Dynamical Systems
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New submissions for Thu, 2 Dec 21
 [1] arXiv:2112.00051 [pdf, other]

Title: Some Generic Properties of Partially Hyperbolic EndomorphismsComments: 17 pagesSubjects: Dynamical Systems (math.DS)
In this work we deal with a notion of partially hyperbolic endomorphism. We explore topological properties of this definition and we obtain, among other results, obstructions to get center leaf conjugacy with the linear part, for a class of partially hyperbolic endomorphism $C^1$sufficient near to a hyperbolic linear endomorphism. Indeed such obstructions are related with the number of center directions of a point. We provide examples illustrating these obstructions and we prove that, when the degree is bigger than two, $C^1$generically (open and dense) partially hyperbolic endomorphisms in dimension three, in fact present properties that make impossible the center leaf conjugacy with their linear parts.
 [2] arXiv:2112.00130 [pdf, other]

Title: Structurally stable nondegenerate singularities of integrable systemsComments: 28 pages, 3 figuresSubjects: Dynamical Systems (math.DS); Differential Geometry (math.DG); Symplectic Geometry (math.SG)
In this paper, we study singularities of the Lagrangian fibration given by a completely integrable system. We prove that a nondegenerate singular fibre satisfying the socalled connectedness condition is structurally stable under (small enough) realanalytic integrable perturbations of the system. In other words, the topology of the fibration in a neighbourhood of such a fibre is preserved after any such perturbation. As an illustration, we show that a saddlesaddle singularity of the Kovalevskaya top is structurally stable under realanalytic integrable perturbations, but structurally unstable under $C^\infty$ smooth integrable perturbations.
 [3] arXiv:2112.00175 [pdf, ps, other]

Title: Entropy of irregular points for some dynamical systemsSubjects: Dynamical Systems (math.DS)
We derive sufficient conditions for a dynamical systems to have a set of irregular points with full topological entropy. Such conditions are verified for some nonuniformly hyperbolic systems such as positive entropy surface diffeomorphisms and rational functions on the Riemann sphere.
 [4] arXiv:2112.00372 [pdf, ps, other]

Title: The rotation number for almost periodic potentials with jump discontinuities and $δ$interactionsComments: 34 pagesSubjects: Dynamical Systems (math.DS); Classical Analysis and ODEs (math.CA)
We consider onedimensional Schr\"odinger operators with generalized almost periodic potentials with jump discontinuities and $\delta$interactions. For operators of this kind we introduce a rotation number in the spirit of Johnson and Moser. To do this, we introduce the concept of almost periodicity at a rather general level, and then the almost periodic function with jump discontinuities and $\delta$interactions as an application.
 [5] arXiv:2112.00469 [pdf, ps, other]

Title: A global isochronous center is linearAuthors: Massimo VillariniComments: noneSubjects: Dynamical Systems (math.DS)
Let $X$ be a polynomial vector field in $\mathbb{R}^2$ which, after onepoint compactification of the plane, has a punctured neighbourhood $\dot U$ of the point at infinity which is foliated by closed orbits of $X$. If the period function of $X$ in $\dot U$ is bounded from below by a positive constant, $X$ is necessarily linear, hence conjugated, up to a nonzero constant factor, to $y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y}$. This result answers to a question posed by J. Llibre \cite{llibre}, proving {\it e.g.} that a global isochronous center is linear.
 [6] arXiv:2112.00526 [pdf, other]

Title: Nonautonomous vector fields on $S^3$: simple dynamics and wild separatrices embeddingComments: 18 pages, 11 figuresSubjects: Dynamical Systems (math.DS)
We construct new substantive examples of nonautonomous vector fields on 3dimensional sphere having a simple dynamics but nontrivial topology. The construction is based on two ideas: the theory of diffeomorpisms with wild separatrix embedding (Pixton, BonattiGrines, etc.) and the construction of a nonautonomous suspension over a diffeomorpism (LermanVainshtein). As a result, we get periodic, almost periodic or even nonrecurrent vector fields which have a finite number of special integral curves possessing exponential dichotomy on $\R$ such that among them there is one saddle integral curve (with an exponential dichotomy of the type (3,2)) having wildly embedded twodimensional unstable separatrix and wildly embedded threedimensional stable manifold. All other integral curves tend, as $t\to \pm \infty,$ to these special integral curves. Also we construct another vector fields having $k\ge 2$ special saddle integral curves with tamely embedded twodimensional unstable separatrices forming mildly wild frames in the sense of DebrunnerFox. In the case of periodic vector fields, corresponding specific integral curves are periodic with the period of the vector field, and they are almost periodic for the case of almost periodic vector fields.
 [7] arXiv:2112.00548 [pdf, ps, other]

