Motion In Phase Space Research Articles

Jeffery’s Orbits and Microswimmers in Flows: A Theoretical Review

In this review, we provide a theoretical introduction to Jeffery’s equations for the orientation dynamics of an axisymmetric object in a flow at low Reynolds number, and review recent theoretical extensions and applications to the motions of self-propelled particles, so-called microswimmers, in external flows. Bacteria colonize human organs and medical devices even with flowing fluid, microalgae occasionally cause huge harmful toxic blooms in lakes and oceans, and recent artificial microrobots can migrate in flows generated in well-designed microfluidic chambers. The Jeffery equations, a simple set of ordinary differential equation, provide a useful building block in modeling, analyzing, and understanding these microswimmer dynamics in a flow current, in particular when incorporating the impact of the swimmer shape since the equations contain a shape parameter as a single scalar, known as the Bretherton parameter. The particle orientation forms a closed orbit when situated in a simple shear, and this non-uniform periodic rotational motion, referred to as Jeffery’s orbits, is due to a constant of motion in the non-linear equation. After providing a theoretical introduction to microswimmer hydrodynamics and a derivation of the Jeffery equations, we discuss possible extensions to more general shapes, including those with rapid deformation. In the latter part of this review, simple mathematical models of microswimmers in different types of flow fields are described, with a focus on constants of motion and their relation to periodic motions in phase space, together with a breakdown of the degenerate orbits, to discuss the stable, unstable, and chaotic dynamics of the system. The discussion in this paper will provide a comprehensive theoretical foundation for Jeffery’s orbits and will be useful to understand the motions of microswimmers under various flows and to analyze and design computer and laboratory experiments, as well as in the active matter and artificial smart swimmer studies.

Independent continuous periodic firing series to chaos in the 3-D Hindmarsh–Rose neuron circuit

In order to study the intricate dynamics of the 3-D Hindmarsh–Rose (HR) model under periodic excitation, a semi-analytical method is proposed. This semi-analytical method shows an implicit mapping relationship by discretizing corresponding continuous nonlinear systems. Subsequently, a new phenomenon of independent continuous periodic firing series is discovered for the first time. We prove that saddle–node, periodic-doubling, and Neimark bifurcations occur when the excitation frequency varies to an appropriate value in this system. The generation of saddle–node bifurcation will lead to the change of limit cycles for periodic motion. The number of limit cycles of periodic motions in phase space increases continuously with excitation frequency increasing. Furthermore, the corresponding circuit is implemented to simulate the periodic stimulus effects, which are confirmed precisely by phase planes and harmonic spectrums. The study provides a better understanding for exploring the mechanism of how neurons code information or how they react to complex excitation and serve as guidelines for the analysis and design of biological circuits.

On the discontinuous dynamics of a class of 2-DOF frictional vibration systems with asymmetric elastic constraints

<abstract><p>In this paper, the discontinuous dynamic behavior of a two-degree-of-freedom frictional collision system including intermediate elastic collision and unilateral elastic constraints subjected to periodic excitation is studied by using flow switching theory. In this system, given that the motion of each object might have a velocity that is either greater than or less than zero and each object experiences a periodic excitation force that has negative feedback, because the kinetic and static friction coefficients differ, the flow barrier manifests when the object's speed is zero. Based on the discontinuity or nonsmoothness of the oscillator's motion generated by elastic collision and friction, the motion states of the oscillator in the system are divided into 16 cases and the absolute and relative coordinates are used to define various boundaries and domains in the oscillator motion's phase space. On the basis of this, the G-function and system vector fields are used to propose the oscillator motion's switching rules at the displacement and velocity boundaries. Finally, some dynamic behaviors for the 2-DOF oscillator are demonstrated via numerical simulation of the oscillator's stick, grazing, sliding and periodic motions and the scene of sliding bifurcation. The mechanical system's optimization designs with friction and elastic collision will benefit from this investigation's findings.</p></abstract>

Stochastic motion in phase space on a surface of constant energy.

