Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
2,482 questions
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Product of two foliations
1.What is an example of a manifold $M$ with two foliations $F$ and $F'$ which are not topological equivalent but the product foliations $F\times F$ and $F'\times F'$, as foliations on $M\times M$, ...
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Asymptotic pseudo orbit of an action
Let $G$ be finitely generated group (i.e $G= <S>$ $S=\{ s_1, ...,
s_n\}$) and $\varphi:G\times M\longrightarrow M$ is an action then
$f:G\longrightarrow M$ is called $\delta$- pseudo orbit if $...
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Rational dynamical system with nonnegative paramaters
let $A$ be a rational system of the form :$\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ z_{n+1}=\frac{\alpha_{3+}+\beta_{3}x_{n}+\sigma_{3}y_{n}+\lambda_{3}z_{...
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Does the number of zero eigenvalues correspond to the dimension of equilibrium manifold for nonlinear system?
Consider nonlinear dynamical system with $m \times m$ Jacobian matrix $J(x)$ that has $k$ zero eigenvalues for all $x$. The rest $m-k$ eigenvalues have negative real part for all $x$. Is that true ...
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Is an odometer action on a product space always conjugate to its inverse by an involution?
This is a follow on question from
Is an non-singualr invertable ergodic transformation on a measure space isomorphic to its inverse?
Given a measure $\mu$ on the product space $X = \prod_{i=1}^\infty ...
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88
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Question on center-stable manifold
Assume that you have a gradient system smooth enough and a fixed point $x_{0}$. Is it true that if $x_{0} \in \omega(x)$ then $\gamma^{+}(x)$ must intersect the local center-stable manifold of $x_{0}$?...
user31317
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excplicit formula of iterates of an interval exchange
Let $f$ be an interval exchange transformation of $[0,1]$. Is there an explicit formula giving $f^k(0)$ in function of $k$?
If not, are there particular cases where this formula is simple? (except ...
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Express measurable entropy in terms of Fourier coefficients of the measure
Let $S^1$ be the unit circle and $T:S^1\to S^1$ be a continuous map. Suppose $\mu$ is a $T$-invariant Borel probability measure on $S^1$, that is, $\mu(T^{-1}A)=\mu(A)$ for every Borel subset $A$ of $...
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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?...
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Why this synchronization error dynamic for Krasovskii-Lyapunov?
I am attempting to work through "Shahverdiev, Sivaprakasam, and Shore (2002) Lag synchronization in time-delayed systems", but I'm missing something basic up front.
The problem is to take a ...
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Application of Morse theory to second order systems
Hello
I'm looking for some applications of Morse theory to second order differential system,( or boundary value problems )
Someone can help me with a pdf or a book which has these applications ?
...
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Degree of freedom restricted by inequalities
Motivational example
Consider a polyhedral graph $G$. A realization of $G$ is given by a convex polyhedron which is - essentially - characterized by the angles between the edges emanating from each ...
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Continuity of the Shadow of a Nondecreasing Function
So I'm working a lot with monotone nondecreasing functions $f : [0,1] \rightarrow [0,1]$, and I'm defining a certain discrete dynamics on them. Here nondecreasing means $x < y \Rightarrow f(x) \leq ...
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cat map re-transformation
Hi,
Is there any way of moving from one cat map transformation to the other without resetting parameters?
For example, suppose you have two matrices '$A$'and '$B$' each permuted with different cat ...
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Linking L function dynamics with behavior close to s = 1 ?
A division, found on a sample set of semi-stable elliptic curves, calls for interpretation regarding the Birch and Swinnerton-Dyer conjecture and the dynamic behavior of the L functions involved.
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Is it a known example of adic transformation ? (2)
Please apologize: the Bratelli-Vershik graph in which I'm interested is not this one but this one: graph2 http://www.freeimagehosting.net/t/6uxds.jpg
At level $n$ there are $n$ vertices, there is one ...
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When are operators extended by linearity bounded?
Greetings.
Suppose that $H$ is a separable infinite-dimensional Hilbert space and that $M$ is an infinite
dimensional closed subspace of $H$. Suppose that {$v_{n}: n\ge 1$} is an infinite linearly ...
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Is an immersed Kronecker join always a multilinear variety on a Hilbert space?
The question asked is:
Is the implicitization of an arbitrary-rank immersed Kronecker join always a multilinear variety on a Hilbert space?
This is related to another MathOverflow question
In ...
