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The aim of this study was to bring forth a model-based control method for trajectory tracking of a normal human-like biped on descending and ascending stairs using the straight knee state and to verify the controller performance in the straight knee state periods. Contemplating a straight state in knee joints does not consent to each leg’s shank link to undertake angular rotations greater than thigh link’s rotations. For this purpose, among suggested methods, the self-impact joint constraint has been recommended for energy-efficient (normal) bipedal walking with the implementation of straight knee constraint. To reach this intention, dynamical motion equations are derived, developed and modified to comprise self-impact joint constraint at the knee joint. For gaining control over this complex dynamical system, accessible anthropometric stair gait cycle data are used to derive desired trajectories of thigh and shank links of self-impact biped. Furthermore, for taking the stair ascent and descent of the biped under control, a nonlinear intelligent controller with adaptive neural network control method was put forward. In proportion to simulation results on stair ascending and descending of biped robot, the biped could track the desired angular positions and velocities of the links very well despite having complex nonlinear terms in its governing dynamical equations due to the presence of self-impact constraints. Moreover, during the activation period of joint self-impact, an error of link position angles was evidently improved in comparison to unconstrained biped.

This paper is concerned with the mean-square exponential stability and performance analysis, stabilization, dissipative control and dissipativity-based sliding mode control (SMC) problems for discrete-time switched stochastic hybrid systems. Such systems can be also regarded as switched hybrid systems with stochastic perturbation. Event detectors, which can save computational resource, are designed to determine whether the current data should be transmitted or not in the design process. Firstly, by proposing an average dwell time method combining with a switched (mode-dependent) Lyapunov function, a sufficient condition is established under the event-triggering scheme, which guarantees that the considered switched stochastic system is mean-square exponentially stable, and then an explicit parametrization of the desired stabilization controller is also given. Secondly, the event-triggering dissipativity analysis and the dissipative control problems are investigated. A dissipativity performance condition is proposed to guarantee the mean-square exponential stability and strict dissipativity for the discrete-time switched stochastic system, and then the dissipative controller is implemented by the state feedback. Thirdly, a switched observer is constructed to estimate unavailable state variables, based on which a differential-type sliding surface function is designed, with which the sliding mode dynamical equations can be easily obtained. The resulting closed-loop system with sliding mode dynamics is an autonomous switched stochastic system. Dissipativity analysis and synthesis are both investigated for the closed-loop system, and consequently sufficient conditions are derived, which pave the way for solving the event-triggering observer-based dissipative sliding mode control problem. Moreover, an event-triggering output-based SMC law is first synthesized to drive the system trajectories to enter a predefined sliding vicinity. Finally, illustrative examples are employed to validate the proposed controller design schemes.

This paper investigates the dynamics of cylindrical collapse with perfect fluid distribution in the background of curvature-matter coupled gravity. Using Misner-Sharp technique, we construct dynamical equations which lead to analyze the effects of correction terms and fluid parameters on the collapse process. A correlation between correction terms, fluid parameters and the Weyl scalar is also formulated. In the case of constant f ( G , T ) model, it is concluded that homogeneity of energy density yields the conformal flatness and vice-versa. We find that the collapse rate is reduced due to anti-gravitational behavior of the correction terms.

The approach to nonholonomic Ricci flows and geometric evolution of regular Lagrange systems [S. Vacaru: J. Math. Phys. \textbf{49} (2008) 043504 \& Rep. Math. Phys. \textbf{63} (2009) 95] is extended to include geometric mechanics and gravity models on Lie algebroids. We prove that such evolution scenarios of geometric mechanics and analogous gravity can be modelled as gradient flows characterized by generalized Perelman functionals if an equivalent geometrization of Lagrange mechanics [J. Kern, Arch. Math. (Basel) \textbf{25} (1974) 438] is considered. The R. Hamilton equations on Lie algebroids describing Lagrange-Ricci flows are derived. Finally, we show that geometric evolution models on Lie algebroids are described by effective thermodynamical values derived from statistical functionals on prolongation Lie algebroids., latex2e, 11pt, 25 pages, v3 substantially modified with a changed title and new co-author - accepted to Can. J. Phys

We consider position-dependent mass (PDM) Lagrangians/Hamiltonians in their standard textbook form, where the long-standing \emph{gain-loss balance} between the kinetic and potential energies is kept intact to allow conservation of total energy (i.e., $L=T-V$, $H=T+V$, and $dH/dt=dE/dt=0$). Under such standard settings, we discuss and report on $n$-dimensional PDM damped harmonic oscillators (DHO). We use some $n$-dimensional point canonical transformation to facilitate the linearizability of their $n$-PDM dynamical equations into some $n$-linear DHOs' dynamical equations for constant mass setting. Consequently, the well know exact solutions for the linear DHOs are mapped, with ease, onto the exact solutions for PDM DHOs. A set of one-dimensional and a set of $n$-dimensional PDM-DHO illustrative examples are reported along with their phase-space trajectories., 15 pages, 20 figures

Isostable reduction is a powerful technique that can be used to characterize behaviors of nonlinear dynamical systems using a basis of slowly decaying eigenfunctions of the Koopman operator. When the underlying dynamical equations are known, previously developed numerical techniques allow for high-order accuracy computation of isostable reduced models. However, in situations where the dynamical equations are unknown, few general techniques are available that provide reliable estimates of the isostable reduced equations, especially in applications where large magnitude inputs are considered. In this work, a purely data-driven inference strategy yielding high-accuracy isostable reduced models is developed for dynamical systems with a fixed point attractor. By analyzing steady-state outputs of nonlinear systems in response to sinusoidal forcing, both isostable response functions and isostable-to-output relationships can be estimated to arbitrary accuracy in an expansion performed in the isostable coordinates. Detailed examples are considered for a population of synaptically coupled neurons and for the one-dimensional Burgers’ equation. While linear estimates of the isostable response functions are sufficient to characterize the dynamical behavior when small magnitude inputs are considered, the high-accuracy reduced order model inference strategy proposed here is essential when considering large magnitude inputs.

