Class Of Dynamical Systems Research Articles

Slow Quantum Thermalization and Many-Body Revivals from Mixed Phase Space

Relaxation of few-body quantum systems can strongly depend on the initial state when the system's semiclassical phase space is mixed, i.e., regions of chaotic motion coexist with regular islands. In recent years, there has been much effort to understand the process of thermalization in strongly interacting quantum systems that often lack an obvious semiclassical limit. Time-dependent variational principle (TDVP) allows to systematically derive an effective classical (nonlinear) dynamical system by projecting unitary many-body dynamics onto a manifold of weakly-entangled variational states. We demonstrate that such dynamical systems generally possess mixed phase space. When TDVP errors are small, the mixed phase space leaves a footprint on the exact dynamics of the quantum model. For example, when the system is initialized in a state belonging to a stable periodic orbit or the surrounding regular region, it exhibits persistent many-body quantum revivals. As a proof of principle, we identify new types of "quantum many-body scars", i.e., initial states that lead to long-time oscillations in a model of interacting Rydberg atoms in one and two dimensions. Intriguingly, the initial states that give rise to most robust revivals are typically entangled states. On the other hand, even when TDVP errors are large, as in the thermalizing tilted-field Ising model, initializing the system in a regular region of phase space leads to slowdown of thermalization. Our work establishes TDVP as a method for identifying interacting quantum systems with anomalous dynamics in arbitrary dimensions. Moreover, the mixed-phase space classical variational equations allow to find slowly-thermalizing initial conditions in interacting models. Our results shed light on a link between classical and quantum chaos, pointing towards possible extensions of classical Kolmogorov-Arnold-Moser theorem to quantum systems.

Classical discrete time crystals

The spontaneous breaking of time-translation symmetry in periodically driven quantum systems leads to a new phase of matter: discrete time crystals (DTC). This phase exhibits collective subharmonic oscillations that depend upon an interplay of non-equilibrium driving, many-body interactions, and the breakdown of ergodicity. However, subharmonic responses are also a well-known feature of classical dynamical systems ranging from predator-prey models to Faraday waves and AC-driven charge density waves. This raises the question of whether these classical phenomena display the same rigidity characteristic of a quantum DTC. In this work, we explore this question in the context of periodically driven Hamiltonian dynamics coupled to a finite-temperature bath, which provides both friction and, crucially, noise. Focusing on one-dimensional chains, where in equilibrium any transition would be forbidden at finite temperature, we provide evidence that the combination of noise and interactions drives a sharp, first-order dynamical phase transition between a discrete time-translation invariant phase and an activated classical discrete time crystal (CDTC) in which time-translation symmetry is broken out to exponentially-long time scales. Power-law correlations are present along a first-order line which terminates at a critical point. We analyze the transition by mapping it to the locked-to-sliding transition of a DC-driven charge density wave. Our work points to a classical limit for quantum time crystals, and raises several intriguing questions concerning the non-equilibrium universality class of the CDTC critical point.

Critical Response of a Quantum van der Pol Oscillator.

Classical dynamical systems close to a critical point are known to act as efficient sensors due to a strongly nonlinear response. We explore such systems in the quantum regime by modeling a quantum version of a driven van der Pol oscillator. We find the classical response survives down to one excitation quantum. At very weak drives, genuine quantum features arise, including diverging and negative susceptibilities. Further, the linear response is greatly enhanced by using a strong incoherent pump. These results are largely generic and can be probed in current experimental platforms suited for quantum sensing.

Classical and Quantum Dynamics of a Particle in a Narrow Angle

We consider the 2D Schrodinger equation with variable potential in the narrow domain diffeomorphic to the wedge with the Dirichlet boundary condition. The corresponding classical problem is the billiard in this domain. In general, the corresponding dynamical system is not integrable. The small angle is a small parameter which allows one to make the averaging and reduce the classical dynamical system to an integrable one modulo exponential small correction. We use the quantum adiabatic approximation (operator separation of variables) to construct the asymptotic eigenfunctions (quasi-modes) of the Schrodinger operator. We discuss the relation between classical averaging and constructed quasi-modes. The behavior of quasi-modes in the neighborhood of the cusp is studied. We also discuss the relation between Bessel and Airy functions that follows from different representations of asymptotics near the cusp.

