Class Of Dynamical Systems Research Articles

Multiphase solutions and their reductions for a nonlocal nonlinear Schrödinger equation with focusing nonlinearity

AbstractA nonlocal nonlinear Schrödinger equation with focusing nonlinearity is considered, which has been derived as a continuum limit of the Calogero–Sutherland model in an integrable classical dynamical system. The equation is shown to stem from the compatibility conditions of a system of linear partial differential equations (PDEs), assuring its complete integrability. We construct a nonsingular N‐phase solution (N: positive integer) of the equation by means of a direct method. The features of the one‐ and two‐phase solutions are investigated in comparison with the corresponding solutions of the defocusing version of the equation. We also provide an alternative representation of the N‐phase solution in terms of solutions of a system of nonlinear algebraic equations. Furthermore, the eigenvalue problem associated with the N‐phase solution is discussed briefly with some exact results. Subsequently, we demonstrate that the N‐soliton solution can be obtained simply by taking the long‐wave limit of the N‐phase solution. The similar limiting procedure gives an alternative representation of the N‐soliton solution as well as the exact results related to the corresponding eigenvalue problem.

Persistent corotation of the large-scale flow of thermal convection and an immersed free body

Inspired by the superrotation of the Earth's solid core, we investigate the dynamics of a free-rotating body as it interacts with the large-scale circulation (LSC) of the Rayleigh-Bénard thermal convection in a cylindrical container. A surprising and persistent corotation of both the free body and the LSC emerges, breaking the axial symmetry of the system. The corotational speed increases monotonically with the intensity of thermal convection, measured by the Rayleigh number Ra, which is proportional to the temperature difference between the heated bottom and cooled top. The rotational direction occasionally and spontaneously reverses, occurring more frequently at higher Ra. The reversal events follow a Poisson process; it is feasible that flow fluctuations randomly interrupt and reestablish the rotation-sustaining mechanism. This corotation is powered by thermal convection alone and promoted by the addition of a free body, enriching the classical dynamical system.

Singular Lagrangians and the Dirac–Bergmann algorithm in classical mechanics

Textbook treatments of classical mechanics typically assume that the Lagrangian is nonsingular; that is, the matrix of second derivatives of the Lagrangian with respect to the velocities is invertible. This assumption ensures that (i) Lagrange's equations can be solved for the accelerations as functions of coordinates and velocities, and (ii) the definitions of the conjugate momenta can be inverted to solve for the velocities as functions of coordinates and momenta. This assumption, however, is unnecessarily restrictive—there are interesting classical dynamical systems with singular Lagrangians. The algorithm for analyzing such systems was developed by Dirac and Bergmann in the 1950s. After a brief review of the Dirac–Bergmann algorithm, several examples are presented using familiar components: point masses connected by massless springs, rods, cords, and pulleys.

Speed limits on deterministic chaos and dissipation

Uncertainty in the initial conditions of classical dynamical systems can cause exponentially fast divergence of trajectories, a signature of deterministic chaos, or be suppressed by the dissipation of energy. A classical uncertainty relation sets a speed limit on the observables underlying these behaviors.

Traveling waves with singularities in a damped hyperbolic MEMS type equation in the presence of negative powers nonlinearity

We consider traveling waves with singularities in a damped hyperbolic MEMS type equation in the presence of negative powers nonlinearity. We investigate how the existence of traveling waves, their shapes, and asymptotic behavior change with the presence or absence of an inertial term. These are studied by applying the framework that combines Poincare compactification, classical dynamical systems theory, and geometric methods for the desingularization of vector fields. We report that the presence of this term causes the shapes to change significantly for sufficiently large wave speeds. For more information see https://ejde.math.txstate.edu/Volumes/2023/05/abstr.html

Controlling Quantum Chaos: Optimal Coherent Targeting.

One of the principal goals of controlling classical chaotic dynamical systems is known as targeting, which is the very weakly perturbative process of using the system's extreme sensitivity to initial conditions in order to arrive at a predetermined target state. It is shown that a generalization to chaotic quantum systems is possible in the semiclassical regime, but requires tailored perturbations whose effects must undo the dynamical spreading of the evolving quantum state. The procedure described here is applied to initially minimum uncertainty wave packets in the quantum kicked rotor, a preeminent quantum chaotic paradigm, to illustrate the method, and investigate its accuracy. The method's error can be made to vanish as ℏ→0.

Is The Modular Dynamic System as Effective as Classical Dynamic Systems in Long Segment Dynamic Thoracolumbar Stabilization?

