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 ...
2
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0answers
30 views
Non-ergodic measure
Is there an easy way to see that if $\mu$ and $\nu$ are $T$-invariant measures on the same space $X$, and $\mu \neq \nu$, then $\frac{1}{2}(\mu+\nu)$ is NOT ergodic?
I know that ergodic measures are ...
1
vote
1answer
22 views
Exp. Stability of perturbed system with temporally vanishing perturbation
I have a perturbation problem for which I can't find a fitting theorem in Khalil's Nonlinear Systems. Maybe someone can point me in the right direction:
Given a nominal system
$\dot x(t) = ...
1
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0answers
22 views
How to compress a linear operator and have the lossless composition property.
Consider a linear operator on $\mathbf{R}^n$ represented by a square matrix of size $n \times n$, call it $A$. The matrix acts on a row vector, call it $x$ and returns a row vector, call it $x'$, so ...
0
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0answers
23 views
Quadratic form of block matrix
If one has a block matrix $\tilde A = \left[ {\begin{array}{*{20}{c}}
D&{{0_{n \times n}}}\\
{{0_{n \times n}}}&{{0_{n \times n}}}
\end{array}} \right]$ where $D\in {R^{n \times n}}$ is a ...
0
votes
0answers
10 views
Phase Plane of Digital Systems
I have a nonlinear digital system which can not become differential equation with subtracting the states and deviding them by the time difference on account of being nonlinear. Therefore, I want to ...
1
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0answers
46 views
Lyapunov Stability of Non-autonomous Nonlinear Dynamical Systems
Let $\mathbf{F}:X\times\mathbb{R}^{+}\to X$ be a non-autonomous dynamical system, which is governed by $\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}, t, u)$, viz,
\begin{equation}
\begin{split}
\dot{x}_1 ...
0
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0answers
13 views
The behavior of the dynamical system (C,c(x)) is very simple with only two types of orbits.
A seed will either converge to zero pr diverge to infinity in magnitude. Do you agree with this statement?
I have identified that (S,f) denotes the dynamical system generated by an iterator function ...
0
votes
1answer
28 views
Complex Dynamics of periodic points
How do I show that e^2PIi/5 as a periodic point for the function of f(z)= z^3. Also what is it's prime period?
1
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1answer
24 views
How do you find the fixed and period-$2$ points of $f(x)=x^2-3x+3$?
I am trying to do this question using the Fixed Point Factor Theorem. I keep getting an answer $>0$ at the end of my long division of $f(x)-x$ into $f^2(x)-x$ therefore I must using the wrong ...
2
votes
0answers
61 views
Why these synchronization error dynamics for Krasovskii-Lyapunov?
I am attempting to work through "Shahverdiev, Sivaprakasam, and Shore (2002) Lag synchronization in time-delayed systems", but I'm missing something basic up front.
The problem is to take a ...
2
votes
1answer
55 views
Eigenfunction of $T^n$ vs. eigenfunction of $T$ when $T$ is ergodic
Let $(X, \mathcal{B}, \mu, T)$ be an ergodic system, and suppose $f \in L^2(X, \mathcal{B}, \mu)$ is an eigenfunction of $T^n$ for some integer $n > 1$. Is it possible to write $f$ as a linear ...
1
vote
3answers
33 views
Fixed points and iterates of an invertible function
Suppose that $g : [0,1] \rightarrow [0,1]$ is a continuous and strictly increasing function such that $g(0)=0$ and $g(1)=1$. Under these hypotheses $g(x)$ has an inverse function $g^{-1} :[0,1] \to ...
0
votes
1answer
16 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
vote
1answer
19 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? ...
0
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1answer
24 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?
0
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2answers
57 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
66 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
$$
...
2
votes
1answer
48 views
Ergodic theory question about the support of a measure.
I am trying to work through some problems from " Ergodic theory with view towards number theory" but I am finding it a tad more difficult than expected. In particular, I have the following question:
...
2
votes
1answer
25 views
Examples of systems conforming the Lorentz Attractor
Might sound like a trivial question but would you please show me some examples of real systems conforming the Lorentz Attractor?
It can be any kind of system, just a little list. It can be a system ...
0
votes
1answer
45 views
How write these two motions into one system
f describe the horizontal motion
g describe the vertical motion
have error when put m(t) in x()
Maple code
...
1
vote
1answer
34 views
Conditions that Roots of a Polynomial be Less than Unity
Is is the case that Samuelson's result is a more specific result of Rouche's Theorem, or the Routh–Hurwitz stability criterion? Is it not the goal for a polynomial to be stable that all of its roots ...
1
vote
0answers
25 views
Strength of attraction of fixed points
Consider a smooth map $f: \mathbb{R} \rightarrow \mathbb{R}$ with an attracting fixed point $F$. Then, we have
if $f'(F) \ne 0$, $F$ is a "simple" attracting fixed point,
if $f'(F) = 0$, $F$ is a ...
1
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0answers
32 views
Normal form of a vector field.
Those are two analogous problems, the first one of which I have already accounted for.
Find the normal form of the vector fields:
a) Solved.
b) $$\dot x_1=x_2$$
$$\dot x_2=-x_1 $$
$$\dot x_3=\sqrt ...
3
votes
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
votes
4answers
115 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 ...
0
votes
1answer
41 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
vote
1answer
47 views
Explicit form of Poincare's map for the following system of Ode’s
Problem: Write in explicit form 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
votes
0answers
67 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
vote
0answers
38 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
votes
0answers
22 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 ...
0
votes
1answer
25 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
votes
1answer
38 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 $f$ ...
0
votes
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 ...
0
votes
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
24 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 ...
0
votes
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
votes
1answer
35 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.
$$\dot x=P(x,y)$$
$$\dot ...
2
votes
1answer
46 views
How important are the following undergrad courses when trying to pursue studies in chaos theory/dynamical systems?
I'm currently a physics major with a year left, and deciding whether to switch into mathematical physics, mathematics or applied mathematics. I'm definitely switching into one of them, as I can meet ...
1
vote
1answer
145 views
An issue with approximations of a recurrence sequence
By trying to give an approximation to a given recurrence sequence I encountered a problem.
To be more precise I have a method but it fails if the right condition is not met and I wonder how I should ...
2
votes
0answers
63 views
Show that this orbit has a zero Lyapunov exponent
I'm using J.Meiss -Differential dynamical systems, and have some trouble to understand a proof about Lyapunov exponents.
We have a dynamical system
$$ \dot{x} = f(x), $$
with the corresponding flow $ ...
7
votes
2answers
80 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
vote
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
36 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
vote
2answers
89 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
votes
1answer
52 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
47 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
37 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 ...