We study closed systems of particles that are subject to stochastic forces in addition to the conservative forces. The stochastic equationsof motion are set up in such a way that the energy is strictly conserved at all times. To ensure this conservation law, the evolution equation for the probability density is derived using an appropriate interpretation of the stochastic equationof motion that is not the Itô nor the Stratonovic interpretation. The trajectories in phase space are restricted to the surface of constant energy. Despite this restriction, the entropy is shown to increase with time, expressing irreversible behavior and relaxation to equilibrium. This main result of the present approach contrasts with that given by the Liouville equation, which also describes closed systems, but does not show irreversibility.

Geometry of Krylov complexity

We develop a geometric approach to operator growth and Krylov complexity in many-body quantum systems governed by symmetries. We start by showing a direct link between a unitary evolution with the Liouvillian and the displacement operator of appropriate generalized coherent states. This connection maps operator growth to a purely classical motion in phase space. The phase spaces are endowed with a natural information metric. We show that, in this geometry, operator growth is represented by geodesics, and Krylov complexity is proportional to a volume. This geometric perspective also provides two novel avenues toward computation of Lanczos coefficients, and it sheds new light on the origin of their maximal growth. We describe the general idea and analyze it in explicit examples, among which we reproduce known results from the Sachdev-Ye-Kitaev model, derive operator growth based on SU(2) and Heisenberg-Weyl symmetries, and generalize the discussion to conformal field theories. Finally, we use techniques from quantum optics to study operator evolution with quantum information tools such as entanglement and Renyi entropies, negativity, fidelity, relative entropy, and capacity of entanglement.

Localization in the Discrete Non-linear Schrödinger Equation and Geometric Properties of the Microcanonical Surface

It is well known that, if the initial conditions have sufficiently high energy density, the dynamics of the classical Discrete Non-Linear Schr\"odinger Equation (DNLSE) on a lattice shows a form of breaking of ergodicity, with a finite fraction of the total charge accumulating on a few sites and residing there for times that diverge quickly in the thermodynamic limit. In this paper we show that this kind of localization can be attributed to some geometric properties of the microcanonical potential energy surface, and that it can be associated to a phase transition in the lowest eigenvalue of the Laplacian on said surface. We also show that the approximation of considering the phase space motion on the potential energy surface only, with effective decoupling of the potential and kinetic partition functions, is justified in the large connectivity limit, or fully connected model. In this model we further observe a synchronization transition, with a synchronized phase at low temperatures.

Motor strategies and adiabatic invariants: The case of rhythmic motion in parabolic flights

The role of gravity in human motor control is at the same time obvious and difficult to isolate. It can be assessed by performing experiments in variable gravity. We propose that adiabatic invariant theory may be used to reveal nearly conserved quantities in human voluntary rhythmic motion, an individual being seen as a complex time-dependent dynamical system with bounded motion in phase space. We study an explicit realization of our proposal: An experiment in which we asked participants to perform ∞- shaped motion of their right arm during a parabolic flight, either at self-selected pace or at a metronome's given pace. Gravity varied between 0 and 1.8 g during a parabola. We compute the adiabatic invariants in the participant's frontal plane assuming a separable dynamics. It appears that the adiabatic invariant in vertical direction increases linearly with g, in agreement with our model. Differences between the free and metronome-driven conditions show that participants' adaptation to variable gravity is maximal without constraint. Furthermore, motion in the participant's transverse plane induces trajectories that may be linked to higher-derivative dynamics. Our results show that adiabatic invariants are relevant quantities to show the changes in motor strategy in time-dependent environments.

Real-time optimal quantum control of mechanical motion at room temperature.

The ability to accurately control the dynamics of physical systems by measurement and feedback is a pillar of modern engineering1. Today, the increasing demand for applied quantum technologies requires adaptation of this level of control to individual quantum systems2,3. Achieving this in an optimal way is a challenging task that relies on both quantum-limited measurements and specifically tailored algorithms for state estimation and feedback4. Successful implementations thus far include experiments on the level of optical and atomic systems5-7. Here we demonstrate real-time optimal control of the quantum trajectory8 of an optically trapped nanoparticle. We combine confocal position sensing close to the Heisenberg limit with optimal state estimation via Kalman filtering to track the particle motion in phase space in real time with a position uncertainty of 1.3times the zero-point fluctuation. Optimal feedback allows us to stabilize the quantum harmonic oscillator to a mean occupation of 0.56±0.02quanta, realizing quantum ground-state cooling from room temperature. Our work establishes quantum Kalman filtering as a method to achieve quantum control of mechanical motion, with potential implications for sensing on all scales. In combination with levitation, this paves the way to full-scale control over the wavepacket dynamics of solid-state macroscopic quantum objects in linear and nonlinear systems.