To evaluate the outcomes of dynamic stabilization in the multilevel degenerative spondylotic spine, and to compare the two dynamic systems (Dynesys® and Orthrus®) to reveal the increasing role of dynamic systems in the management of the degenerative spine. A total of 74 patients who received dynamic stabilization for degenerative pathologies were retrospectively analyzed. Demographic details: preoperative data including neurological status, pain scores, and radiology; and intraoperative data including blood loss, duration of surgery, complications, and postoperative data including the neurologic status, duration of hospital stay, and pain scores were examined. Patients in both groups showed statistically significant improvements in their Visual Analog Scale and Oswestry Disability Index scores. Significant corrections of thoracic kyphosis (T2-T12), the sagittal vertical axis, and T10-L2 thoracolumbar kyphosis were obtained in our cohort (p < 0.05). A total of 4 patients received revision surgery due to screw loosening. Patients were discharged after 3-4 days and mobilized on the first postoperative day. We did not experience any serious issues in terms of stabilization in the cases in which we employed both systems. Our patients' clinical results were satisfactory in both systems. Existing systems can be used safely even in long-segment stabilization surgeries.

The 𝒫(𝜑)₂ Model on de Sitter Space

In 1975 Figari, Høegh-Krohn and Nappi constructed theP(φ)2{\mathscr P}(\varphi )_2model on the de Sitter space. Here we complement their work with new results, which connect this model to various areas of mathematics. In particular, i.) we discuss the causal structure of de Sitter space and the induces representations of the Lorentz group. We show that the UIRs ofSO0(1,2)SO_0(1,2)for both the principal and the complementary series can be formulated on Hilbert spaces whose functions are supported on a Cauchy surface. We describe the free classical dynamical system in both its covariant and canonical form, and present the associated quantum one-particle KMS structures in the sense of Kay (1985). Furthermore, we discuss the localisation properties of one-particle wave functions and how these properties are inherited by the algebras of local observables.ii.) we describe the relations between the modular objects (in the sense of Tomita-Takesaki theory) associated to wedge algebras and the representations of the Lorentz group. We connect the representations of SO(1,2) to unitary representations ofSO(3)SO(3)on the Euclidean sphere, and discuss how theP(φ)2{\mathscr P}(\varphi )_2interaction can be represented by a rotation invariant vector in the Euclidean Fock space. We present a novel Osterwalder-Schrader reconstruction theorem, which shows that physicalinfrared problemsare absent on de Sitter space. As shown in Figari, Høegh-Krohn, and Nappi (1975), the ultraviolet problems are resolved just like on flat Minkowski space. We state the Haag–Kastler axioms for theP(φ)2{\mathscr P}(\varphi )_2model and we explain how thegeneratorsof the boosts and the rotations for the interacting quantum field theory arise from thestress-energy tensor. Finally, we show that the interacting quantum fields satisfy theequations of motionin their covariant form. In summary, we argue that the de SitterP(φ)2{\mathscr P}(\varphi )_2model is the simplest and most explicit relativistic quantum field theory, which satisfies basic expectations, like covariance, particle creation, stability and finite speed of propagation.

Classification of nonnegative traveling wave solutions for the 1D degenerate parabolic equations

&lt;p style='text-indent:20px;'&gt;Traveling wave solutions for the one-dimensional degenerate parabolic equations are considered. The purpose of this paper is to classify the nonnegative traveling wave solutions including sense of weak solutions of these equations and to present their existence, information about their shape and asymptotic behavior. These are studied by applying the framework that combines Poincaré compactification and classical dynamical systems theory. We also aim to use these results to generalize the results of our previous studies. The key to this is the introduction of a transformation, which overcomes the generalization difficulties faced by these studies.&lt;/p&gt;

A New Slack Lyapunov Functional for Dynamical System with Time Delay

The traditional method of constructing a Lyapunov functional for dynamical systems with time delay is usually dependent on positive definite matrices in the quadratic form. In this paper, a new Lyapunov functional is proposed and the corresponding proof is given. It do not require that all matrices in the quadratic form of Lyapunov functionals are positive definite, while the quadratic form is still positive definite, which makes the estimate more relaxed due to special construction of matrices. It is a general form and can be used in the performance analysis of a variety of dynamical systems. Moreover, a lemma concerning the quadratic function is applied to deal with the quadratic term of time-varying delay. Lastly, in the case of classical dynamical systems with time delay, the verification results are given to illustrate the usefulness of the new slack Lyapunov functional.