Analogs are nearest neighbors of the state of a system. By using analogs and their successors in time, one is able to produce empirical forecasts. Several analog forecasting methods have been used in atmospheric applications and tested on well-known dynamical systems. Although efficient in practice, theoretical connections between analog methods and dynamical systems have been overlooked. Analog forecasting can be related to the real dynamical equations of the system of interest. This study investigates the properties of different analog forecasting strategies by taking local approximations of the system's dynamics. We find that analog forecasting performances are highly linked to the local Jacobian matrix of the flow map, and that analog forecasting combined with linear regression allows to capture projections of this Jacobian matrix. The proposed methodology allows to estimate analog forecasting errors, and to compare different analog methods. These results are derived analytically and tested numerically on two simple chaotic dynamical systems., Article submitted to the Journal of Atmospheric Sciences

10 pags., In this article, we reexamine the derivation of the dynamical equations of the Ashtekar-Olmedo-Singh black hole model in order to determine whether it is possible to construct a Hamiltonian formalism where the parameters that regulate the introduction of quantum geometry effects are treated as true constants of motion. After arguing that these parameters should capture contributions from two distinct sectors of the phase space that had been considered independent in previous analyses in the literature, we proceed to obtain the corresponding equations of motion and analyze the consequences of this more general choice. We restrict our discussion exclusively to these dynamical issues. We also investigate whether the proposed procedure can be reconciled with the results of Ashtekar, Olmedo, and Singh, at least in some appropriate limit., This work has been supported by Project. No. MICINN FIS 2017–86497-C2-2-P from Spain (with extension Project. No. MICINN PID 2020-118159GB-C41 under evaluation). The project that gave rise to these results received the support of a fellowship from “la Caixa” Foundation (ID 100010434). The fellowship code is LCF/BQ/DR19/11740028. Partial funds for open access publication have been received from CSIC

The evolution of multidrug resistance (MDR) is a pressing public health concern. Yet many aspects, such as the role played by population structure, remain poorly understood. Here we argue that studying MDR evolution by focusing upon the dynamical equations for linkage disequilibrium (LD) can greatly simplify the calculations, generate more insight, and provide a unified framework for understanding the role of population structure. We demonstrate how a general epidemiological model of MDR evolution can be recast in terms of the LD equations. These equations reveal how the different forces generating and propagating LD operate in a dynamical setting at both the population and metapopulation levels. We then apply these insights to show how the LD perspective: (i) explains equilibrium patterns of MDR, (ii) provides a simple interpretative framework for transient evolutionary dynamics, and (iii) can be used to assess the consequences of different drug prescription strategies for MDR evolution.

We study nonlinear dynamical equations for coupled conserved and non-conserved fields describing nanoparticle concentration and liquid crystal order parameter, respectively, and solve them numerically over bidimensional domains. These equations model the rapid segregation of nanoparticles away from nematic domains, which has been observed experimentally in a suspension of gold nanoparticles in 5CB below the isotropic-nematic transition temperature. We contrast the different behaviors obtained when the LC order parameter is treated as a scalar or a tensor, as well as the different rates of evolution observed with each of these. We find, after an instantaneous quench lowering the temperature below the transition one, an initial linear regime where the ordering of the nematic phase proceeds exponentially with time. Only after a lag period the nanoparticle material couples effectively to the LC order parameter and segregates to regions that are less orientationally ordered (extended domain walls for a scalar order parameter, but point disclinations for a tensor one). The lag period is followed by the onset of nonlinear dynamics and saturation of the order parameter. The choice of a scalar or tensor LC order parameter does not change this sequence but results in a clear overshooting of the nonlinear saturation level for the tensor order parameter case. These results are found to be insensitive to weak anchoring due to coupling of gradients of the conserved and non-conserved variables, for the nanoparticle concentrations and anchoring parameters studied. Our modeling approach can be extended in a straightforward manner to cases where the cooling rate is finite and to other systems where a locally conserved concentration is coupled to a orientation field, such as active Langmuir monolayers, and possibly to other examples of nonlinear dynamics in ecological or excitable media problems.

In this paper, we have investigated the non-adiabatic spherical gravitational collapse in the framework of the $f(R,T)$ theory of gravity with a locally anisotropic fluid that undergoes dissipation in the form of heat flux, free-streaming radiation, and shearing viscosity. The dynamical equations are analyzed in detail, both in the Newtonian and post-Newtonian regimes. Finally we couple the dynamical equations to the full causal transport equation in the context of Israel-Stewart theory of dissipative systems. This yields us a better understanding of the collapse dynamics and may be connected to various astrophysical consequences., 22 pages, accepted for publication in EPJP