Can Signal Delay be Functional? Including Delay in Evolved Robot Controllers.

Engineers, control theorists, and neuroscientists often view the delay imposed by finite signal propagation velocities as a problem that needs to be compensated for or avoided. In this article, we consider the alternative possibility that in some cases, signal delay can be used functionally, that is, as an essential component of a cognitive system. To investigate this idea, we evolve a minimal robot controller to solve a basic stimulus-distinction task. The controller is constrained so that the solution must utilize a delayed recurrent signal. Different from previous evolutionary robotics studies, our controller is modeled using delay differential equations, which (unlike the ordinary differential equations of conventional continuous-time recurrent neural networks) can accurately capture delays in signal propagation. We analyze the evolved controller and its interaction with its environment using classical dynamical systems techniques. The analysis shows what kinds of invariant sets underlie the various successful and unsuccessful performances of the robot, and what kinds of bifurcations produce these invariant sets. In the second phase of our analysis, we turn our attention to the parameter θ, which describes the amount of signal delay included in the model. We show how the delay destabilizes certain attractors that would exist if there were no delay and creates other stable attractors, resulting in an agent that performs well at the target task.

Real numbers are the hidden variables of classical mechanics

Do scientific theories limit human knowledge? In other words, are there physical variables hidden by essence forever? We argue for negative answers and illustrate our point on chaotic classical dynamical systems. We emphasize parallels with quantum theory and conclude that the common real numbers are, de facto, the hidden variables of classical physics. Consequently, real numbers should not be considered as “physically real” and classical mechanics, like quantum physics, is indeterministic.

Qualitative control of undesired oscillations in a genetic negative feedback loop with uncertain measurements

In the context of gene regulatory networks, a negative feedback loop is modeled by N-coupled ordinary differential equations. The resulting system is highly non-linear due to the use of smooth Hill functions. This classical dynamical system properly captures the two main biological behaviors arising from this type of recurrent network motif: homeostasis under global stability of the unique fixed point, and biochemical oscillations otherwise. When homeostatic conditions are disrupted, undesired sustained oscillations can appear. In this context, a biologically relevant control strategy is designed in order to suppress these undesirable oscillations. As biological measurement techniques do not provide a quantitative knowledge of the system, the control law is chosen piecewise constant and dependent on specific regions of the state space. Moreover, due to biological devices inaccuracies, the measurements are considered uncertain leading to regions in which the control law is undefined. Under appropriate conditions on the control inputs, successive repelling regions of the state space are determined in order to prove the global convergence of the system towards an adjustable zone around the fixed point. These results are illustrated with the well-known p53-Mdm2 genetic feedback loop.

Indeterminism in Physics, Classical Chaos and Bohmian Mechanics: Are Real Numbers Really Real?

It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can’t contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant. Moreover, a better terminology for the so-called real numbers is “random numbers”, as their series of bits are truly random. I propose an alternative classical mechanics, which is empirically equivalent to classical mechanics, but uses only finite-information numbers. This alternative classical mechanics is non-deterministic, despite the use of deterministic equations, in a way similar to quantum theory. Interestingly, both alternative classical mechanics and quantum theories can be supplemented by additional variables in such a way that the supplemented theory is deterministic. Most physicists straightforwardly supplement classical theory with real numbers to which they attribute physical existence, while most physicists reject Bohmian mechanics as supplemented quantum theory, arguing that Bohmian positions have no physical reality.

