Tagged Questions

Dynamics of differential equations and flows, mechanics, classical few-body problems, iterations, complex dynamics, delayed differential equations.

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2
votes
1answer
124 views

Is there a one-dimensional subshift of positive entropy s, all of whose sub-subshifts also have entropy s?

A subshift is a subset $X$ of $A^\mathbb{N}$ or $A^\mathbb{Z}$ (with $A$ finite), such that $X$ is topologically closed and closed under the shift operation. The shift operation is defined by ...
9
votes
1answer
256 views

Random circle rotations

Weyl's equidistribution theorem states that the orbit of a point on the circle under rotation by $\alpha$ becomes asymptotically equidistributed with respect to Lebesgue (Haar) measure whenever ...
9
votes
2answers
426 views

How to draw a Zoll surface?

I take into account that lots of questions on Zoll surfaces have already been asked on the forum. But I will stubbornly continue asking. Are there any chances to draw explicitely at least one Zoll ...
3
votes
1answer
46 views

Hopf bifurcation for systems where the dynamics is homogeneous of degree 1

Consider dynamical system in dimension 3 $$x'(t)=f(x(t),d)$$ where the dynamics f is homogeneous of degree 1 and there is exactly one line of equilibrium points. This line is independent of the ...
2
votes
3answers
195 views

Non-linear state-space model system stability using Lyapunov?

I have a non-linear system modelled in state-space as follow: $$ \mathbf {\dot x} = \mathbf A(x) \mathbf x $$ I need to find out if this system is stable, so I was thinking in using the Lyapunov ...
1
vote
0answers
79 views

A argument related measurable partitions in dynamic system

$X$ is a compact metric space, and $T:X\rightarrow X$ be a continuous map, which is finite to one. Denoted by$ X_{0}$ the set of all points $x\in X$, such that for all sufficiently small ...
0
votes
3answers
379 views

Good books on Geometric Theory of Dynamical Systems

I am looking for a good book on Geometric Theory of Dynamical Systems . I found Geometric Theory of Dynamical Systems by Jr. Palis myself,but it's very old, anyway i would like to find a pure ...
1
vote
1answer
92 views

Stability of a system of ODEs

It is well known that for a system of ODEs, $\dot{\boldsymbol{y}} = \boldsymbol{Ay}$, the global stable equlibrium point is given by the eigenvector that correponds to the largest eigenvalue of ...
3
votes
0answers
188 views

The $\Omega$-Stability Theorem

I'm currently studying the $\Omega$-Stability Theorem: Theorem: If $\mathcal{R}(f)$ has a hyperbolic structure then $f$ is $\mathcal{R}$-stable. Some explanations about the statement: $f$ is a $C^1$ ...
16
votes
2answers
527 views

“Derived” polyhedra and polytopes

The notion of derived polygon is natural and leads to remarkable convergence. Start with a polygon, and replace it by locating a point on every edge a fraction $\alpha$ between the two endpoints. For ...
1
vote
0answers
41 views

Discrete Optimal Control and Monotone Policies

Let $x = (x_1,x_2) \in \mathbb{N}^2$ be the state, $u$ be the control, and the dynamics be given by $x^{(k+1)} = f(x^{(k)}, u^{(k)}, w^{(k)})$ where $w^{(k)}$ is an IID noise source. For some stage ...
9
votes
1answer
365 views

Do quantum “Sure-Shor separators” have a natural Veronese/Segre classification? (question inspired by Gil Kalai and Aram Harrow)

Aram Harrow asked: "Is there any place this is written up?" Update  Partly in answer to Aram's question, the thermodynamical properties of varietal dynamical systems now are written-up in ...
5
votes
0answers
115 views

Using topological pressure to determine a subshift of finite type

I am interested in recognising graphs (or matrices, or subshifts of finite type) using topological pressure. Suppose that we play the following game: ${\bf Step 1:}$ I write down an irreducible n x n ...
1
vote
0answers
79 views

Structural stability of the Chen system

Structural stability of the Chen system The concept of structural stability was introduced by Andronov and Pontryagin in 1937, and it has a crucial role in dynamical systems theory. Conditions for ...
5
votes
1answer
133 views

Entire functions with a null real escaping set

Let $f$ be a entire function (stable on $\mathbb{R}$), and $E_{\mathbb{R}}$ its real escaping set : $$E_{\mathbb{R}} = \{ x \in \mathbb{R} : f^{(k)}(x) \rightarrow_{k \to \infty} \infty \} $$ We put ...
4
votes
1answer
175 views

Contractibility of connected holomorphic dynamics?

