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Magnetic skyrmion bags are composite topological spin textures with arbitrary topological charges. Here, we computationally study the transient rotational motion of skyrmion bags, which is characterized by a global rotation of inner skyrmions around the central point. Distinct from conventional rotational modes found in skyrmions, the observed rotation is a forced motion associated with the breathing mode induced solely by vertical microwave fields. The driving force behind this rotation originates from the interactions between outer and inner skyrmions, with the angular velocity determined by the phase difference resulting from their asynchronous breathing behaviors. It is also found that skyrmion bags with larger skyrmion numbers are more conducive to the occurrence of the rotation. Our results are useful for understanding the cluster dynamics of complex topological spin textures driven by dynamic fields. [ABSTRACT FROM AUTHOR]

Given a partial action α of a groupoid G on a ring R, we study the associated partial skew groupoid ring R ⋊ α G , which carries a natural G-grading. We show that there is a one-to-one correspondence between the G-invariant ideals of R and the graded ideals of the G-graded ring R ⋊ α G . We provide sufficient conditions for primeness, and necessary and sufficient conditions for simplicity of R ⋊ α G . We show that every ideal of R ⋊ α G is graded if and only if α has the residual intersection property. Furthermore, if α is induced by a topological partial action θ, then we prove that minimality of θ is equivalent to G-simplicity of R, topological transitivity of θ is equivalent to G-primeness of R, and topological freeness of θ on every closed invariant subset of the underlying topological space is equivalent to α having the residual intersection property. As an application, we characterize condition (K) for an ultragraph in terms of topological properties of the associated partial action and in terms of algebraic properties of the associated ultragraph algebra. [ABSTRACT FROM AUTHOR]

We investigate the ultrafast spin dynamics of the prototypical topological insulator Bi2Se3 using time-resolved Kerr rotation (polarization-change) measurements across near-infrared wavelengths. The Kerr rotation angle Δ θ K of Bi2Se3 was found to significantly depend on the photon energy around a resonance transition (∼ 1.0 eV) of bulk states, as well as the ellipticity of the pump light, in the presence of spin excitation. The observed photon-energy dependence of Δ θ K can be well simulated by assuming spin-dependent refractive-index changes in the presence of bandgap renormalization and state-filling effect upon photoexcitation. Our study delivers comprehensive insights into the opto-spintronic properties of bulk Bi2Se3 and the fundamental physical processes underlying polarization changes. These findings are expected to be crucial in developing ultrafast magneto-optical memory devices, which can perform read-and-write operations in the terahertz regime. [ABSTRACT FROM AUTHOR]

Advancements in photonics technologies have significantly enhanced their capability to facilitate experiments involving quantum light, even at room temperature. Nevertheless, fully integrating photonic chips that include quantum light sources, effective manipulation and transport of light minimizing losses, and appropriate detection systems remains an ongoing challenge. Topological photonic systems have emerged as promising platforms to protect quantum light properties during propagation, beyond merely preserving light intensity. In this work, we delve into the dynamics of non-classical light traversing a Su-Schrieffer-Heeger photonic lattice with topological domain walls. Our focus centers on how topology influences the quantum properties of light as it moves across the array. By precisely adjusting the spacing between waveguides, we achieve dynamic repositioning and interaction of domain walls, facilitating effective beam-splitting operations. Our findings demonstrate high-fidelity transport of non-classical light across the lattice, replicating known results that are now safeguarded by the topology of the system. This protection is especially beneficial for quantum communication protocols with continuous variable states. Our study enhances the understanding of light dynamics in topological photonic systems and paves the way for high-fidelity, topology-protected quantum communication. [ABSTRACT FROM AUTHOR]

The abundant topological magnetic textures in three-dimensional systems provide opportunities to investigate the fundamental spin dynamics and realize spintronic applications. The dynamics of such magnetic textures have however rarely been studied, especially for those driven by spin waves, which allow applications with ultralow energy consumption and ease of implementation even in insulating systems. Here, we report our micromagnetic simulations on the spin-wave-driven dynamics of a skyrmion tube (SkT) and chiral bobber (ChB) in a thick magnetic film. We predict tornado-like dynamics in both SkT and ChB, where the topological centers present a lateral rotation with the rotation centers forming a distorted profile in the thickness direction. While the velocity of SkT scales with the driving power, the ChB motion presents a threshold in the driving field, which is found to depend linearly on its penetration length. This distinct behavior could be useful to differentiate ChB from SkT and estimate its penetration length experimentally. The magnetic dynamics driven by spin waves can be useful for low-consumption devices, which is however still largely unexplored in three-dimensional system. Here, the authors find, using micromagnetic simulations, spin-wave-driven tornado-like dynamics of skyrmion tube and chiral bobber in thick magnetic films. [ABSTRACT FROM AUTHOR]

Any affine map on the (n + 1)-dimensional Euclidean space gives rise to a natural map on the n-dimensional sphere whose dynamical aspects are not so well-studied in the literature. We explore the dynamical aspects of these maps by investigating about their distality and expansivity. [ABSTRACT FROM AUTHOR]

