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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124 views

What can you say about the periods of a function with uniformly bounded periodic orbits?

Assume that all prime periods of periodic orbits of a continuous map $f:[0:1]\to [0:1]$ are uniformly bounded (i.e. there exists N such that the prime period of every periodic orbit of f is smaller ...
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1answer
251 views

Discrete rational rotations on the two dimensional torus

It is well known (Kronecker's Theorem) that "irrational rotations" are dense on $[0,1)$. That is, the set $$ \{ x+nr\mod 1 : n \in \mathbb{N} \} $$ is dense on $[0,1)$, provided that $r$ is ...
22
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2answers
811 views

Why is the topological pressure called pressure?

Let us consider a compact topological space $X$, and a continuous function $f$ acting on $X$. One of the most important quantities related to such a topological dynamical system is the entropy. For ...
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0answers
56 views

Show that zero attractor basin is not $\mathbb{R}^2$

I have a dynamical system $$ \left\{ \begin{array}{rcl} \dot {x_1} & = & -\frac{2x_{1}}{(1+x_{1}^2)^2} - \frac{2x_2}{(1+x_2^2)^2} \\ \dot {x_2} & = & 2x_1 - \frac{2 ...
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1answer
198 views

Plot bifurcation diagram from time series chaotic data

I have equations for Chua's circuit and need to plot bifurcation diagram. From the things I have read so far, I need to use 1-dimensional map to get the bifurcation diagram, but I have trouble ...
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1answer
345 views

Discrete dynamical systems fixed points

I'm trying to understand this question which asked to find the fixed points of this tent map. However, I was under the impression that the fixed points are the points that hit the y=x line. ...
4
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0answers
450 views

Why is unique ergodicity important or interesting?

I have a very simple motivational question: why do we care if a measure-preserving transformation is uniquely ergodic or not? I can appreciate that being ergodic means that a system can't really be ...
2
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1answer
58 views

An abstract $\alpha$-contracting dynamical system

$\newcommand{\f}{\phi}$$\newcommand{\ep}{\varepsilon}$$\newcommand{\R}{\mathbb R}$ Suppose $(\f_t)_{t\ge0}$ is an abstract dynamical system in a Banach space $(X,\|\mathord\cdot\|)$. Let $C(x,\ep)$ ...
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1answer
70 views

Perturbing an abstract discrete dynamical system

Let $X$ be a Banach space. Denote for $x_0\in X$ and $r>0$ the closed ball centered at $x_0$ by $B(x_0,r)=\lbrace x\in X:\|x-x_0\|\le r\rbrace$. Suppose $f:X\to X$ a bounded map with a fixed point ...
25
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1answer
2k views

Why isn't the 3 body problem solvable?

I'm new to this "integrable system" stuff, but from what I've read, if there are as many linearly independent constants of motion that are compatible with respect to the poisson brackets as degrees of ...
9
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1answer
267 views

The Mandelbrot Set Membership

To define the Mandelbrot Set we consider a sequence of complex numbers $z_0$, $z_1$, $z_2$, $z_3$, with the following conditions: $$ \begin{cases} z_{n+1} &= &z_n^2 + c &\text{ for }n\geq ...
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1answer
63 views

Gliders, static structures in various (dynamic) systems

Structures, i.e. symmetries over time, appear in various systems: gliders in cellular automata, like Game of Life or Rule 110, unmatched string's parts in rewrite systems – unchanged in multiple ...
10
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1answer
387 views

Is a Markov process a random dynamic system?

A random dynamic system is defined in Wikipedia. Its definition, which is not included in this post for the sake of clarity, reminds me how similar a Markov process is to a random dynamic system just ...
3
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1answer
87 views

inequality in a differential equation

Let $u:\mathbb{R}\to\mathbb{R}^3$ where $u(t)=(u_1(t),u_2(t), u_3(t))$ be a function that satisfies $$\frac{d}{dt}|u(t)|^2+|u|^2\le 1,\tag{1}$$where $|\cdot|$ is the Euclidean norm. According to ...
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1answer
62 views

Dynamical system: hypothesis on metric functions

Let $E$ be a completely metrizable separable topological space and $\mathscr E$ be its Borel $\sigma$-algebra. Consider a measurable map $F:E\to E$ such that if $f:E\to \mathbb R$ is continuous and ...
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1answer
194 views

