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SUMMARYIn this study, we introduce a novel three-dimensional micro-scale robot capable of swimming in low Reynolds number. The proposed robot consists of three rotating disks linked via three perpendicular adjustable rods, actuated by three rotary and three linear motors, respectively. The robot mechanism has an important property which makes it superior to the previously designed micro swimmers. In fact, our goal is designing a micro swimmer which its controllability matrix has full rank and hence it will be capable to perform any desired maneuver in space. After introducing the mechanism and derivation of the dynamical equations of motion, we propose a control method to steer the robot to the desired position and orientation in the presence of external disturbances in the low Reynolds number flow. Simulation results confirm the successful performance of the proposed mechanism and employed control method demonstrating high maneuverability of microrobot.

In this paper, the problem of modeling and anti-disturbance control is studied for lightweight personal robotics (P-Robs) with a six-degree-of-freedom robot manipulator to solve the movement instability phenomenon caused by time-varying uncertain disturbances during the movement of the robot manipulator. The detailed dynamical equations of the P-Rob system are solved based on the Lagrange energy equation, and the actual dynamical model of the robot manipulator system is obtained. The disturbance observer is designed to estimate the disturbance effectively, and an integral sliding mode control algorithm is proposed to realize tracking control. Stability analysis of the system is carried out using the Lyapunov function. Finally, experiments are conducted on an actual P-Rob system model, and the experimental results show that the robot manipulator system tracks the desired trajectory effectively, which validates the effectiveness of the proposed control algorithm.

We investigate the effective Dirac equation, corrected by merging two scenarios that are expected to emerge towards the quantum gravity scale. Namely, the existence of a minimal length, implemented by the generalized uncertainty principle, and exotic spinors, associated with any non-trivial topology equipping the spacetime manifold. We show that the free fermionic dynamical equations, within the context of a minimal length, just allow for trivial solutions, a feature that is not shared by dynamical equations for exotic spinors. In fact, in this coalescing setup, the exoticity is shown to prevent the Dirac operator to be injective, allowing the existence of non-trivial solutions., Comment: 9 pages

There are three types of dynamical equations for a post-Newtonian Lagrangian formulation of three black holes. They are approximate post-Newtonian Lagrangian equations of motion, coherent post-Newtonian Lagrangian equations of motion, and post-Newtonian Hamiltonian canonical equations. The energy integral is conserved approximately in the first model, but exactly in the second and third models. The three formulations are not exactly equivalent, while they are approximately related in the orbital dynamical behavior. Figure-eight orbits in the three problems may have different initial conditions. Imai, Chiba, and Asada adjusted the initial velocities satisfying the figure-eight orbits for given initial positions in the first model. The initial velocities are still suitable for the figure-eight orbits of the latter two models when the initial separations of the first body are enough large. However, they are not generally applicable to relatively small initial separations in the three systems. We can obtain the desired initial velocities by scanning the initial velocities of the third body in the three systems. The figure-eight orbits in these models exhibit sensitive dependence on the scanned initial velocities for smaller initial separations. Although the differences in the scanned initial velocities among the three problems are minor for the small initial separations, the scanned initial velocities for any one of the three systems do not yield the figure-eight orbits for the other two systems.

The prediction of climate change and its impact on extreme weather events is one of the great societal and intellectual challenges of our time. The first part of the problem is to make the distinction between weather and climate. The second part is to understand the dynamics of the fluctuations of the physical variables. The third part is to predict how the variances of the fluctuations are affected by statistical correlations in their fluctuating dynamics. This paper investigates a framework called LA SALT which can meet all three parts of the challenge for the problem of climate change. As a tractable example of this framework, we consider the Euler–Boussinesq (EB) equations for an incompressible stratified fluid flowing under gravity in a vertical plane with no other external forcing. All three parts of the problem are solved for this case. In fact, for this problem, the framework also delivers global well-posedness of the dynamics of the physical variables and closed dynamical equations for the moments of their fluctuations. Thus, in a well-posed mathematical setting, the framework developed in this paper shows that the mean field dynamics combines with an intricate array of correlations in the fluctuation dynamics to drive the evolution of the mean statistics. The results of the framework for 2D EB model analysis define its climate, as well as climate change, weather dynamics, and change of weather statistics, all in the context of a model system of SPDEs with unique global strong solutions.