We experimentally and theoretically study the dynamics of a one-dimensional array of pendula with a mild spatial gradient in their self-frequency and where neighboring pendula are connected with weak and alternating coupling. We map their dynamics to the topological Su-Schrieffer-Heeger model of charged quantum particles on a lattice with alternating hopping rates in an external electric field. By directly tracking the dynamics of a wave-packet in the bulk of the lattice, we observe Bloch oscillations, Landau-Zener transitions, and coupling between the isospin (i.e., the inner wave function distribution within the unit cell) and the spatial degrees of freedom (the distribution between unit cells). We then use Bloch oscillations in the bulk to directly measure the nontrivial global topological phase winding and local geometric phase of the band. We measure an overall evolution of 3.1 ± 0.2 radians for the geometrical phase during the Bloch period, consistent with the expected Zak phase of π. Our results demonstrate the power of classical analogs of quantum models to directly observe the topological properties of the band structure and shed light on the similarities and the differences between quantum and classical topological effects. [ABSTRACT FROM AUTHOR]

Skyrmion bags are spin structures with arbitrary topological degrees. They are expected to be promising next-generation information carriers due to their inherent high topological degrees. Here, we report the dynamics of the topological transition process when a skyrmion bag passes through a voltage gate driven by spin current in a synthetic antiferromagnetic racetrack with voltage-controlled magnetic anisotropy. The topological degrees of skyrmion bags controlled by voltage gate and driving current density are investigated. It is found that the different topological degrees of skyrmion bags transformed in this process are related to the interaction between antiskyrmions inside skyrmion bags, and the energy of each inner antiskyrmion after topological transformation is on the order of $10^{{-}19}\;J$. Furthermore, we have realized the successive transition of a skyrmion bag from a high topological degree to a low topological degree on a racetrack with three voltage gates. This work is helpful for designing high-density racetrack memory and logical devices based on skyrmion bags. [ABSTRACT FROM AUTHOR]

Features of the dynamics of topological defects in an electroconvective structure formed in a π/2-twisted nematic liquid crystal have been studied. The electroconvective structure (Williams domains) is a system of rolls similar to the system of convective cells in thermal convection. Hydrodynamic flows in rolls of the twisted nematic liquid crystal are helicoidal because the axial velocity component existing in addition to the tangential component has opposite directions in the neighboring rolls. This feature is responsible for the formation of stable localized extended objects—linear defects—oriented normally to Williams domains. The continuity condition for a helicoidal flow of an anisotropic liquid in twisted nematic liquid crystals prevents the decay of a linear defect into single dislocations. The length of the linear defect and the number of dislocations in it are controlled by an ac voltage applied to a liquid crystal cell. In contrast to the case of planar orientation, where linear defects decay into single dislocations with increasing applied voltage, zigzag oscillations appear in this case, but the structure of domains remains stationary. The boundaries between zig and zag regions in the core of the linear defect are dislocations with the topological charges S = +1 and –1. An "elementary" decay of a dislocation with the topological charge S = +1 (kink) into the dislocation with S = –1 (antikink) and two dislocations the topological charges S = +1 has been found for the first time in the linear defect of a certain length. The decay of the topological defect is possibly due to the appearance of the local instability of the orientational twist mode of the director n in the defect core caused by the critical growth of hydrodynamic fluctuations with increasing applied voltage. It has been shown that the detected decay of the topological soliton is qualitatively well described by the perturbed sine-Gordon eq-uation. [ABSTRACT FROM AUTHOR]

In this paper we build a geometric model for the renormalisation of irrationally indifferent fixed points. The geometric model incorporates the fine arithmetic properties of the rotation number at the fixed point. Using this model for the renormalisation, we build a topological model for the dynamics of a holomorphic map near an irrationally indifferent fixed point. We also explain the topology of the maximal invariant set for the model, and also explain the dynamics of the map on the maximal invariant set. [ABSTRACT FROM AUTHOR]

In this paper we study the static structure and the dynamics of topological defects associated with isotropic droplets in nematic environment. Investigations were made in confined geometry of optical cells when the droplet size was of the order of or larger than the gap of the cell. We observed the coexistence of point boojums and Saturn ring or modified Saturn ring defects. We found transformation of the Saturn ring defect to two localized broad defects at increasing the droplet size. At droplet coalescence antipodes of point and localized broad defects were born and the dynamics of their annihilation with existing defects was investigated. We found strong difference in the process of annihilation of point and localized broad defects. [ABSTRACT FROM AUTHOR]

[Display omitted] Atomically thin van der Waals magnetic materials have not only provided a fertile playground to explore basic physics in the two-dimensional (2D) limit but also created vast opportunities for novel ultrafast functional devices. Here we systematically investigate ultrafast magnetization dynamics and spin wave dynamics in few-layer topological antiferromagnetic MnBi 2 Te 4 crystals as a function of layer number, temperature, and magnetic field. We find laser-induced (de)magnetization processes can be used to accurately track the distinct magnetic states in different magnetic field regimes, including showing clear odd–even layer number effects. In addition, strongly field-dependent AFM magnon modes with tens of gigahertz frequencies are optically generated and directly observed in the time domain. Remarkably, we find that magnetization and magnon dynamics can be observed in not only the time-resolved magneto-optical Kerr effect but also the time resolved reflectivity, indicating strong correlation between the magnetic state and electronic structure. These measurements present the first comprehensive overview of ultrafast spin dynamics in this novel 2D antiferromagnet, paving the way for potential applications in 2D antiferromagnetic spintronics and magnonics as well as further studies of ultrafast control of both magnetization and topological quantum states. [ABSTRACT FROM AUTHOR]

