Topological Mixing Research Articles

Flow topology and mixing in alveolar edema: Unsteady flow in interconnected cavities with moving walls

The study of alveolar fluid mechanics is critical for comprehending respiratory function and lung diseases, particularly in cases of alveolar lesions that result in significant structural and fluid dynamic changes. This study investigates the flow topology and chaotic mixing within both normal and edematous alveoli, where the alveoli in the edematous model are interconnected by pores. To numerically simulate alveolar flow, a mathematical model is developed to ascertain the key parameters of Reynolds number (Re) and alveolar expansion ratio. Subsequently, the flow fields are analyzed to determine wall shear stress (WSS) and to identify WSS critical points and critical points of velocity vector, with a thorough presentation of the various flow topologies corresponding to these critical points. Moreover, a dynamic mode decomposition-based method is introduced to track particle trajectories, and the exploration of chaotic mixing is conducted through tracer advection, Poincare map, and the calculation of finite-time Lyapunov exponents. Results indicate that the edematous model exhibits a higher Re and higher WSS due to the fluid properties. Within the alveoli, high WSS is usually localized at the pores. The pores increase critical points and alter flow topologies, significantly changing chaotic mixing. Additionally, Re and alveolar locations also affect mixing patterns. These findings are crucial for understanding alveolar physiology and designing inhaled drugs for lung diseases, considering the role of chaos in particle transport in the lung acini.

Topological mixing of the geodesic flow on convex projective manifolds

Topological mixing of the geodesic flow on convex projective manifolds

Topological mixing of positive diagonal flows

Topological mixing of positive diagonal flows

Exponential decay of random correlations for random Anosov systems mixing on fibers

In this paper, we study the statistical property of Anosov systems on surface driven by an external force. By utilizing the Birkhoff cone method, we show that if the systems on surface satisfying the Anosov and topological mixing on fibers property, then the quenched random correlation for Hölder observables with respect to the unique random SRB measures decays exponentially.

Giant room-temperature nonlinearities in a monolayer Janus topological semiconductor

Nonlinear optical materials possess wide applications, ranging from terahertz and mid-infrared detection to energy harvesting. Recently, the correlations between nonlinear optical responses and certain topological properties, such as the Berry curvature and the quantum metric tensor, have attracted considerable interest. Here, we report giant room-temperature nonlinearities in non-centrosymmetric two-dimensional topological materials—the Janus transition metal dichalcogenides in the 1 T’ phase, synthesized by an advanced atomic-layer substitution method. High harmonic generation, terahertz emission spectroscopy, and second harmonic generation measurements consistently show orders-of-the-magnitude enhancement in terahertz-frequency nonlinearities in 1 T’ MoSSe (e.g., > 50 times higher than 2H MoS2 for 18th order harmonic generation; > 20 times higher than 2H MoS2 for terahertz emission). We link this giant nonlinear optical response to topological band mixing and strong inversion symmetry breaking due to the Janus structure. Our work defines general protocols for designing materials with large nonlinearities and heralds the applications of topological materials in optoelectronics down to the monolayer limit.

Chaos and weak mixing on uniform spaces

Let f0,∞={fn}n=0∞ be a sequence of uniformly continuous self-maps on a uniform space X. We prove that under some natural conditions, topological weak mixing implies dense distributional η-chaos in a sequence, and generic η-chaos is equivalent to dense η′-chaos for (X,f0,∞). We give several equivalent characterizations of sensitivity for (X,f0,∞), and as applications, we get the relationships between sensitivity and Li-Yorke sensitivity, and further obtain that topological weak mixing implies Li-Yorke sensitivity for a class of abelian (X,f0,∞). We show that (X×Y,f0,∞×g0,∞) is sensitive (resp. Li-Yorke sensitive and multi-sensitive) if and only if (X,f0,∞) or (Y,g0,∞) is sensitive (resp. Li-Yorke sensitive and multi-sensitive). We also prove that distributional chaos (resp. Li-Yorke chaos, sensitivity and Li-Yorke sensitivity) is equivalent between (X,f0,∞) and its N-th iteration system. Finally, we confirm that distributional chaos in a sequence, Li-Yorke chaos, sensitivity and Li-Yorke sensitivity are all preserved under topological equi-conjugacy.

