Interesting Dynamical Systems Research Articles

On control intervals of chaos with Fréchet derivative of certain iterations as discrete dynamical system in Banach spaces

Dynamical systems are one of the interesting concepts where iteration algorithms and chaos can be considered together. In iteration algorithms, one of the basic concepts of fixed point theory, it is well known that the behavior of the iteration mechanism is chaotic if the original transformation is taken as chaotic. One of the natural ways to transform a chaotic system into a dynamical system is through control mechanisms. In this paper, we first consider an iteration class defined on Banach spaces, which is prominent in the literature in terms of both speed and convergence rate. Then, we consider the transformation constituting the iteration class as chaotic and obtain the stability and unstability behaviors of the iterations according to the operator norm by using Gâteaux and Fréchet derivatives representing the direction‐dependent derivative. In this way, we aimed to obtain the chaos control intervals obtained with functions in real space with the help of operators in Banach spaces. Using the operator norm obtained with the Fréchet derivative, we derive interesting dynamical system intervals from a chaotic system with parameter variables of iteration algorithms. It is noteworthy that the parameter variables of the studied iteration classes are the same in some transformation classes. In addition, analytical proofs are followed by computer simulations of parameter‐dependent control intervals considering the logistic operator with a chaotic structure. At the same time, the periodic behavior of the iteration algorithms used in our study is illustrated by the Lyapunov exponent with parameter‐dependent control intervals according to operator norm. Finally, as a real‐life problem, it has been shown that the chaos of the logistic population growth model with dynamic features can be controlled by chaos control mechanisms established by fixed point iteration methods.

Verification of Diagnosability for Cyber-Physical Systems via Hybrid Barrier Certificates

In this paper, we provide an automata-based framework for verifying diagnosability property of Cyber-Physical Systems leveraging a notion of so-called hybrid barrier certificates. Concretely, we first construct a so-called (δ,K)-deterministic finite automata ((δ,K)-DFA) associated with the desired diagnosability property, which captures the occurrence of the fault to be diagnosed. Having a (δ,K)-DFA, we show that the verification of diagnosability properties is equivalent to a safety verification problem over a product system between this DFA and the dynamical system of interest. We further show that such a verification problem can be solved via computing hybrid barrier certificates for the product system. To compute the hybrid barrier certificates, we provide a systematic technique leveraging a counter-example guided inductive synthesis framework. Finally, we showcase the effectiveness of our results through a case study.

An Innate Moving Frame on Parametric Surfaces: The Dynamics of Principal Singular Curves

This article reports an experimental work that unveils some interesting yet unknown phenomena underneath all smooth nonlinear maps. The findings are based on the fact that, generalizing the conventional gradient dynamics, the right singular vectors of the Jacobian matrix of any differentiable map point in directions that are most pertinent to the infinitesimal deformation of the underlying function and that the singular values measure the rate of deformation in the corresponding directions. A continuous adaption of these singular vectors, therefore, constitutes a natural moving frame that carries indwelling information of the variation. This structure exists in any dimensional space, but the development of the fundamental theory and algorithm for surface exploration is an important first step for immediate application and further generalization. In this case, trajectories of these singular vectors, referred to as singular curves, unveil some intriguing patterns per the given function. At points where singular values coalesce, curious and complex behaviors occur, manifesting specific landmarks for the function. Upon analyzing the dynamics, it is discovered that there is a remarkably simple and universal structure underneath all smooth two-parameter maps. This work delineates graphs with this interesting dynamical system and the possible new discovery that, analogous to the double helix with two base parings in DNA, two strands of critical curves and eight base pairings could encode properties of a generic and arbitrary surface. This innate structure suggests that this approach could provide a unifying paradigm in functional genetics, where all smooth surfaces could be genome-sequenced and classified accordingly. Such a concept has sparked curiosity and warrants further investigation.

New role of null lagrangians in derivation of equations of motion for dynamical systems

The space of null Lagrangians is the least investigated territory in dynamics as these Lagrangians are identically sent to zero by their Euler–Lagrange operator, and thereby they are having no effects on equations of motion. A procedure that significantly generalizes the previous work, which appeared in (2022, Physica Scripta 97, 125213), is developed and used to construct null Lagrangians and then the corresponding non-standard Lagrangians, which represent a range of interesting dynamical systems. By using the generalized procedure, derivation of equations of motion for a harmonic oscillator as well as for the Bateman and Duffing oscillators is presented. The obtained results demonstrate a new role played by the null Lagrangians and their corresponding non-standard Lagrangians in describing linear and nonlinear, and dissipative and non-dissipative dynamical systems.

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.