In this paper, we look into the dynamic motion planning and control of an unmanned aerial vehicle, namely, the quadrotor, governed by its dynamical equations. It is shown for the first time that the Direct or the Second Method of Lyapunov is an effective tool to derive a set of continuous nonlinear control laws that not only provide smooth trajectories from a designated initial position to a designated target, but also continuously minimise the roll and pitch of the quadrotor en route to its targets. The latter successfully addresses the challenging problem of a quadrotor autonomously transporting valuable and fragile payloads safely to the designated target. Computer simulations are used to illustrate the effectiveness of the proposed control laws.

In this work, we have investigated the dynamical instability of spherically symmetric gravitating object under expansion-free condition in Einstein Gauss–Bonnet gravity. In this context, the field equations and dynamical equations have been established in the Gauss–Bonnet gravity. The linear perturbation scheme has been used on the dynamical equations to construct the collapse equation. The Newtonian, post Newtonian and post Newtonian approximations have been applied to investigate the general dynamical (in)stability equations. It has been observed that the instability range of the collapsing source is independent of adiabatic index Γ (stiffness of the fluid does not play any role). The instability range can be determined by the pressure anisotropy, energy density profile, Gauss–Bonnet parameter α and some constraints at Newtonian, post Newtonian and post Newtonian order.

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.

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

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

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.

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.

For theoretical study and engineering application, it is necessary to provide an accurate and simple dynamical model to simulate the multibody mechanical systems with clearance joints and it is also the subject of this article. Based on Lagrange equations of the first kind, a different numerical methodology, the length and rotation angle of the clearance joints are looked as independent coordinates for the first time, is presented in detail. The slider–crank mechanism, with a single or double adjustable revolute clearance joints, is used as a numerical model. A test rig and a simulink model, fully in accordance with the numerical model, are used to measure the velocity, displacement, and acceleration. The numerical results tally with experimental and simulink results reveal that the new methodology, presented in this paper, provides a correct approach to build the dynamical equations of mechanism with clearance joints. Lyapunov exponent is used to analyze the motion status, chaotic or periodic, of the slider. Based on data points, mean absolute deviation (MAD) is applied to judge the dynamical errors, displacement, velocity, and acceleration, of the slider due to clearance joints. With the help of Lyapunov exponent and MAD, the results indicated that various clearance sizes and drive speeds can change the dynamical behaviors of the slider, which is complex but can be predicted in some way.

Aiming to dynamic modeling of a three-link manipulator subjected to motion constraints, a novel explicit approach to the dynamical equations based on Udwadia-Kalaba (UK) theory is established. The motion constraints on the three-link manipulator can be regarded as external constraints of the system. However, it is not easy to obtain explicit equations for the dynamic modeling of constrained systems. For a multibody system subjecting to motion constraints, it is common to introduce Lagrange multipliers, but obtaining an explicit dynamical equation using traditional Lagrange multipliers is difficult. In order to obtain such equations more simply, motion constraints are handled using the UK equation. Compared with the Lagrange method, the UK approach can simplify the analysis and solution of a constrained system, without the need to introduce additional auxiliary variables to solve the Lagrange equation. Based on a more real-life nominal system (whose parameters are known) model considering the uncertain environment, this paper develops a nonlinear controller that satisfies the required trajectory. This controller allows the nonlinear nominal system to track the desired trajectory exactly without linearizations or approximations. These continuous controllers compensate extra force to eliminate the errors caused by uncertainties. The controllers are based on a generalization of sliding surfaces. Error bounds on tracking caused by uncertainties are analytically obtained. The numerical results show the simplicity and efficacy of the proposed methodology, and the reliability of the error bounds.

The physics of Markovian open quantum systems can be described by quantum master equations. These are dynamical equations, that incorporate the Hamiltonian and jump operators, and generate the system's time evolution. Reconstructing the system's Hamiltonian and and its coupling to the environment from measurements is important both for fundamental research as well as for performance-evaluation of quantum machines. In this paper we introduce a method that reconstructs the dynamical equation of open quantum systems, directly from a set of expectation values of selected observables. We benchmark our technique both by a simulation and experimentally, by measuring the dynamics of a trapped $^{88}\text{Sr}^+$ ion under spontaneous photon scattering., 6 pages, 6 figures. Comments are welcome!

The study presents a new method to model the degradation and reliability of multi-function complex systems using the hazard rate matrix and Markovian approximation. The system is hierarchically decomposed into a set of states according to the states of its functions, and the hazard rate matrix is proposed to describe the failure rates of the functions. By using Markovian approximation, the elements of the hazard rate matrix can be expressed as a set of dynamical equations involving the failure information about the functions. Consequently, the dynamical behavior of the degradation for the whole system and its functions can be determined using the observed failure data of those functions instead of the lifetime data of the whole system, eliminating the need for lifetime testing data of the whole system in statistical-based reliability assessment. The overall method is illustrated using an example problem, and is further applied to a power system degradation problem to demonstrate its usage in realistic engineering applications.

