Class Of Dynamical Systems Research Articles

Bilinear Approximate Model-Based Robust Lyapunov Control for Parabolic Distributed Collectors

This brief addresses the control problem of distributed parabolic solar collectors in order to maintain the field outlet temperature around a desired level. The objective is to design an efficient controller to force the outlet fluid temperature to track a set reference despite the unpredictable varying working conditions. In this brief, a bilinear model-based robust Lyapunov control is proposed to achieve the control objectives with robustness to the environmental changes. The bilinear model is a reduced order approximate representation of the solar collector, which is derived from the hyperbolic distributed equation describing the heat transport dynamics by means of a dynamical Gaussian interpolation. Using the bilinear approximate model, a robust control strategy is designed applying Lyapunov stability theory combined with a phenomenological representation of the system in order to stabilize the tracking error. On the basis of the error analysis, simulation results show good performance of the proposed controller, in terms of tracking accuracy and convergence time, with limited measurement even under unfavorable working conditions. Furthermore, the presented work is of interest for a large category of dynamical systems knowing that the solar collector is representative of physical systems involving transport phenomena constrained by unknown external disturbances.

Quantum Entropy and Complexity

We study the relations between the recently proposed machine-independent quantum complexity of P. Gacs [1] and the entropy of classical and quantum systems. On one hand, by restricting Gacs complexity to ergodic classical dynamical systems, we retrieve the equality between the Kolmogorov complexity rate and the Shannon entropy rate derived by A. A. Brudno [2]. On the other hand, using the quantum Shannon-McMillan theorem [3], we show that such an equality holds densely in the case of ergodic quantum spin chains.

Microscopic statistical description of incompressible Navier-Stokes granular fluids

Based on the recently established Master kinetic equation and related Master constant H-theorem which describe the statistical behavior of the Boltzmann-Sinai classical dynamical system for smooth and hard spherical particles, the problem is posed of determining a microscopic statistical description holding for an incompressible Navier-Stokes fluid. The goal is reached by introducing a suitable mean-field interaction in the Master kinetic equation. The resulting Modified Master Kinetic Equation (MMKE) is proved to warrant at the same time the condition of mass-density incompressibility and the validity of the Navier-Stokes fluid equation. In addition, it is shown that the conservation of the Boltzmann-Shannon entropy can similarly be warranted. Applications to the plane Couette and Poiseuille flows are considered showing that they can be regarded as final decaying states for suitable non-stationary flows. As a result, it is shown that an arbitrary initial stochastic 1-body PDF evolving in time by means of MMKE necessarily exhibits the phenomenon of Decay to Kinetic Equilibrium (DKE), whereby the same 1-body PDF asymptotically relaxes to a stationary and spatially uniform Maxwellian PDF.

Gross-Pitaevski map as a chaotic dynamical system.

The Gross-Pitaevski map is a discrete time, split-operator version of the Gross-Pitaevski dynamics in the circle, for which exponential instability has been recently reported. Here it is studied as a classical dynamical system in its own right. A systematic analysis of Lyapunov exponents exposes strongly chaotic behavior. Exponential growth of energy is then shown to be a direct consequence of rotational invariance and for stationary solutions the full spectrum of Lyapunov exponents is analytically computed. The present analysis includes the "resonant" case, when the free rotation period is commensurate to 2π, and the map has countably many constants of the motion. Except for lowest-order resonances, this case exhibits an integrable-chaotic transition.

