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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4
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67 views

Recommended second textbook for dynamical systems?

I recently finished a course on dynamical systems supplemented by Strogatz's textbook. There are a few parts of the book that we didn't cover (in particular, the material on fractals), but the ...
1
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1answer
48 views

Angle-doubling map is mixing

Let $$ T:\mathbb{S}^1\to\mathbb{S}^1\\ x\mapsto 2x $$ be the angle-doubling map on the circle. We know that this transformation is ergodic. We want to prove that is mixing. I have to show that $$ ...
2
votes
1answer
30 views

What is the difference between a trajectory and an orbit in dynamical systems?

In dynamical systems, the integral curves for a differential equation that governs a system are referred to as trajectories or orbits, then what is the difference bewteen they?
12
votes
1answer
254 views

Equality by iteratively applying $(a,b)\rightarrow [(a+1,2b)\text{ or }(2a,b+1)]$?

I play a game starting with $2$ positive integers $a$ and $b$. At each step of the game I can double one of the integers and add $1$ to the other integer. Is there always a procedure for any ...
6
votes
1answer
123 views

Properties of the space of $ T $-invariant probability measures over a compact topological space.

Let $T: X \to X$ be a continuos map defined on the compact space (Maybe Haussdorff) $X$. Denote by $M_T$ the space of $T$-invariant probability measures defined over the Sigma Algebra of Borel sets. ...
1
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0answers
55 views

What's the difference between $Df$ and $Tf$?

I'm reading Michael Shub's Global Stability of Dynamical Systems. In chapter 4, he defined hyperbolic set and said the splitting $E^s$ and $E^u$ are $Tf$ invariant. So I assume this $Tf$ is the ...
2
votes
1answer
50 views

A differentiable manifold of class $\mathcal{C}^{r}$ tangent to $E^{\pm}$ and representable as graph

The stable manifold theorem tell us: A local stable manifold $W^{s}_{loc}(x^{*})$, which is a differentiable manifold of class $\mathcal{C}^{r}$ and dimension $n_{-}, $ tangent to the ...
0
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0answers
11 views

How to understand this verbal description of the dynamics?

Let $T$ map $\left\{0,r,l\right\}^{\mathbb{Z}}$ to itself by having the r's move right, the l's move left and an r and an l annhihilate each other when they meet or cross. How would you understand ...
0
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0answers
37 views

Terminology for functions such that $f(x)\ge x$ for all $x$

Is there a common terminology for a real function $f$ such that $$f(x)\ge x$$ for all $x$? same question for the conditions $\forall x,f(x)>x$; $\forall x,f(x)\le x$; $\forall x,f(x)<x$. (I'm ...
2
votes
0answers
46 views

Question related to the ballistic motion

A point mass will move in the gravitational field of the Earth according to the equation $$\ddot R =-\frac{GM_eR}{|R|^3},$$ where $R$ is the position vector of the point mass measured from the ...
1
vote
1answer
63 views

Conditions for Deriving $R_0$ for SIR Model Using Survival Function Method

I'm taking a look at the SIR model given by the system of differential equations \begin{align} \frac{dS}{dt} & = - \beta S I \\ \frac{dI}{dt} & = \beta S I - \gamma I \\ \frac{dR}{dt}& ...
2
votes
1answer
83 views

Eigenvectors question

$x'=x-2y$ $y'=4x-x^3$ Equilibrium points are $(2,1),(-2,-1),(0,0)$ Consider equilibrium point $(2,1)$: Let $X=x-2$ and $Y=y-1$. Subbing this into the main and eliminating all the nonlinear terms ...
0
votes
1answer
45 views

Sketching phase portrait

$\dot{x}=-2x-2y$ $\dot{y}=-x-3y$ Equilibrium point is $(0,0)$. Eigenvalues are $\lambda_+=-1$ and $\lambda_-=-4$ which have corresponding eigenvectors $2\choose -1$ and $1 \choose 1$ respectively. ...
-1
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1answer
33 views