We studied the impact of field heterogeneity on entrainment in a system of uniformly interacting phase oscillators. Field heterogeneity is shown to induce dynamical heterogeneity in the system. In effect, the heterogeneous field partitions the system into interacting groups of oscillators that feel the same local field strength and phase. Based on numerical and analytical analysis of the explicit dynamical equations derived from the periodically forced Kuramoto model, we found that the heterogeneous field can disrupt entrainment at different field frequencies when compared to the homogeneous field. This transition occurs when the phase- and frequency-locked synchronization between groups of oscillators is broken at a critical field frequency, causing each group to enter a new dynamical state (disrupted state). Strikingly, it is shown that disrupted dynamics can differ between groups., 10 pages, 7 figures

Self-gravitating kink solutions of a two-dimensional dilaton gravity are revisited in this work. Analytical kink solutions are derived from a concise superpotential formalism of the dynamical equations. A general analysis on the linear stability is conducted for an arbitrary static solution of the model. After gauge fixing, a Schr\"odinger-like equation with factorizable Hamiltonian operator is obtained, which ensures the linear stability of the solution., Comment: 17 pages, 4 figures

A study of a complete cosmological model based on an asymmetric scalar doublet represented by the classical and phantom scalar Higgs fields is carried out. At the same time, the assumption about the nonnegativity of the expansion rate of the Universe, which in some cases contradicts the complete system of Einstein's equations, was removed. A closed system of dynamic equations describing the evolution of the cosmological model is formulated, and the dependence of the topology of the Einstein - Higgs hypersurface of the 5-dimensional phase space of the dynamical system that determines the global properties of the cosmological model on the fundamental constants of the model is investigated. A qualitative analysis of the dynamical system of the corresponding cosmological model is carried out, asymptotic phase trajectories are constructed, and the results of numerical modeling are presented, illustrating various types of behavior of the cosmological model., 30 pages, 66 figures, 52 references

We present a framework for predicting the interactions between motion at a single scale and the underlying stress fluctuations in wall turbulence, derived from approximations to the Navier–Stokes equations. The dynamical equations for an isolated scale and stress fluctuations at the same scale are obtained from a decomposition of the governing equations and formulated in terms of a transfer function between them. This transfer function is closely related to the direct correlation coefficient of Duvvuri & McKeon (J. Fluid Mech., vol. 767, 2015, R4), and approximately to the amplitude modulation coefficient described in Mathis et al. (J. Fluid Mech., vol. 628, 2009, pp. 311–337), by consideration of interactions between triadically consistent scales. In light of the agreement between analysis and observations, the modelling approach is extended to make predictions concerning the relationship between very-large motions and small-scale stress in the logarithmic region of the mean velocity. Consistent with experiments, the model predicts that the zero-crossing height of the amplitude modulation statistic coincides with the wall-normal location of the very large-scale peak in the one-dimensional premultiplied spectrum of streamwise velocity fluctuations, the critical layer location for the very large-scale motion. Implications of fixed phase relationships between small-scale stresses and larger isolated scales for closure schemes are briefly discussed.

Erosion can dramatically change the dynamics and deposition morphology and escalate the destructive power of a landslide by rapidly amplifying its volume, turning it into a catastrophic event. Mobility is the direct measure of the thread posed by an erosive landslide as it plays a dominant role in controlling the enormous impact energy. However, no clear-cut mechanical condition has been presented so far for when and how the erosive landslide gains or loses energy resulting in enhanced or reduced mobility. We pioneer a mechanical model for the energy budget of an erosive landslide that delineates the enhanced or reduced mobility. A fundamentally new understanding is that the increased inertia due to the increased mass is not related to the landslide velocity, but it is associated with the distinctly different entrainment velocity emerging from the inertial frame of reference. The true inertia can be much less than incorrectly proposed previously. We eliminate the existing erroneous perception and make a breakthrough in correctly determining the mobility of the erosive landslide. We reveal that the erosion velocity plays an outstanding role in appropriately determining the energy budget of the erosive landslide. Crucially, whether the erosion related mass flow mobility will be enhanced, reduced or remains unaltered depends exclusively on whether the newly constructed energy generator is positive, negative or zero. This provides a first-ever explicit mechanical quantification of the state of energy, and thus, the precise description of mobility. This becomes a game-changer and fully addresses the long-standing scientific question of why and when some erosive landslides have higher mobility, while others have their mobility reduced. By introducing three important novel mechanical concepts: erosion-velocity, entrainment-velocity and energy-velocity, we demonstrate that the erosion and entrainment are essentially different processes. With this, we draw a central inference: that the landslide gains energy and enhances its mobility if the erosion velocity is greater than the entrainment velocity. The energy velocity delineates the three excess energy regimes: positive, negative and zero. We establish a mechanism of landslide-propulsion that emerges from the net momentum production, providing the erosion-thrust to the landslide. Analytically obtained velocity quantifies the effect of erosion in landslide mobility and indicates the fact that erosion can have the major control on the landslide dynamics. We have also presented a full set of dynamical equations in conservative form in which the momentum balance correctly includes the erosion induced change in inertia and the momentum production. This is a great advancement in legitimate simulation of landslide motion with erosion.

The purpose of this paper is to introduce an active strategy for designing a fault tolerant control system based on the virtual actuator. The proposed method is the fault hiding idea to compensate for the flexible robot links with hysteresis system faults composed of nonlinear dynamical equations. The control loop is configured so that any existing nominal controller designed for the fault-free system can continue to work in the presence of faults. The proposed technique is an active fault tolerant control strategy that reconfigures the virtual actuator with faults and operational point changes. The controller maintains the stability and performance of the closed-loop system at the time of the fault. The simulation results show that the control performance of the flexible robot links with the hysteresis system is desirable. Using the designed controller for the faulty system that deviates from the stable state, the output is stable.

