Stochastic oscillators play a prominent role in different fields of science. Their simplified description in terms of a phase has been advocated by different authors using distinct phase definitions in the stochastic case. One notion of phase that we put forward previously, the asymptotic phase of a stochastic oscillator, is based on the eigenfunction expansion of its probability density. More specifically, it is given by the complex argument of the eigenfunction of the backward operator corresponding to the least-negative eigenvalue. Formally, besides the “backward” phase, one can also define the “forward” phase as the complex argument of the eigenfunction of the forward Kolomogorov operator corresponding to the least-negative eigenvalue. Until now, the intuition about these phase descriptions has been limited. Here we study these definitions for a process that is analytically tractable, the two-dimensional Ornstein-Uhlenbeck process with complex eigenvalues. For this process, (i) we give explicit expressions for the two phases; (ii) we demonstrate that the isochrons are always the spokes of a wheel but that (iii) the spacing of these isochrons (their angular density) is different for backward and forward phases; (iv) we show that the isochrons of the backward phase are completely determined by the deterministic part of the vector field, whereas the forward phase also depends on the noise matrix; and (v) we demonstrate that the mean progression of the backward phase in time is always uniform, whereas this is not true for the forward phase except in the rotationally symmetric case. We illustrate our analytical results for a number of qualitatively different cases.

Technological progress is leading to proliferation and diversification of trading venues, thus increasing the relevance of the long-standing question of market fragmentation versus consolidation. To address this issue quantitatively, we analyze systems of adaptive traders that choose where to trade based on their previous experience. We demonstrate that only based on aggregate parameters about trading venues, such as the demand-to-supply ratio, we can assess whether a population of traders will prefer fragmentation or specialization towards a single venue. We investigate what conditions lead to market fragmentation for populations with a long memory and analyze the stability and other properties of both fragmented and consolidated steady states. Finally, we investigate the dynamics of populations with finite memory; when this memory is long the true long-time steady states are consolidated but fragmented states are strongly metastable, dominating the behavior out to long times.

Two identical autonomous dynamical systems unidirectionally coupled in a sender-receiver configuration can exhibit anticipated synchronization (AS) if the receiver neuron also receives a delayed negative self-feedback. Recently, AS was shown to occur in a three-neuron motif with standard chemical synapses where the delayed inhibition was provided by an interneuron. Here, we show that a two-neuron model in the presence of an inhibitory autapse, which is a massive self-innervation present in the cortical architecture, may present AS. The GABAergic autapse regulates the internal dynamics of the receiver neuron and acts as the negative delayed self-feedback required by dynamical systems in order to exhibit AS. In this biologically plausible scenario, a smooth transition from the usual delayed synchronization (DS) to AS typically occurs when the inhibitory conductance is increased. The phenomenon is shown to be robust when model parameters are varied within a physiological range. For extremely large values of the inhibitory autapse the system undergoes to a phase-drift regime in which the receiver is faster than the sender. Furthermore, we show that the inhibitory autapse promotes a faster internal dynamics of the free-running Receiver when the two neurons are uncoupled, which could be the mechanism underlying anticipated synchronization and the DS-AS transition.

Most researchers derived the arterial pressure-wave equation by taking the axial blood flow as the major mechanism and assumed the flow motion was governed by the Navier-Stokes equation. However, only realistic hemodynamic theory can help to develop method for future healthcare. Here, we pointed out a different rigorous hemodynamic model that explained many physiological facts. Our previous work started directly from Newton's law by deriving the radial and the axial momentum equations of the elastic arterial wall system and of the enclosed blood system, with the contact forces between the two systems being counted explicitly. Our model has some important applications in future healthcare, such as providing a basis for studying the collective behavior of the cardiovascular system and developing quantitative methods for disease prevention and diagnosis.

We perform an analytic study on the stochastic thermodynamics of a small classical particle trapped in a time-dependent single-well potential in the highly underdamped limit. It is shown that the nonequilibrium probability density function for the system's energy is a Maxwell-Boltzmann distribution (as in equilibrium) with a closed form time-dependent effective temperature and fractional degrees of freedom. We also find that the solvable model satisfies the Crooks fluctuation theorem, as expected. Moreover, we compute the average work in this isothermal process and characterize analytically the optimal protocol for minimum work. The optimal protocol presents an initial and a final jump which correspond to adiabatic processes linked by a smooth exponential time-dependent part for all kinds of single-well potentials. Furthermore, we argue that this result connects two distinct relevant experimental setups for trapped nanoparticles, the levitated particle in a harmonic trap, and the free particle in a box, as they are limiting cases of the general single-well potential and display the time-dependent optimal protocols. Finally, we highlight the connection between our system and an equivalent model of a gas of Brownian particles.