Title: Bifurcations in asymptotically autonomous Hamiltonian systems under multiplicative noiseAuthors: O. A. SultanovComments: 22 pages, 6 figuresSubjects: Dynamical Systems (math.DS)
The effect of multiplicative stochastic perturbations on Hamiltonian systems on the plane is investigated. It is assumed that perturbations fade with time and preserve a stable equilibrium of the limiting system. The paper investigates bifurcations associated with changes in the stability of the equilibrium and with the appearance of new stochastically stable states in the perturbed system. It is shown that depending on the structure and the parameters of the decaying perturbations the equilibrium can remain stable or become unstable. In some intermediate cases, a practical stability of the equilibrium with estimates for the length of the stability interval is justified. The proposed stability analysis is based on a combination of the averaging method and the construction of stochastic Lyapunov functions.
 [8] arXiv:2112.00701 [pdf, ps, other]

Title: Julia sets of hyperbolic rational maps have positive Fourier dimensionAuthors: Gaétan LeclercComments: 35 pagesSubjects: Dynamical Systems (math.DS); Classical Analysis and ODEs (math.CA)
Let $f:\widehat{\mathbb{C}}\rightarrow \widehat{\mathbb{C}}$ be a hyperbolic rational map of degree $d \geq 2$, and let $J \subset \mathbb{C}$ be its Julia set. We prove that $J$ always has positive Fourier dimension. The case where $J$ is included in a circle follows from a recent work of Sahlsten and Stevens, see arXiv:2009.01703. In the case where $J$ is not included in a circle, we prove that a large family of probability measures supported on $J$ exhibit polynomial Fourier decay: our result applies in particular to the measure of maximal entropy and to the conformal measure.
Crosslists for Thu, 2 Dec 21
 [9] arXiv:2112.00073 (crosslist from mathph) [pdf, other]

Title: On the discrete Dirac spectrum of a point electron in the zerogravity KerrNewman spacetimeComments: 62 pages, 13 figuresSubjects: Mathematical Physics (mathph); General Relativity and Quantum Cosmology (grqc); Analysis of PDEs (math.AP); Dynamical Systems (math.DS); Quantum Physics (quantph)
The discrete spectrum of the Dirac operator for a point electron in the maximal analytically extended KerrNewman spacetime is determined in the zero$G$ limit (z$G$KN), under some restrictions on the electrical coupling constant and on the radius of the ringsingularity of the z$G$KN spacetime. The spectrum is characterized by a triplet of integers, associated with winding numbers of orbits of dynamical systems on cylinders. A dictionary is established that relates the spectrum with the known hydrogenic Dirac spectrum. Numerical illustrations are presented. Open problems are listed.
 [10] arXiv:2112.00104 (crosslist from nlin.AO) [pdf, other]