In this present paper, we study the cosmological evolution of the cubic galileon along with modified teleparallel gravity at perturbed and non-perturbed levels. We show the dynamical equations of motion and investigate the evolution of different cosmological parameters by using the dynamical variables analysis. In addition, a detailed analysis of different cosmological evolution in the matter, radiation and de Sitter eras is presented by solving the dynamical equations numerically. In our analysis, we find that the equations of motion in the Friedmann–Robertson–Walker (FRW) background metric is characterized by a stable de Sitter era and a tracker solution in which $$H{\dot{\varphi }}$$ H φ ˙ is always constant. We find also that the equation of state of dark energy associated to the proposed model in this work can deviate from − 2 at the matter era. Moreover, the conditions of avoiding ghost and Laplacian instabilities in our model are derived; then we show that the model is free of these instabilities. Furthermore we place an observational constraint on the parameters of the model through Monte Carlo numerical method using growth rate and observational Hubble data. Finally, using the best-fit values of parameters in the model we compare our growth rate of matter perturbation with the prediction of $$\varLambda $$ Λ CDM model and the latest measurement.

I argue that a particle ontology naturally emerges from the basic dynamical equations of non-relativistic quantum mechanics, when the quantum continuity equation is realistically interpreted. This was recognized by J.J. Sakurai in his famous textbook "Modern Quantum Mechanics", and then dismissed on the basis of the Heisenberg position-momentum uncertainty principle. In this paper, I show that the reasons of this rejection are based on a misunderstanding of the physical import of the uncertainty principle. As a consequence, a particle ontology can be derived from the quantum formalism without the need of additional ad hoc assumptions, and therefore it cannot be regarded as "extra-structure".

A fully analytical treatment of the base-pair and codon dynamics in double-stranded DNA molecules is introduced, by means of a realistic treatment that considers different mass values for G, A, T, and C nucleotides and takes into account the intrinsic three-dimensional, helicoidal geometry of DNA in terms of a Hamitonian in cylindrical coordinates. Within the framework of the Peyrard&ndash, Dauxois&ndash, Bishop model, we consider the coupling between stretching and stacking radial oscillations as well as the twisting motion of each base pair around the helix axis. By comparing the linearized dynamical equations for the angular and radial variables corresponding to the bp local scale with those of the longer triplet codon scale, we report an underlying hierarchical symmetry. The existence of synchronized collective oscillations of the base-pairs and their related codon triplet units are disclosed from the study of their coupled dynamical equations. The possible biological role of these correlated, long-range oscillation effects in double standed DNA molecules containing mirror-symmetric codons of the form XXX, XX&rsquo, X, X&rsquo, XX&rsquo, YXY, and XYX is discussed in terms of the dynamical equations solutions and their related dispersion relations.

In this study, we construct the new solitary wave solutions for two well known nonlinear wave equations whose describe the wave propagation in nonlinear media. We implemented a new technique which is extension of modified rational expansion method on nonlinear three-dim modified Kortewege–de Vries Zakharov–Kuznetsov and extended Zakharov–Kuznetsov equations. The new results are elliptic, hyperbolic, trigonometric, rational type which are representing to solitary waves, periodic waves, traveling waves, kink-antikink solitons, dark-bright solitons. The physical behavior of some calculated results represented graphical by using the software Mathematica. The determined results are new, more general and have numerous applications in the area of nonlinear sciences including nonlinear plasma, fiber optics, mechanics, quantum physics, laser physics, fluid physics, engineering and other types of physical sciences. The demonstrated results show that new technique is efficient, fruitful, powerful and reliable to study analytically different types of higher order complex NLPDEs arises in various types of applied sciences.

The framework of a kind of noncanonical warm inflation is introduced, and the dynamical equations of this scenario are presented. We propose the slow roll approximations and give some redefining slow roll parameters in this scenario which remain dimensionless. Performing systemic stability analysis, we calculate the slow roll conditions to guarantee that slow roll approximations hold. The slow roll conditions suggest slow roll inflation in general noncanonical warm inflationary scenario can still exist, and in addition, the slow roll approximations are more easily to be satisfied. Then, a concrete Dirac-Born-Infeld warm inflationary model is studied., 6 pages, 0figures

The objective of this paper is to study the propagation of Love waves in a pre-stressed anisotropic porous layer which is a sandwich between a rigid layer and a non-homogeneous elastic half-space. Dynamical equations of motion are based on Biot’s incremental deformation theory. Dispersion equations of phase velocity of Love waves have been derived. The impact of anisotropy, pre-stresses, and porosity on the phase velocity are examined in details. It has been observed that the velocity of Love waves increases when the porosity of the layer decreases. Also, velocity of Love waves increases when anisotropy in the layer and compressive initial stresses both are increased. Numerical computations for the velocity of Love waves have been carried out and the results are plotted in different graphs.

Snake robot is a complicated dynamical system comprised of large number of inputs and generalized coordinates. Moreover, the dynamical equations of motion are highly nonlinear and coupled. The aforementioned complexities make path planning and control system design particularly difficult for this kind of system. The control approaches presented in Chaps. 1, 2, 3 and 4 confirm this observation as they employ a multi-layer control methodology to compute the torque input which is not directly mapped to the output space trajectory of the robot. Furthermore, dynamically capable path planning is still beyond the reach for snake robots. Hence, to address these issues and go beyond the conventional approach, it is required to map the control inputs directly to the output variables. This can be accomplished by adopting a differential-flatness-based approach. Differential flatness is a method of transforming a nonlinear system into a flat system, i.e. a corresponding system of flat outputs by establishing a diffeomorphic relation between the states and the outputs [1, 2, 3]. The flat outputs chosen should be either measurable or be computed from the measured variables. Therefore, trajectories designed as functions of flat outputs become convenient and the tracking control problem becomes simpler as well.