Multi-hump solutions with small oscillations at infinity for stationary Swift–Hohenberg equation

The paper considers the stationary Swift–Hohenberg equationcw−(∂x2+k02)2w−w3=0,where c > 0 is a constant, , and is a small parameter. In this case, the linear operator has a pair of real eigenvalues and a pair of purely imaginary eigenvalues. It can be proved that the equation has homoclinic (or single hump) solutions approaching to periodic solutions as (called single-hump generalized homoclinic solutions). This paper provides the first rigorous proof of existence of homoclinic solutions with two humps which tend to periodic solutions at infinity (or two-hump generalized homoclinic solutions) by pasting two appropriate single-hump generalized homoclinic solutions together. The dynamical system approach is used to reformulate the problem into a classical dynamical system problem and then the solution is decomposed into a decaying part and an oscillatory part at positive infinity. By adjusting some free constants and modifying the single-hump generalized homoclinic solution near negative infinity, it is shown that the solution is reversible with respect to a point near negative infinity. Therefore, the translational invariant and reversibility properties of the system yield a two-hump generalized homoclinic solution. The method may be applied to prove the existence of 2k-hump solutions for any positive integer k.

Hybrid simulation theory for a classical nonlinear dynamical system

Hybrid simulation is an experimental and computational technique which allows one to study the time evolution of a system by physically testing a subset of it while the remainder is represented by a numerical model that is attached to the physical portion via sensors and actuators. The technique allows one to study large or complicated mechanical systems while only requiring a subset of the complete system to be present in the laboratory. This results in vast cost savings as well as the ability to study systems that simply can not be tested due to scale. However, the errors that arise from splitting the system in two requires careful attention, if a valid simulation is to be guaranteed. To date, efforts to understand the theoretical limitations of hybrid simulation have been restricted to linear dynamical systems. In this work we consider the behavior of hybrid simulation when applied to nonlinear dynamical systems. As a model problem, we focus on the damped, harmonically-driven nonlinear pendulum. This system offers complex nonlinear characteristics, in particular periodic and chaotic motions. We are able to show that the application of hybrid simulation to nonlinear systems requires a careful understanding of what one expects from such an experiment. In particular, when system response is chaotic we advocate the need for the use of multiple metrics to characterize the difference between two chaotic systems via Lyapunov exponents and Lyapunov dimensions, as well as correlation exponents. When system response is periodic we advocate the use of L2 norms. Further, we are able to show that hybrid simulation can falsely predict chaotic or periodic response when the true system has the opposite characteristic. In certain cases, we are able to show that control system parameters can mitigate this issue.

Parameter Identification of a Chaotic Circuit with a Hidden Attractor Using Krill Herd Optimization

Parameter estimation plays an important role in modeling and system identification. However, parameter estimation of chaotic systems has some basic differences with other dynamical systems due to butterfly effect. In this paper, we apply a new cost function for parameter estimation in a very interesting chaotic system, a system with a plane of equilibrium which belongs to a newly introduced category of dynamical systems: systems with hidden attractor. The nonlinear dynamics of this system is described in terms of equilibria and its stability, phase portraits, bifurcation diagram and Lyapunov exponents. In order to minimize the proposed cost function and obtain the correct parameters, we use a new efficient optimization method, Krill Herd algorithm. The results show the success of proposed procedures.

Störmer problem restricted to a spherical surface and the Euler and Lagrange tops

In a recent work, Cortés and Poza (2015 Eur. J. Phys. 36 055009) analysed, in full, the dynamics of a charged particle in the field of a magnetic dipole restricted to a spherical surface with the dipole at its centre. This model can be considered as the classical non-relativistic Störmer problem on a sphere. Here, we started from a Lagrangian approach: we derived the Hamilton equations of motion and observed that in this restricted case the equations can be reduced to quadratures, and they were integrated numerically. From the Hamiltonian function we found, for the polar angle, an equivalent one-dimensional system of a particle in the presence of an effective potential. In the present work we start from a change of variable to the cosine of the polar angle. In terms of this variable we obtain an equation that turns out to be the same as the one of a particle in a quartic potential. Then, we can actually solve the equations of motion for the polar angle using Jacobi elliptic functions, and for the azimuthal angle we use the same integrals which were expressed by Jacobi in terms of theta functions, both in the Euler and Lagrange tops. In this restricted Störmer problem, the student at undergraduate or graduate level will have a good example of an integrable nonlinear physical system in which, after analysis of its complex dynamics, one can obtain an analytical solution by means of some special functions of mathematical physics. Additionally, one discovers that the equations of motion of this restricted case of a charge in a magnetic dipole field have the same mathematical structure as the corresponding equations of other well known integrable classical dynamical systems.