Title: Sandpile cascades on oscillator networksComments: 12 pages, 11 figuresSubjects: Adaptation and SelfOrganizing Systems (nlin.AO); Disordered Systems and Neural Networks (condmat.disnn); Dynamical Systems (math.DS); Chaotic Dynamics (nlin.CD)
The BTW sandpile model of cascading dynamics forms a cornerstone for our understanding of failures in systems ranging from avalanches and forest fires to power grids and brain networks. The latter two are examples of oscillator networks, yet the BTW model does not account for this. Here we establish that the interplay between the oscillatory and sandpile dynamics can lead to emergent new behaviors by considering the BTW sandpile model on a network of Kuramoto oscillators. Inspired by highlevel objectives in the power grids, we aim to leverage this interaction to maximize synchronization, maximize load, and minimize large cascades. We assume that the more outofsync a node is with its neighbors, the more vulnerable it is to failure so that a node's capacity is a function of its local level of synchronization. And when a node topples, its phase is reset at random. This leads to a novel longtime oscillatory behavior at an emergent timescale. The bulk of an oscillatory cycle is spent in a buildup phase where oscillators are fully synchronized, and cascades are largely avoided while the overall load in the system increases. Then the system reaches a tipping point where, in contrast to the BTW model, a large cascade triggers an even larger cascade, leading to a "cascade of cascades," which can be classified as a Dragon King event, after which the system has a short transient dynamic that restores full synchrony. This coupling between capacity and synchronization gives rise to endogenous cascade seeds in addition to exogenous ones, and we show their respective roles in the Dragon King events. We establish the phenomena from numerical studies and develop the accompanying meanfield theory to locate the tipping point, calculate the amount of load in the system, determine the frequency of the emergent longtime oscillations and find the distribution of cascade sizes during the buildup phase.
 [11] arXiv:2112.00135 (crosslist from astroph.EP) [pdf, ps, other]

Title: The spatial Hill fourbody problem I  An exploration of basic invariant setsSubjects: Earth and Planetary Astrophysics (astroph.EP); Dynamical Systems (math.DS)
In this work, we perform a first study of basic invariant sets of the spatial Hill's fourbody problem, where we have used both analytical and numerical approaches. This system depends on a mass parameter mu in such a way that the classical Hill's problem is recovered when mu = 0. Regarding the numerical work, we perform a numerical continuation, for the Jacobi constant C and several values of the mass parameter mu by applying a classical predictorcorrector method, together with a highorder Taylor method considering variable step and order and automatic differentiation techniques, to specific boundary value problems related with the reversing symmetries of the system. The solution of these boundary value problems defines initial conditions of symmetric periodic orbits. Some of the results were obtained departing from periodic orbits within Hill's threebody problem. The numerical explorations reveal that a second distant disturbing body has a relevant effect on the stability of the orbits and bifurcations among these families. We have also found some new families of periodic orbits that do not exist in the classical Hill's threebody problem; these families have some desirable properties from a practical point of view.
 [12] arXiv:2112.00316 (crosslist from math.AP) [pdf, ps, other]

Title: On the KadomtsevPetviashvili equation with combined nonlinearitiesSubjects: Analysis of PDEs (math.AP); Dynamical Systems (math.DS)
In this paper, we study the generalized KP equation with combined nonlinearities. First we show the existence of solitary waves of this equation. Then, we consider the associated Cauchy problem and obtain conditions under which solutions blow up in finite time or are uniformly bounded in the energy space. We also prove the strong instability of the ground states
 [13] arXiv:2112.00371 (crosslist from qbio.CB) [pdf, other]

Title: A multistage model of cell proliferation and death: tracking cell divisions with Erlang distributionsComments: 26 pages, 11 figuresSubjects: Cell Behavior (qbio.CB); Dynamical Systems (math.DS)
Lymphocyte populations, stimulated in vitro or in vivo, grow as cells divide. Stochastic models are appropriate because some cells undergo multiple rounds of division, some die, and others of the same type in the same conditions do not divide at all. If individual cells behave independently, each can be imagined as sampling from a probability density of times to division. The most convenient choice of density in mathematical and computational work, the exponential density, overestimates the probability of short division times. We consider a multistage model that produces an Erlang distribution of times to division, and an exponential distribution of times to die. The resulting dynamics of competing fates is a type of cyton model. Using Approximate Bayesian Computation, we compare our model to published cell counts, obtained after CFSElabelled OTI and F5 T cells were transferred to lymphopenic mice. The death rate is assumed to scale linearly with the generation (number of divisions) and the number of stages of undivided cells (generation $0$) is allowed to differ from that of cells that have divided at least once (generation greater than zero). Multiple stages are preferred in posterior distributions, and the mean time to first division is longer than the mean time to subsequent divisions.
 [14] arXiv:2112.00483 (crosslist from math.SG) [pdf, ps, other]