In recent decades, many stochastic processes have been proposed as models for real world time series data with anomalous spreading, highly persistent correlations, and transient distributional characteristics. We introduce the higher order fractional tempered stable motion as the stochastic integral of the tempered stable motion with respect to a generalized higher order moving average kernel, which provides an analytic model for stochastic processes possessing these characteristics. This stochastic process provides a mathematical model for anomalous diffusion with a transient distribution resembling higher order fractional stable motion on short timescales and higher order fractional Brownian motion in the long run. The specifics of the crossover dynamics from the Lévy stable anomalous diffusion to the Gaussian anomalous diffusion are controlled by explicit parameter values that correspond to physical attributes of the process. It is well suited to modeling anomalous diffusion of any “type” (sub-, super-, regular, or hyperdiffusion) under appropriate parametrizations due to its power-law scaling of variance with respect to time. It is also a useful model for position-velocity-acceleration triples due to its convenient path differentiability and integrability properties. To highlight the potential physical relevance of this model for real world data, we outline its key statistical properties including its covariance structure, memory, and second order self-similarity. We also give an easy to implement elementary method for sample path generation which may be used as a basis for simulation and Monte Carlo studies.

Applying the theory of self-adjoint extensions of Hermitian operators to Koopman von Neumann classical mechanics, the most general set of probability distributions is found for which entropy is conserved by Hamiltonian evolution. A new dynamical phase associated with such a construction is identified. By choosing distributions not belonging to this class, we produce explicit examples of both free particles and harmonic systems evolving in a bounded phase-space in such a way that entropy is nonconserved. While these nonconserving states are classically forbidden, they may be interpreted as states of a quantum system tunneling through a potential barrier boundary. In this case, the allowed boundary conditions are the only distinction between classical and quantum systems. We show that the boundary conditions for a tunneling quantum system become the criteria for entropy preservation in the classical limit. These findings highlight how boundary effects drastically change the nature of a system.

A two-dimensional version of a chaotic thermostat is investigated. Its structure follows the concept previously introduced by the author [G. J. Morales, Phys. Rev. E 97, 032203 (2018)] to generate a one-dimensional chaotic thermostat, namely, the usual friction force of a deterministic thermostat is supplemented with a self-consistent fluctuating force that depends on the drag coefficient associated with coupling to the heat bath. Azimuthal symmetry requires the thermostat to have two internal degrees of freedom, thus the Martyna-Klein-Tuckerman [G. J. Martyna et al., J. Chem. Phys. 97, 2635 (1992)] model is chosen for the heat bath. The unmagnetized system exhibits two-dimensional diffusive behavior, achieves symmetric Maxwellian velocity distributions in the absence of an external potential, and satisfies the Einstein relation when an external force is applied. The velocity fluctuations display the characteristic exponential frequency spectrum associated with chaotic systems. The model is used to explore the diffusive motion of a thermalized charge in a weak magnetic field and the associated Hall and Pedersen mobilities. Over a range of magnetic field strengths the charge exhibits absolute negative mobility.

High-order frequency locking phenomena were recently observed using semiconductor lasers subject to large delayed feedbacks. Specifically, the relaxation oscillation (RO) frequency and a harmonic of the feedback-loop round-trip frequency coincided with the ratios 1:5 to 1:11. By analyzing the rate equations for the dynamical degrees of freedom in a laser subject to a delayed optoelectronic feedback, we show that the onset of a two-frequency train of pulses occurs through two successive bifurcations. While the first bifurcation is a primary Hopf bifurcation to the ROs, a secondary Hopf bifurcation leads to a two-frequency regime where a low frequency, proportional to the inverse of the delay, is resonant with the RO frequency. We derive an amplitude equation, valid near the first Hopf bifurcation point, and numerically observe the frequency locking. We mathematically explain this phenomenon by formulating a closed system of ordinary differential equations from our amplitude equation. Our findings motivate experiments with particular attention to the first two bifurcations. We observe experimentally (1) the frequency locking phenomenon as we pass the secondary bifurcation point and (2) the nearly constant slow period as the two-frequency oscillations grow in amplitude. Our results analytically confirm previous observations of frequency locking phenomena for lasers subject to a delayed optical feedback.