The realization of Majorana zero modes in condensed matter have been attracting enormous interests from fundamental science such as topological quantum computation. Recently iron based superconductors were identified as a high-temperature platform for realizing topological superconductivity and Majorana modes. As unconventional superconductors, one of the most important characteristics of them is that they are in the vicinity of magnetic states due to the strong Hund's coupling in iron atoms. Here we propose that the line defects with missing Te/Se anions in Fe(Se, Te) superconductors provide the realization of intrinsic antiferromagnetic (AFM) chains with Rashba spin-orbit coupling. Against conventional wisdom, Majorana zero modes (MZMs) can be robustly generated at these AFM chain ends. These results can consistently explain the recent experimental observation of zero-energy end states in line defects of monolayer Fe(Te, Se)/SrTiO3 by scanning tunneling microscopy (STM) measurements. Our research not only demonstrates an unprecedented interplay among native line defect, emergent magnetism and topological superconductivity but also explores a high-temperature platform for Majorana fermions. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of the solution curves of the dynamics of spacetimes of the topological type Σ p × R , p > 1, where Σp is a closed Riemann surface of genus p, in the regime of 2 + 1 dimensional classical general relativity. The configuration space of the gauge fixed dynamics is identified with the Teichmüller space (T Σ p ≈ R 6 p − 6 ) of Σp. Utilizing the properties of the Dirichlet energy of certain harmonic maps, estimates derived from the associated elliptic equations in conjunction with a few standard results of the theory of the compact Riemann surfaces, we prove that every non-trivial solution curve runs off the edge of the Teichmüller space at the limit of the big bang singularity and approaches the space of projective measured laminations/foliations (P M L P M F) , the Thurston boundary of the Teichmüller space. [ABSTRACT FROM AUTHOR]

Laser-driven coherent phonons can act as modulated strain fields and modify the adiabatic ground state topology of quantum materials. Here we use time-dependent first-principles and effective model calculations to simulate the effect of the coherent phonon induced by strong terahertz electric field on electronic carriers in the topological insulator ZrTe5. We show that a coherent A1g Raman mode modulation can effectively pump carriers across the band gap, even though the phonon energy is about an order of magnitude smaller than the equilibrium band gap. We reveal the microscopic mechanism of this effect which occurs via Landau–Zener-Stückelberg tunneling of Bloch electrons in a narrow region in the Brillouin zone center where the transient energy gap closes when the system switches from strong to weak topological insulator. The quantum dynamics simulation results are in excellent agreement with recent pump-probe experiments in ZrTe5 at low temperature. Coherent phonons can modify the wavefunction topology of quantum materials, yet the implications for electron dynamics remain to be addressed. The authors use time-dependent approaches to simulate the effect of coherent phonons, induced by strong terahertz laser field, on the electronic carrier dynamics in the topological insulator Zirconium Pentatelluride. [ABSTRACT FROM AUTHOR]

We give a new categorical approach to the Halmos–von Neumann theorem for actions of general topological groups. As a first step, we establish that the categories of topological and measure-preserving irreducible systems with discrete spectrum are equivalent. This allows to prove the Halmos–von Neumann theorem in the framework of topological dynamics. We then use the Pontryagin and Tannaka–Krein duality theories to obtain classification results for topological and then measure-preserving systems with discrete spectrum. As a byproduct, we obtain a complete isomorphism invariant for compactifications of a fixed topological group. [ABSTRACT FROM AUTHOR]

In this paper, we develop the theory of \mathbb {Z}_p-index which has been introduced by Tsukamoto, Tsutaya and Yoshinaga [ G-index, Topological Dynamics and Marker Property , arXiv: 2012.15372 , 2020]. As an application, we show that given any positive number, there exists a dynamical system with mean dimension equal to such number such that it does not have the marker property. [ABSTRACT FROM AUTHOR]

We give a review of some recent developments in embeddings of time series and dynamic networks. We start out with traditional principal components and then look at extensions to dynamic factor models for time series. Unlike principal components for time series, the literature on time‐varying nonlinear embedding is rather sparse. The most promising approaches in the literature is neural network based, and has recently performed well in forecasting competitions. We also touch on different forms of dynamics in topological data analysis (TDA). The last part of the article deals with embedding of dynamic networks, where we believe there is a gap between available theory and the behavior of most real world networks. We illustrate our review with two simulated examples. Throughout the review, we highlight differences between the static and dynamic case, and point to several open problems in the dynamic case. [ABSTRACT FROM AUTHOR]

The Besicovitch pseudometric is a shift-invariant pseudometric over the set of infinite sequences, that enjoys interesting properties and is suitable for studying the dynamics of cellular automata. It corresponds to the asymptotic behavior of the Hamming distance on longer and longer prefixes. Though dynamics of cellular automata were already studied in the literature, we propose the first study of the dynamics of substitutions. We characterize those that yield a well-defined dynamical system as essentially the uniform ones. We also explore a variant of this pseudometric, the Feldman–Katok pseudometric, where the Hamming distance is replaced by the Levenshtein distance. Like in the Besicovitch space, cellular automata are Lipschitz in this space, but here also all substitutions are Lipschitz. In both spaces, we discuss equicontinuity of these systems, give a number of examples, and generalize our results to the class of dill maps, that embeds both cellular automata and substitutions. [ABSTRACT FROM AUTHOR]

