Natural Phase Space Research Articles

Hausdorff and fractal dimensions of attractors for functional differential equations in Banach spaces

The main objective of this paper is to obtain estimations of Hausdorff dimension as well as fractal dimension of global attractors and pullback attractors for both autonomous and nonautonomous functional differential equations (FDEs) in Banach spaces. New criterions for the finite Hausdorff dimension and fractal dimension of attractors in Banach spaces are firslty proposed by combining the squeezing property and the covering of finite subspace of Banach spaces, which generalize the method established in Hilbert spaces. In order to surmount the barrier caused by the lack of orthogonal projectors with finite rank, which is the key tool for proving the squeezing property of partial differential equations in Hilbert spaces, we adopt the state decomposition of phase space based on the exponential dichotomy of the studied FDEs to obtain similar squeezing property. The theoretical results are applied to a retarded nonlinear reaction-diffusion equation and a non-autonomous retarded functional differential equation in the natural phase space, for which explicit bounds of dimensions that do not depend on the entropy number but only depend on the spectrum of the linear parts and Lipschitz constants of the nonlinear parts are obtained.

ω-Limit Sets of Impulsive Semigroups for Hyperbolic Equations

In this paper, we investigate the qualitative behavior of an evolutionary problem consisting of a hyperbolic dissipative equation whose trajectories undergo instantaneous impulsive discontinuities at the moments when the energy functional reaches a certain threshold value. The novelty of the current study is that we consider the case in which the entire infinite-dimensional phase vector undergoes an impulsive disturbance. This substantially broadens the existing results, which admit discontinuities for only a finite subset of phase coordinates. Under fairly general conditions on the system parameters, we prove that such a problem generates an impulsive dynamical system in the natural phase space, and its trajectories have nonempty compact ω-limit sets.

Diagnostics of Bolted Joints in Vibrating Screens Based on a Multi-Body Dynamical Model.

The condition-based maintenance of vibrating screens requires new methods of their elements' diagnostics due to severe disturbances in measured signals from vibrators and falling pieces of material. The bolted joints of the sieving deck, when failed, require a lot of time and workforce for repair. In this research, the authors proposed the model-based diagnostic method based on modal analysis of the 2-DOF system, which accounts for the interaction of the screen body and the upper deck under conditions of bolted joint degradation. It is shown that the second natural mode with an out-of-phase motion of the upper deck against the main screen housing may coincide with the excitation frequency or its higher harmonics, which appear when vibrators' bearings are in bad condition. This interaction speeds up bolt loosening and joint opening by the dynamical loading of higher amplitude. The proposed approach can be used to detune the system from resonance and anti-resonance to reduce maintenance costs and energy consumption. To prevent abrupt failures, such parameters as second natural mode frequency, damping factor, and phase space plot (PSP) distortion measures are proposed as bolt health indicators, and these are verified on the laboratory vibrating screen. Also, the robustness is tested by the impulsive non-Gaussian noise addition to the measurement data. A special diagram was proposed for the bolted joints' strength capacity assessment and maintenance actions planning (tightening, replacement), depending on clearance in the joints.

Optimal Floquet Stationkeeping under the Relative Dynamics of the Three-Body Problem

Deep space missions, and particularly cislunar endeavors, are becoming a major field of interest for the space industry, including for the astrodynamics research community. While near-Earth missions may be completely covered by perturbed Keplerian dynamics, deep space missions require a different modeling approach, where multi-body gravitational interactions play a major role. To this end, the Restricted Three-Body Problem stands out as an insightful first modeling strategy for early mission design purposes, retaining major dynamical transport structures while still being relatively simple. Dynamical Systems Theory and classical Hamiltonian Mechanics have proven themselves as remarkable tools to analyze deep-space missions within this context, with applications ranging from ballistic capture trajectory design to stationkeeping. In this work, based on this premise, a Hamiltonian derivation of the Restricted Three-Body Problem co-orbital dynamics between two spacecraft is introduced in detail. Thanks to the analytical and numerical models derived, connections between the relative and classical Keplerian and CR3BP problems are shown to exist, including first-order linear solutions and an inherited Hamiltonian normal form. The analytical linear and higher-order models derived allow the theoretical finding and unveiling of natural co-orbital phase space structures, including relative periodic and quasi-periodic orbital families, which are further exploited for general proximity operation applications. In particular, a novel reduced-order, optimal low-thrust stationkeeping controller is derived in the relative Floquet phase space, hybridizing the classical State Dependent Ricatti Equation (SDRE) with Koopman control techniques for efficient unstable manifold regulation. The proposed algorithm is demonstrated and validated within several end-to-end low-cost stationkeeping missions, and comparison against classical continuous stationkeeping algorithms presented in the literature is also addressed to reveal its enhanced performance. Finally, conclusions and open lines of research are discussed.