A general equation of state is considered for analysing the possible behaviors for (Early) dark energy that alleviates the Hubble parameter tension problem. By departing from the possible evolution for the (Early) dark energy density and the corresponding dynamical equations, the equation of state is obtained, which allow us to analyze qualitatively the cosmological evolution and the dominance of each term in the equation of state along the cosmic expansion, which show some interesting consequences as the occurrence of (past) future singularities. Then, by considering two general models, their free parameters are fit with different sources of data, showing the goodness of the fits in comparison to more standard models. Results might be considered as a promising starting point to get a better understanding of the cosmological evolution as a whole., 19 pages, 2 figures, analysis extended, version accepted for publication in Phys. of the Dark Universe

The present work aims to extend the hybrid non zonal RANS/LES partially integrated transport modeling (PITM) method to turbulent flows in the presence of passive scalar contaminant for simulating large scales of turbulent flows. Focussing on the methodological aspects, we derive the basic transport equations both for the scalar variance of fluctuations and dissipation-rate of the variance. The basis of the method was introduced in references [R. Schiestel and A. Dejoan, “Towards a new partially integrated transport model for coarse grid and unsteady turbulent flow simulations ”, Theor. Comput. Fluid Dyn. 18, 443 (2005)] and [B. Chaouat and R. Schiestel, “A new partially integrated transport model for subgrid-scale stresses and dissipation rate for turbulent developing flows ”, Phys. Fluids 17, 065106 (2005)]. It provides a continuous approach for hybrid Reynolds averaged Navier–Stokes equations-large eddy simulation (RANS-LES) with seamless coupling between RANS and LES regions. The main motivation is to simulate accurately in LES mode performed on coarse grids, scalar fluctuation fields, in addition to mean scalar fields. As known, the knowledge of the rms scalar fluctuations is often involved in handling practical engineering and geophysical flows. Like in dynamical equations, it is found that the coefficient appearing in the destruction term of the dissipation-rate of the scalar variance is a function of the cutoff-wave number and also of the Prandtl number and the Reynolds number. Depending on the value of the Prandtl-number, different expressions of this function have been derived according to the relevant physics in the wave number space. Finally, numerical simulations of fully turbulent flows including passive scalar transport fields have been performed on several meshes of medium and coarse grid resolutions at the Reynolds number R τ = 395 for the Prandtl numbers P r = 0.1 , 1 and 10, respectively associated with heat transfer of liquid metals, gas and water for illustrating the theoretical development made on PITM. As expected, the PITM method provides satisfactory results in good agreement with data of direct numerical simulations. From a general point of view, this work opens new routes of modeling and simulation of turbulent flows including a passive scalar with a drastic reduction of the computational time and memory in term of number of grid points in comparison with the demanding resources of highly resolved LES.

The Einstein equations applied to the FLRW metric give basic dynamical equations for cosmology, specifically for the scale factor. The dynamical equations depend on the physical properties of the constituents of the cosmic fluid, which we take to be vacuum or dark energy, cold matter, radiation, and an effective curvature. Together with the behavior of the constituents the dynamical equations lead to what we here call the Friedmann master equation for the scale factor of the universe; it is remarkably useful.

We perform numerical simulations of a long-range spherical spin glass with two and three body interaction terms. We study the gradient descent dynamics and the inherent structures found after a quench from initial conditions, well thermalized at temperature $T_{in}$. In large systems, the dynamics strictly agrees with the integration of the mean-field dynamical equations. In particular, we confirm the existence of an onset initial temperature, within the liquid phase, below which the energy of the inherent structures undoubtedly depends on $T_{in}$. This behavior is in contrast with that of pure models, where there is a 'threshold energy' that attracts all the initial configurations in the liquid. Our results strengthen the analogy between mean-field spin glass models and supercooled liquids., 15 pages, 8 figures

We present a detailed study of the dynamics associated with the ghost sector of quenched QCD in the Landau gauge, where the relevant dynamical equations are supplemented with key inputs originating from large-volume lattice simulations. In particular, we solve the coupled system of Schwinger-Dyson equations that governs the evolution of the ghost dressing function and the ghost-gluon vertex, using as input for the gluon propagator lattice data that have been cured from volume and discretization artifacts. In addition, we explore the soft gluon limit of the same system, employing recent lattice data for the three-gluon vertex that enters in one of the diagrams defining the Schwinger-Dyson equation of the ghost-gluon vertex. The results obtained from the numerical treatment of these equations are in excellent agreement with lattice data for the ghost dressing function, once the latter have undergone the appropriate scale-setting and artifact elimination refinements. Moreover, the coincidence observed between the ghost-gluon vertex in general kinematics and in the soft gluon limit reveals an outstanding consistency of physical concepts and computational schemes., Comment: 34 pages, 12 figures