Decidability of CPC-irreducibility of subshifts of finite type over free groups

This paper attempts to study the irreducibility on complete prefix code (CPC-irreducibility) of a Markov shift over a free group, a topological mixing property first considered for shift spaces over free semigroups that induces chaotic behavior such as the existence of a dense set of periodic points. An example shows that the ({textsf{d}},{textsf{c}})-reduction, an effective algorithm of determination of CPC-irreducibility of Markov shifts over free semigroups (Ban et al. in J Stat Phys 177:1043–1062, 2019), fails for general Markov shifts over free groups. This paper reveals an algorithm for determining the CPC-irreducibility of Markov shifts over both free semigroups and groups. Furthermore, such an examination is finitely checkable, and an upper bound for the complexity of the algorithm is provided.

Disjoint topological transitivity for cosine operator functions on weighted Orlicz spaces

In this note, we give sufficient conditions for sequences of operators to be disjoint topologically transitive on the weighted Orlicz spaces of locally compact groups. This condition also ensures the disjoint blow-up/collapse property holds. In our case, these sequences of operators can be regarded as cosine operator functions. Disjoint topological mixing for such cosine operator functions is investigated as well.

Topological dynamics of Markov multi-maps of the interval

We study Markov multi-maps of the interval from the point of view of topological dynamics. Specifically, we define the forward trajectory system of a Markov multi-map, and we investigate whether it has properties such as topological transitivity, topological mixing, density of periodic points, and specification. To each Markov multi-map we associate a shift of finite type (SFT), and our main results relate the properties of the SFT with those of the forward trajectory system. Under a general coding condition, we establish necessary and sufficient conditions for topological transitivity and topological mixing of the forward trajectory system. These results complement existing work showing a relationship between the topological entropy of a Markov multi-map and its associated SFT. We also characterize when the inverse limit systems associated to the Markov multi-maps have the properties mentioned above.

Neural Networks as Geometric Chaotic Maps.

The use of artificial neural networks (NNs) as models of chaotic dynamics has been rapidly expanding. Still, a theoretical understanding of how NNs learn chaos is lacking. Here, we employ a geometric perspective to show that NNs can efficiently model chaotic dynamics by becoming structurally chaotic themselves. We first confirm NN's efficiency in emulating chaos by showing that a parsimonious NN trained only on few data points can reconstruct strange attractors, extrapolate outside training data boundaries, and accurately predict local divergence rates. We then posit that the trained network's map comprises sequential geometric stretching, rotation, and compression operations. These geometric operations indicate topological mixing and chaos, explaining why NNs are naturally suitable to emulate chaotic dynamics.

Topological mixing of random substitutions

We investigate topological mixing of compatible random substitutions. For primitive random substitutions on two letters whose second eigenvalue is greater than one in modulus, we identify a simple, computable criterion which is equivalent to topological mixing of the associated subshift. This generalises previous results on deterministic substitutions. In the case of recognisable, irreducible Pisot random substitutions, we show that the associated subshift is not topologically mixing. Without recognisability, we rely on more specialised methods for excluding mixing and we apply these methods to show that the random Fibonacci substitution subshift is not topologically mixing.

On Topological Shadowing and Chain Properties of IFS on Uniform Spaces

We define notions such as pseudo-orbit, topological shadowing, and topological chain transitivity of iterated function systems on compact uniform spaces. We prove that these notions are invariant under topological conjugacy on a compact uniform space. For an IFS on a compact uniform space with topological shadowing property, we show that the topological chain transitivity implies topological transitivity. We also show that in a connected compact uniform space, notions such as topological chain mixing, totally topological chain transitive, topological chain transitive, and topological chain recurrent are equivalent.

Chaos and Shadowing in General Systems

In this paper we describe some basic notions of topological dynamical systems for maps of type f : X × X → X named general systems. This is proved that every uniformly expansive general system has the shadowing property and every uniformly contractive general system has the (asymptotic) average shadowing and shadowing properties. In the rest, Devaney chaos for general systems is considered. Also, we show that topological transitivity and density of periodic points of a general systems imply topological ergodicity. We also obtain some results on the topological mixing and sensitivity for general systems.

Topological mixing notions on Turing machine dynamical systems

Over the past few decades, Turing machines have been studied as dynamical systems, with the focus being on their behavior rather than their results. Noteworthy results concerning topological and dynamical properties, such as the existence and undecidability of topological transitivity in TMH and topological minimality in TMT, were established. Both properties are related to reaching finite windows from some or any possible configuration. Nonetheless, both properties exhibit no restriction over the time a machine takes to reach these finite windows. In this article, we focus on the mixing notions: weak mixing, total transitivity and topological mixing. These properties are related to a time window or gap where finite configurations must reach one another. In this article, we analyze the SMART machine to prove that its TMT dynamical model is topologically weak mixing (and therefore totally transitive) and that all mixing notions are undecidable.

