested in labeling all the odd numbers. This can be represented as the infinite string $1010 \dots$ This also corresponds to a dynamical system quadruplet $(T,X,c,x_0)$ where $c: X ...
1
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0answers
100 views
Proof Strategy for a Dynamical System of Points on the Plane
I have a rather simple-looking system which exhibits a particular behaviour in simulation, and I would now like to attempt to prove this formally. The problem is, I don't really know where to start, ...
0
votes
2answers
53 views
Is $g(z)=\frac{1}{z}+\frac{1}{z^2+1}+\frac{1}{(z^2+1)^2 +1}+…$ analytic for $|z|>2$?
Let $z$ be a complex number. Let |.| denote be the absolute value. Let $n$ be a positive integer.
Let $f_1(z)=z^2+1$. Let $f_n(z)=f_1(f_{n-1}(z)).$
Is ...