We present a complete theory of higher-order autonomous contact mechanics, which allows us to describe higher-order dynamical systems with dissipation. The essential tools for the theory are the extended higher-order tangent bundles, ${\rm T}^kQ\times{\mathbb R}$, whose geometric structures are previously introduced in order to state the Lagrangian and Hamiltonian formalisms for these kinds of systems, including their variational formulation. The variational principle, the contact forms, and the geometric dynamical equations are obtained by using those structures and generalizing the standard formulation of contact Lagrangian and Hamiltonian systems. As an alternative approach, we develop a unified description that encompasses the Lagrangian and Hamiltonian equations as well as their relationship through the Legendre map; all of them are obtained from the contact dynamical equations and the constraint algorithm that is implemented because, in this formalism, the dynamical systems are always singular. Some interesting examples are finally analyzed using these geometric formulations., 39 pp. Minor changes. References updated

Recent years have seen the flourishing of research devoted to quantum effects on mesoscopic and macroscopic scales. In this context, in Entropy 2019, 21, 705, a formalism aiming at describing macroscopic quantum fields, dubbed Reduced State of the Field (RSF), was envisaged. While, in the original work, a proper notion of entropy for macroscopic fields, together with their dynamical equations, was derived, here, we expand thermodynamic analysis of the RSF, discussing the notion of heat, solving dynamical equations in various regimes of interest, and showing the thermodynamic implications of these solutions., Comment: 10 pages, 1 figure

A complete bifurcation analysis of explicit dynamical equations for the periodically forced Kuramoto model was performed in [L. M. Childs and S. H. Strogatz. Chaos 18 , 043128 (2008)], identifying all bifurcations within the model. We show that the phase diagram predicted by this analysis is incomplete. Our numerical analysis of the equations reveals that the model can also undergo an abrupt phase transition from oscillations to wobbly rotations of the order parameter under increasing field frequency or decreasing field strength. This transition was not revealed by bifurcation analysis because it is not caused by a bifurcation, and can neither be classified as first nor second-order since it does not display critical phenomena characteristic of either transition. We discuss the topological origin of this transition and show that it is determined by a singular point in the order-parameter space., 5 pages, 4 figures

In this chapter, the most important mathematical tools related to fractional calculus are presented. Besides, it is presented an introduction to the theory of dynamical systems with dynamical equations that present fractional-order integrals and derivatives, as well as some fractional-order controllers that have been proposed, and some stability results developed for this kind of systems.

We investigate the linear dynamics of an electromagnetic field propagating in curved spacetime in the presence of plasma. The dynamical equations are generically more involved and richer than the effective Proca equation adopted as a model in previous work. We discuss the general equations and focus on the case of a cold plasma in the background of a spherically-symmetric black hole, showing that the system admits plasma-driven, quasibound electromagnetic states that are prone to become superradiantly unstable when the black hole rotates. The quasibound states are different from those of the Proca equation and have some similarities with the case of a massive scalar field, suggesting that the linear instability can be strongly suppressed compared to previous estimates. Our framework provides the first step towards a full understanding of the plasma-photon interactions around astrophysical black holes., 11 pages, 4 figures. Matches the published version

We introduce an improved form for the anisotropic hydrodynamics distribution function which explicitly takes into account the free-streaming and equilibrating contributions separately. We demonstrate that with this improvement one can better reproduce exact results available in the literature for the evolution of moments of the distribution function, in particular, for moments which contain no powers of the longitudinal momentum in their definition (m=0 moments). Using the resulting dynamical equations, we extract the non-equilibrium attractor associated with our improved aHydro ansatz and demonstrate that the improvement also allows one to better reproduce the exact dynamical attractor obtained using kinetic theory in the relaxation time approximation, particularly at early rescaled times and for m=0 moments., 22 pages, 4 figures

Using the phase field crystal model (PFC model), an analysis of slow and fast dynamics of solid-liquid interfaces in solidification and melting processes is presented. Dynamical regimes for cubic lattices invading metastable liquids (solidification) and liquids propagating into metastable crystals (melting) are described in terms of the evolving amplitudes of the density field. Dynamical equations are obtained for body-centered cubic (bcc) and face-centered cubic (fcc) crystal lattices in one- and two-mode approximations. A universal form of the amplitude equations is obtained for the three-dimensional dynamics for different crystal lattices and crystallographic directions. Dynamics of the amplitude's propagation for different lattices and PFC mode's approximations is qualitatively compared. The traveling-wave velocity is quantitatively compared with data of molecular dynamics simulation previously obtained by Mendelev et al. [Modell. Simul. Mater. Sci. Eng. 18, 074002 (2010)MSMEEU0965-039310.1088/0965-0393/18/7/074002] for solidification and melting of the aluminum fcc lattice.

In this manuscript, two mathematical approaches have been functionalized to discover novel wave results of 3rd-order Klein–Gordon and Tzitzeica equations. With the alliance of Mathematica, the competency of these methods for discovering these exact solutions have been more exhibited. As a result, several solitary solutions are constructed and indicated by hyperbolic solutions, diverse combinations of trigonometric and exponential results. Furthermore, employed techniques are more efficient techniques for exploring essential nonlinear waves that enhance a variety of dynamic models that arises in nonlinear fields. All drafting is given out to express the properties of the innovative explicit analytic solutions. Hence our proposed schemes are directed, succinct, and reasonably good for the various nonlinear evaluation equations (NLEEs) related treatment and mathematical physics also.