Radially symmetric stationary solutions for a MEMS type reaction–diffusion equation with fringing field

Radially symmetric stationary solutions for a micro-electro-mechanical system (MEMS) type reaction–diffusion equation with fringing field are considered. This equation arises in the study of the MEMS devices. This paper is devoted to the study of the existence of these solutions, information about their shape, and their asymptotic behaviour. These are studied by applying the framework that combines Poincaré type compactification, classical dynamical systems theory, and geometric methods for desingularization of vector fields called the blow-up technique.

Density matrix formulation of dynamical systems.

Physical systems that are dissipating, mixing, and developing turbulence also irreversibly transport statistical density. However, predicting the evolution of density from atomic and molecular scale dynamics is challenging for nonsteady, open, and driven nonequilibrium processes. Here, we establish a theory to address this challenge for classical dynamical systems that is analogous to the density matrix formulation of quantum mechanics. We show that a classical density matrix is similar to the phase-space metric and evolves in time according to generalizations of Liouville's theorem and Liouville's equationfor non-Hamiltonian systems. The traditional Liouvillian forms are recovered in the absence of dissipation or driving by imposing trace preservation or by considering Hamiltonian dynamics. Local measures of dynamical instability and chaos are embedded in classical commutators and anticommutators and directly related to Poisson brackets when the dynamics are Hamiltonian. Because the classical density matrix is built from the Lyapunov vectors that underlie classical chaos, it offers an alternative computationally tractable basis for the statistical mechanics of nonequilibrium processes that applies to systems that are driven, transient, dissipative, regular, and chaotic.

Large Fronts in Nonlocally Coupled Systems Using Conley–Floer Homology

In this paper, we study travelling front solutions for nonlocal equations of the type ∂tu=N∗S(u)+∇F(u),u(t,x)∈Rd.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\partial _t u = N * S(u) + \ abla F(u), \\qquad u(t,x) \\in {{\ extbf {R}}}^d. \\end{aligned}$$\\end{document}Here, N * denotes a convolution-type operator in the spatial variable x in {{textbf {R}}}, either continuous or discrete. We develop a Morse-type theory, the Conley–Floer homology, which captures travelling front solutions in a topologically robust manner, by encoding fronts in the boundary operator of a chain complex. The equations describing the travelling fronts involve both forward and backward delay terms, possibly of infinite range. Consequently, these equations lack a natural phase space, so that classic dynamical systems tools are not at our disposal. We therefore develop, from scratch, a general transversality theory, and a classification of bounded solutions, in the absence of a phase space. In various cases the resulting Conley–Floer homology can be interpreted as a homological Conley index for multivalued vector fields. Using the Conley–Floer homology, we derive existence and multiplicity results on travelling front solutions.

Quantum mechanical approach to bifurcation point detection in Hamiltonian dynamical systems

Abstract The energy level statistics of a bounded quantum system whose classical dynamical system exhibits bifurcations is investigated using the two-point correlation function (TPCF), which at the bifurcation points exhibits periodic spike oscillations owing to the accumulation of levels called the shell effect. The spike oscillations of the TPCF are analyzed by the reduced chi-squared value, which exhibits abrupt increases at bifurcation points, thereby yielding a novel detection approach. Using this method, we attempt to numerically detect the bifurcation points of a lemon-shaped billiard.

Detection and Control of Epileptiform Regime in the Hodgkin–Huxley Artificial Neural Networks via Quantum Algorithms

The problem of detection and the following suppression of epileptiform dynamics in artificial neural networks (ANN) still is a hot topic in modern theoretical and applied neuroscience. For the purpose of such modeling, the Hodgkin–Huxley (HH) elements are important due to the variety of their behavior such as resting, singular spikes, and spike trains and bursts. This dynamical spectrum of individual HH neurons can cause an epileptiform regime originated in the hyper-synchronization of the cell outcomes. Our model covers the detection and suppression of ictal behavior in a small ANN consisting of HH cells. The model follows our approach [Borisenok et al., 2018] for the HH neurons as a classical dynamical system driving the collective neural bursting, but here we use a quantum paradigm-based algorithm emulated with the pair of HH neurons. Such emulation becomes possible due to the complexity of the individual 4d HH dynamics. The linear chain of two HH neurons is connected to the rest of ANN and works autonomously. The first neuron plays a role of the detecting element for the hyper-synchronization in the ANN and the quantum algorithm emulator; while the second one works as a measuring element (emulation of the quantum measurement converting the signals into the classical domain) and the trigger for the feedback suppressing the epileptiform regime. We use here the speed gradient algorithm for controling the emulating neuron and discuss its pros and cons to compare with our classical model of epileptiform suppression.