Active-Matter models commonly consider particles with overdamped dynamics subject to a force (speed) with constant modulus and random direction. Some models also include random noise in particle displacement (a Wiener process), resulting in diffusive motion at short time scales. On the other hand, Ornstein–Uhlenbeck processes apply Langevin dynamics to the particles’ velocity and predict motion that is not diffusive at short time scales. Experiments show that migrating cells have gradually varying speeds at intermediate and long time scales, with short-time diffusive behavior. While Ornstein–Uhlenbeck processes can describe the moderate-and long-time speed variation, Active-Matter models for over-damped particles can explain the short-time diffusive behavior. Isotropic models cannot explain both regimes, because short-time diffusion renders instantaneous velocity ill-defined, and prevents the use of dynamical equations that require velocity time-derivatives. On the other hand, both models correctly describe some of the different temporal regimes seen in migrating biological cells and must, in the appropriate limit, yield the same observable predictions. Here we propose and solve analytically an Anisotropic Ornstein–Uhlenbeck process for polarized particles, with Langevin dynamics governing the particle’s movement in the polarization direction and a Wiener process governing displacement in the orthogonal direction. Our characterization provides a theoretically robust way to compare movement in dimensionless simulations to movement in experiments in which measurements have meaningful space and time units. We also propose an approach to deal with inevitable finite-precision effects in experiments and simulations.

Reasons and realities such as being non-linear of dynamical equations, being lightweight and unstable nature of quadrotor, along with internal and external disturbances and parametric uncertainties, have caused that the controller design for these quadrotors is considered the challenging issue of the day. In this work, an adaptive sliding mode controller based on neural network is proposed to control the altitude of a quadrotor. The error and error derivative of the altitude of a quadrotor are the inputs of neural network and altitude sliding surface variable is its output. Neural network estimates the sliding surface variable adaptively according to the conditions of quadrotor and sets the altitude of a quadrotor equal to the desired value. The proposed controller stability has been proven by Lyapunov theory and it is shown that all system states reach to sliding surface and are remaining in it. The superiority of the proposed control method has been proven by comparison and simulation results.

Based on Lagrangian mechanics, we present a novel and computationally efficient set of equations of motion in the matrix notation, for unconstrained or constrained mechanical systems including ignorable coordinates. The equations are applicable to multibody systems including holonomic or nonholonomic constraints. It is shown that by appropriate selection of generalized speeds as a new set of motion variables, the constraint reaction forces can be automatically eliminated from the set of developed reduced dynamical equations in a straightforward manner, resulting in a minimal set of dynamic equations. We present simulation results on one constrained and one unconstrained system to demonstrate the advantages of the proposed dynamic modeling approach for these different classes of dynamical systems. According to the results, the proposed dynamic equations not only reduce the number of equations and computational effort, but also better satisfy the invariance of constraints and conserved quantities of systems in comparison to the other conventional methods.

It is well known that the projective Newton–Euler equation and the Lagrange equation of second kind lead to the same result when deriving the dynamical equations of motion for holonomic rigid multibody systems. It can be shown that both approaches follow from the principles of d’Alembert or Jourdain. However, as to the author’s knowledge, no direct rigorous proof for the equivalence of these approaches is given in literature so far when it comes to spatial systems of rigid bodies. In this paper, we present a novel proof that directly addresses the projective Newton–Euler equation and the Lagrange equation of second kind without the detour via variational principles. The proof is mainly based on vector and matrix manipulations and elementary concepts of differential geometry. Although the mathematical framework is thereby kept simple, the argumentation is considerably more complex compared to the case of planar systems of rigid bodies or spatial systems of particles. An illustrative example is presented.

A framework for describing the causal, topology changing, evolution of interacting biomolecules is developed. The quantum dynamical manifold equations (QDMEs) derived from this framework can be related to the causality restrictions implied by a finite speed of light and to Planck's constant to set a transition frequency scale. The QDMEs imply conserved stress-energy, angular-momentum and Noether currents. The functional whose extremisation leads to this result provides a causal, time-dependent, non-equilibrium generalisation of the Hohenberg–Kohn theorem. The system of dynamical equations derived from this functional and the currents J derived from the QDMEs are shown to be causal and consistent with the first and second laws of thermodynamics. This has the potential of allowing living systems to be quantum mechanically distinguished from non-living ones.