Critical dynamics of classical systems under slow quench

We study the slow quench dynamics of a one-dimensional nonequilibrium lattice gas model which exhibits a phase transition in the stationary state between a fluid phase with homogeneously distributed particles and a jammed phase with a macroscopic hole cluster. Our main result is that in the critical region (i.e., at the critical point and in its vicinity) where the dynamics are assumed to be frozen in the standard Kibble-Zurek argument, the defect density exhibits an algebraic decay in the inverse annealing rate with an exponent that can be understood using critical coarsening dynamics. However, in a part of the critical region in the fluid phase, the standard Kibble-Zurek scaling holds. We also find that when the slow quench occurs deep into the jammed phase, the defect density behavior is explained by the rapid quench dynamics in this phase.

Complex Activity Recognition Via Attribute Dynamics

The problem of modeling the dynamic structure of human activities is considered. Video is mapped to a semantic feature space, which encodes activity attribute probabilities over time. The binary dynamic system (BDS) model is proposed to jointly learn the distribution and dynamics of activities in this space. This is a non-linear dynamic system that combines binary observation variables and a hidden Gauss---Markov state process, extending both binary principal component analysis and the classical linear dynamic systems. A BDS learning algorithm, inspired by the popular dynamic texture, and a dissimilarity measure between BDSs, which generalizes the Binet---Cauchy kernel, are introduced. To enable the recognition of highly non-stationary activities, the BDS is embedded in a bag of words. An algorithm is introduced for learning a BDS codebook, enabling the use of the BDS as a visual word for attribute dynamics (WAD). Short-term video segments are then quantized with a WAD codebook, allowing the representation of video as a bag-of-words for attribute dynamics. Video sequences are finally encoded as vectors of locally aggregated descriptors, which summarize the first moments of video snippets on the BDS manifold. Experiments show that this representation achieves state-of-the-art performance on the tasks of complex activity recognition and event identification.

A hybrid systems model for simple manipulation and self-manipulation systems

Rigid bodies, plastic impact, persistent contact, Coulomb friction, and massless limbs are ubiquitous simplifications introduced to reduce the complexity of mechanics models despite the obvious physical inaccuracies that each incurs individually. In concert, it is well known that the interaction of such idealized approximations can lead to conflicting and even paradoxical results. As robotics modeling moves from the consideration of isolated behaviors to the analysis of tasks requiring their composition, a mathematically tractable framework for building models that combine these simple approximations yet achieve reliable results is overdue. In this paper we present a formal hybrid dynamical system model that introduces suitably restricted compositions of these familiar abstractions with the guarantee of consistency analogous to global existence and uniqueness in classical dynamical systems. The hybrid system developed here provides a discontinuous but self-consistent approximation to the continuous (though possibly very stiff and fast) dynamics of a physical robot undergoing intermittent impacts. The modeling choices sacrifice some quantitative numerical efficiencies while maintaining qualitatively correct and analytically tractable results with consistency guarantees promoting their use in formal reasoning about mechanism, feedback control, and behavior design in robots that make and break contact with their environment.

Hamiltonian Structure of the Schrödinger Classical Dynamical System

The connection between quantum mechanics and classical statistical mechanics has motivated in the past the representation of the Schrodinger quantum-wave equation in terms of “projections” onto the quantum configuration space of suitable phase-space asymptotic kinetic models. This feature has suggested the search of a possible exact super-dimensional classical dynamical system (CDS), denoted as Schrodinger CDS, which uniquely determines the time-evolution of the underlying quantum state describing a set of N like and mutually interacting quantum particles. In this paper the realization of the same CDS in terms of a coupled set of Hamiltonian systems is established. These are respectively associated with a quantum-hydrodynamic CDS advancing in time the quantum fluid velocity and a further one the RD-CDS, describing the relative dynamics with respect to the quantum fluid.

