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Dynamical Systems

New submissions

[ total of 14 entries: 1-14 ]
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New submissions for Fri, 12 Apr 24

[1]  arXiv:2404.07233 [pdf, other]
Title: Structure of the codimesion one gradient flows with at most six singular points on the Möbius strip
Comments: 14 pages, 14 figures. arXiv admin note: substantial text overlap with arXiv:2303.14975
Subjects: Dynamical Systems (math.DS)

We describe all possible topological structures of Morse flows and typical one-parametric gradient bifurcation on the M\"obius strip in the case that the number of singular point of flows is at most six. To describe structures, we use the separatrix diagrams of flows. The saddle-node bifurcation is specified by selecting a separatrix in the diagram of the Morse flow befor the bifurcation and the saddle connection is specified by a separatrix, which connect two saddles on the diagram.

[2]  arXiv:2404.07247 [pdf, other]
Title: Thermodynamic formalism for subsystems of expanding Thurston maps II
Comments: 46 pages. This is a sequel to arXiv:2312.15822
Subjects: Dynamical Systems (math.DS); Complex Variables (math.CV)

Expanding Thurston maps were introduced by M. Bonk and D. Meyer with motivation from complex dynamics and Cannon's conjecture from geometric group theory via Sullivan's dictionary. In this paper, we study subsystems of expanding Thurston maps motivated via Sullivan's dictionary as analogs of some subgroups of Kleinian groups. We prove the uniqueness and various ergodic properties of the equilibrium states for strongly primitive subsystems and real-valued H\"older continuous potentials, and establish the equidistribution of preimages of subsystems with respect to the equilibrium states. Here, the sphere $S^{2}$ is equipped with a natural metric, called a visual metric, introduced by M. Bonk and D. Meyer. As a result, for strongly primitive subsystems of expanding Thurston maps without periodic critical points, we obtain a level-$2$ large deviation principle for Birkhoff averages and iterated preimages.

[3]  arXiv:2404.07288 [pdf, ps, other]
Title: Topological entropy of Turing complete dynamics
Comments: 18 pages, 3 figures
Subjects: Dynamical Systems (math.DS); Computational Complexity (cs.CC)

We explore the relationship between Turing completeness and topological entropy of dynamical systems. We first prove that a natural class of Turing machines (which includes most of the examples of universal Turing machines) has positive topological entropy. We deduce that any Turing complete dynamics with a continuous encoding that simulates a universal machine in this class is chaotic. This applies to our previous constructions of Turing complete area-preserving diffeomorphisms of the disk and 3D stationary Euler flows.

[4]  arXiv:2404.07326 [pdf, ps, other]
Title: On an extension of a theorem by Ruelle to long-range potentials
Subjects: Dynamical Systems (math.DS); Probability (math.PR)

Ruelle's transfer operator plays an important role in understanding thermodynamic and probabilistic properties of dynamical systems. In this work, we develop a method of finding eigenfunctions of transfer operators based on comparing Gibbs measures on the half-line $\mathbb Z_+$ and the whole line $\Z$. For a rather broad class of potentials, including both the ferromagnetic and antiferromagnetic long-range Dyson potentials, we are able to establish the existence of integrable, but not necessarily continuous, eigenfunctions. For a subset thereof we prove that the eigenfunction is actually continuous.

[5]  arXiv:2404.07480 [pdf, other]
Title: Geometric Aspects of Observability of Hypergraphs
Comments: Accepted to IFAC LHMNC 2024, 6 pages, 2 figures. arXiv admin note: text overlap with arXiv:2304.04883
Subjects: Dynamical Systems (math.DS)

In this paper we consider aspects of geometric observability for hypergraphs, extending our earlier work from the uniform to the nonuniform case. Hypergraphs, a generalization of graphs, allow hyperedges to connect multiple nodes and unambiguously represent multi-way relationships which are ubiquitous in many real-world networks including those that arise in biology. We consider polynomial dynamical systems with linear outputs defined according to hypergraph structure, and we propose methods to evaluate local, weak observability.

[6]  arXiv:2404.07536 [pdf, other]
Title: EKF-SINDy: Empowering the extended Kalman filter with sparse identification of nonlinear dynamics
Comments: 20 pages, 10 figures
Subjects: Dynamical Systems (math.DS); Systems and Control (eess.SY)

Observed data from a dynamic system can be assimilated into a predictive model by means of Kalman filters. Nonlinear extensions of the Kalman filter, such as the Extended Kalman Filter (EKF), are required to enable the joint estimation of (possibly nonlinear) system dynamics and of input parameters. To construct the evolution model used in the prediction phase of the EKF, we propose to rely on the Sparse Identification of Nonlinear Dynamics (SINDy). The numerical integration of a SINDy model leads to great computational savings compared to alternate strategies based on, e.g., finite elements. Indeed, SINDy allows for the immediate definition of the Jacobian matrices required by the EKF to identify system dynamics and properties, a derivation that is usually extremely involved with physical models. As a result, combining the EKF with SINDy provides a computationally efficient, easy-to-apply approach for the identification of nonlinear systems, capable of robust operation even outside the range of training of SINDy. To demonstrate the potential of the approach, we address the identification of a linear non-autonomous system consisting of a shear building model excited by real seismograms, and the identification of a partially observed nonlinear system. The challenge arising from applying SINDy when the system state is not accessible has been relieved by means of time-delay embedding. The great accuracy and the small uncertainty associated with the state identification, where the state has been augmented to include system properties, underscores the great potential of the proposed strategy, paving the way for the development of predictive digital twins in different fields.