We investigate the different types of matter-wave solitons in spin-orbit-coupled spin-2 spinor Bose-Einstein condensates. Using mean-field theory and adopting the multiscale perturbation method, the original five-component Gross-Pitaevskii spin-orbit-coupled spin-2 spinor Bose-Einstein condensate model can be reduced to a single effective nonlinear Schrödinger equation, which allows us to find analytical soliton solutions of this system. In this way, for different regimes of the spin-orbit coupling, Raman coupling, and interatomic interactions, we find approximate bright and dark soliton solutions. Particularly, the type of solitons depends on the dispersion properties of the system. When the lowest-energy band has a single-well structure, we find there only exist positive mass bright or dark solitons due to the dispersion coefficient of effective nonlinear Shrödinger equation always positive. However, when the lowest-energy band has a double-well structure, there will appear positive (negative) mass bright or dark solitons because the sign of the dispersion coefficient can be positive (negative) under different momentum. We employ direct numerical simulation of the original five-component Gross-Pitaevskii equations to confirm the analytical results.

Complex systems and relational data are often abstracted as dynamical processes on networks. To understand, predict, and control their behavior, a crucial step is to extract reduced descriptions of such networks. Inspired by notions from control theory, we propose a time-dependent dynamical similarity measure between nodes, which quantifies the effect a node-input has on the network. This dynamical similarity induces an embedding that can be employed for several analysis tasks. Here we focus on (i) dimensionality reduction, i.e., projecting nodes onto a low-dimensional space that captures dynamic similarity at different timescales, and (ii) how to exploit our embeddings to uncover functional modules. We exemplify our ideas through case studies focusing on directed networks without strong connectivity and signed networks. We further highlight how certain ideas from community detection can be generalized and linked to control theory, by using the here developed dynamical perspective.

Typically the motion of self-propelled active particles is described in a quiescent environment establishing an inertial frame of reference. Here we assume that friction, self-propulsion, and fluctuations occur relative to a noninertial frame and thereby the active Brownian motion model is generalized to noninertial frames. First, analytical solutions are presented for the overdamped case, both for linear swimmers and for circle swimmers. For a frame rotating with constant angular velocity (“carousel”), the resulting noise-free trajectories in the static laboratory frame are trochoids if these are circles in the rotating frame. For systems governed by inertia, such as vibrated granulates or active complex plasmas, centrifugal and Coriolis forces become relevant. For both linear and circling self-propulsion, these forces lead to out-spiraling trajectories which for long times approach a spira mirabilis. This implies that a self-propelled particle will typically leave a rotating carousel. A navigation strategy is proposed to avoid this expulsion, by adjusting the self-propulsion direction at will. For a particle, initially quiescent in the rotating frame, it is shown that this strategy only works if the initial distance to the rotation center is smaller than a critical radius $R_c$ which scales with the self-propulsion velocity. Possible experiments to verify the theoretical predictions are discussed.

Fascinating crack and wrinkle patterns driven by stresses are ubiquitous in natural and artificial systems. It is of great interest to control the morphologies of stress-driven patterns by using facile techniques. Here we report on the spontaneous formation of hierarchical wrinkles and oscillatory cracks in metal films deposited on liquid (or soft polymer) stripes. It is found that the metal film is under a tensile stress during deposition owing to the thermal expansion of the liquid substrate. As the film thickness is beyond a critical value, oscillatory cracks with sawtoothlike shapes form on the liquid stripes. The ratio of crack oscillation period to amplitude is independent of the stripe width and film material, which can be well explained by the “brittle adhesive joints” model. After deposition, the metal film is under a compressive stress, which is relieved by formation of various wrinkle patterns. Hierarchical wrinkles with changing wavelengths form near the stripe edge while labyrinth or wavy wrinkles form at the center. Energy analysis was adopted to explain the formation and evolution of the wrinkle patterns. This study could promote better understanding of the formations of crack and wrinkle patterns in constrained film structures and controllable fabrication of stress-driven patterns by prefabricating liquid (or soft polymer) interlayer arrays.

Coagulation growth kinetics of nanoparticles in plasma is affected by interparticle electrostatic forces due to the charging phenomenon. In stationary plasmas, unipolar charging of particles results in retardation of particle growth and may result in a limitation on particle size. We demonstrate the opposite effect of enhanced particle growth in atmospheric pressure nonstationary arc discharge. Modeling of the nanoparticle growth kinetics reveals the formation of a bipolar charge distribution. As a result, reversed (attractive) Coulomb forces promote the formation of micrometer-size particles in a millisecond timescale as observed in experiment.