Difference between state and parameters of a system

In my previous question, based on the information I have so far, my understanding about a system is that a system transforms an input function to an output function. So I think all the things ...
6
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2answers
245 views

A recursive sequence generated by a polynomial

Let $P(x)$ be a polynomial in one variable with integer coefficients, and define the sequence $a_0, a_1, a_2, a_3,\cdots$ $$a_0 = 0, \ a_n = P(a_{n-1})$$ If there exists a $m$ natural number such ...
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1answer
107 views

Matrix-valued ODE - nonsingularity of solution

I have a matrix-valued inhomogenous linear ODE $X' = F(t)X + G(t)$, $X(0) = I_{n \times n}$, $F(t),G(t) \in \mathbb{R}^{n \times n}$, and the entries of $f$ and $g$ are continuous functions. What ...
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1answer
217 views

diff.eq.: linear stability of ODE

I'm stuck with a little exercise and cannot find out where I'm wrong. Maybe you can help me. So we have a differential equation (modified logistic growth): $$\frac{dN}{dt}=k \ N ...
9
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3answers
359 views

Quadratic Julia sets and periodic cycles

Consider the function $f_c(z) = z^2 + c$. Applying this function repeatedly, we get the familiar quadratic Julia sets that fractal enthusiasts burn compute cycles plotting. Infinity is always one ...
4
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1answer
321 views

Every basin of attraction contains a critical point?

Years and years ago, back when I first became interested in fractals [but didn't know much about anything], I vaguely remember coming across an interesting theorem. The gist of it was that "every ...
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1answer
140 views

Dynamics Question

Let be $T_{\beta}:[0,1]\to [0,1]$ defined by $T_{\beta}(x)=\beta x \bmod 1$ where $\beta \in (1,2).$ Questions: $T_{\beta}$ is topologically transitive? What about the periodic points? ...
2
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1answer
160 views

Symplectic submanifolds and first integrals

I was working with symplectic submanifolds when I posed the following question: Suppose I have a Hamiltonian system with the phase space $\mathcal{M}$, a symplectic manifold with the standard ...
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1answer
74 views

Stability of the origin as parameter varies

Since it's quite a long time I've gone through mathematical physics problems, I'm quite rusted with those topics, so I welcome cheerfully all your answers: For every $\alpha\in[0,1]$ we consider ...
2
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1answer
67 views

Product of Transitive Systems

Let be $M$ a topological space, and $f:M\to M$ a danymical system, i.e, a continuous map between from $M$ to $M$. We say that a dynamical system, $f:M\to M$ is topologically transitive when, ...
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1answer
117 views

System, dynamic system and feedback system

Can a system be defined as a mapping from a set of mappings, called input signals, to another set of mappings, called output signals, where the two sets of mappings may or may not have the same ...
2
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0answers
174 views

Ergodicity and mixing

From MathOverflow, R W said: Unfortunately, the way the term "ergodic" is used in the theory of (finite) Markov chains is completely misleading from the point of view of general ergodic ...
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1answer
87 views

Logistic map - like processes

Logistic map is a dynamic system based on real world processes in nature. Thus, it is possible to assign a meaning to multiplicands r, x, 1-x. Using that, it is possible to construct abstract ...
2
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1answer
667 views

Direction of arrows in a phase portrait.

If I have a system like: $\frac{dx}{dt} = x, \frac{dy}{dt} = -y+x^2$ and I am asked to draw a phase portrait, once I have found the type of portrait (saddle point, node, spiral, etc.) from the ...
6
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1answer
297 views

is the geodesic flow on Hyperbolic Plane completely integrable?

I'm looking for examples of completely integrable systems and specifically geodesic flows. We remember that when we have a symplectic manifold $(M,\omega)$ (with $M$ of dimension $2n$) and ...
1
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1answer
367 views

Baker's map in dynamical systems

The baker's map can be defined as $E_{2}:S^{1}\rightarrow S^{1}$, $E_{2}x=2x \mod 1$. This map has various properties, for example, denoting Lebesgue measure by $\lambda$, we have ...
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2answers
148 views

Having trouble understanding the concept of “mixing” in dynamical systems.

