Tagged Questions

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

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0
votes
3answers
362 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
187 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
523 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
360 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
111 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
78 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
132 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
173 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
214 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
164 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
122 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
306 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
697 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
286 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
327 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
173 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
213 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
373 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
424 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
201 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 ...
5
votes
2answers
212 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
202 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
99 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
84 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
206 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
304 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
177 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
332 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 ...
14
votes
3answers
609 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
468 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
196 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
283 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 ...
5
votes
1answer
195 views

Is Taylor expansion related to Helmholtz decomposition?

The Taylor expansion of a vector field $f(x)$ to the order of one is $$f(x)=f(x_0)+Jf(x_0)\cdot\Delta x+o(\Delta x)$$ where $Jf$ is Jacobian of the vector field and $\Delta x=x-x_0$. Suppose we ...
3
votes
1answer
248 views

Examples of transformations that are totally ergodic but not weakly mixing?

The most natural examples are irrational rotations. Are there other examples that are fundamentally different from irrational rotations? By the way, if $T$ is totally ergodic, but not weakly mixing, ...
2
votes
1answer
121 views

Is autonomous dynamical system equivalent to one single higher-order ode?

We know that a higher-order ode can be converted to dynamical system by replacing each higher-order derivative by a new variable. What about inverse problem? Does a dynamical system convert to a ...
10
votes
2answers
451 views

Birkhoff ergodic theorem and the measure of the bad points

In the Birkhoff ergodic theorem we have a PMPS $(X,B,\mu,T)$ and that for any $f\in L^1(X,\mu)$ $\frac{1}{N}\sum_{n=0}^{N-1}f(T^n x)\to \int f \, d\mu,$ in measure, in $L^1$-norm and $\mu$-a.e. My ...
2
votes
1answer
280 views

Liouville's theorem: How to get an invariant measure?

Liouville's theorem states that the `natural' 2-from is preserved under the Hamiltonian flow. Apparently this leads to an invariant measure $\mu$ as follows \begin{equation} d\mu = \frac{d\sigma}{|| ...
2
votes
3answers
242 views

Fatou sets and topological entropy

Let us consider a diffeomorphism of a compact real manifold (complex manifold defined over the reals), and let us say that the diffeomorphism is birational. Hence, it extends to a birational map from ...