Approximate controllability of non-instantaneous impulsive semi-linear neutral differential equations with non-local conditions and unbounded delay

In this work, we study the approximate controllability for retarded neutral differential equations with unbounded delay, non-instantaneous impulses, and non-local conditions. First, we set the problem in a natural Banach phase space satisfying Hale-Kato axiomatic Theory about the phase space for retarded ordinary equations with unbounded delay. Second, we assume some conditions on the nonlinear terms to achieve the approximate controllability using the technique developed by Bashirov and Ghahramanlou [6], which avoids the use of fixed point theorems.

Natural phase space reconstruction-based broad learning system for short-term wind speed prediction: Case studies of an offshore wind farm

Accurate prediction of wind speed can not only help to develop strategies for wind turbine operation, but also reduce impact on power systems when wind energy is integrated into the grid. However, it is difficult to predict speed accurately due to the stochastic nature of wind. To address this issue, this paper presents a novel wind speed prediction model based on phase space reconstruction and broad learning system (BLS). First, phase spaces under various delay dimensions and phase scales are reconstructed. Afterwards, natural neighbor spectrum is constructed without parameter setting based on phase vectors for selecting the optimal phase space. Then, the optimal inputting number of BLS is decided, elastic-net regularization is introduced to alleviate overfitting and BLS is trained in an incremental way. Finally, predicting values are given by output layer. Two cases about an offshore wind farm are used to demonstrate the effectiveness of the proposed model and five traditional models are used for comparison. Results show that compared with the other models, proposed model not only achieves higher predicting accuracy, but also has faster learning speed, meeting the requirement of online prediction for scale-growing wind speed and leaving more time for making strategies about grid planning.

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.

Well-posedness of solutions for a class of quasilinear wave equations with structural damping or strong damping

This paper deals with a class of quasilinear wave equations with structural damping or strong damping. By virtue of the improved Faedo–Galerkin method and some technical efforts, we first establish the local well-posedness of weak solutions, especially the continuity of weak solutions with respect to time in the natural phase space. Then we investigate the global existence, asymptotic behavior and finite time blow-up of weak solutions with subcritical or critical initial energy. As for the supercritical initial energy case, we show that the weak solutions may blow up in finite time with arbitrarily high initial energy. Last but not least, the upper and lower bounds of the blow-up time for blow-up solutions are derived.

Asymptotic Regularity and Attractors for Slightly Compressible Brinkman–Forchheimer Equations

Slightly compressible Brinkman–Forchheimer equations in a bounded 3D domain with Dirichlet boundary conditions are considered. These equations model fluids motion in porous media. The dissipativity of these equations in higher order energy spaces is obtained and regularity and smoothing properties of the solutions are studied. In addition, the existence of a global and an exponential attractors for these equations in a natural phase space is verified.

Parameterization Method for State-Dependent Delay Perturbation of an Ordinary Differential Equation

We consider state-dependent delay equations (SDDEs) obtained by adding delays to a planar ODE with a limit cycle. Situations of this type appear in models of several physical processes, where small delay effects are added. Even if the delays are small, they are very singular perturbations since the natural phase space of an SDDE is an infinite-dimensional space. We show that for the SDDE, there are initial values which lead to solutions similar to those of the ODE. That is, there exist a periodic solution and a two parameter family of solutions whose evolution converges to the periodic solution (in the ODE case, these are called the isochrons). The method of proof bypasses the theory of existence, uniqueness, dependence on parameters of SDDE. We consider the class of functions of time that have a well defined behavior (e.g., periodic, or asymptotic to periodic) and derive functional equations which impose that they are solutions of the SDDE. These functional equations are studied using functional analysis methods. We provide a result in “a posteriori” format: given an approximate solution of the functional equation, which has some good condition numbers, we prove that there is a true solution close to the approximate one. Thus, our result can be used to validate the results of numerical computations or formal expansions. The method of proof also leads to practical algorithms. In a companion paper, we present the implementation details and representative results. One feature of the method presented here is that it allows us to obtain smooth dependence on parameters for the periodic solutions and their slow stable manifolds without studying the smoothness of the flow (which seems to be problematic for SDDEs, for now the optimal result on smoothness of the flow is $C^1$).

Periodic Solutions and Torsional Instability in a Nonlinear Nonlocal Plate Equation

A thin and narrow rectangular plate having the two short edges hinged and the two long edges free is considered. A nonlinear nonlocal evolution equation describing the deformation of the plate is introduced: well-posedness and existence of periodic solutions are proved. The natural phase space is a particular second order Sobolev space that can be orthogonally split into two subspaces containing, respectively, the longitudinal and the torsional movements of the plate. Sufficient conditions for the stability of periodic solutions and of solutions having only a longitudinal component are given. A stability analysis of the so-called prevailing mode is also performed. Some numerical experiments show that instabilities may occur. This plate can be seen as a simplified and qualitative model for the deck of a suspension bridge, which does not take into account the complex interactions between all the components of a real bridge.

Well-posedness and attractors for a super-cubic weakly damped wave equation with H−1 source term

This work is devoted to Dirichlet problem for the sup-cubic semi-linear wave equation with H−1 forcing in a bounded domain of R3. The global well-posedness of the weak solution with some translational regularity is obtained. Moreover, the existence of a global attractor as well as exponential attractor in the natural phase space are also considered. The results are crucially based on the Strichartz estimates for the linear wave equation in bounded domain of R3.