This paper concerns the study of the linear threshold model in random networks, specifically in Erdös-Rényi networks. In our approach, we consider an activation threshold defined by the expected value for the node degree and the associated influence activation mapping. According to these assumptions, we present a theoretical procedure for the linear threshold model, under fairly general conditions, regarding the topological structure of the networks and the activation threshold. Aiming at the dynamics of the influence maximization process, we analyze and discuss different choices for the seed set based on several centrality measures along with the state conditions for the procedure to trigger. The topological entropy established for Erdös-Rényi networks defines a topological order for this type of random networks. Sufficient conditions are presented for this topological entropy to be characterized by the spectral radius of the associated adjacency matrices. Consequently, a number of properties are proved. The threshold dynamics are analyzed through the relationship between the activation threshold and the topological entropy. Numerical studies are included to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Amplifiers based on Josephson junctions allow for a fast and noninvasive readout of superconducting qubits. Motivated by the ongoing progress toward the realization of fault-tolerant qubits based on Majorana bound states, we investigate the topological counterpart of the Josephson bifurcation amplifier. We predict that the bifurcation dynamics of a topological Josephson junction driven in the appropriate parameter regime may be used as an additional tool to detect the emergence of Majorana bound states. [ABSTRACT FROM AUTHOR]

In recent years, there has been significant progress in the understanding of the dynamics of transcendental entire functions with bounded postsingular set. In particular, for certain classes of such functions, a complete description of their topological dynamics in terms of a simpler model has been given inspired by methods from polynomial dynamics. In this paper, and for the 1st time, we give analogous results in cases when the postsingular set is unbounded. More specifically, we show that if |$f$| is of finite order, has bounded criticality on its Julia set |$J(f)$|⁠ , and its singular set consists of finitely many critical values that escape to infinity and satisfy a certain separation condition, then |$J(f)$| is a collection of dynamic rays or hairs , which split at critical points, together with their corresponding landing points. In fact, our result holds for a much larger class of functions with bounded singular set. Moreover, this result is a consequence of a significantly more general one: we provide a topological model for the action of |$f$| on its Julia set. [ABSTRACT FROM AUTHOR]

How the network topology drives the response dynamic is a basic question that has not yet been fully answered in neural networks. Elucidating the internal relation between topological structures and dynamics is instrumental in our understanding of brain function. Recent studies have revealed that the ring structure and star structure have a great influence on the dynamical behavior of neural networks. In order to further explore the role of topological structures in the response dynamic, we construct a new tree structure that differs from the ring structure and star structure of traditional neural networks. Considering the diffusion effect, we propose a diffusion neural network model with binary tree structure and multiple delays. How to design control strategies to optimize brain function has also been an open question. Thus, we put forward a novel full-dimensional nonlinear state feedback control strategy to optimize relevant neurodynamics. Some conditions about the local stability and Hopf bifurcation are obtained, and it is proved that the Turing instability does not occur. Moreover, for the formation of the spatially homogeneous periodic solution, some diffusion conditions are also fused together. Finally, several numerical examples are carried out to illustrate the results' correctness. Meanwhile, some comparative experiments are rendered to reveal the effectiveness of the proposed control strategy. • A diffusion neural network with binary tree structure and delays is established. • A novel full-dimensional nonlinear state feedback control strategy is proposed. • Some bifurcation conditions integrated with diffusion are obtained and validated. • The feasibility of the full-dimensional control strategy is revealed. • The relationships among some parameters and the bifurcation threshold are provided. [ABSTRACT FROM AUTHOR]

A unified method for analyzing the dynamics and topological structure associated with a class of Floquet topological insulators is presented. The method is applied to a system that describes the propagation of electromagnetic waves through the bulk of a two-dimensional lattice that is helically driven in the direction of propagation. Tight-binding approximations are employed to derive reduced dynamical systems. Further asymptotic approximations, valid in the high-frequency driving regime, yield a time-averaged system which governs the leading order behavior of the wave. From this follows an analytic calculation of the Berry connection, curvature, and Chern number via analyzing the local behavior of the eigenfunctions near the critical points of the spectrum. Examples include honeycomb, Lieb, kagome, and 1/5-depleted lattices. In the nonlinear regime, novel equations governing slowly varying wave envelopes are derived. For the honeycomb lattice, numerical simulations show that for relatively small nonlinear effects a striking spiral pattern occurs; as nonlinearity increases, localized structures emerge, and for somewhat higher nonlinearity the waves appear to collapse. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the dynamics of regular curve homeomorphisms without periodic points. We prove mainly that they are extensions of irrational rotations of the circle via a monotone factor map collapsing all proximal pairs and we prove also the absence of Li-Yorke pairs. Furthermore, we give a full characterization of minimal sets. In particular, we get that the circle is the only regular curve supporting a minimal (or a transitive) Z -action. At the end of the paper, we give some counter-examples on rational curves. [ABSTRACT FROM AUTHOR]

We give a model-theoretic treatment of the fundamental results of Kechris-Pestov-Todorčević theory in the more general context of automorphism groups of not necessarily countable structures. One of the main points is a description of the universal ambit as a certain space of types in an expanded language. Using this, we recover results of Kechris et al. (Funct Anal 15:106–189, 2005), Moore (Fund Math 220:263–280, 2013), Ngyuen Van Thé (Fund Math 222: 19–47, 2013), in the context of automorphism groups of not necessarily countable structures, as well as Zucker (Trans Am Math Soc 368, 6715–6740, 2016). [ABSTRACT FROM AUTHOR]