Quantum state tomography across the exceptional point in a single dissipative qubit

Open systems with gain and loss, described by non-trace-preserving, non-Hermitian Hamiltonians, have been a subject of intense research recently. The effect of exceptional-point degeneracies on the dynamics of classical systems has been observed through remarkable phenomena such as the parity-time symmetry breaking transition, asymmetric mode switching, and optimal energy transfer. On the other hand, consequences of an exceptional point for quantum evolution and decoherence are hitherto unexplored. Here, we use post-selection on a three-level superconducting transmon circuit with tunable Rabi drive, dissipation, and detuning to carry out quantum state tomography of a single dissipative qubit in the vicinity of its exceptional point. Quantum state tomography reveals the PT symmetry breaking transition at zero detuning, decoherence enhancement at finite detuning, and a quantum signature of the exceptional point in the qubit relaxation state. Our observations demonstrate rich phenomena associated with non-Hermitian physics such as non-orthogonality of eigenstates in a fully quantum regime and open routes to explore and harness exceptional point degeneracies for enhanced sensing and quantum information processing.

Complete Integrability of Quantum and Classical Dynamical Systems

It is proved that the Schrodinger equation with any self-adjoint Hamiltonian is unitary equivalent to a set of non-interacting classical harmonic oscillators and in this sense any quantum dynamics is completely integrable. Integrals of motion are presented. A similar statement is proved for classical dynamical systems in terms of Koopman’s approach to dynamical systems. Examples of explicit reduction of quantum and classical dynamics to the family of harmonic oscillators by using direct methods of scattering theory and wave operators are given.

Stability Analysis of a Model for Phenol and Cresol Mixture Degradation

We propose a mathematical model for phenol and p-cresol mixture degradation in a continuously stirred bioreactor. The model is described by three nonlinear ordinary differential equations. The novel idea in the model design is the biomass specific growth rate, known as sum kinetics with interaction parameters (SKIP). We establish existence and uniform boundedness of positive solutions. Then we determine the equilibrium points of the model and study their asymptotic stability. The theoretical investigations are based on the classical dynamic systems theory. Numerical simulations are included to confirm and fill out the theoretical studies.

Vadere: An Open-Source Simulation Framework to Promote Interdisciplinary Understanding

Pedestrian dynamics is an interdisciplinary field of research. Psychologists, sociologists, traffic engineers, physicists, mathematicians and computer scientists all strive to understand the dynamics of a moving crowd. In principle, computer simulations offer means to further this understanding. Yet, unlike for many classic dynamical systems in physics, there is no universally accepted locomotion model for crowd dynamics. On the contrary, a multitude of approaches, with very different characteristics, compete. Often only the experts in one special model type are able to assess the consequences these characteristics have on a simulation study. Therefore, scientists from all disciplines who wish to use simulations to analyze pedestrian dynamics need a tool to compare competing approaches. Developers, too, would profit from an easy way to get insight into an alternative modeling ansatz. Vadere meets this interdisciplinary demand by offering an open-source simulation framework that is lightweight in its approach and in its user interface while offering pre-implemented versions of the most widely spread models.