[7]  arXiv:2404.07552 [pdf, ps, other]
Title: Correspondence Research of the Most Probable Transition Paths between a Stochastic Interacting Particle System and its Mean Field Limit System
Subjects: Dynamical Systems (math.DS); Probability (math.PR)

This paper derived the indirect approximation theorem of the most probable transition pathway of a stochastic interacting particle system in the mean field sense. This paper studied the problem of indirect approximation of the most probable transition pathway of an interacting particle system (i.e., a high-dimensional stochastic dynamic system) and its mean field limit equation (McKean-Vlasov stochastic differential equation). This study is based on the Onsager-Machlup action functional, reformulated the problem as an optimal control problem. With the stochastic Pontryagin's Maximum Principle, this paper completed the derivation. This paper proved the existence and uniqueness theorem of the solution to the mean field optimal control problem of McKean-Vlasov stochastic differential equations, and also established a system of equations satisfying the control parameters $\theta^{*}$ and $\theta^{N}$ respectively. There are few studies on the most probable transition pathways of stochastic interacting particle systems, it is still a great challenge to solve the most probable transition pathways directly or to approximate it with the mean field limit system. Therefore, this paper first gave the proof of correspondence between the core equation of Pontryagin's Maximum Principle, that is, Hamiltonian extreme condition equation. That is to say, this correspondence indirectly explain the correspondence between the most probable transition pathways of stochastic interacting particle systems and the mean field systems.

[8]  arXiv:2404.07614 [pdf, ps, other]
Title: On homotopy properties of solutions of some differential inclusions in the $W^{1,p}$-topology
Comments: AMS-LaTex, 17 pages
Subjects: Dynamical Systems (math.DS); Differential Geometry (math.DG)

We consider a differential inclusion on a manifold, defined by a field of open half-spaces whose boundary in each tangent space is the kernel of a one-form. We make the assumption that the corank one distribution associated to the kernel is completely nonholonomic. We identify a subset of solutions of the differential inclusion, satisfying two endpoints and periodic boundary conditions, which are homotopically equivalent in the $W^{1,p}$-topology, for any $p\in [1,+\infty)$, to the based loop space and the free loop space respectively.

[9]  arXiv:2404.07752 [pdf, ps, other]
Title: Singular linear forms over global function fields
Comments: 32 pages
Subjects: Dynamical Systems (math.DS); Number Theory (math.NT)

In this paper, we consider singular linear forms over global function fields of class number one and give an upper bound for the Hausdorff dimension of the set of singular linear forms by constructing an appropriate Margulis function over global function fields.

[10]  arXiv:2404.07876 [pdf, other]
Title: Joint transitivity for linear iterates
Comments: Comments welcome!
Subjects: Dynamical Systems (math.DS)

We establish sufficient and necessary conditions for the joint transitivity of linear iterates in a minimal topological dynamical system with commuting transformations. This result provides the first topological analogue of the classical Berend and Bergelson joint ergodicity criterion in measure-preserving systems.

[11]  arXiv:2404.07907 [pdf, other]
Title: On orthogonality to uniquely ergodic systems
Comments: 1 figure
Subjects: Dynamical Systems (math.DS)

We solve Boshernitzan's problem of characterization (in terms of so called Furstenberg systems) of bounded sequences that are orthogonal to all uniquely ergodic systems. Some variations of Boshernitzan's problem involving characteristic classes are considered. As an application, we characterize sequences orthogonal to all uniquely ergodic systems whose (unique) invariant measure yields a discrete spectrum automorphism as those satisfying an averaged Chowla property.

Replacements for Fri, 12 Apr 24

[12]  arXiv:2107.09407 (replaced) [pdf, other]
Title: The Parabolic Mandelbrot Set
Comments: This revised version has 80 pages, 26 illustrations
Subjects: Dynamical Systems (math.DS)
[13]  arXiv:2403.09132 (replaced) [pdf, other]
Title: Quantitative Reducibility of $C^k$ Quasi-Periodic Cocycles
Subjects: Dynamical Systems (math.DS); Mathematical Physics (math-ph)
[14]  arXiv:2403.19453 (replaced) [pdf, ps, other]
Title: The submodularity of the covolume function in global function fields
Authors: Gukyeong Bang
Comments: 19 pages
Subjects: Number Theory (math.NT); Dynamical Systems (math.DS)
[ total of 14 entries: 1-14 ]
[ showing up to 2000 entries per page: fewer | more ]

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