Langevin dynamical simulations of two-dimensional (2D) Yukawa liquids are performed to investigate the shear modulus of 2D solid dusty plasmas. Using the known transverse sound speeds, we obtain a theoretical expression of the shear modulus of 2D Yukawa crystals as a function of the screening parameter $\kappa$, which can be used as the candidate of their shear modulus. The shear relaxation modulus $G\left(t\right)$ of 2D Yukawa liquids is calculated from the shear stress autocorrelation function, consisting of the kinetic, potential, and cross portions. Due to their viscoelasticity, 2D Yukawa liquids exhibit the typical elastic property when the time duration is much less than the Maxwell relaxation time. As a result, the infinite frequency shear modulus $G_{\infty }$, i.e., the shear relaxation modulus $G\left(t\right)$ when $t=0$, of a 2D Yukawa liquid should be related to the shear modulus of the corresponding quenched 2D Yukawa solid (with the same $\kappa$ value), with all particles suddenly frozen at their locations of the liquid state. It is found that the potential portion of the infinite frequency shear modulus for 2D Yukawa liquids at any temperature well agrees with the shear modulus of 2D Yukawa crystals with the same $\kappa$ obtained from the transverse sound speeds. Thus, we find that the shear modulus of 2D Yukawa solids can be obtained from the motion of individual particles of the corresponding Yukawa liquids using their viscoelastic property.

We present a data-driven method for separating complex, multiscale systems into their constituent timescale components using a recursive implementation of dynamic mode decomposition (DMD). Local linear models are built from windowed subsets of the data, and dominant timescales are discovered using spectral clustering on their eigenvalues. This approach produces time series data for each identified component, which sum to a faithful reconstruction of the input signal. It differs from most other methods in the field of multiresolution analysis (MRA) in that it (1) accounts for spatial and temporal coherencies simultaneously, making it more robust to scale overlap between components, and (2) yields a closed-form expression for local dynamics at each scale, which can be used for short-term prediction of any or all components. Our technique is an extension of multi-resolution dynamic mode decomposition (mrDMD), generalized to treat a broader variety of multiscale systems and more faithfully reconstruct their isolated components. In this paper we present an overview of our algorithm and its results on two example physical systems, and briefly discuss some advantages and potential forecasting applications for the technique.

Standard statistical analysis is unable to provide reliable confidence intervals on expectation values of probability distributions that do not satisfy the conditions of the central limit theorem. We present a regression-based estimator of an arbitrary moment of a probability distribution with power-law heavy tails that exploits knowledge of the exponents of its asymptotic decay to bypass this issue entirely. Our method is applied to synthetic data and to energy and atomic force data from variational and diffusion quantum Monte Carlo calculations, whose distributions have known asymptotic forms [J. R. Trail, Phys. Rev. E 77, 016703 (2008); A. Badinski et al., J. Phys.: Condens. Matter 22, 074202 (2010)]. We obtain convergent, accurate confidence intervals on the variance of the local energy of an electron gas and on the Hellmann-Feynman force on an atom in the all-electron carbon dimer. In each of these cases the uncertainty on our estimator is 45% and 60 times smaller, respectively, than the nominal (ill-defined) standard error.

We use the dynamic mode decomposition (DMD) methodology to study weakly turbulent flows in two-dimensional Bose-Einstein condensates modeled by a Gross-Pitaevskii equation subject to band-limited stochastic forcing. The forcing is balanced by the removal of energy at both ends of the energy spectrum through phenomenological hypoviscosity and hyperviscosity terms. Using different combinations of these parameters, we simulate three different regimes corresponding to weak-wave turbulence, and high- and low-frequency saturation. By extracting and ranking the primary DMD modes carrying the bulk of the energy, we are able to characterize the different regimes. In particular, the proposed DMD mode projection is able to seamlessly extract the vortices present in the condensate. This is achieved despite the fact that we do not use any phase information of the condensate as it is usually not directly available in realistic atomic BEC scenarios. Being model independent, this DMD methodology should be portable to other models and experiments involving complex flows. The DMD implementation could be used to elucidate different types of turbulent regimes as well as identifying and pinpointing the existence of delicate and hidden coherent structures within complex flows.

We investigate the dynamics and pattern formation of two ring dark solitons in a two-dimensional binary Bose-Einstein condensate with tunable intercomponent interaction via numerical simulation of the time-dependent Gross-Pitaevskii equation. Both the black and gray ring dark solitons are considered for cases where the modulation frequency of the intercomponent interaction is resonant or nonresonant with the one of the trapping potential. Our results show that in the presence of periodic modulation of the intercomponent interaction not only are the lifetimes of the ring dark solitons largely extended but also their decaying dynamics are dramatically affected. Before snaking instability sets in, new ring dark solitons are formed, and both the numbers and depths of the ring dark solitons exhibit collective oscillations. With the development of instability, the system exhibits different decaying processes, and a variety of decay profiles, such as vortex necklace, distorted octagon, vortex-antivortex ring, and cross, are formed, showing a strong dependence on the modulation frequency of the intercomponent interaction and the initial depth of the ring dark soliton.