Let $f$ be a function, holomorphic in $\mathbb{C}$, and $K(f)$ its non-escaping set : $$K(f) = \{ z \in \mathbb{C} : f^{(k)}(z) \nrightarrow_{k \to \infty} \infty \} $$ Question : If $K(f)$ ...
4
votes
1answer
217 views

Katok's conjecture on entropy and periodic orbits for generic $C^1$ diffeomorphisms

Let $M$ be a compact finite-dimensional manifold and $f\colon M\to M$ be a diffeomrphism. By $P_n(f)$ we denote the number of periodic points of $f$ with period $n$, that is, the number of fixed ...
1
vote
0answers
53 views

On global attraction of a stable node for a four dimensional nonlinear system

Consider the dynamical system on ${\mathbb R}^2\times{\mathbb I}^2$ (or ${\mathbb T}^2\times{\mathbb I}^2$) described by $$\left\{ \begin{array}{l} \dot{\theta}_1 = \omega_1 - ...
6
votes
1answer
231 views

Why aren't operator semigroups studied from a dynamical perspective?

Often times one talks about iterating a continuous map to get discrete topological dynamics, or having a 1-parameter family of continuous maps to get continuous topological dynamics. When studying ...
3
votes
0answers
107 views

Approximating solutions of non-linear differential equations

I have met a system of non-linear equations as follows, $$\frac{\mathbb{d}y_k}{\mathbb{d}t}=-(1-\alpha)y_k\sum_s{s^az_s}-\alpha y_kz_k,$$ ...
27
votes
2answers
2k views

Dynamical properties of injective continuous functions on $\mathbb{R}^d$

Let $\varphi:\mathbb{R}^d\to\mathbb{R}^d$ be an injective continuous function. Denote by $\varphi_n$ the $n$-th iterate of $\varphi$, i.e. $\varphi_n(x)=\varphi_{n-1}(\varphi(x))$ for all ...
3
votes
2answers
206 views

Dynamic of $SL_2(\mathbb{Z}$) on $\mathbb{C}^2$

I imagine the dynamic of $SL(2,\mathbb{Z}$) on $\mathbb{C}^2$ has been studied. Does one know if it is recurrent or ergodic (with respect to the Lebesgue measure) ? Is there any explicit description ...
2
votes
2answers
165 views

Regularity of a nonlinear ODE [Traveling wave solutions of parabolic systems]

In the book of Volpert on Traveling wave solutions of Parabolic Systems (AMS), one reads "the following assertion is readily proved and we shall not discuss it in detail". The same result is tacitely ...
2
votes
0answers
124 views

Reference for and Properties of the $alpha$-entropy

Let $T : X \to X$ be a continuous map on, say, a compact metric space $X$. Let $\mu$ be an invariant borel measure. Under suitable conditions, a result of Brin and Katok states that $\mu$-almost ...
6
votes
3answers
318 views

Poincare recurrence theorem and convergence on compact metric spaces

I am looking for a proof (or a reference to a proof) of the following theorem: Let $X$ be a compact metric space with metric $d$, endow $X$ with the Borel $\sigma$-algebra and a probability measure ...
22
votes
3answers
700 views

Tetrahedron insphere iteration

I know that iterating the following incircle construction approaches an equilateral triangle in the limit:       Starting with any triangle $T$, one forms $T'$ by connecting ...
2
votes
1answer
289 views

Some puzzles about the three conditions in a paper of D.Berend

Recently, I am reading a paper titled "multi-invariant sets on tori" by D.Berend. I am puzzled by the three necessary and sufficient conditions given there. Could you provide me with some concrete ...
6
votes
2answers
330 views

Rotation numbers for amenable group actions on the circle

Given an orientation-preserving homeomorphism $f: S^1 \to S^1$, one can define its rotation number $\rho(f) \in \mathbb{R}/\mathbb{Z}$, as $\rho(f) = (\lim_{n \to \infty} \tilde{f}^n(0)/n) + ...
2
votes
1answer
175 views

Is it possible to approximate an area-preserving diffeomorphism by a sequence of conjugates of periodic rotations?

Is it possible to approximate an area-preserving diffeomorphism $T$ of the disk $\mathbb{D}^2$ by a sequence of conjugates of periodic rotations $B_n^{-1} S_{\frac{p_n}{q_n}} B_n$, where $ ...
0
votes
1answer
118 views

Stability analysis of ODE

My questions concerns the stability analysis of the following dynamical system : $\dfrac{d}{dt} a_{i}(t) = D_{i} + \displaystyle{\sum_{j=1}^{n}L_{ij}a_{j}(t) + \sum_{j=1}^{n}\sum_{k=1}^{n} C_{ijk} ...
4
votes
1answer
174 views

Under what conditions can interval exchanges be approximated by periodic maps?