An efficient ensemble Kalman Filter implementation via shrinkage covariance matrix estimation: exploiting prior knowledge

In this paper, we propose an efficient and practical implementation of the ensemble Kalman filter via shrinkage covariance matrix estimation. Our filter implementation combines information brought by an ensemble of model realizations, and that based on our prior knowledge about the dynamical system of interest. We perform the combination of both sources of information via optimal shrinkage factors. The method exploits the rank-deficiency of ensemble covariance matrices to provide an efficient and practical implementation of the analysis step in EnKF based formulations. Localization and inflation aspects are discussed, as well. Experimental tests are performed to assess the accuracy of our proposed filter implementation by employing an Advection Diffusion Model and an Atmospheric General Circulation Model. The experimental results reveal that the use of our proposed filter implementation can mitigate the impact of sampling noise, and even more, it can avoid the impact of spurious correlations during assimilation steps.

Visualising mechanics: washing machine dynamics

Mechanical engineering students often find the formulation and analysis of dynamical systems difficult. The response of some mechanical engineering undergraduates is that as much as possible courses on mechanics are best avoided. The aim of this paper is to produce some interesting dynamical systems that may help to change the opinions of the students. A dynamic model for the washing in a washing machine and the washing machine are presented. At mid-range, spin speeds the washing machine slides across the ground and at higher spin speeds; the washing machine both slides and rocks from side to side. The models are formulated using concepts taught in the early years of an undergraduate course in mechanics and where appropriate simplifying assumptions are made to ensure it is a realistic goal for undergraduate students to understand the model basis. This allows a student’s focus to be on the influence of different parameters on the system dynamics. The simulation tool could be used as a dynamics demonstrator in a lecture or as an educational tool driven by the imagination of the student. In this paper, several simulations are presented and analysed.

Numerical integration in Celestial Mechanics: a case for contact geometry

Several dynamical systems of interest in celestial mechanics can be written in the form of a Newton equation with time-dependent damping, linear in the velocities. For instance, the modified Kepler problem, the spin-orbit model and the Lane-Emden equation all belong to such class. In this work we start an investigation of these models from the point of view of contact geometry. In particular we focus on the (contact) Hamiltonisation of these models and on the construction of the corresponding geometric integrators.

Numerical dynamics of integrodifference equations: Basics and discretization errors in a C0-setting

Besides being interesting infinite-dimensional dynamical systems in discrete time, integrodifference equations successfully model growth and dispersal of populations with nonoverlapping generations, and are often illustrated by simulations. This paper points towards and initiates a mathematical foundation of such simulations using generic methods to numerically discretize (and solve) integral equations. We tackle basic properties of a flexible class of integrodifference equations, as well as of their collocation and degenerate kernel semi-discretizations on the state space of continuous functions over a compact domain. Moreover, various estimates for the global discretization error are provided. Numerical simulations illustrate and confirm our theoretical results.

Membrane shape-mediated wave propagation of cortical protein dynamics

Immune cells exhibit stimulation-dependent traveling waves in the cortex, much faster than typical cortical actin waves. These waves reflect rhythmic assembly of both actin machinery and peripheral membrane proteins such as F-BAR domain-containing proteins. Combining theory and experiments, we develop a mechanochemical feedback model involving membrane shape changes and F-BAR proteins that render the cortex an interesting dynamical system. We show that such cortical dynamics manifests itself as ultrafast traveling waves of cortical proteins, in which the curvature sensitivity-driven feedback always constrains protein lateral diffusion in wave propagation. The resulting protein wave propagation mainly reflects the spatial gradient in the timing of local protein recruitment from cytoplasm. We provide evidence that membrane undulations accompany these protein waves and potentiate their propagation. Therefore, membrane shape change and protein curvature sensitivity may have underappreciated roles in setting high-speed cortical signal transduction rhythms.

Approaches for Creation and Evaluation of Computationally Efficient Thermofluid System Models

Useful simulation of complex systems for real-time or otherwise computationally constrained applications often requires the creation of simplified system models from existing analytical models and data. In this work, the creation of candidate models through machine learning and heuristic-informed model simplification methods are explored, and the resulting candidate models are evaluated and compared through an optimal experiment design process. The dynamic system of interest is a counterflow air-to-water heat exchanger.