Due to the increasing complexity of dynamic systems, it is increasingly difficult for traditional mathematical methods to meet the modeling requirements of complex dynamic systems. With the continuous innovation of computer and big data technologies, massive data can be easily obtained and stored. Therefore, studies of dynamic system modeling through data-driven approaches have attracted more and more researchers’ attention. This paper compares the dynamic mode decomposition method and dynamic equation reconstruction. Taking Lorenz and nonlinear Helmholtz resonant systems as examples, the two methods show the ability to reconstruct and describe the evolution characteristics of the dynamic system. Specifically, the dynamic mode decomposition method can describe the characteristics of the dynamic system more intuitively; however, it cannot provide physical insights. On the other hand, the discovery of dynamic equations from data can more accurately express the physical evolution characteristics of the dynamic system; however, it is easily affected by random noise. Because the dynamic mode decomposition method can obtain a reduced-order model, which can not only retain useful information of the original data but also reduce the noise disturbances, it can effectively improve the noise attenuation and finally reconstructions of differential dynamical equations.

In this study, the dynamic stability of double-walled carbon nanotubes (DWCNTs) conveying pulsating viscous fluid located on visco-Pasternak medium is investigated. The effects of small scale are considered using Eringen's non-local theory. The surrounding medium of DWCNT is anticipated as a visco-Pasternak foundation including the normal, shear, and damping forces. The van der Waals (vdWs) force is considered among the inner and outer layers of DWCNT. Three types of temperature changes, i.e. uniform, linear and sinusoidal temperature rise, through the thickness are studied. The equations of motion are obtained using higher order sinusoidal shear deformation shell theory (SSDT), in which the surface effects are included. Dynamical equations of DWCNT are extracted using the energy method and Hamilton's principle. Due to orthogonal conditions, the governing equations are solved based on Bolotin method. The effects of different parameters such as surface stress, non-local parameter, fluid velocity, Knudsen's number, fluid density, dimensions ratio and various temperature loadings on the dynamic stability regions of DWCNTs are discussed.

The main objectives of this paper are five-fold. The first is to introduce a general dynamic law for all physical motion systems, based on a new variational principle with constraint-infinitesimals. The second is to postulate the potential-descending principle (PDP). We show that PDP is a more fundamental principle than the first and second laws in thermodynamics, and gives rise to dynamical equations for non-equilibrium systems. The third is to demonstrate that the PDP is the first principle to describe irreversibility of all thermodynamic systems, with ther-modynamic potential as the basic physical quantity, rather than entropy. The fourth objective is to examine the problems faced by the Boltzmann equation. We show that the Boltzmann is not a physical law, is created as a mathematical model to obey the entropy-increasing principle (for dilute gases), and consequently is unable to faithfully describe Nature. The fifth objective is to prove an orthogonal-decomposition theorem and a theorem on variation with constraint-infinitesimals, providing the needed mathematical foundations of the dynamical law of physical motion.

This paper contributes to the simulation of transients and stability properties of singular states in engineering and environmental gas–solid systems. Qualitative properties of paths around equilibrium, quasi-equilibrium and disequilibrium states in closed and open systems are investigated by Lyapunov functions or functionals, V, determined by an original method. The method requires first to assume a constant-sign time derivative of V with respect to perturbed dynamical equations, V ˙ , and next to determine V itself from a matrix formula. Such functions V, which may be of an indefinite sign, describe qualitative properties of system paths, and may be derived from the derivatives V ˙ attributed to the constant-sign rates of entropy production or related exergy sink. Examples confirming the effectiveness of the approach for testing stability and qualitative properties of paths in non-reacting and reacting gas–solid systems are presented. A nontrivial example of the research benefit is presented in the paper in the form of the original classification of drying–moistening paths, based on the nontrivial saddle form of V in countercurrent systems and the focus form of V in co-current systems.

In the framework of the Darmois-Israel formalism, the dynamical equations of motion of spherically-symmetric thin-shell wormholes supported by a polytropic phantom energy in Einstein- Born-Infeld theory are constructed. A stability analysis of the spherically-symmetric thin-shell wormhole by using the standard potential method is carried out. The existence of stable, static solutions depends on the values of some parameters.