Anisotropic Psychophysical Trends in the Discrimination of Tactile Direction in a Precision Grip.

Tactile cues arising from interactions with objects have a sense of directionality which affects grasp. Low latency responses to varied grip perturbations indicate that grasp safety margins are exaggerated in certain directions and conditions. In a grip with the ulnar-radial axis vertical, evidence suggests that distal and downward directions are more sensitive to task parameters and have larger safety margins. This suggests that, for the purpose of applying forces with the fingers, reference frames with respect to the hand and gravity are both in operation. In this experiment, we examined human sensitivities to the direction of tactile movement in the context of precision grip in orientations either orthogonal to or parallel to gravity. Subjects performed a two-alternative-forced-choice task involving a textured cube which moved orthogonal to their grip axis. Subjects’ arms were placed in a brace that allowed for finger movement but minimized arm movement. Movement of thumb and index joints were monitored via PhaseSpace motion capture. The subject was presented with a textured cube and instructed to lightly grasp the cube, as if it were slipping. In each trial the object was first translated 1 cm in 0° (proximal), 90° (radial), 180° (distal), or 270° (ulnar) and returned to its origin. This primary stimulus was immediately followed by a 10 mm secondary stimulus at a random 5° interval between −30° and 30° of the primary stimulus. Response from the subject after each pair of stimuli indicated whether the test direction felt the same as or different from the primary stimulus. Traditional bias and sensitivity analyses did not provide conclusive results but suggested that performance is best in the ulnar-radial axis regardless of gravity. Modeling of the response curve generated a detection threshold for each primary stimulus. Lower thresholds, indicating improved detection, persisted in the ulnar-radial axis. Anisotropic thresholds of increased detection appear to coincide with digit displacement and appear to be independent of the grasp orientation.

Discontinuous dynamical behaviors in a 2-DOF friction collision system with asymmetric damping

Abstract By using the flow switchability theory in discontinuous dynamical systems, this paper deals with the discontinuous dynamical behaviors of a two degrees of freedom system with asymmetric damping, where considering that friction and impact coexist and the static and dynamic friction coefficients are different. Because of the particularity of friction force, the flow barriers on the velocity boundary that affect the leaving flow are considered in this paper. Based on discontinuity that is caused by the sudden change of friction force or the collision between two objects, the phase space of motion for the object is divided into several different domains and boundaries; and with the help of the analysis of vector fields and G-functions on the corresponding discontinuous boundaries or in domains, the analytical conditions for all possible motions are obtained, which is used to determine the switching of motion state in this system. Finally, numerical simulations are presented to better understand the analytical conditions of the stick, grazing, impact, stuck and periodic motions.

Geometric Aspects of the Isentropic Liquid Dynamics and Vorticity Invariants.

We review a modern differential geometric description of fluid isentropic motion and features of it including diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. We analyze the adiabatic liquid dynamics, within which, following the general approach, the nature of the related Poissonian structure on the fluid motion phase space as a semidirect Banach groups product, and a natural reduction of the canonical symplectic structure on its cotangent space to the classical Lie-Poisson bracket on the adjoint space to the corresponding semidirect Lie algebras product are explained in detail. We also present a modification of the Hamiltonian analysis in case of a flow governed by isothermal liquid dynamics. We study the differential-geometric structure of isentropic magneto-hydrodynamic superfluid phase space and its related motion within the Hamiltonian analysis and related invariant theory. In particular, we construct an infinite hierarchy of different kinds of integral magneto-hydrodynamic invariants, generalizing those previously constructed in the literature, and analyzing their differential-geometric origins. A charged liquid dynamics on the phase space invariant with respect to an abelian gauge group transformation is also investigated, and some generalizations of the canonical Lie-Poisson type bracket is presented.

Experimental Observation of a Time-Driven Phase Transition in Quantum Chaos.