Title: Microlocal Projector for Complete FlowAuthors: ShengFu ChiuComments: 18 pages, no figureSubjects: Symplectic Geometry (math.SG); Category Theory (math.CT); Dynamical Systems (math.DS)
In this note we generalize the construction of microlocal projector to the sublevel set of autonomous function with complete Hamiltonian flow under some mild conditions. Furthermore, we mention that the condition of being complete is optimal, by providing a nonexample that violates the projector property.
 [15] arXiv:2112.00603 (crosslist from math.AG) [pdf, ps, other]

Title: LEFgroups and endomorphisms of symbolic varietiesAuthors: Xuan Kien PhungSubjects: Algebraic Geometry (math.AG); Category Theory (math.CT); Dynamical Systems (math.DS); Group Theory (math.GR); Cellular Automata and Lattice Gases (nlin.CG)
Let $G$ be a group and let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic zero. Denote $A=X(k)$ the set of rational points of $X$. We investigate invertible algebraic cellular automata $\tau \colon A^G \to A^G$ whose local defining map is induced by some morphism of algebraic varieties $X^M \to X$ where $M$ is a finite memory. When $G$ is locally embeddable into finite groups (LEF), then we show that the inverses of reversible algebraic cellular automata are automatically algebraic cellular automata. Generalizations are also obtained for finite product Hopfian pointed object alphabets in concrete categories.
 [16] arXiv:2112.00670 (crosslist from math.ST) [pdf, ps, other]

Title: Dynamical hypothesis tests and Decision Theory for Gibbs distributionsSubjects: Statistics Theory (math.ST); Dynamical Systems (math.DS); Probability (math.PR)
We consider the problem of testing for two Gibbs probabilities $\mu_0$ and $\mu_1$ defined for a dynamical system $(\Omega,T)$. Due to the fact that in general full orbits are not observable or computable, one needs to restrict to subclasses of tests defined by a finite time series $h(x_0), h(x_1)=h(T(x_0)),..., h(x_n)=h(T^n(x_0))$, $x_0\in \Omega$, $n\ge 0$, where $h:\Omega\to\mathbb R$ denotes a suitable measurable function. We determine in each class the NeymanPearson tests, the minimax tests, and the Bayes solutions and show the asymptotic decay of their risk functions as $n\to\infty$. In the case of $\Omega$ being a symbolic space, for each $n\in \mathbb{N}$, these optimal tests rely on the information of the measures for cylinder sets of size $n$.
Replacements for Thu, 2 Dec 21
 [17] arXiv:2104.00110 (replaced) [pdf, ps, other]

Title: Renormalization in Lorenz maps  completely invariant sets and periodic orbitsComments: 28 pages, 4 figuresSubjects: Dynamical Systems (math.DS)
 [18] arXiv:2109.04531 (replaced) [pdf, ps, other]

Title: Some Remarks on the Notion of Bohr Chaos and Invariant MeasuresAuthors: Matan TalSubjects: Dynamical Systems (math.DS)
 [19] arXiv:2110.05989 (replaced) [pdf, ps, other]

Title: Furstenberg's Times 2, Times 3 Conjecture (a Short Survey)Authors: Matan TalSubjects: Dynamical Systems (math.DS)
 [20] arXiv:2111.12254 (replaced) [pdf]

Title: Emergence of dynamic properties in network hypermotifsSubjects: Dynamical Systems (math.DS)
 [21] arXiv:2102.04877 (replaced) [pdf, other]

Title: Noisy Recurrent Neural NetworksComments: 38 pagesJournalref: NeurIPS 2021 (https://proceedings.neurips.cc/paper/2021/hash/29301521774ff3cbd26652b2d5c95996Abstract.html)Subjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Dynamical Systems (math.DS); Probability (math.PR)
 [22] arXiv:2104.12006 (replaced) [pdf, ps, other]

Title: Functional limits for "tied down" occupation time processes of infinite ergodic transformationsComments: A proposition corrected and a theorem upgraded. 36 pagesSubjects: Probability (math.PR); Dynamical Systems (math.DS)
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