Tangled active filaments are ubiquitous in nature, from chromosomal DNA and cilia carpets to root networks and worm collectives. How activity and elasticity facilitate collective topological transformations in living tangled matter is not well understood. We studied California blackworms (Lumbriculus variegatus), which slowly form tangles in minutes but can untangle in milliseconds. Combining ultrasound imaging, theoretical analysis, and simulations, we developed and validated a mechanistic model that explains how the kinematics of individual active filaments determines their emergent collective topological dynamics. The model reveals that resonantly alternating helical waves enable both tangle formation and ultrafast untangling. By identifying generic dynamical principles of topological self-transformations, our results can provide guidance for designing classes of topologically tunable active materials. [ABSTRACT FROM AUTHOR]

A Möbius strip, fascinating for its unique topological property of being a one-side nonorientable surface, has inspired mathematicians, physicists, engineers, and artists for many centuries. In a coherent system, coherent waves on the nonorientable surfaces reveal rich topological dynamics due to the interplay of coherence and topology. Here we experimentally observe topological resonances in a Möbius ring resonator formed in a twisted optical fiber loop. The twisted polarization-maintaining fiber ring encourages the hybridization of two polarization states, giving rise to the crucial Berry phase. This geometrical phase leads to the frequency shifts of fiber resonant modes with a non-trivial fractional mode number. Moreover, the resonant modes are topological, only resonating with certain polarized modes with circular chirality. These topological features introduce geometrical factors into coherent wave resonances, paving the way for topological information processing for quantum information, and coherent wave dynamics. The Möbius strip exhibits unique topological resonances, which are fundamental for topology-based signal processing. The authors realise a Möbius ring resonator via polarization-maintaining fibers, observing topological resonances emerging from the coupling between the two polarization modes along the fast and slow axes of the ring. [ABSTRACT FROM AUTHOR]

We study ultrafast population dynamics in the topological surface state of Sb 2 Te 3 in two-dimensional momentum space with time- and angle-resolved two-photon photoemission spectroscopy. Linearly polarized mid-infrared pump pulses are used to permit a direct optical excitation across the Dirac point. We show that this resonant excitation is strongly enhanced within the Dirac cone along three of the six Γ ¯ – M ¯ directions and results in a macroscopic photocurrent when the plane of incidence is aligned along a Γ ¯ – K ¯ direction. Our experimental approach makes it possible to disentangle the decay of transiently excited population and photocurent by elastic and inelastic electron scattering within the full Dirac cone in unprecedented detail. This is utilized to show that doping of Sb 2 Te 3 by vanadium atoms strongly enhances inelastic electron scattering to lower energies, but only scarcely affects elastic scattering around the Dirac cone. [ABSTRACT FROM AUTHOR]

We study ultrafast population dynamics in the topological surface state of Sb 2 Te 3 in two-dimensional momentum space with time- and angle-resolved two-photon photoemission spectroscopy. Linearly polarized mid-infrared pump pulses are used to permit a direct optical excitation across the Dirac point. We show that this resonant excitation is strongly enhanced within the Dirac cone along three of the six Γ ¯ – M ¯ directions and results in a macroscopic photocurrent when the plane of incidence is aligned along a Γ ¯ – K ¯ direction. Our experimental approach makes it possible to disentangle the decay of transiently excited population and photocurent by elastic and inelastic electron scattering within the full Dirac cone in unprecedented detail. This is utilized to show that doping of Sb 2 Te 3 by vanadium atoms strongly enhances inelastic electron scattering to lower energies, but only scarcely affects elastic scattering around the Dirac cone. [ABSTRACT FROM AUTHOR]

Analogues of Ramsey's Theorem for infinite structures such as the rationals or the Rado graph have been known for some time. In this context, one looks for optimal bounds, called degrees, for the number of colors in an isomorphic substructure rather than one color, as that is often impossible. Such theorems for Henson graphs however remained elusive, due to lack of techniques for handling forbidden cliques. Building on the author's recent result for the triangle-free Henson graph, we prove that for each k ≥ 4 , the k-clique-free Henson graph has finite big Ramsey degrees, the appropriate analogue of Ramsey's Theorem. We develop a method for coding copies of Henson graphs into a new class of trees, called strong coding trees, and prove Ramsey theorems for these trees which are applied to deduce finite big Ramsey degrees. The approach here provides a general methodology opening further study of big Ramsey degrees for ultrahomogeneous structures. The results have bearing on topological dynamics via work of Kechris, Pestov, and Todorcevic and of Zucker. [ABSTRACT FROM AUTHOR]