We report the first experimental observation of the time-driven phase transition in a canonical quantum chaotic system, the quantum kicked rotor. The transition bears a firm analogy to a thermodynamic phase transition, with the time mimicking the temperature and the quantum expectation of the rotor's kinetic energy mimicking the free energy. The transition signals a sudden change in the system's memory behavior: before the critical time, the system undergoes chaotic motion in phase space and its memory of initial states is erased in the course of time; after the critical time, quantum interference enhances the probability for a chaotic trajectory to return to the initial state, and thus the system's memory is recovered.

On quasi-satellite periodic motion in asteroid and planetary dynamics

Applying the method of analytical continuation of periodic orbits, we study quasi-satellite motion in the framework of the three-body problem. In the simplest, yet not trivial model, namely the planar circular restricted problem, it is known that quasi-satellite motion is associated with a family of periodic solutions, called family $f$, which consists of 1:1 resonant retrograde orbits. In our study, we determine the critical orbits of family $f$ that are continued both in the elliptic and in the spatial model and compute the corresponding families that are generated and consist the backbone of the quasi-satellite regime in the restricted model. Then, we show the continuation of these families in the general three-body problem, we verify and explain previous computations and show the existence of a new family of spatial orbits. The linear stability of periodic orbits is also studied. Stable periodic orbits unravel regimes of regular motion in phase space where 1:1 resonant angles librate. Such regimes, which exist even for high eccentricities and inclinations, may consist dynamical regions where long-lived asteroids or co-orbital exoplanets can be found.

Experimental investigation of perpetual points in mechanical systems

In dissipative dynamical systems, equilibrium (stationary) points have a dominant organizing effect on transient motion in phase space, especially in nonlinear systems. These time-independent solutions are readily defined in the context of ordinary differential equations, that is, they occur when all the time derivatives are simultaneously zero. However, there has been some recent interest in perpetual points: points at which the higher time derivatives are zero, but not necessarily the first. Previous work has focused on analytic work (including simulation) and some experimental studies of electric circuits. This paper focuses attention on the occurrence of these points in a simple mechanical system, including experimental verification. Thus, points of zero acceleration can be found in which the corresponding velocity is a maximum or minimum, but not zero. Specifically, the rigid-arm pendulum is used to generate data for which acceleration (and its derivative) can be evaluated. In this paper an experimental (mechanical) setup is described, specifically designed to investigate perpetual points, including a description of the data analysis approaches developed to identify their location.

Steady state obliquity of a rigid body in the spin–orbit resonant problem: application to Mercury

We investigate the stable Cassini state 1 in the p : q spin–orbit resonant problem. Our study includes the effect of the gravitational potential up to degree and order 4 and p : q spin–orbit resonances with \(p,q\le 8\) and \(p\ge q\). We derive new formulae that link the gravitational field coefficients with its secular orbital elements and its rotational parameters. The formulae can be used to predict the orientation of the spin axis and necessary angular momentum at exact resonance. We also develop a simple pendulum model to approximate the dynamics close to resonance and make use of it to predict the libration periods and widths of the oscillatory regime of motions in phase space. Our analytical results are based on averaging theory that we also confirm by means of numerical simulations of the exact dynamical equations. Our results are applied to a possible rotational history of Mercury.

Advances made by Belgrade's group in research of oscillatory reactions

Oscillatory dynamic states as one form of selforganization of nonlinear systems can be found in almost all sciences, like mechanics, physical chemistry or biomedicine. Although origin of these oscillations is different, computational challenges in modelling oscillatory phenomena remain similar in all fields. Since 1979 researchers from Belgrade's group perform systematic examinations of oscillatory reactions. As stability of steady states is the central point in modelling oscillatory reactions, in last 10 years they have adapted and improved powerful tool of the Stoichiometric Network Analysis for this goal. Moreover, bifurcations of few types were identified in several models of oscillatory reactions. Even very complex chaotic motions in phase space were characterized and quantified by several numerical techniques. Multiple time scale behaviour is found within the core of the complex dynamic behaviour of mixed-mode oscillations. Analytical applications were developed, too.