On impulsive semidynamical systems: minimal, recurrent and almost periodic motions

This paper concerns results about minimal, recurrent and almost periodic motions in impulsive semidynamical systems. In the first part, we investigate general properties of minimal sets. In the sequel, we study some relations among minimal, recurrent and almost periodic motions. Some important results from the classical dynamical systems theory are generalized to the impulsive case, as Birkhoff's theorem for instance.

Mixing of discontinuously deforming media.

Mixing of materials is fundamental to many natural phenomena and engineering applications. The presence of discontinuous deformations-such as shear banding or wall slip-creates new mechanisms for mixing and transport beyond those predicted by classical dynamical systems theory. Here, we show how a novel mixing mechanism combining stretching with cutting and shuffling yields exponential mixing rates, quantified by a positive Lyapunov exponent, an impossibility for systems with cutting and shuffling alone or bounded systems with stretching alone, and demonstrate it in a fluid flow. While dynamical systems theory provides a framework for understanding mixing in smoothly deforming media, a theory of discontinuous mixing is yet to be fully developed. New methods are needed to systematize, explain, and extrapolate measurements on systems with discontinuous deformations. Here, we investigate "webs" of Lagrangian discontinuities and show that they provide a template for the overall transport dynamics. Considering slip deformations as the asymptotic limit of increasingly localised smooth shear, we also demonstrate exactly how some of the new structures introduced by discontinuous deformations are analogous to structures in smoothly deforming systems.

Tracking trajectory in the workspace of rigid manipulators using distributed adaptive control strategy

This paper discusses the tracking trajectory in the workspace of rigid manipulators using distributed adaptive control strategy. This control strategy consists of two steps; first, the classical MIMO dynamical system is decomposed into a set of nonlinear interconnected subsystems. Each subsystem has one joint. Second, the distributed adaptive control strategy is introduced. This control strategy consists of controlling the last subsystem while assuming that the remaining subsystems are stable. Then, going backward to the second last subsystem, the same strategy is applied and so on until the first one. The system parameters are assumed to be unknown. An adaptive control is used to estimate these parameters. Indeed, the unknown parameters existing in the equation of motion of the last subsystem are first estimated and the control law is developed based on these estimated parameters. Then, going backward to the before last joint, the control law is developed using its own estimated parameters and the ones already estimated in the upper level subsystem. Asymptotical stability of the error dynamics is proved using Lyapunov approach. The developed algorithm is experimented on a 4 DOF hyper redundant articulated nimble adaptable trunk robot and compared with the classical computed torque approach. Good tracking in the workspace and joint space is obtained and effectiveness of the results is shown.

A novel method for detecting abrupt dynamic change based on the changing Hurst exponent of spatial images

The climate system is a classical spatiotemporal evolutionary dynamic system with spatiotemporal correlation characteristics. Based on this, two-dimensional detrended fluctuation analysis (TD-DFA) is used to estimate the Hurst exponent of two-dimensional images. Then, we monitored the change of the Hurst exponent of the images to identify an abrupt dynamic change. We tested the performance of this method with a coupled spatiotemporal dynamic model and found that it works well. The changes in the Hurst exponents of the spatial images are stable when there is no dynamic change in the system, but there will be a clear non-stationary change of the Hurst exponents; for example, the abrupt mean values change if the dynamics of the system change. Thus, the TD-DFA method is suitable for detecting an abrupt dynamic change from natural and artificial images. The spatial images of the NCEP reanalysis of the daily average temperature exhibited fractality. Based on this, we found three non-stationary changes in the Hurst exponents for the NCEP reanalysis of the daily average temperature or for the annual average temperature in the region (60°S–60°N). It can be concluded that the climate system may have incurred three dynamic changes since 1961 on decadal timescales, i.e., in approximately the mid-1970s, the mid-1980s, and between the late 1990s and the early 2000s.