In this paper, we explore local dynamics with topological classifications, bifurcation analysis, and chaos control in a discrete‐time COVID‐19 epidemic model in the interior of ℝ+4$$ {\mathbb{R}}_{+}^4 $$. It is explored that for all involved parametric values, discrete‐time COVID‐19 epidemic model has boundary equilibrium solution and also it has an interior equilibrium solution under definite parametric condition. We have explored the local dynamics with topological classifications about boundary and interior equilibrium solutions of the discrete‐time COVID‐19 epidemic model by linear stability theory. Further, for the discrete‐time COVID‐19 epidemic model, existence of periodic points and convergence rate are also investigated. It is also studied the existence of possible bifurcations about boundary and interior equilibrium solutions and proved that there exists no flip bifurcation about boundary equilibrium solution. Moreover, it is proved that about interior equilibrium solution, there exist Hopf and flip bifurcations, and we have studied these bifurcations by utilizing explicit criterion. Moreover, by feedback control strategy, chaos in the discrete COVID‐19 epidemic model is also explored. Finally, theoretical results are verified numerically. [ABSTRACT FROM AUTHOR]

The interplay between system dynamics and topological structure plays a crucial role in nonlinear system and modern network science. In this paper, we intend to investigate the effects of both heterogeneity and asymmetric coupling on the collective behaviors of Kuramoto oscillators, and reveal the processes how the synchronization transits to multi-cluster or disorder states. We find that the complete synchronization and anti-phase synchronization states in a star motif change into the remote synchronization state, owing to the disparity in the values of oscillators' natural frequencies, and further become into multi-cluster due to the existence of multiple hubs in complete bipartite network, and finally achieve disorder state in complex networks. These findings are helpful to understand the forming processes of remote synchronization and general multi-cluster in coupled systems. [ABSTRACT FROM AUTHOR]

In this article, a discretized two-dimensional chemostat model is investigated. By using the algebraic technique, it is proved that the discrete chemostat model has semitrivial equilibrium solution for all involved parametric values, but it has an interior equilibrium solution under certain parametric conditions. We have studied the local dynamics with topological classifications about the semitrivial and interior equilibrium solution on the basis of the theory of the discrete dynamical system. By bifurcation theory, we studied the existence of bifurcations for the understudied discrete chemostat model and proved that at a semitrivial equilibrium solution, it did not undergo flip bifurcation, but it undergoes flip bifurcation at interior equilibrium solution, and no other bifurcation occurs at this point. It has also explored the existence of periodic points and chaos control by the state feedback method. Finally, with the help of MATLAB software, flip bifurcation diagrams and corresponding maximum Lyapunov exponents were also presented. These numerical simulations not only show the correctness of theoretical results but also reveal the complex dynamics such as the orbits of period − 7 , − 8 , − 13 , and − 16. [ABSTRACT FROM AUTHOR]

The high separation of crude oil supply and demand markets has led to the formation of a global crude oil trading system. This paper constructs global crude oil trade networks, integrates macro, meso, and micro network analysis methods, combines geospatial visualization techniques, and then portrays the spatiotemporal patterns and topological evolution of the global crude oil trade networks. Thus, it attempts to dig deeper into the world crude oil competition and cooperation links and evolution laws and provides a scientific reference for a comprehensive understanding of the global crude oil market dynamics. The results show that: (1) After three fluctuations of increase and decrease since 2000, the global crude oil trade volume is entering the adjustment period, and the scale of the crude oil market is rising slowly. (2) The international crude oil trade has formed trade network patterns with complex structures, clear hierarchy and unbalanced distribution. The "rich club" phenomenon is significant, with large trading countries dominating the trade network. (3) The scale and density of the global crude oil trade network show a trend of increasing and then decreasing, the network agglomeration pattern becoming more obvious, the inter-nodal links continuously strengthening, and the network connectivity improving. (4) The global crude oil trade networks are characterized by core–periphery structures, and the polarization effect is significant. The US, Russia, China, Japan, the Netherlands, and South Korea hold the core positions in the crude oil trade network, and the major importing countries have become the dominant forces in the trade network. In addition, we present policy suggestions for different types of countries for energy transformation and security in the global trade market system, which can be used as a reference for policymakers. [ABSTRACT FROM AUTHOR]

• High resolution traffic speed temporospatial distribution and propagation dynamics prediction • Deep learning framework integrating traffic flow dynamics and topological dependency • Dynamic programming capturing traffic flow interdependency The information of network-wide future traffic speed distribution and its propagation is beneficial to develop proactive traffic congestion management strategies. However, predicting network-wide traffic speed propagation is non-trivial. This study develops a t raffic flow d ependency and d ynamics based deep learning aided approach (TD2-DL), which predict network-wide high resolution traffic speed propagation by explicitly integrating temporal-spatial flow dependency, traffic flow dynamics with deep learning method techniques. Specifically, we first develop a graph theory-based method to identify the local temporal-spatial traffic dependency of each road among neighboring roads adaptive to the prediction horizon and traffic delay. Then, traffic speed propagation on every road is mathematically described by v -CTM based on traffic initial and boundary conditions. Next, the long short-term memory (LSTM) model is employed to predict boundary conditions factoring the traffic temporal-spatial dependency and historical data predicted by v -CTM. In this way, we well couple the physical models (traffic dependency and v -CTM) with the deep learning approach, and further make them coevolution under this framework. Last, an EKF is used to assimilate predicted traffic speed predicted by v -CTM coupled with the LSTMs and the field traffic data; an FNN is introduced to impute missing and corrupted data for improving the traffic speed prediction accuracy. The numerical experiments indicated that the TD2-DL predicted the network-wide traffic speed propagation in 30 minutes with accuracy varying from 85%-98%. It outperformed the tested models recently developed in literature. The ablation experimental results confirmed the significance of factoring traffic dependency and integrating data imputation and assimilation techniques for improving the prediction accuracy. [ABSTRACT FROM AUTHOR]