Under what conditions can an interval exchange be approximated by periodic maps? (in the weak topology for the Lebesgue measure on $[0,1]$ ). Are there non-trivial examples of periodically ...
9
votes
0answers
368 views

Poincaré recurrence and symplectic packings

Question. Is there any example of a path connected symplectic manifold $(M,\omega)$ that has infinite volume, but which cannot be packed by an infinite number of symplectic balls of a fixed radius ...
5
votes
1answer
215 views

Dynamics of $3^x$ mod 1

Consider the map $f(x)=3^x$ mod 1. Using the the iterated function system $T_{0}x=\log_{3}(x+1), T_{1}x=\log_{3}(x+2)$ we see that $f$ is dynamical conjugated to a full shift on two symbols. Moreover ...
3
votes
1answer
157 views

Extending the hyperbolic splitting on $\Lambda$ to a neighborhood of $\Lambda$

Let $M$ be a compact Riemannian manifold and let $f:M→M$ a diffeomorphism. Let $\Lambda\subset M$ be a compact invariant subset of $M$. We say that $\Lambda $ is a hyperbolic set for $f$ when there ...
6
votes
2answers
392 views

“is topologically mixing” vs. “is topologically transitive” in the defition of chaos

This question is cross-posted from MSE, since it hasn't gotten an answer there for over 72 hours. Wikipedia gives essentially "is topologically mixing and has dense periodic periodic orbits" as the ...
10
votes
2answers
438 views

Vector field on 3-sphere

Let $V$ be a vector field on $S^3$ such that its singularity points, namely the points at which the vector field vanishes, are only sinks or sources (i.e. the field is converging or diverging). Is ...
1
vote
0answers
203 views

Weakened jacobian conjecture for entire functions

A rudin's theorem is the assertion that any polynomial injection between affine spaces of the same dimension has a polynomial inverse, and the inverse is also given by polynomials. The jacobian ...
6
votes
2answers
216 views

What one really can do with fractals built from L-systems?

For any L-system one can naturally associate a fractal. Why these fractals are (mathematically) useful apart that they are a source of nice pictures?
4
votes
1answer
203 views

What is the probability of an arbitrary nonlinear dynamical system to be chaotic?

Particularly, how to characterize a set of chaotic nonlinear dynamical systems as a subset of nonlinear dynamical systems with respect to the set cardinality? To explain the question more, a simple ...
1
vote
1answer
100 views

whether there are some books and original papers ergodic theory approach to ODE

Recently I become more and more interested in the field of ergodic theory, especially in the dimension theory and thermal formalism and its applications. People always said that most of the ideas in ...
1
vote
0answers
85 views

Volume Function on Banach Spaces

I'm looking for a reference for the following so-called Volume Function $V_n$, which is intended to be a Banach/normed vector space generalization of the determinant. Let $X$ be a Banach space with ...
5
votes
1answer
208 views

inverse problem for ergodic measures

It is a basic fact in the weak-* topology, the set of invariant measures for a dynamical system is closed, compact, and convex in the weak-* topology. Furthermore, the set of ergodic measures is equal ...
1
vote
0answers
117 views

Extension of diffeomorphisms preserving bilateral bounds of the derivatives

Suppose $f$ is a $C^k (1\leq k\leq\infty)$ function from the unit ball $\mathcal{B}$ in $\mathbb{R}^n$ to itself, which is a diffeomorphism from the domain to its image, with the upper and lower ...
6
votes
1answer
305 views

What is known about dynamics on Grassmannians?

I have found myself becoming interested in dynamical systems given by homeomorphisms acting on $G(r,d)$, the space of $r$-dimensional subspaces of $\mathbb{R}^d$. I tried to do a literature search ...
1
vote
1answer
182 views

Construction of a Denjoy Counterexample

John Milnor proofed the existence of a Denjoy counterexample in http://www.math.sunysb.edu/~jack/DYNOTES/dn15.pdf page 15-4. I could follow his arguments until "It is now reasonably straightforward to ...
4
votes
1answer
341 views

Intuition of Kolmogorov-Sinai entropy

For a measurable entropy of measurable transformation $T$ from $(X,\mathcal{B},m)$ to itself. For each finite measurable partition $\mathcal{A}=\{A_i\}_{i=1}^{m}$ of $X$, we can define ...
15
votes
3answers
621 views

fixed point property for maps of compacts

Definition. A topological space $X$ has the Fixed Point Property (FPP) if every continuous self-map $X\to X$ has a fixed point. Question. If $X$ and $Y$ are homotopy-equivalent compact metrizable ...
7
votes
3answers
472 views

Knots and Dynamics. Recent breakthroughs?

I recently started reading Étienne Ghys slides on knots and dynamics <http://www.umpa.ens-lyon.fr/~ghys/articles/icm.pdf> which seem very interesting. I know this approach to knots and dynamics is ...
3
votes
1answer
197 views

Boundary flux maximizing drift (velocity) vector fields for 2D heat equation

Looking for literature / known results on the following class of problems: Consider the domain bounded, open $\Omega\in \mathbb R^2$ with smooth boundary, divergence free drift $u=u(x,t)$, scalar ...
3
votes
2answers
286 views

How to understand a solenoid?

Consider the invertible extension of the circle-doubling map T(x)=2x (mod 1), the new system can be represented as X={(x_k)|x_{k+1}=T(x_k)}(see GTM 259 ...