Topological Weak Specification and Distributional Chaos on Noncompact Spaces

In this paper, we relate the topological definitions of specification property and distributional chaos defined for uniformly continuous self-maps on noncompact, nonmetrizable spaces. We prove that a uniformly continuous surjective self-map acting on a uniformly locally compact Hausdorff uniform space with topologically weak specification property and a pair of distal points is topologically distributionally chaotic of type 1. This extends the result due to Oprocha and Štefánková [2008]. As a consequence, we get that uniformly continuous surjective self-map on a uniformly locally compact totally bounded Hausdorff uniform space with topological shadowing, topological mixing, and a distal pair is topologically distributionally chaotic of type 1.

Disjoint topological transitivity for weighted translations generated by group actions

Abstract In this note, we give a sufficient and necessary condition for weighted translations, generated by group actions, to be disjoint topologically transitive in terms of the weights, the group element and the measure. The characterization of disjoint topological mixing is obtained as well. Moreover, we apply the results to the quotient spaces of locally compact groups and hypergroups.

Topologically mixing tiling of generated by a generalized substitution

AbstractThis paper presents sufficient conditions for a substitution tiling dynamical system of $\mathbb {R}^2$ , generated by a generalized substitution on three letters, to be topologically mixing. These conditions are shown to hold on a large class of tiling substitutions originally presented by Kenyon in 1996. This problem was suggested by Boris Solomyak, and many of the techniques that are used in this paper are based on the work by Kenyon, Sadun, and Solomyak [Topological mixing for substitutions on two letters. Ergod. Th. & Dynam. Sys.25(6) (2005), 1919–1934]. They studied one-dimensional tiling dynamical systems generated by substitutions on two letters and provided similar conditions sufficient to ensure that one-dimensional substitution tiling dynamical systems are topologically mixing. If a tiling dynamical system of $\mathbb {R}^2$ satisfies our conditions (and thus is topologically mixing), we can construct additional topologically mixing tiling dynamical systems of $\mathbb {R}^2$ . By considering the stepped surface constructed from a tiling $T_\sigma $ , we can get a new tiling of $\mathbb {R}^2$ by projecting the surface orthogonally onto an irrational plane through the origin.

Constructing Higher-Dimensional Digital Chaotic Systems via Loop-State Contraction Algorithm

This paper aims to refine and expand the theoretical and application framework of higher-dimensional digital chaotic system (HDDCS). Topological mixing for HDDCS is strictly proved theoretically at first. Topological mixing implies Devaney's definition of chaos in a compact space, but not vice versa. Therefore, the proof of topological mixing promotes the theoretical research of HDDCS. Then, a general design method for constructing HDDCS via loop-state contraction algorithm is given. The construction of the iterative function uncontrolled by random sequences (hereafter called iterative function) is the starting point of this research. On this basis, this paper put forward a general design method to solve the construction problem of HDDCS, and several examples illustrate the effectiveness and feasibility of this method. The adjacency matrix corresponding to the designed HDDCS is used to construct the chaotic Echo State Network (ESN) for predicting Mackey-Glass time series. Compared with other ESNs, the chaotic ESN has better prediction performance and is able to accurately predict a much longer period of time.

Verification of mixing properties in two-dimensional shifts of finite type

This work introduces constructive and systematic methods for verifying the topological mixing and strong specification (or strong irreducibility) of two-dimensional shifts of finite type. First, we define transition matrices on infinite strips of width n for all n ≥ 2. To determine the primitivity of the transition matrices, we introduce the connecting operators that reduce the high-order transition matrices to lower-order transition matrices. Then, two sufficient conditions for primitivity are provided; they are invariant diagonal cycles and primitive commutative cycles of connecting operators. Then, the primitivity, corner-extendability, and crisscross-extendability are used to demonstrate the topological mixing. Finally, we show that the hole-filling condition yields the strong specification property. The application of all the above-mentioned conditions can be verified in a finite number of steps.

Topological weak mixing and diffusion at all times for a class of Hamiltonian systems

AbstractWe present examples of nearly integrable analytic Hamiltonian systems with several strong diffusion properties: topological weak mixing and diffusion at all times. These examples are obtained by AbC constructions with several frequencies.