The oxygen depletion problem is an implicit moving boundary value problem. The dynamics allow topological changes in the moving boundary. We show several mathematical formulations of this model from the literature and give a new formulation based on a gradient flow with constraint. All formulations are shown to be equivalent. We explore the possibilities for the numerical approximation of the problem that arise from the different formulations. We show a convergence result for an approximation based on the gradient flow with constraint formulation that applies to the general dynamics including topological changes. More general (vector, higher order) implicit moving boundary value problems are discussed. Several open problems are described. [ABSTRACT FROM AUTHOR]

The problem of topological classification is fundamental in the study of dynamical systems. However, when we consider systems without well-posedness, it is unclear how to generalize the notion of equivalence. For example, when a system has trajectories distinguished only by parametrization, we cannot apply the usual definition of equivalence based on the phase space, which presupposes the uniqueness of trajectories. In this study, we formulate a notion of "topological equivalence" using the axiomatic theory of topological dynamics proposed by Yorke [7], where dynamical systems are considered to be shift-invariant subsets of a space of partial maps. In particular, we study how the type of problems can be regarded as invariants under the morphisms between systems and how the usual definition of topological equivalence can be generalized. This article is intended to also serve as a brief introduction to the axiomatic theory of ordinary differential equations (or topological dynamics) based on the formalism presented in [6]. [ABSTRACT FROM AUTHOR]

We define a new family of spectral invariants associated to certain Lagrangian links in compact and connected surfaces of any genus. We show that our invariants recover the Calabi invariant of Hamiltonians in their limit. As applications, we resolve several open questions from topological surface dynamics and continuous symplectic topology: We show that the group of Hamiltonian homeomorphisms of any compact surface with (possibly empty) boundary is not simple; we extend the Calabi homomorphism to the group of hameomorphisms constructed by Oh and Müller, and we construct an infinite-dimensional family of quasi-morphisms on the group of area and orientation preserving homeomorphisms of the two-sphere. Our invariants are inspired by recent work of Polterovich and Shelukhin defining and applying spectral invariants, via orbifold Floer homology, for links composed of parallel circles in the two-sphere. A particular feature of our work is that it avoids the orbifold setting and relies instead on 'classical' Floer homology. This not only substantially simplifies the technical background but seems essential for some aspects (such as the application to constructing quasi-morphisms). [ABSTRACT FROM AUTHOR]

The persistent dynamics in systems out of equilibrium, particularly those characterized by annihilation and creation of topological defects, is known to involve complicated spatiotemporal processes and is deemed difficult to control. Here the complex dynamics of defects in active smectic layers exposed to strong confinements is explored, through self-propulsion of active particles and a variety of confining geometries with different topology, ranging from circular, flower-shaped epicycloid, to hypocycloid cavities, channels, and rings. We identify a wealth of dynamical behaviors during the evolution of complex spatiotemporal defect patterns as induced by the confining shape and topology, particularly a perpetual creation-annihilation dynamical state at intermediate activity with large fluctuations of topological defects and a controllable transition from oscillatory to damped time correlation of defect number density via mechanisms governed by boundary cusps. Our results are obtained by using an active phase field crystal approach. Possible experimental realizations are also discussed. It is intriguing but challenging to control the complex dynamics of topological defects that emerge in nonequilibrium spatiotemporal systems. Through an effective modelling this work explores a viable way to controllably vary the properties of persistent defect dynamics in self-propelled active smectics via the competition of activity and boundary confinements with different topology and geometry. [ABSTRACT FROM AUTHOR]

Elucidating collective dynamics in crystalline systems is a common scientific question in multiple fields. In this work, by combination of high-precision numerical approach and analytical normal mode analysis, we systematically investigate the dynamical response of two-dimensional Lennard–Jones crystal as a purely classical mechanical system under random disturbance of varying strength, and reveal rich microscopic dynamics. Specifically, we observe highly symmetric velocity field composed of sharply divided coherent and disordered regions, and identify the order–disorder dynamical transition of the velocity field. Under stronger disturbance, we reveal the vacancy-driven shattering of the crystal. This featured disruption mode is fundamentally different from the dislocation-unbinding scenario in two-dimensional melting. We also examine the healing dynamics associated with vacancies of varying size. The results in this work advance our understanding about the formation of collective dynamics and crystal disruption, and may have implications in elucidating relevant non-equilibrium behaviors in a host of crystalline systems. Microscopic dynamics and underlying topological defects in the disruption of the Lennard–Jones lattice under random disturbance. [ABSTRACT FROM AUTHOR]

Topological defects—locations of local mismatch of order—are a universal concept playing important roles in diverse systems studied in physics and beyond, including the universe, various condensed matter systems, and recently, even life phenomena. Among these, liquid crystal has been a platform for studying topological defects via visualization, yet it has been a challenge to resolve three-dimensional structures of dynamically evolving singular topological defects. Here, we report a direct confocal observation of nematic liquid crystalline defect lines, called disclinations, relaxing from an electrically driven turbulent state. We focus in particular on reconnections, characteristic of such line defects. We find a scaling law for in-plane reconnection events, by which the distance between reconnecting disclinations decreases by the square root of time to the reconnection.Moreover, we show that apparently asymmetric dynamics of reconnecting disclinations is actually symmetric in a comoving frame, in marked contrast to the twodimensional counterpart whose asymmetry is established.We argue, with experimental supports, that this is because of energetically favorable symmetric twist configurations that disclinations take spontaneously, thanks to the topology that allows for rotation of the winding axis. Our work illustrates a general mechanism of such spontaneous symmetry restoring that may apply beyond liquid crystal, which can take place if topologically distinct asymmetric defects in lower dimensions become homeomorphic in higher dimensions and if the symmetric intermediate is energetically favorable. [ABSTRACT FROM AUTHOR]