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Is this submonoid of the isometry group on $\Bbb Q_2$ closed to inverses? [closed]
Let $\textrm{aff}(ax+b)$ be the affine group on $\Bbb Z_2^\times$
i.e. the set of linear polynomials over 2-adic numbers with $a\in\Bbb Z_2^\times, b\in\Bbb Z_2$
Now let $X$ be the restriction of its ...
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homoclinic points of toral automorphisms
Hi!
Can you help me with solving this problem:
Suppose that A is hyperbolic toral automorphism( represented with matrix A) with only one real eigenvalue \lambda >1 with geometric multiplicity 1. ...
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How do we approximate the pressure in the Boussinesq equations of hydrodynamics? [closed]
How do we approximate the pressure or the gradient of it in the Boussinesq equations of hydrodynamics ?
Is the pressure limited or can it be any amount?
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Ergodicity of a measure preserving Anosov flow
Let $M$ be a Riemannian manifold and $\phi^t$ an Anosov flow on $M$.
If $\phi^t$ is measure preserving (with respect to any Borel-measure on $M$), it is ergodic. Does anybody have a proof of that ...
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transforming a Ricatti equation into a generalised Ricatti equation [closed]
C̶o̶n̶s̶i̶d̶e̶r̶ ̶a̶ ̶R̶i̶c̶a̶t̶t̶i̶ ̶e̶q̶u̶a̶t̶i̶o̶n̶ ̶o̶f̶ ̶t̶h̶e̶ ̶f̶o̶r̶m̶
$$ y' + y^2 = S(x), \qquad \qquad \qquad (1)$$
w̶h̶e̶r̶e̶ ̶$̶S̶(̶x̶)̶$̶ ̶i̶s̶ ̶a̶ ̶m̶e̶r̶o̶m̶o̶r̶p̶h̶i̶c̶ ...
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Does this function belong to $L^2(\mathbb{D})$?
Edit: After the answer of Prof. Eremenko to the previous version, I realized that a weaker assumption works for the main motivation of this post. so I revise the question.
The unit ...
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An example of a matrix whose eigenvalues fullfill 'No-resonance' condition
No-resonance for a matrix is defined in terms of its eigenvalue as (last para page-3 in ref.):
$$\lambda_i \neq \sum_{j=1}^N m_j\lambda_j;\ \forall m_j\in \mathbb{Z}\ \ and\ m_j\geq 0$$
$$such\ that\ \...
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Invariant ergodic measure Volterra operator
Define the Volterra operator $V$ on $C_0([0,1])\triangleq \{g \in C([0,1]):g(0)=0\}$ by
$$
f \mapsto \int_0^{\cdot} f(s)ds.
$$
Is there an example of an ergodic and $V$-invariant Borel probability ...
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Find $x,y,z \in \mathbb{Z}$ with $|x^2 + y^2 - \sqrt{3} z^2| < 10^{-6}$ [closed]
does Oppenheim conjecture hold for specific quadratic forms? or for generic quadratic forms with a set of measure 1.
for example can we find $x,y,z \in \mathbb{Z}$ with
$$|x^2 + y^2 - \sqrt{3} z^2| &...
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Reference request on dynamics and hyperbolic dynamics (hyperbolicity in absence of periodic orbits)
I would appreciate if you introduce me a reference (paper or book) who address the concept of hyperbolic dynamics but with emphasis on absence of periodic orbits. a possible ...
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Quantum dynamics on varieties and Salmon Prizes
Concluding Progressive Remarks
A new finding is Bates and Oeding's preprint "Toward a salmon conjecture" (arXiv:1009.6181), with its reference to the Salmon Prize.
The Salmon Prize (photo of the ...
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1
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Proof of im/possibility of constructing any fractal by iterated function systems? [closed]
Well the question is as simple as that and what I really want to see is if there is a mathematical proof that can tell whether every fractal(object of any non-integer dimension) can be constructed by ...
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Existence and uniqueness of solutions for a system of first order PDEs [closed]
Which results can be applied and which conditions are needed, to ensure the existence and uniqueness of the solutions of the first order of PDEs:
A$\dfrac{\partial}{\partial t}\pmb{v}(t,x)=B(t,x,\pmb{...
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Central limit theorem for irrational rotations
Let $\alpha$ be an algebraic integer of modulus 1, and $ R_\alpha z=\alpha z$. Is
$$\lim_{n\to\infty}\frac{\log|\sum_{k=1}^n \Re R_\alpha^k z|}{\log n}=\frac12$$ for all $z\in S^1$?
Birkhoff's ergodic ...
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