Thermodynamics of the general diffusion process: Equilibrium supercurrent and nonequilibrium driven circulation with dissipation

Unbalanced probability circulation, which yields cyclic motions in phase space, is the defining characteristics of a stationary diffusion process without detailed balance. In over-damped soft matter systems, such behavior is a hallmark of the presence of a sustained external driving force accompanied with dissipations. In an under-damped and strongly correlated system, however, cyclic motions are often the consequences of a conservative dynamics. In the present paper, we give a novel interpretation of a class of diffusion processes with stationary circulation in terms of a Maxwell-Boltzmann equilibrium in which cyclic motions are on the level set of stationary probability density function thus non-dissipative, e.g., a supercurrent. This implies an orthogonality between stationary circulation $J^{ss}(x)$ and the gradient of stationary probability density $f^{ss}(x)>0$. A sufficient and necessary condition for the orthogonality is a decomposition of the drift $b(x)=j(x)+ D(x)\nabla\varphi(x)$ where $\nabla\cdot j(x)=0$ and $j(x)$ $\cdot\nabla\varphi(x)=0$. Stationary processes with Maxwell-Boltzmann equilibrium has an underlying conservative dynamics $\dot{x}= j(x)\equiv$ $\big(f^{ss}(x)\big)^{-1}J^{ss}(x)$, and a first integral $\varphi(x)\equiv-\ln f^{ss}(x)=$ const, akin to a Hamiltonian system. At all time, an instantaneous free energy balance equation exists for a given diffusion system; and an extended energy conservation law among a family of diffusion processes with different parameter $\alpha$ can be established via a Helmholtz theorem. For the general diffusion process without the orthogonality, a nonequilibrium cycle emerges, which consists of external driven $\varphi$-ascending steps and spontaneous $\varphi$-descending movements, alternated with iso-$\varphi$ motions. The theory presented here provides a rich mathematical narrative for complex mesoscopic dynamics.

Stochastic Bridges of Linear Systems

We consider particles obeying Langevin dynamics while being at known positions and having known velocities at the two end-points of a given interval. Their motion in phase space can be modeled as an Ornstein-Uhlenbeck process conditioned at the two end-points-a generalization of the Brownian bridge. Using standard ideas from stochastic optimal control we construct a stochastic differential equation (SDE) that generates such a bridge that agrees with the statistics of the conditioned process, as a degenerate diffusion. Higher order linear diffusions are also considered. In general, a time-varying drift is sufficient to modify the prior SDE and meet the end-point conditions. When the drift is obtained by solving a suitable differential Lyapunov equation, the SDE models correctly the statistics of the bridge. These types of models are relevant in controlling and modeling distribution of particles and the interpolation of density functions.

Возрастная эволюция организма человека как движение квазиаттракторов

Today the formalization of the description of the evolving system doesn’t exist in biology. But if a biosystem is described by a state vector x=x(t)=(x1, x2, …, xm)T in multidimensional phase space, notions of speed and acceleration for the description of quasi-attractor’s motion can be introduced. Vector x(t) moves constantly and chaotically inside the quasi-attractor, i.e. dx/dt≠0 is constant. With age these quasi-attractors show translational motion in phase space for which the model as an dx/dt=(a-bx)x is created and speed V=dx/dt and acceleration a=dV/dt are determined for evolution of biosystems. The current paper presents concrete examples of age-related changes of quasi-attractor’s parameters (two-dimensional phase space) that should be considered as an evolution of vector of cardiorespiratory system in six-dimensional phase space. Models of such dynamics are discussed according to quasi-attractor’s parameters that allow to calculating speed and acceleration of evolution in some integrative values.

Poisson theory and integration method for a dynamical system of relative motion

This paper focuses on studying the Poisson theory and the integration method of dynamics of relative motion. Equations of a dynamical system of relative motion in phase space are given. Poisson theory of the system is established. The Jacobi last multiplier of the system is defined, and the relation between the Jacobi last multiplier and the first integrals of the system is studied. Our research shows that for a dynamical system of relative motion, whose configuration is determined by n generalized coordinates, the solution of the system can be found by using the Jacobi last multiplier if (2n−1) first integrals of the system are known. At the end of the paper, an example is given to illustrate the application of the results.