Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
2,388
questions
2
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The centralizer of Lienard equation
Consider the lienard vector field $\cases{
x'=y -F(x) \\
y'=-x }
$ in $\mathbb{R}^{2}$, where $F$ is a polynomial fuction with $F(0)=0$. Assume that $Y$ is a smooth vector field globally defined ...
24
votes
2
answers
2k
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Periodic orbit property
A topological space $X$ satisfies the "Periodic orbit property", briefly POP, if for every continuous map $f:X \to X$, there exist a natural number $n$ and a point $x_{0}\in X$ such that $...
4
votes
1
answer
261
views
Volume-preserving mappings in the torus $T^n$
Let $T^n$ be the $n$-dimensional torus and let $F$ be the set of all volume preserving continuous mappings $f:T^n\to T^n$. I would like to know if $F$ is connected in the sense that for any $f\in F$ ...
10
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2
answers
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Getting unique ergodicity from minimality
It is known that minimality does not imply unique ergodicity (Furstenberg example). I ask whether the implication holds in following particular situation:
Suppose $X$ is a compact space, $f:X \to X$...
3
votes
1
answer
342
views
Pointwise ergodic theorem for amenable semigroups
Using tempered Følner sequences one may show a pointwise ergodic theorem for amenable groups.
(see http://www.aimsciences.org/journals/pdfsnews.jsp?paperID=2413&mode=full)
Is there a ...
8
votes
4
answers
2k
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Uniform convergence of Birkhoff averages and unique ergodicity
I am looking for a proof or a reference for the following two facts (which appear proofless in my notes from an ergodic theory course- they might be easy but i am no expert in ET):
Let $T$ be a ...
0
votes
0
answers
315
views
Prove that origin is globally exponentially stable with Lyapunov Indirect Method
I'm wondering, if we have a nonlinear system governed by
$\dot{x} = Ax + g(x)$ where $||g(x)|| \leq \gamma ||x||^2$ and A is Hurwitz
how can we show that the origin is globally exponentially stable?...
1
vote
1
answer
204
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Number of solutions of a system of equation!
Let $\Theta =(\theta_1,\ldots, \theta_n)\in {\mathbb T}^n$. I want to show that the system of equations
$$
\sum_j 2\sin(\theta_i -\theta_j)+\sin(2\theta_i -2\theta_j) =0,\ \ i=1,\ldots, n,
$$
has ...
2
votes
1
answer
339
views
Dynamics of Master Equation
I'm going to do research on dynamics of master equation of $n$ states
$$\dot p_i=A_{ij}p_j\qquad i=1\ldots n$$
where $p_i$ is the $i$-th component of probability vector and $A_{ij}$ is transition rate ...
2
votes
0
answers
100
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Properties of algebraic vector fields which generates a $\mathbb{C^*}$ action
My question is rather vague and I apologize. Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$. I am interested in whether there are homological properties which distinguish algebraic ...
9
votes
0
answers
365
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Periodic orbits of a spinning ball in a square
Periodic orbits of a billiard ball bouncing in a square have been well-studied.
I am seeking similar analysis of what is sometimes called a rough ball, one
whose high friction causes it to pick up ...
5
votes
1
answer
415
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 $\sigma(...
11
votes
2
answers
885
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 $\...
17
votes
3
answers
2k
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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 ...
2
votes
1
answer
89
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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
3
answers
990
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
0
answers
105
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 $\epsilon>...
4
votes
6
answers
4k
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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
1
answer
228
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 $\...
4
votes
0
answers
407
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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 $C^1$ ...
18
votes
2
answers
953
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
0
answers
54
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 ...
11
votes
1
answer
665
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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
0
answers
223
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 ...
6
votes
1
answer
175
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 ...
3
votes
1
answer
230
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 $K(f)$ is ...
3
votes
1
answer
611
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 ...
7
votes
1
answer
312
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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
0
answers
130
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Approximating solutions of non-linear differential equations
I have met a system of non-linear equations as follows,
$$\frac{\mathbb{d}y_k}{\mathbb{d}t}=-(1-\alpha)y_k\sum_s{s^az_s}-\alpha y_kz_k,$$
$$\frac{\mathbb{d}z_k}{\mathbb{d}t}=(1-\alpha)y_k\sum_s{s^az_s}...
28
votes
2
answers
2k
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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 $x\in\...
3
votes
2
answers
286
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 ...
3
votes
2
answers
353
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 ...
5
votes
0
answers
203
views
Reference for and Properties of the $\alpha$-entropy
Let $T \colon X \to X$ 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
3
answers
754
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 $\...
24
votes
3
answers
1k
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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
1
answer
325
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 ...
7
votes
2
answers
569
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) + \mathbb{Z}...
2
votes
1
answer
342
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 $ S_{\frac{...
0
votes
1
answer
171
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} a_{...
4
votes
1
answer
225
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 ...
10
votes
0
answers
532
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 $r$,...
5
votes
1
answer
470
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 ...
4
votes
1
answer
368
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 ...
10
votes
2
answers
2k
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“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 ...
15
votes
3
answers
1k
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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
0
answers
396
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 ...
9
votes
2
answers
900
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?
5
votes
1
answer
395
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
1
answer
223
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
0
answers
220
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 ...