We obtain an exact correspondence between the dynamical equations in Israel-Stewart (IS) theory and first-order causal and stable (FOCS) hydrodynamics for a boost-invariant system with an ideal gas equation of state at finite baryon chemical potential. Explicit expressions for the temperature and chemical potential dependence of the regulators in the FOCS theory are given in terms of the kinetic coefficients and constant relaxation time of the IS theory. Using the correspondence between the IS and FOCS theory, stability conditions for a charged fluid which are known in the FOCS approach are applied and one finds that the IS theory considered is unstable., 8 pages, no figures

Energy consumption prediction is important for improving the energy efficiency of underwater gliders. This paper establishes a novel prediction method for energy consumption of underwater gliders based on Least Squares Support Vector Machine (LSSVM). To improve the performance of the LSSVM model, the Particle Swarm Optimization (PSO) algorithm is introduced to optimize its parameters. Sea trial data obtained by a glider are used to train and validate the model. The results demonstrate that the LSSVM-PSO model has a higher prediction accuracy than the general energy consumption model derived from the dynamical equations in our previous study.

Tropical cyclone (TC) motion has an important impact on both human lives and infrastructure. Predicting TC intensity is crucial, especially within the 24 h warning time. TC intensity change prediction can be regarded as a problem of both regression and classification. Statistical forecasting methods based on empirical relationships and traditional numerical prediction methods based on dynamical equations still have difficulty in accurately predicting TC intensity. In this study, a prediction algorithm for TC intensity changes based on deep learning is proposed by exploring the joint spatial features of three-dimensional (3D) environmental conditions that contain the basic variables of the atmosphere and ocean. These features can also be interpreted as fused characteristics of the distributions and interactions of these 3D environmental variables. We adopt a 3D convolutional neural network (3D-CNN) for learning the implicit correlations between the spatial distribution features and TC intensity changes. Image processing technology is also used to enhance the data from a small number of TC samples to generate the training set. Considering the instantaneous 3D status of a TC, we extract deep hybrid features from TC image patterns to predict 24 h intensity changes. Compared to previous studies, the experimental results show that the mean absolute error (MAE) of TC intensity change predictions and the accuracy of the classification as either intensifying or weakening are both significantly improved. The results of combining features of high and low spatial layers confirm that considering the distributions and interactions of 3D environmental variables is conducive to predicting TC intensity changes, thus providing insight into the process of TC evolution.

An autonomous holonomic dynamical system is described by a system of second order differential equations whose solution gives the trajectories of the system. The solution is facilitated by the use of first integrals (FIs) that are used to reduce the order of the system of differential equations and, if there are enough of them, to determine the solution. Therefore, in the study of dynamical systems, it is important that there exists a systematic method to determine the FIs of second order differential equations. On the other hand, a system of second order differential equations defines (as a rule) a kinetic energy (or Lagrangian), which provides a symmetric second order tensor that we call the kinetic metric. This metric via its symmetries (or collineations) brings into the scene the differential geometry that provides numerous results and methods concerning the determination of these symmetries. It is apparent that if one manages to provide a systematic way that will relate the determination of the FIs of a given dynamical system to the symmetries of the kinetic metric defined by this very system, then one will have at his/her disposal the powerful methods of differential geometry in the determination of the FIs and, consequently, the solution of the dynamical equations. This was also a partial aspect of Lie’s work on the symmetries of differential equations. The subject of this work is to provide a theorem that realizes this scenario. The method we follow has been considered previously in the literature and consists of the following steps: Consider the generic quadratic FI of the form I=Kab(t,qc)qaqb+Ka(t,qc)qa+K(t,qc), where Kab(t, qc), Ka(t, qc), and K(t, qc) are unknown tensor quantities and require dI/dt = 0. This condition leads to a system of differential equations involving the coefficients Kab(t, qc), Ka(t, qc), and K(t, qc) whose solution provides all possible quadratic FIs of this form. We demonstrate the application of the theorem in the classical cases of the geodesic equations and the generalized Kepler potential in which we obtain all the known results in a systematic way. We also obtain and discuss the time-dependent FIs that are as important as the autonomous FIs determined by other methods.

Utilising Onsager's variational formulation, we derive dynamical equations for the relaxation of a fluid membrane tube in the limit of small deformation, allowing for a contrast of solvent viscosity across the membrane and variations in surface tension due to membrane incompressibility. We compute the relaxation rates, recovering known results in the case of purely axis-symmetric perturbations and making new predictions for higher order (azimuthal) $m$-modes. We analyse the long and short wavelength limits of these modes by making use of various asymptotic arguments. We incorporate stochastic terms to our dynamical equations suitable to describe both passive thermal forces and non-equilibrium active forces. We derive expressions for the fluctuation amplitudes, an effective temperature associated with active fluctuations, and the power spectral density for both the thermal and active fluctuations. We discuss an experimental assay that might enable measurement of these fluctuations to infer the properties of the active noise. Finally we discuss our results in the context of active membranes more generally and give an overview of some open questions in the field., 14 pages, 9 figures

The Kuramoto model serves as a paradigm to study the phenomenon of spontaneous collective synchronization. We study here a nontrivial generalization of the Kuramoto model by including an interaction that breaks explicitly the rotational symmetry of the model. In an inertial frame (e.g., the laboratory frame), the Kuramoto model does not allow for a stationary state, that is, a state with time-independent value of the so-called Kuramoto (complex) synchronization order parameter $z\equiv re^{i\psi}$; Note that a time-independent $z$ implies $r$ and $\psi$ both time independent, with the latter fact corresponding to a state in which $\psi$ rotates at zero frequency (no rotation). In this backdrop, we ask: Does the introduction of the symmetry-breaking term suffice to allow for the existence of a stationary state in the laboratory frame? Compared to the original model, we reveal a rather rich phase diagram of the resulting model, with the existence of both stationary and standing wave phases. While in the former the synchronization order parameter $r$ has a long-time value that is time independent, one has in the latter an oscillatory behavior of the order parameter as a function of time that nevertheless yields a non-zero and time-independent time average. Our results are based on numerical integration of the dynamical equations as well as an exact analysis of the dynamics by invoking the so-called Ott-Antonsen ansatz that allows to derive a reduced set of time-evolution equations for the order parameter., Comment: 10 pages, 7 figures

