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Register a new userRemarks on the paper Non-existence of Shilnikov chaos in continuous-time systems
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The present note refers to a result proposed in [Elhadj and Sprott, 2012], and shows that the Theorem therein is not correct. We explain that a proof of that Theorem cannot be given, as the statement is not correct, and we underline a mistake occurring in their proof. Since this note is supplementary to [Elhadj and Sprott, 2012], the reader should consult this paper for further explanations of the matter and the symbols used.
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Let $(X,mathcal{B},mu)$ be a standard probability space. We give new fundamental results determining solutions to the coboundary equation: begin{eqnarray*} f = g - g circ T end{eqnarray*} where $f in L^p$ and $T$ is ergodic invertible measure preserving on $(X, mathcal{B}, mu )$. We extend previous results by showing for any measurable $f$ that is non-zero on a set of positive measure, the class of measure preserving $T$ with a measurable solution $g$ is meager (including the case where $int_X f dmu = 0$). From this fact, a natural question arises: given $f$, does there always exist a solution pair $T$ and $g$? In regards to this question, our main results are: (i) Given measurable $f$, there exists an ergodic invertible measure preserving transformation $T$ and measurable function $g$ such that $f(x) = g(x) - g(Tx)$ for a.e. $xin X$, if and only if $int_{f > 0} f dmu = - int_{f < 0} f dmu$ (whether finite or $infty$). (ii) Given mean-zero $f in L^p$ for $p geq 1$, there exists an ergodic invertible measure preserving $T$ and $g in L^{p-1}$ such that $f(x) = g(x) - g( Tx )$ for a.e. $x in X$. (iii) In some sense, the previous existence result is the best possible. For $p geq 1$, there exist mean-zero $f in L^p$ such that for any ergodic invertible measure preserving $T$ and any measurable $g$ such that $f(x) = g(x) - g(Tx)$ a.e., then $g otin L^q$ for $q > p - 1$. Also, we show this situation is generic for mean-zero $f in L^p$. Finally, it is shown that we cannot expect finite moments for solutions $g$, when $f in L^1$. In particular, given any $phi : mathbb{R} to mathbb{R}$ such that $lim_{xto infty} phi (x) = infty$, there exist mean-zero $f in L^1$ such that for any solutions $T$ and $g$, the transfer function $g$ satisfies: begin{eqnarray*} int_{X} phi big( | g(x) | big) dmu = infty. end{eqnarray*}
We show that there is a natural restriction on the smoothness of spaces where the transfer operator for a continuous dynamical system has a spectral gap. Such a space cannot be embedded in a Holder space with Holder exponent greater than 1/2 unless it consists entirely of coboundaries.
Many systems of interest in science and engineering are made up of interacting subsystems. These subsystems, in turn, could be made up of collections of smaller interacting subsystems and so on. In a series of papers David Spivak with collaborators formalized these kinds of structures (systems of systems) as algebras over presentable colored operads. It is also very useful to consider maps between dynamical systems. This amounts to viewing dynamical systems as objects in an appropriate category. This is the point taken by DeVille and Lerman in the study of dynamics on networks. The goal of this paper is to describe an algebraic structure that encompasses both approaches to systems of systems. To this end we replace the monoidal category of wiring diagrams by a monoidal double category whose objects are surjective submersions. This allows us, on one hand, build new large open systems out of collections of smaller open subsystems and on the other keep track of maps between open systems. As a special case we recover the results of DeVille and Lerman on fibrations of networks of manifolds.
We introduce a new method to establish time-quantitative density in flat dynamical systems. First we give a shorter and different proof of our earlier result that a half-infinite geodesic on an arbitrary finite polysquare surface P is superdense on P if the slope of the geodesic is a badly approximable number. We then adapt our method to study time-quantitative density of half-infinite geodesics on algebraic polyrectangle surfaces.
Certain dynamical models of competition have a unique invariant hypersurface to whichevery nonzero tractory is asymptotic, having simple geometry and topology.
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