ABOUT THE UNEXPECTED IN MATHIMATICS AND THE CAUSES OF ACCIDENTS/CATASTROPHES RELATED TO IT

Theoretical background. A number of researches claim that the classical theory of dynamic systems ignores spe-cial cases of incomplete equivalence of mathematic transformations descriptions. Sometimes it is even argued that (con-trary to a prevailing paradigm) the study of purely discriminatory polynomial of control system (the system of differen-tial equations) fails to guarantee the correct judgments about the parametrical stability and system’s stability factors as the probably wrong interpretation of stability may result in accidents and even catastrophes caused by a defectively designed object. Such conclusion obviously ensues from the fact that there are examples of the systems that have the same discriminatory polynomial but differ substantially in the parametrical stability and stability factors under the vari-Bulletin of Lviv State University of Li fe Safety, №19, 2019 35 able parameters. These researches are concerned about the fact that generally used packages of applied programs – for they usually require the equivalent in the classical sense consolidation of differential equations system to a single “standard” form – are not able to secure the veracity of dynamic systems computation and to guarantee the correctness of their characteristics analysis without the application of additional controlling subprograms. For example, there may exist the risks of stability losses in the initial system, however being brought to the differential equations of first order, as a common practice, these risk will become absolutely imperceptible, and, as a result, the source of dangerous casual-ties may occur – accidents and catastrophes in case of the system material embodiment. Thus it is categorically declared the necessity of substantial researches in correctness of the results of engineers and IT specialists and of relevant amendments of bachelors and masters degrees curriculum. The purpose of the research. Thus, it is natural that there is a necessity to find out whether the previously imper-ceptible risks of accidents and catastrophes do exist and whether the classical dynamic systems theory does not take into consideration the unexpected possibilities of its problems correctness losses as a result (in the process) of their equiva-lent transformations. The aim of this article is to substantiate the essence and content of this kind “discoveries”. The paper provides a comprehensive analysis of the system’s simple examples that are to prove the possible risks from the equivalent, in classical sense, transformations of mathematical descriptions.Results and discussion. It has been found out that after the equivalent transformations instability as well as incor-rectness in fact do not “hide”, they do not become invisible and untraceable. The researchers rather consciously do not pay attention to the possible substantial deformations of the system. For indeed, in case of the reduction of the system description to the form of the normal system of differential equations of first order the possibilities of the stability loss become invisible not because the transformations were nonequivalent but because the variability of the system order is not prognosticated, and, therefore, the treatments of initial (where the change of order is obvious) and final systems differ considerably. Here at, the controller equation — the defined first integral — is the manifestation of one more possible system order which cannot be ignored. Actually, much depends on how we define, see, read, interpret the ana-lytical description of a certain phenomenon or process. Different characteristic determinants that identify, materially, different dynamic systems may correspond to the same characteristic polynomial. The determinant may be consciously equivalently transformed (deformed), and any transformed (deformed) determinant will identify a new system. Thus, any transformation – is, without exaggeration, the creation of something new, something different. The process of solving simple linear differential equations with fixed factor and their variation with the aim of so-lutions stability or analytical descriptions correctness evaluation is reduced to the solution of a relevant algebraic prob-lem and the research of its properties and characteristics. Consequently, there is no point in expecting any enigmatic or dramatic unexpectancies when the research is sophisticated and profound. Conclusions. The characteristic determinant reflects the properties of any system more deeply than the character-istic polynomial does. Any equivalent transformations of the system are always visible in the structure of the determi-nant, even if they are not defined in its equation roots (zeroes). In the result of equivalent transformations there certainly emerges a new formation – it looks like the same system but with new properties (otherwise there will be no necessity in any transformations). The loss of robustness is treated as an unexpectancy occurring as a result of motivated defor-mation of the system which is easy prognosticated. Nonrobust systems could have their own perspective. Their exten-sive application is advancing.

The Three-body Problem in Riemannian Geometry. Hidden Irreversibility of the Classical Dynamical System

The classical three-body problem is formulated as a problem of geodesic flows on a Riemannian manifold. It is proved that a curved space allows to detect new hidden symmetries of the internal motion of a dynamical system and reduces the three-body problem to the system of 6th order. It is shown that the equivalence of the original Newtonian three-body problem and the developed representation provides coordinate transformations together with an underdetermined system of algebraic equations. The latter makes the system of geodesic equations relative to the evolution parameter (internal time), i.e. to the arc length of the geodesic curve, irreversible.

Bifurcation and anomalous spectral accumulation in an oval billiard

Abstract The spectral statistics of a quantum oval billiard whose classical dynamical system shows bifurcations is numerically investigated in terms of the two-point correlation function (TPCF), which is defined as the probability density of finding two levels at a specific energy interval. The eigenenergy levels at the bifurcation point are found to show anomalous accumulation, which is observed as a periodic spike oscillation of the TPCF. We analyzed the eigenfunctions localizing onto the various classical trajectories in the phase space and found that the oscillation is supplied from a limited region in the phase space that contains the bifurcating orbit. We also show that the period of the oscillation is in good agreement with the period of a contribution from the bifurcating orbit to the semiclassical TPCF obtained by Gutzwiller’s trace formula [J. Math. Phys. 12, 343 (1971)].