I'm trying to understand the concept of mixing in dynamical systems theory, especially when the system in question has a measure-preserving flow. Here's how the condition is expressed mathematically: ...
0
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1answer
190 views

Reference for Unstable/Stable Manifold Theorem

I'm looking for a reference to a proof of the theorem of stable / unstable manifolds , using the Hadamard method, ie, characterizing the stable / unstable manifolds locally as the graph of a ...
1
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1answer
100 views

Relationship between homogenous and inhomogenous ODEs

If I'm given the homogenous first-order linear ODE in one dimension $\frac{dx}{dt} = ax(t), \ \ x_0 = 1$, a straightforward calculation shows that $x(t) = e^{a t}$. Now if I add some driving ...
9
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1answer
415 views

Number of limit points of a continued exponential

Inspired by the work of C. Bender, I recently played with continued exponentials (like continued fractions but with exponential functions ;) ). Given all prefactors are equal to 1, the continued ...
21
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3answers
669 views

Self Study in Dynamical Systems

I'm trying to get into the field of dynamical systems by (self) studying one-dimensional dynamics and circle homeomorphisms; for my guidance, I'm trying to assemble materials in this field that obey ...
4
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1answer
144 views

Maximal Ergodic Theorem

Does the maximal ergodic theorem have any dynamical or qualitative interpretations, or is it just a custom-made theorem to leave the demonstration of the Birkhoff ergodic theorem more elegant?
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1answer
132 views

Van Der Waerden Theorem

Can someone explain me what's the meaning of the term "l-equivalent" in the following paper: http://www.math.ucsd.edu/~ronspubs/74_01_van_der_waerden.pdf ? I saw the definition at the first lines, ...
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2answers
184 views

Feedback characteristics of nonlinear dynamical systems

I am trying to understand the following article: A. Lahellec, S. Hallegatte, J.-Y. Grandpeix, P. Dumas and S. Blanco, Europhys. Lett. 81, 60001 (2008). In the beginning, it was quite easy to follow: ...
0
votes
1answer
120 views

Strong (topological) mixing and cofinite set

For a topological dynamical system $(X,T)$ (X is a compact Hausdorff space, and T is a continuous map from X to X), it is called strong mixing if For any nonempty open set U and V, $$N(U,V):=\{n\in ...
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2answers
164 views

Why this is not a differential equation?

On the exam I was asked the question about Transcritical bifurcation. I gave the equation $$ \dot x = rx - x^2 $$ Then I was asked why it is not a differential equation and I couldn't answer. I ...
3
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0answers
126 views

The measures in Furstenberg's correspondence

In the paper Inverting the Furstenberg correspondence (IFC), the author defines a function $D_{A}(\sigma)$ on the Basic clopens of Cantor space, $2^{\mathbb{N}}$, where $A$ is a finite binary string ...
0
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1answer
41 views

In which cases iterative equations can be reduced to finite-difference equations?

In which cases iterative equations can be reduced to finite-difference equations and when they can't?
3
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1answer
181 views

Hölder continuity

Reading a paper, they use the fact that the function $x \mapsto \log |\det Df(x)|$ is $v$-Hölder, where $Df$ is the derivative of some map. Then, they state that the function $f$ is $C^{1+v}$ and ...
5
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2answers
391 views

Example of a process that is ergodic but not mixing?

I recently heard a talk which depended on the dynamics of a system being mixing. I was told, and Wikipedia confirms, that "mixing" is a stronger condition than "ergodic", but after comparing the ...
2
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0answers
75 views

A lower positive bound on the number of closed orbits with given energy for a mechanical system

Let be given a mechanical system with configuration manifold $M,$ potential energy $V$ and kinetic energy $K$ corresponding to a riemannian metric on $M.$ Its dynamics is determined by the ...
5
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3answers
620 views

The golden ratio and the logistic equation

I have this bonus question in an assignment: How is the golden ratio related to the logistic equation in discrete time? The logistic equation is $x_{n+1}=rx_n(1-x_n)$. The professor suggested ...
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0answers
138 views

Confusion with periodic orbits, classical dynamical systems

I have a question related to two theorems in the book Differential Equations, Dynamical Systems, and Linear Algebra [Hirsch & Smale,1974]. First let me describe the framework. Let us consider a ...
3
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1answer
165 views

Why can you determine the stability of a system by taking the eigenvalues of the Jacobian?

Why can you determine the stability of a system by taking the eigenvalues of the Jacobian? I know it's an elementary question but it's been a while. Thank you!
2
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0answers
277 views

Characterization of Asymptotic Stability via KL-class functions

Let us adopt the following definition of stability and asymptotic stability of a dynamical system of the form: $$ \dot{x}=f(x) $$ The trajectory of this system starting from the initial point ...