Stubborn dynamical system state

It is rather common to use matrices to represent the relationship of the states of dynamical systems. It is very natural to use matrices because of their ease in analysis. Stability, convergence ...
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0answers
21 views

Characterizing equicontinuity via ultrafilters

We have a compact metric space $(X,d)$ and a homeomorphism $T:X\to X$. For any ultrafilter $p\in\beta\mathbb{Z}$ we can define the map $T^p:X\to X$ given by $T^p(x):=\lim_{n\to p}T^n x$ (which can ...
9
votes
1answer
164 views

Bifurcations in the Duffing oscillator

I'm trying to describe all the bifurcations in the two parameter Duffing oscillator: $$\ddot{x} + ax + bx^3 = 0$$ In phase space with $y = \dot{x}$ I've found the origin to be a centre for $a>0$ ...
2
votes
1answer
41 views

How to proof that bracket of two vector field can be computed by second derivation

Can some one give a hint how can I proof that where $\phi$ indicated the flow of vector fields.
1
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1answer
83 views

Show system of ODEs has a periodic solution by finding smallest annular trapping region

Find the smallest annular trapping region of the following: $r'=r(1-2r^2+sin(2\theta)r^2)$ $\theta ' = -1$ I really do not understand how to do this. I have been trying to figure it out from things ...
0
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0answers
17 views

A question regarding space-state representation

First of all I am not sure if this is the right place to ask this. Lets say we have a system in a form of a harmonic oscillator desribed by a second order DE. There will be 2 state variables - x ...
1
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0answers
25 views

How to diagonalise this pentadiagonal pseudo-Toeplitz matrix?

How can one diagonalise this N-by-N pentadiagonal matrix (where $r$ is some real constant)? $$ \tiny \begin{pmatrix} r^2 +r & -2r -1 & 1 & & & & & & ...
0
votes
0answers
40 views

Topological entropy of circle homeomorphism is zero. True or false?

may I know if it is true that $\ f: S^1 \to S^1$ a homeomorphism, then $h_{top}(f) = 0$, where $h_{top}$ stands for topological entropy. I believe this statement is true, but I cannot prove it.
2
votes
0answers
77 views

We all know about compositions of functions, but what about decomposition. Is there a way with math, not just heuristics?

The composition operator is a well know and quite often used method in integration and differentiation, think u-substitution. However, given a composition like $$f(f(f(...f(x)...)))$$ Where there are ...
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0answers
30 views

T invariant probability measure on a compact space $X$

Let $T:X\to X$ be a continuous map defined on the compact space $X$. I read that if $X$ is a metric space then the set of T-invariant probability measure is not empty. I want to know if this result ...
0
votes
0answers
41 views

the $C^2$ structural stability of maps on $S^1$

Let $f, g:S^1\rightarrow S^1$ be two $C^2$ maps, $q\in S^1$ be such that $\inf_{n\geq 0}(f^n(q), C_f)>0$ where $C_f$ is the critical points of $f$, i.e., $C_f=\{x\in S^1:f'(x)=0\}.$ Assume that all ...
5
votes
1answer
59 views

Beta transformation is Ergodic.

Let $\beta \in \mathbb{R}$ with $\beta >1$. Define $T_{\beta}:[0,1)\to [0,1)$ by: $$T_{\beta}(x)=\beta x-[\beta x]=\{\beta x\} $$. Consider: $$ ...
2
votes
1answer
35 views

Expanding maps of the circle and their coding

A continuous smooth ($C^{10}$-smooth, for example) map $f:S^1\to S^1$ of the circle is called expanding if $\inf_{x\in S^1} f'(x) > 1 $. Here $S^1 = [0,1]/\sim$, the segment with identified ...
1
vote
0answers
18 views

Question about history of Entropy

I have started to study Ergodic theory and entropy by some books and lecture notes more than three months but unfortunately I'm not familiar with history of Entropy (I know some thing about name of ...
1
vote
1answer
61 views

A dynamical system of differential equations - periodic solutions?