International audience; We study experimentally the early stages of integrable turbulence of unidirectional deep water gravity waves. By generating partially coherent waves in a 148 m long wave flume, we observe the emergence of high-amplitude structures formed by nonlinear focusing, commonly referred to as rogue waves. This work confronts the experiment with two recent results obtained in the framework of the nonlinear Schrödinger equation (NLSE), namely that (i) these structures can be locally fitted by a Peregrine soliton and (ii) their emergence leaves a visible trace on the evolution of statistical parameters such as kurtosis. Although Peregrine solitons have been observed for almost ten years in experiments using a deterministic forcing, we report their first systematic study in hydrodynamics with a random forcing. We show that (i) yields accurate results as long as the wave steepness remains moderate, whereas (ii) is very robust and remains valid beyond the assumption of integrability. Numerical simulations of the NLSE and of the fully nonlinear dynamical equations are also performed to support these results.

In this chapter, we provide a brief introduction to time scale calculus and introduce fundamental concepts that we need throughout this book. In Sect. 1.2 based on the work of Peterson and Tisdell (J Differ Equ Appl 10(13–15):1295–1306, 2004), we introduce the concept of Lyapunov functions for ordinary dynamical equations on time scales and prove general theorems in terms of wedges in which we derive specific conditions for stability of the zero solution and the boundedness of all solutions. We proceed to prove new and general theorems in terms of wedges and the existence of Lyapunov functions that satisfy certain conditions and give necessary conditions for stability and instability of the zero solution. In Sect. 1.4 we furnish all the details in proving the existence of the resolvent of Volterra integral dynamic equations by appealing to the results of Adivar and Raffoul (Bull Aust Math Soc 82(1):139–155, 2010). In addition, we will introduce the notion of shift that we make use of in Chaps. 5 and 8, and utilize the notion of resolvent and develop new results concerning Volterra dynamic equations. We end the chapter by introducing the notion of periodicity. In the famous paper (Kuchment, Floquet theory for partial differential equations, Birkhauser Verlag, Basel, 1993) of Raffoul and Kaufmann, the notion of periodicity on time scales was first introduced. Later on, Adivar (Math Slovaca 63(4):817–828, 2013) generalized the definition of periodicity to general time scales by introducing the concept of periodic shift operator. We end the chapter with some interesting and meaningful open problems that should be somewhat challenging.

This section casts stochastic dynamics into the previously developed language of field theory. The resulting formulation is advantageous in several respects. First, it expresses the dynamical equations into a path-integral, where the dynamic equations give rise to the definition of an “action.” In this way, the perturbation expansion with the help of Feynman diagrams or the loopwise expansions to obtain a systematic treatment of fluctuations (in Chap. 13; Zinn-Justin, Quantum Field Theory and Critical Phenomena (Clarendon Press, Oxford, 1996)) can be applied. Within neuroscience, the recent review (Chow and Buice (J Math Neurosci 5:8, 2015)) illustrates the first, the work by Buice and Cowan (Phys Rev E 75:051919, 2007) the latter approach. Moreover, this formulation will be essential for the treatment of disordered systems in Chap. 10, following the spirit of the work by De Dominicis and Peliti (Phys Rev B 18:353, 1978) to obtain a generating functional that describes an average system belonging to an ensemble of systems with random parameters.

Extrinsic noise-induced transitions to bimodal dynamics have been largely investigated in a variety of chemical, physical, and biological systems. In the standard approach in physical and chemical systems, the key properties that make these systems mathematically tractable are that the noise appears linearly in the dynamical equations, and it is assumed Gaussian and white. In biology, the Gaussian approximation has been successful in specific systems, but the relevant noise being usually non-Gaussian, non-white, and nonlinear poses serious limitations to its general applicability. Here we revisit the fundamental features of linear Gaussian noise, pinpoint its limitations, and review recent new approaches based on nonlinear bounded noises, which highlight novel mechanisms to account for transitions to bimodal behaviour. We do this by considering a simple but fundamental gene expression model, the repressed gene, which is characterized by linear and nonlinear dependencies on external parameters. We then review a general methodology introduced recently, so-called nonlinear noise filtering, which allows the investigation of linear, nonlinear, Gaussian and non-Gaussian noises. We also present a derivation of it, which highlights its dynamical origin. Testing the methodology on the repressed gene confirms that the emergence of noise-induced transitions appears to be strongly dependent on the type of noise adopted, and on the degree of nonlinearity present in the system., Review paper, 17 pages, 8 figures