Almost periodic fuzzy multidimensional dynamic systems and applications on time scales

In this paper, we introduce a new division of fuzzy vectors depending on a determinant algorithm and develop a theory of almost periodic fuzzy multidimensional dynamic systems on time scales. Moreover, several applications are provided. In particular, a new type of fuzzy dynamic systems called fuzzy q-dynamic systems (i.e., fuzzy quantum dynamic systems) is proposed and studied. Our results are not only effective on classical fuzzy dynamic systems including their continuous and discrete situations (i.e., T=Z or R) but are also valid for other fuzzy multidimensional dynamic systems on various hybrid domains like qZ¯,±N12, ∪k=1+∞[2k,2k+1] etc.

Combinatorial vs. classical dynamics: Recurrence

Establishing the existence of periodic orbits is one of the crucial and most intricate topics in the study of dynamical systems, and over the years, many methods have been developed to this end. On the other hand, finding closed orbits in discrete contexts, such as graph theory or in the recently developed field of combinatorial dynamics, is straightforward and computationally feasible. In this paper, we present an approach to study classical dynamical systems as given by semiflows or flows using techniques from combinatorial topological dynamics. More precisely, we present a general existence theorem for periodic orbits of semiflows which is based on suitable phase space decompositions, and indicate how combinatorial techniques can be used to satisfy the necessary assumptions. In this way, one can obtain computer-assisted proofs for the existence of periodic orbits and even certain chaotic behavior.

Algebraic and geometric structures inside the Birkhoff polytope

The Birkhoff polytope Bd consisting of all bistochastic matrices of order d assists researchers from many areas, including combinatorics, statistical physics, and quantum information. Its subset Ud of unistochastic matrices, determined by squared moduli of unitary matrices, is of particular importance for quantum theory as classical dynamical systems described by unistochastic transition matrices can be quantized. In order to investigate the problem of unistochasticity, we introduce the set Ld of bracelet matrices that forms a subset of Bd, but a superset of Ud. We prove that for every dimension d, this set contains the set of factorizable bistochastic matrices Fd and is closed under matrix multiplication by elements of Fd. Moreover, we prove that both Ld and Fd are star-shaped with respect to the flat matrix. We also analyze the set of d × d unistochastic matrices arising from circulant unitary matrices and show that their spectra lie inside d-hypocycloids on the complex plane. Finally, applying our results to small dimensions, we fully characterize the set of circulant unistochastic matrices of order d ≤ 4 and prove that such matrices form a monoid for d = 3.

Criteria stochastic filtering of max-plus discrete event systems with bounded random variables

A wide range of Discrete Event Systems (DES) such as manufacturing systems, telecommunications networks, transportation networks, and parallel computing, etc., can be modelled as max-plus systems in which maximization and addition are the main operations. In this paper, we use a nonlinear approach to deal with the error-estimation of nondeterministic max-plus systems with bounded random variables. This estimation is carried out following the ideas of the stochastic filtering theory for classical time-driven dynamic systems. The probability densities for the entries of the system matrices are assumed to be known, and a prediction-correction filtering scheme is used to compute the estimated state. The filtering algorithm is based on the minimization of a criterion, which is capable to evaluate the estimation-error, and to deal with the trade-off between prediction and measurement.

From dual-unitary to quantum Bernoulli circuits: Role of the entangling power in constructing a quantum ergodic hierarchy

Deterministic classical dynamical systems have an ergodic hierarchy, from ergodic through mixing, to Bernoulli systems that are "as random as a coin-toss". Dual-unitary circuits have been recently introduced as solvable models of many-body nonintegrable quantum chaotic systems having a hierarchy of ergodic properties. We extend this to include the apex of a putative quantum ergodic hierarchy which is Bernoulli, in the sense that correlations of single and two-particle observables vanish at space-time separated points. We derive a condition based on the entangling power $e_p(U)$ of the basic two-particle unitary building block, $U$, of the circuit, that guarantees mixing, and when maximized, corresponds to Bernoulli circuits. Additionally we show, both analytically and numerically, how local-averaging over random realizations of the single-particle unitaries, $u_i$ and $v_i$ such that the building block is $U^\prime = (u_1 \otimes u_2 ) U (v_1 \otimes v_2 )$ leads to an identification of the average mixing rate as being determined predominantly by the entangling power $e_p(U)$. Finally we provide several, both analytical and numerical, ways to construct dual-unitary operators covering the entire possible range of entangling power. We construct a coupled quantum cat map which is dual-unitary for all local dimensions and a 2-unitary or perfect tensor for odd local dimensions, and can be used to build Bernoulli circuits.