Large time behavior of a third grade fluid system

We consider the large time behavior of a non-autonomous third grade fluid system, which could be viewed as a perturbation of the classical Navier-Stokes system. Under proper assumptions, we firstly prove that the family of processes generated by the problem admits a uniform attractor in the natural phase space. Then we prove the upper-semicontinuity of the uniform attractor when the perturbation tends to zero.

On the Existence of Strongly Continuous Physical Solutions for Classes of Autonomous Evolutionary Variational Inequalities

UDC 517.9 Abstract. The concept of a physical solution (on a finite time interval) is introduced on the basis of natural energy equalities and continuous dependence of state functions on time variables in the phase space for classes of autonomous evolutionary variational inequalities on convex cones with nonlinear nonmonotone (in the general case) mappings. For approximate searching for such solutions, the classical penalty method is used. For the solutions obtained, this article substantiates the possibility of a global description of the behavior of such systems in the natural phase space with respect to the topology of strong convergence by finite algorithms up to an arbitrary small parameter.

A stochastic model of inward diffusion in magnetospheric plasmas

The inward diffusion of particles, often observed in magnetospheric plasmas (either naturally created stellar ones or laboratory devices), creates a spontaneous density gradient, which seemingly contradicts the entropy principle. We construct a theoretical model of diffusion that can explain the inward diffusion in a dipole magnetic field. The key is the identification of the proper coordinates on which an appropriate diffusion operator can be formulated. The effective phase space is foliated by the adiabatic invariants; on the symplectic leaf, the invariant measure (by which the entropy must be calculated) is distorted, by the inhomogeneous magnetic field, with respect to the conventional Lebesgue measure of the natural phase space. The collision operator is formulated to be consistent to the ergodic hypothesis on the symplectic leaf, i.e., the resultant diffusion must diminish gradients on the proper coordinates. The non-orthogonality of the cotangent vectors of the configuration space causes a coupling between the perpendicular and parallel diffusions, which is derived by applying Ito’s formula of changing variables. The model has been examined by numerical simulations. We observe the creation of a peaked density profile that mimics radiation belts in planetary magnetospheres as well as laboratory experiments.

Global φ-attractor for a modified 3D Bénard system on channel-like domains

AbstractIn this paper we prove the existence of a global '-attractor in the weak topology of the natural phase space for the family of multi-valued processes generated by solutions of a nonautonomous modified 3D Bénard system in unbounded domains for which Poincaré inequality takes place

Quantum particle on a Möbius strip, coherent states and projection operators

The coherent states for a quantum particle on a Möbius strip are constructed and their relation to the natural phase space for fermionic fields is shown. The explicit comparison of the obtained states with previous works where the cylinder quantization was used and the spin-1/2 was introduced by hand is given, and the relation between the geometrical phase space, constraints and projection operators is analyzed and discussed.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

Global dynamics of delayed reaction–diffusion equations in unbounded domains

We consider a nonlocal delayed reaction–diffusion equation in an unbounded domain that includes some special cases arising from population dynamics. Due to the non-compactness of the spatial domain, the solution semiflow is not compact. We first show that, with respect to the compact open topology for the natural phase space, the solutions induce a compact and continuous semiflow $${\Phi}$$ on a bounded and positively invariant set Y in C + = C([−1, 0], X +) that attracts every solution of the equation, where X + is the set of all bounded and uniformly continuous functions from $${\mathbb{R}}$$ to [0, ∞). Then, to overcome the difficulty in describing the global dynamics, we establish a priori estimate for nontrivial solutions after describing the delicate asymptotic properties of the nonlocal delayed effect and the diffusion operator. The estimate enables us to show the permanence of the equation with respect to the compact open topology. With the help of the permanence, we can employ standard dynamical system theoretical arguments to establish the global attractivity of the nontrivial equilibrium. The main results are illustrated with the diffusive Nicholson’s blowfly equation and the diffusive Mackey–Glass equation.

Piecewise linear difference equations and convexity

This note presents a new method for analysing piecewise linear difference equations. The equations are considered in their natural phase space and interpreted via their associated semigroups and number theoretic graphs.

Quantum mechanics onSO(3)via noncommutative dual variables

We formulate quantum mechanics on SO(3) using a non-commutative dual space representation for the quantum states, inspired by recent work in quantum gravity. The new non-commutative variables have a clear connection to the corresponding classical variables, and our analysis confirms them as the natural phase space variables, both mathematically and physically. In particular, we derive the first order (Hamiltonian) path integral in terms of the non-commutative variables, as a formulation of the transition amplitudes alternative to that based on harmonic analysis. We find that the non-trivial phase space structure gives naturally rise to quantum corrections to the action for which we find a closed expression. We then study both the semi-classical approximation of the first order path integral and the example of a free particle on SO(3). On the basis of these results, we comment on the relevance of similar structures and methods for more complicated theories with group-based configuration spaces, such as Loop Quantum Gravity and Spin Foam models.