Limiting Phase Trajectories as an Alternative to Nonlinear Normal Modes

We discuss a recently developed concept of limiting phase trajectories (LPTs) allowing a unified description of resonance, highly non-stationary processes for a wide range of classical and quantum dynamical systems with constant and varying parameters. This concept provides a far going extension and adequate mathematical description of the well-known linear beating phenomenon to a diverse variety of nonlinear systems ranging from classical multi-particle models to nonlinear quantum tunneling. While stationary (and non-stationary, but non-resonant) oscllations can be described in the framework of non-linear normal modes (NNMs) concept, it is not so in the considered case of resonant non-stationary processes. In the latter case which is characterized by intense energy exchange between different parts of a system, an additiional slow time scale appears. The energy exchange proceeds in this time scale and can be identified as strong modulation of the fast oscillations. The aforementioned resonant non-staionary prcesses include, e.g., targeted energy transfer, non-stationary vibrations of carbon nanotubes, quantum tunneling, auto-resonance and non-conventional synchronization. Besides the non-linear beating, the LPT concept allows one to find the conditions of transition from intense energy exchange to strongly localized (e.g. breather-like) excitations. A special mathematical technique based on the non- smooth temporal transformations leads to the clear and simple description of strongly modulated regimes. The role of LPTs in the theory of resonance non-stationary processes turns out to be similar to that of NNMs in stationary case.As an example we present results of analytical and numerical study of planar dynamics of a string with uniformly distributed discrete masses without a preliminary stretching. Each mass is also affected by grounding support with cubic characteristic (which is equivalent to transversal unstretched string). We consider the most important case of low-energy transversal dynamics. This example is especially instructive because the considered system cannot be linearized. Adequate analytical description of resonance non-stationary processes which correspond to intensive energy exchange between different parts of the system (clusters) in low frequency domain was obtained in terms of LPTs. We have revealed also in these terms the conditions of energy localization on the initially excited cluster. Analytical results are in agreement with the results of numerical simulations. It is shown that the considered system can be used as an efficient energy sink.

Implementation of a Hyperchaotic System with Hidden Attractors into a Microcontroller

In this work, the implementation of a hyperchaotic oscillator by using a microcontroller is proposed. The dynamical system, which is used, belongs to the recently new proposed category of dynamical systems with hidden attractors. By programming the microcontroller, the three most useful tools of nonlinear theory, the phase portrait, the Poincare map and the bifurcation diagram can be produced. The comparison of these with the respective simulation results, which are produced by solving the continuous dynamical system with Runge-Kutta, verified the feasibility of the proposed method. The algorithms could be easily modified to add or substitute the hyperchaotic system.

Detectability and Invariance Properties for Set Dynamical Systems

Abstract: Invariance properties and convergence of solutions of set dynamical systems are studied. Using a framework for systems with set-valued states, notions of stability and detectability, similar to the existing results for classical dynamical systems, are defined and used to obtain information about the convergence properties of solutions. In particular, it is shown that local stability, detectability, and boundedness can be combined to conclude convergence of set-valued solutions. Under the assumption of bounded solutions and outer semicontinuity of the set-valued maps that define the system’s dynamics, invariance properties for set dynamical systems are also presented along with an invariance principle. The invariance principle involves the use of Lyapunov-like functions to locate invariant sets. Examples illustrate the results.

Geodesic Transport Barriers in Jupiter's Atmosphere: A Video-Based Analysis

Jupiter's zonal jets and Great Red Spot are well known from still images. Yet the planet's atmosphere is highly unsteady, which suggests that the actual material transport barriers delineating its main features should be time-dependent. Rare video footages of Jupiter's clouds provide an opportunity to verify this expectation from optically reconstructed velocity fields. Available videos, however, provide short-time and temporally aperiodic velocity fields that defy classical dynamical systems analyses focused on asymptotic features. To this end, we use here the recent theory of geodesic transport barriers to uncover finite-time mixing barriers in the wind field extracted from a video captured by NASA's Cassini space mission. More broadly, the approach described here provides a systematic and frame-invariant way to extract dynamic coherent structures from time-resolved remote observations of unsteady continua.