Chaotic Oscillation via Edge of Chaos Criteria

In this paper, we show that nonlinear dynamical systems which satisfy the edge of chaos criteria can bifurcate from a stable equilibrium point regime to a chaotic regime by periodic forcing. That is, the edge of chaos criteria can be exploited to engineer a phase transition from ordered to chaotic behavior. The frequency of the periodic forcing can be derived from this criteria. In order to generate a periodic or a chaotic oscillation, we have to tune the amplitude of the periodic forcing. For example, we engineer chaotic oscillations in the generalized Duffing oscillator, the FitzHugh–Nagumo model, the Hodgkin–Huxley model, and the Morris–Lecar model. Although forced oscillators can exhibit chaotic oscillations even if the edge of chaos criteria is not satisfied, our computer simulations show that forced oscillators satisfying the edge of chaos criteria can exhibit highly complex chaotic behaviors, such as folding loci, strong spiral dynamics, or tight compressing dynamics. In order to view these behaviors, we used high-dimensional Poincaré maps and coordinate transformations. We also show that interesting nonlinear dynamical systems can be synthesized by applying the edge of chaos criteria. They are globally stable without forcing, that is, all trajectories converge to an asymptotically-stable equilibrium point. However, if we apply a forcing signal, then the dynamical systems can oscillate chaotically. Furthermore, the average power delivered from the forced signal is not dissipated by chaotic oscillations, but on the contrary, energy can be generated via chaotic oscillations, powered by locally-active circuit elements inside the one-port circuit [Formula: see text] connected across a current source.

Mid-Infrared Gas Sensor Based on Mutually Injection Locked Quantum Cascade Lasers

We present experimental investigations of a novel mid-infrared gas sensor relying on a laser system consisting of two quantum cascade sources mutually coupled with each other. The free-space path between the two lasers contains the sensing cell which is filled with the gas of interest. This laser system proves to be a very interesting dynamical system which can offer enhanced detectivity when compared to a conventional direct absorption spectrometer utilizing a laser, the gas cell, and a detector. We present comprehensive investigations of the operation regimes under which the laser system behaves as an efficient sensor and demonstrate its dependence on critical parameters such as the coupling strength between the two lasers and their bias currents. The experimental study is supported by numerical simulations, which provide a theoretical understanding of the underlying mechanisms. A six fold improvement of the sensitivity in a N2O absorption spectroscopy experiment is demonstrated in comparison to a conventional direct absorption spectrometer. This sensitivity improvement is exhibited for a wide range of N2O concentrations.

A Statically Unstable Passive Hopper: Design Evolution

I have designed a sequence of gravity-powered passive-dynamic toys. These explore locomotion in general and hopping in particular. As with walking, running, crawling, etc., for animals, locomotion in these devices is a horizontal translation by means of approximately periodic patterns of motion. These toys were developed using intuitively guided trial-and-error design iteration based on live viewing, sound sequences, and review of slow motion video. A series of statically stable mechanisms is described. A progression of designs led to the central result: a monopod hopper that repeatedly hops more than 70 steps down a ramp, without conventional feedback control, fast spinning parts, or sensing means, yet unlike the previously statically stable designs, it cannot stand still stably. This free hopping was facilitated by a special mass distribution, and a spring that allowed relative translation and rotation between the body and leg. A retrospective evaluation reveals similarities to the morphology and gaits of hopping bipeds. These toys, interesting dynamical systems in any case, highlight the possibility of a significant role of mechanical structure in locomotion.

Escape and collision dynamics in the planar equilateral restricted four-body problem

We consider the planar circular equilateral restricted four body-problem where a test particle of infinitesimal mass is moving under the gravitational attraction of three primary bodies which move on circular orbits around their common center of gravity, such that their configuration is always an equilateral triangle. The case where all three primaries have equal masses is numerically investigated. A thorough numerical analysis takes place in the configuration (x,y) as well as in the (x,y) space in which we classify initial conditions of orbits into four main categories: (i) bounded regular orbits, (ii) trapped chaotic orbits, (iii) escaping orbits and (iv) collision orbits. Interpreting the collision motion as leaking in the phase space we related our results to both chaotic scattering and the theory of leaking Hamiltonian systems. We successfully located the escape and the collision basins and we managed to correlate them with the corresponding escape and collision times of orbits. We hope our contribution to be useful for a further understanding of the escape and collision properties of motion in this interesting dynamical system.

Investigation of bearing clearance effects in dynamics of turbochargers

Turbochargers are modern and very interesting dynamical systems used in various engines in order to increase their power. They are operated at hundreds of thousands revolutions per minute and tend to be fatigue and stability prone. For this reason, a turbocharger rotor has to be designed carefully with respect to its dynamic properties. The following article deals with effects of radial bearing clearances on the dynamical response of the turbocharger rotor. The influence of a bearing clearance on stiffness and damping of a single-film journal bearing is well known and documented. Turbochargers, however, are often supported by floating ring bearings. Such bearings have two bearing clearances—between a journal and a floating ring and between a floating ring and a housing—which are determined by different temperatures of oil films. Turbocharger analysed in the article is modelled by means of flexible multibody dynamics approaches. Bearings’ behaviour is described using Reynolds equation, which is solved numerically. It is shown, but not mathematically proved, how the outer clearance and the ratio between the inner and the outer clearance affect amplitudes of sub-synchronous components in rotor's response.