Katznelson's Question is a long-standing open question concerning recurrence in topological dynamics with strong historical and mathematical ties to open problems in combinatorics and harmonic analysis. In this article, we give a positive answer to Katznelson's Question for certain towers of skew-product extensions of equicontinuous systems, including systems of the form (x,t) \mapsto (x + \alpha, t + h(x)). We describe which frequencies must be controlled for in order to ensure recurrence in such systems, and we derive combinatorial corollaries concerning the difference sets of syndetic subsets of the natural numbers. [ABSTRACT FROM AUTHOR]

The self-organising global dynamics underlying brain states emerge from complex recursive nonlinear interactions between interconnected brain regions. Until now, most efforts of capturing the causal mechanistic generating principles have supposed underlying stationarity, being unable to describe the non-stationarity of brain dynamics, i.e. time-dependent changes. Here, we present a novel framework able to characterise brain states with high specificity, precisely by modelling the time-dependent dynamics. Through describing a topological structure associated to the brain state at each moment in time (its attractor or 'information structure'), we are able to classify different brain states by using the statistics across time of these structures hitherto hidden in the neuroimaging dynamics. Proving the strong potential of this framework, we were able to classify resting-state BOLD fMRI signals from two classes of post-comatose patients (minimally conscious state and unresponsive wakefulness syndrome) compared with healthy controls with very high precision. Author summary: Brain states emerge through continuously evolving dynamics of brain networks. The usual way of modelling these dynamics is by using stationary systems: there is one structure (attractor) which is responsible of the brain dynamics. We adopt a different approach by characterising the brain activity through a landscape of informational structures (IS) changing in time. We use a model transformation procedure to produce these structures and look at several properties related to how the different brain networks interact not in the observed resting-state fMRI signal but in the information structure underlying it. These properties provide measures strongly related with relevant characteristics of conscious activity, such as metastability, information integration or synchronisation. The distribution of IS measures is studied for healthy controls (HC) and two groups of post-comatose patients with disorders of consciousness (DOC): minimally conscious state (MCS) and unresponsive wakefulness syndrome (UWS). Based on IS measures, machine learners classifiers identify the state of consciousness with an outstanding discrimination (precision of 95.6% por HC/DOC and 86.6% for MCS/UWS). [ABSTRACT FROM AUTHOR]

We discuss special properties of the spaces of characters and positive definite functions, as well as their associated dynamics, for arithmetic groups of product type. Axiomatizing these properties, we define the notions of charmenability and charfiniteness and study their applications to the topological dynamics, ergodic theory and unitary representation theory of the given groups. To do that, we study singularity properties of equivariant normal ucp maps between certain von Neumann algebras. We apply our discussion also to groups acting on product of trees. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of systems with the Coulomb friction expressed as Lagrange's equations of the second kind with solutions in the sense of Filippov. The friction coefficients are assumed to have dependence on generalized states, velocities, and time which can arise for various reasons such as temperature change, surface roughness, and other characteristics of the rubbing bodies. Combining the direct method of Lyapunov functions with semidefinite derivatives and the method of limit equations stemming from the works of Sell (1967) and Artstein (1977, 1978) on the topological dynamics of nonautonomous systems, we construct the limit differential inclusions by the methods of set-valued analysis and the theory of discontinuous systems. [ABSTRACT FROM AUTHOR]

For polynomials, local connectivity of Julia sets is a much‐studied and important property. Indeed, when the Julia set of a polynomial of degree d⩾2$d\geqslant 2$ is locally connected, the topological dynamics can be completely described as a quotient of a much simpler system: angle d$d$‐tupling on the circle. For a transcendental entire function, local connectivity is less significant, but we may still ask for a description of the topological dynamics as the quotient of a simpler system. To this end, we introduce the notion of docile functions (Definition 1.1): a transcendental entire function with bounded postsingular set is docile if it is the quotient of a suitable disjoint‐type function. Moreover, we prove docility for the large class of geometrically finite transcendental entire functions with bounded criticality on the Julia set. This is an analogue of the local connectivity of Julia sets for geometrically finite polynomials, first proved by Douady and Hubbard, and extends previous work of the second author and of Mihaljević for more restrictive classes of entire functions. We deduce a number of further results, concerning bounded Fatou components and the local connectivity of Julia sets. In particular, we show that the Julia set of the sine function is locally connected, answering a question raised by Osborne. [ABSTRACT FROM AUTHOR]

We prove that the minimal chain recurrence classes of a holomorphic endomorphism of \mathbb {P}^k have finitely many connected components. We also obtain results on arbitrary classes. These strong constraints on the topological dynamics in the phase space are all deduced from the associated action on a space of currents. [ABSTRACT FROM AUTHOR]