I am solving a physical problem with a known periodic solution. When I simulate the behaviour of the system numerically, with full blown differential equations, I get stable, but rather complicated ...
2
votes
0answers
57 views

Is there an elegant proof of this elementary bifurcation theory result?

Let's suppose I have a $C^1$ function $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$, $(x,\lambda)\mapsto f(x,\lambda)$. Suppose there is a unique solution of the equation $f(x,\lambda_1)=0$, ...
2
votes
2answers
49 views

Can there be an interval where $F(x)=4 x^2-\frac{1}{2}$ is chaotic?

The function $$F(x)=4 x^2-\frac{1}{2}$$ has two repelling fixed points. Now, I wonder, can there be an interval $I$ where it is chaotic? I think not, because of the repulsiveness of the fixed points. ...
27
votes
0answers
852 views

This one weird thing that bugs me about summation and the like

Most of us know $$\sum_{n=a}^b c_n=c_a+c_{a+1}...+c_{b-1}+c_b$$ Some of us know $$\prod_{n=a}^b c_n=c_a \cdot c_{a+1}...c_{b-1} \cdot c_{b}$$ A few of us know ...
1
vote
1answer
48 views

Prove that if $x<0$ then $F_\mu^n(x)\rightarrow -\infty$ as $n\rightarrow \infty$, where $F_\mu=\mu x(1-x)$.

Logistic map is given as $$x_{n+1}=\mu x_n(1-x_n)$$ Let $F_\mu=\mu x(1-x)$. Therefore, for $\mu>1$, prove that if $x<0$ then $F_\mu^n(x)\rightarrow -\infty$ as $n\rightarrow \infty$.
0
votes
0answers
26 views

Recursion and splitting into even and odd parts

Given the relations $v_{2i}=v_i$ and $v_{2i+1}=-v_{i}$ and $\eta(m)=\lim_{n\to\infty}\frac{1}{N}\sum_{i=0}^{N-1}v_{i}v_{i+m}$. I want to show $\eta(m)=$\eta(2m)$. $\textnormal{Part of the hint is to ...
1
vote
1answer
25 views

Introducing noise and time lag between two coupled Rössler systems

I have two Rössler systems mutually coupled by the second component. I want to introduce some small noise and a slight time lag of the coupling between the systems. I'm not sure 1. what the best ...
1
vote
0answers
27 views

Qualitative properties of eigenvalues that can be inferred from matrix structure?

I am doing a linear stability analysis of a 6-dimensional system, what I want to know is if the system is stable at numerically solved steady states by looking at the eigenvalues of the jacobian ...
2
votes
0answers
162 views

System of first order ODEs with coherent sinusoidal time varying coefficient

I have encountered equations of the form $$\frac{{d{\bf{y}}(t)}}{{dt}} = \left( {{A_0} + {A_1}\cos (\omega t)} \right){\bf{y}}(t)$$where ${\bf{y}}$ is a vector and ${{A_0}}$ and ${{A_1}}$ are square ...
3
votes
0answers
34 views

Topology equivalence in dynamical system

my name is Eric. I've got trouble when proofing that system $\dot{x}=\alpha+x^2+O(x^3)$ is topological equivalence with system $\dot{x}=\alpha+x^2$. I don't understand how to build the homeomorphism ...
0
votes
1answer
17 views

How to change the variables so $x_{n+3}=f(x_{n},x_{n+1},x_{n+2})$ becomes of the form $g(y_n)=y_{n+1}$?