Learning nonlinear input–output maps with dissipative quantum systems

In this paper, we develop a theory of learning nonlinear input-output maps with fading memory by dissipative quantum systems, as a quantum counterpart of the theory of approximating such maps using classical dynamical systems. The theory identifies the properties required for a class of dissipative quantum systems to be {\em universal}, in that any input-output map with fading memory can be approximated arbitrarily closely by an element of this class. We then introduce an example class of dissipative quantum systems that is provably universal. Numerical experiments illustrate that with a small number of qubits, this class can achieve comparable performance to classical learning schemes with a large number of tunable parameters. Further numerical analysis suggests that the exponentially increasing Hilbert space presents a potential resource for dissipative quantum systems to surpass classical learning schemes for input-output maps.

On the nonlinearity of quantum dynamical entropy

Linearity of a dynamical entropy means that the dynamical entropy of the n-fold composition of a dynamical map with itself is equal to n times the dynamical entropy of the map for every positive integer n. We show that the quantum dynamical entropy introduced by Slomczynski and Zyczkowski is nonlinear in the time interval between successive measurements of a quantum dynamical system. This is in contrast to Kolmogorov-Sinai dynamical entropy for classical dynamical systems, which is linear in time. We also compute the exact values of quantum dynamical entropy for the Hadamard walk with varying Luders-von Neumann instruments and partitions.

Mathematical modelling and analysis of love dynamics: A fractional approach

A range of new fractional-order love dynamics is modelled in this paper using the Caputo (power-law), Caputo–Fabrizio (exponential decay-law) and Atangana–Baleanu (Mittag-Leffler-law) fractional operators. We utilize the new fractional Adams–Bashforth schemes for the approximation of these derivatives. This method was developed with the standard differentiation technique by applying the fundamental theorem of calculus and taking the difference of two times levels at t=(tn,tn+1). The stability analysis applies the theory of classical order differential equations and dynamical system. The Existence and uniqueness of solution of fractional dynamics is proved by adopting the fixed-point theorem. Numerical results presented for various α− values justify the theoretical findings. It was observed that modelling of interpersonal and romantic love affairs with fractional derivative could exhibit some strange emotional attractors.

Thermal Emission from Semiclassical Dynamical Systems.

Recently the bound on the Lyapunov exponent λ_{L}≤2πT/ℏ in thermal quantum systems was conjectured by Maldacena, Shenker, and Stanford. If we naïvely apply this bound to a system with a fixed Lyapunov exponent λ_{L}, it might predict the existence of the lower bound on temperature T≥ℏλ_{L}/2π. Particularly, it might mean that chaotic systems cannot be zero temperature quantum mechanically. Even classical dynamical systems, which are deterministic, might exhibit thermal behaviors once we turn on quantum corrections. We elaborate this possibility by investigating semiclassical particle motions near the hyperbolic fixed point and show that indeed quantum corrections may induce energy emission, which obeys a Boltzmann distribution. We also argue that this emission is related to acoustic Hawking radiation in quantum fluid. Besides, we discuss when the bound is saturated, and show that a particle motion in an inverse harmonic potential and c=1 matrix model may saturate the bound, although they are integrable.

Model selection for hybrid dynamical systems via sparse regression.

Hybrid systems are traditionally difficult to identify and analyse using classical dynamical systems theory. Moreover, recently developed model identification methodologies largely focus on identifying a single set of governing equations solely from measurement data. In this article, we develop a new methodology, Hybrid-Sparse Identification of Nonlinear Dynamics, which identifies separate nonlinear dynamical regimes, employs information theory to manage uncertainty and characterizes switching behaviour. Specifically, we use the nonlinear geometry of data collected from a complex system to construct a set of coordinates based on measurement data and augmented variables. Clustering the data in these measurement-based coordinates enables the identification of nonlinear hybrid systems. This methodology broadly empowers nonlinear system identification without constraining the data locally in time and has direct connections to hybrid systems theory. We demonstrate the success of this method on numerical examples including a mass–spring hopping model and an infectious disease model. Characterizing complex systems that switch between dynamic behaviours is integral to overcoming modern challenges such as eradication of infectious diseases, the design of efficient legged robots and the protection of cyber infrastructures.