Preface

Stellar mass loss is now widely recognized to have a significant impact on stellar evolution. Mass loss on the asymptotic giant branch (AGB) allows stars with initial masses under 8 solar masses to avoid the fate of going supernovae. Over 95% of stars in our Galaxy will evolve through the planetary nebulae phase to end up as white dwarfs instead of neutron stars or black holes. Massive stars undergo mass loss both in the blue and red phases of evolution and create new classes of stars such as Wolf-Rayet stars and luminous blue variables. The circumstellar matter ejected by these mass loss processes becomes new laboratories to study the physical and chemical processes of interstellar matter. The interaction between different phases of mass loss (with variable mass loss rates, ejection speeds, and directions) leads to spectacular morphological transformation of the circumstellar nebulae. The circumstellar nebulae are also sites of molecular and solid-state synthesis. Close to 100 molecular species and a variety of solids, including minerals and complex organics, have been detected in circumstellar envelopes. Since the dynamical time scale of the ejection puts an upper limit on the chemical time scale, we are witnessing a rapid synthesis of chemical species in an extremely low-density environment, creating new challenges to our understanding of chemical reactions.Effects of mass loss are not limited to single stars. Mass loss by one component of a binary system allows mass transfer to occur at separations beyond the Roche Lobe limit. Accreted wind materials on the surface of a degenerate star can lead to periodic outbursts through H-shell burning. When both components are losing mass, we have interesting dynamical systems such as symbiotic novae.The theme of this conference is “Physics and Chemistry of the Late Stages of Stellar Evolution”. We try to bring together experts in different fields to exchange ideas in the hope of solving the many unsolved problems in the late stages of stellar evolution. The size of the conference was limited to about 100 participants to allow for sufficient time for discussions and personal exchanges of opinions. Although the conference is part of the Pacific Rim series, astronomers from Asia, Europe, North and South America participated in the conference. We hope that these proceedings will serve as a record of our current understanding of the late stages of stellar evolution and be a useful resource for future generation of scientists working in the field.

Role of Computational Modeling in Understanding Cell Cycle Oscillators.

The periodic oscillations in the activity of the cell cycle regulatory program, drives the timely activation of key cell cycle events. Interesting dynamical systems, such as oscillators, have been investigated by various theoretical and computational modeling methods. Thanks to the insights achieved by these modeling efforts we have gained considerable insights about the underlying molecular regulatory networks that can drive cell cycle oscillations. Here we review the basic features and characteristics of biological oscillators, discussing from a computational modeling point of view their specific architectures and the current knowledge about the dynamics that the life evolution selected to drive cell cycle oscillations.

Integrable quantum hydrodynamics in two-dimensional phase space

Quantum liquids in two dimensions represent interesting dynamical quantum systems for several reasons, among them the possibility of the existence of infinite hidden symmetries, such as conformal symmetry or the symmetry associated with area preserving diffeomorphisms. It is known that when the symmetry algebra is large enough, symmetry may fully prescribe the dynamics. However, the way this is borne out in two-dimensional hydrodynamics, both classical and quantum, is not fully understood. Here we take a step towards clarifying this issue, by focusing on a particular example, namely that of a two-dimensional phase space liquid which emerges when one considers the Calogero model, a many-body one-dimensional system interacting through an inverse square law potential. We demonstrate how the symmetry algebra of conserved quantities of the one-dimensional system is expressed in terms of the incompressible Euler hydrodynamics of point vortices in phase space. Due to a formal relation between quantum hydrodynamics and classical stochastic hydrodynamics, which is inherent in the method of stochastic quantization, the ideas and methods developed here may also have application in the study of stochastic classical hydrodynamics, after suitable modifications.

Squirals and beyond: substitution tilings with singular continuous spectrum

AbstractThe squiral inflation rule is equivalent to a bijective block substitution rule and leads to an interesting lattice dynamical system under the action of${ \mathbb{Z} }^{2} $. In particular, its balanced version has purely singular continuous diffraction. The dynamical spectrum is of mixed type, with pure point and singular continuous components. We present a constructive proof that admits a generalization to bijective block substitutions of trivial height on${ \mathbb{Z} }^{d} $.