How to change the variables so $x_{n+3}=f(x_{n},x_{n+1},x_{n+2})$ becomes of the form $g(\underline{y}_n)=\underline{y}_{n+1}$
1
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0answers
16 views

Flow Notation over Interval

Given a section of the flow $\Phi^t(x_0)$ (for finite $t$), I'd like to denote a subsection of this flow from times $\tau^{i-1}$ to time $\tau^{i}$ using similar notation. I was considering using ...
0
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0answers
47 views

Normal form calculation

I am working on a problem involves 4 dimensional dynamical system. Is there any ready package (for maple ,matlab...) which calculate the normal form of nonlinear continuous dynamical systems? The ...
1
vote
1answer
37 views

Closed Form Solutions To Simple Iterated Polynomial Building Blocks

I've been doing some work on fractals and simple iterated polynomials lately. I admit, I've only taken classes up through Calc 2, although I've done a decent bit of reading on many topics over the ...
4
votes
1answer
70 views

Dynamical system $x_{n+1} = \frac{1}{2}(x_n - \frac{1}{x_n}) \ \ , \ \ n = 0, 1 , 2,…$

Consider the dynamical system $$ x_{n+1} = \frac{1}{2}(x_n - \frac{1}{x_n}) \ \ , \ \ n = 0, 1 , 2,... $$ So by using the substitution $x_n = \cot(y_n)$, I have found: $$ x_n = \cot(\cot^{-1} ...
0
votes
0answers
36 views

For which $r$ is this system dissipative $\ddot{q}+rq^2\dot{q}+\sin(q)\cos(q)=u$

Could someone please help me to understand the following: Having the differential equation: $\ddot{q}+rq^2\dot{q}+\sin(q)\cos(q)=u$ that models the electrical charge $q$ of a particle in an ...
0
votes
2answers
137 views

approximate vanishing in Pontryagin dual

Let $\{n_k\}\subseteq \mathbb{Z}$ to be any given sequence of integers, and suppose it satisfies the following property: (*) For any $\lambda\in A\subseteq \mathbb{T}$(the unit circle), ...
0
votes
2answers
36 views

How to modify this bump function so that the “bump” is at $y=1$?

$$f(x) = \begin{cases} e^{-1/(1 - x^2)} & -1 < x < 1\\ 0 & \text{otherwise} \end{cases} $$ I noticed that when I multiply the denominator of the fractional part of this ...
3
votes
2answers
48 views

Lotka-Volterra model with two predators

In this, Lotka-Volterra model, we have two predators: $$\frac{dp}{dt} = ap\left(1-\frac{p}{K}\right) - (b_1q_1+b_2q_2)p$$ $$\frac{dq_1}{dt}=e_1b_1pq_1-m_1q_1$$ ...
0
votes
2answers
58 views

How to construct a diffeomorphic function using another function with certain properties

A $C^{\infty}$ function $f(x)$ on the interval $[a, b]$ satisfies the following 3 properties: 1) $f(x) = 1$ for $a \leq x \leq b$ 2) $f(x) = 0$ for $x < \alpha$ and $x > \beta$ where $\alpha ...
0
votes
1answer
21 views

How to modify a function to meet certain properties?

I want to modify $$B(x) = \left\{ \begin{array}{lr} e^{-\frac{1}{x^2}} & : x > 0\\ 0 & : x \leq 0 \end{array} \right.$$ so that the new function $$C(x) = \left\{ ...
1
vote
1answer
27 views

prove $H(x)=x^\top x$ is constant along solutions of the sytem if $A(x)^\top + A(x)=0$

Could someone please help me to understand the following: Having the differential equation: $\dot{x} = A(x)x$ where $A(x)$ is a real -valued matrix of dimension $n\times n$ How can I prove ...
-1
votes
1answer
30 views

Phase portrait in 2 dimensions

I am trying to plot the phase portrait for the system: $\dot{x} = 1+y -e^{-x}$ $\dot{y} = x^3-y$ Now I worked out my eigenvalues to be $\lambda_1 = 2, \lambda_2 = -1$ and these correspond to 2 ...