Double Rotary inverted pendulum (DRIP) is an important member of nonlinear, unstable, non-minimum phase, and under-actuated mechanical systems. The DRIP is known widely as experimental setup for testing different kind of control algorithms. This paper, described a development of nonlinear dynamical equations of the DRIP system using Euler-Lagrange methods. Euler-Lagrange methods does not requisite complicated and tedious formulation since DRIP is not large multi-body system. The linear model and state space representation was also presented. The Simulink model of DRIP was developed based on the derived equations. Simulation study was carried out and the results indicated that, the DRIP system is inherently nonlinear and unstable. It is realized that the difficulties and limitations in the previous dynamic equation of DRIP proposed in literature are eliminated. Euler-Lagrange methods can be regarded as an alternative method for finding the dynamic model of the systems.

Deterministic epidemic models, such as the Susceptible-Infected-Recovered (SIR) model, are immensely useful even if they lack the nuance and complexity of social contacts at the heart of network science modeling. Here we present a simple modification of the SIR equations to include the heterogeneity of social connection networks. A typical power-law model of social interactions from network science reproduces the observation that individuals with a high number of contacts, “hubs” or “superspreaders”, can become the primary conduits for transmission. Conversely, once the tail of the distribution is saturated, herd immunity sets in at a smaller overall recovered fraction than in the analogous SIR model. The new dynamical equations suggest that cutting off the tail of the social connection distribution, i.e., stopping superspreaders, is an efficient non-pharmaceutical intervention to slow the spread of a pandemic, such as COVID-19.

We show, for the first time, the explicit form of the nonlinear partial differential equations (PDEs) subject to the correct boundary conditions that have to be satisfied by transport coefficients having a vanishing skew-symmetric piece. We also report, for the first time, the nonlinear PDEs (with the appropriate boundary conditions) for transport coefficients when the thermodynamic system is subject to two thermodynamic forces. Since the proposed PDEs have been derived without neglecting any term present in the dynamical equations (i.e., the energy, mass, and momentum balance equations), we propose them as a good candidate for describing transport in thermodynamic systems also far from equilibrium (e.g., in the turbulent regime). The preliminary test was carried out by analyzing a concrete example where Onsager’s relationships manifestly disagree with experience: magnetically confined Tokamak-plasmas. More specifically, we focus our calculations to compute mass and energy transports in Frascati Tokamak Upgrade-plasmas subject to two thermodynamic forces. We show a good agreement between the theoretical predictions and the experimental data. The aim of this study is to apply our approach to the Divertor Tokamak Test Facility, to be built in Italy, and to the International Thermonuclear Experimental Reactor.

The dynamics of a two-dimensional Bose-Einstein condensate in a presence of quantum fluctuations is studied. The properties of localized density distributions, quantum droplets (QDs), are analyzed by means of the variational approach. It is demonstrated that the super-Gaussian function gives a good approximation for profiles of fundamental QDs and droplets with non-zero vorticity. The dynamical equations for parameters of QDs are obtained. Fixed points of these equations determine the parameters of stationary QDs. The period of small oscillations of QDs near the stationary state is estimated. It is obtained that periodic modulations of the strength of quantum fluctuations can actuate different processes, including resonance oscillations of the QD parameters, an emission of waves and a splitting of QDs into smaller droplets., 7 pages, 7 figures. Phys. Rev. E, accepted (2020)

The research represented in this thesis pertains to applying digital control to that of a Quadruple-Tank Process via implementation of State-Space theory. Having arranged the tank apparatus into the desired configurations of both the Coupled and Quadruple-Tank Processes, a mathematical model was applied that properly represented each systems’ dynamics. From these dynamical equations linearized state-space models were formed, which would be the basis for computer aided control and simulation. Through software application of a linear state feedback tracking system both tank processes were able to be successfully controlled. Complex control issues regarding plant saturation and integrator wind-up are addressed giving way to a more responsive system that is stable under a larger range of unanticipated modeling errors. Analysis of the systems performance has shown good results in both simulation and experimental application to the Quanser Lab Tank Systems.

A nematic liquid crystal confined to the surface of a sphere exhibits topological defects of total charge $+2$ due to the topological constraint. In equilibrium, the nematic field forms four $+1/2$ defects, located at the corners of a regular tetrahedron inscribed within the sphere, since this minimizes the Frank elastic energy. If additionally the individual nematogens exhibit self-driven directional motion, the resulting active system creates large-scale flow that drives it out of equilibrium. In particular, the defects now follow complex dynamic trajectories which, depending on the strength of the active forcing, can be periodic (for weak forcing) or chaotic (for strong forcing). In this paper we derive an effective particle theory for this system, in which the topological defects are the degrees of freedom, whose exact equations of motion we subsequently determine. Numerical solutions of these equations confirm previously observed characteristics of their dynamics and clarify the role played by the time dependence of their global rotation. We also show that Onsager's variational principle offers an exceptionally transparent way to derive these dynamical equations, and we explain the defect mobility at the hydrodynamics level., 16 pages, 7 figures