We study the quantum mechanical many-body problem of $$N \ge 1$$ non-relativistic electrons with spin interacting with their self-generated classical electromagnetic field and $$K \ge 0$$ static nuclei. We model the dynamics of the electrons and their self-generated electromagnetic field with the so-called many-body Maxwell–Pauli equations. Here we construct time global, finite-energy, weak solutions to the many-body Maxwell–Pauli equations under the assumption that the fine structure constant $$\alpha $$ and the nuclear charges are not too large. The particular assumptions on the size of $$\alpha $$ and the nuclear charges ensure that we have energetic stability of the many-body Pauli Hamiltonian, i.e., the ground state energy is finite and uniformly bounded below with lower bound independent of the magnetic field and the positions of the nuclei. This work serves as an initial step towards understanding the connection between the energetic stability of matter and the well-posedness of the corresponding dynamical equations.

The homoclinic phenomena in double-layered nanoplates (DLNP) are investigated. Based on the nonlocal continuum theory, the nonlinear dynamical equations for DLNP subjected to in-plane excitation are derived by double-mode Galerkin truncation. The extended Melnikov method is utilized to discuss the homoclinic phenomena and chaotic motion for the buckled DLNP system. The criterions for the existence of transverse homoclinic orbits are established under different four buckling cases (i.e., the first- and second-type synchronous buckling as well as asynchronous buckling). And the results derived by the above-mentioned analysis are verified by molecular dynamics and Lyapunov exponent spectrum. Small-scale effect on the homoclinic motion is mainly inspected. From the result, it is rather novel that the transversality condition is independent of the nonlinear terms in the equations. This fact means that it is not necessary to distinguish whether the boundaries are movable or immovable for the in-plane parametric excitation DLNP. The parametric regime where homoclinic phenomena appear shrinks with the augment of nonlocal parameter for the first-type synchronous buckling case. However, this trend is just opposite for the other three buckling cases. Finally, it can be seen that homoclinic phenomena more likely occur on Mode I (i.e., lower mode) for the second-type synchronous and asynchronous buckling cases. However, the homoclinic motion for the first-type asynchronous buckling most likely takes place on Mode III.

In this paper, our objective is to analyze integrability of three famous nonlinear models, namely unstable nonlinear Schrödinger equation (UNLSE), modified UNLSE (MUNLSE) as well as (2+1)-dimensional cubic NLSE (CNLSE) by utilizing Painlevé test ([Formula: see text]-test). The non-appearance of some sort of singularities such as moveable branch points indicates a sound probability of complete integrability of the concerned NLSE. In case an NLSE passes the [Formula: see text]-test, the studied model can be solved by implementing inverse scattering transformation (IST).

To explain the nutation phenomenon, Euler chose a geocentric frame of coordinates to present his dynamical equations. In accordance with his formulas, the nutation is caused by a momentum relative to the Earth's center, due to certain external forces. Further, Poinsot presented a special case of the Euler's equations, when any momentum of those certain external forces does not exist. Due to a problematic approximation, Poinsot announced that in this case may take place a nutation period of 10 month, named “free period”, to distinguish it from the Euler's dynamical solution, known as “forced nutation”. After Poinsot, the notion of “free nutation” was extended also to those nutation phenomena which have only a geophysical origin, without a momentum of certain external forces. Usually, the daily nutation is considered as being a free nutation phenomenon. In order to search for a forced daily nutation component, the daily trajectory of the Earth-Moon barycenter inside the Earth is used (a simple scheme of this barycenter trajectory is presented in a geocentric system of axes). Finally, if the ecliptic line is accepted to be described by the Earth-Moon barycenter, it must be accepted, too, that a torque due the Sun and the Moon acts during a sideral day interval on diurnal Earth rotation around its axis. Due to the small value of the nutation constant, a period of 18,6 years is needed to correctly and completely detect the forced daily nutation; this phenomenon permanently presents very fine variations depending on the longitude of the ascending lunar node, the Moon’s declination and the Sun’s ecliptic longitude.

In this paper we give a physical explanation to the accelerated expansion of the Universe, alleviating the tension between the discrepancy of Hubble constant measurements. By the Euler Cauchy stress principle, we identify a controversy on the lack of consideration of the surface forces contemplated in the study of the expansion of the Universe. We distinguish a new effect that modifies the spacetime fabric by means of the energy conservation equation. The resulting dynamical equations from the proposed hypothesis are contrasted to several testable astrophysical predictions. This paper also explains why we have not found any particle or fluid responsible for dark energy and clarifies the Cosmological Coincidence Problem. These explanations are achieved without assuming the existence of exotic matter of unphysical meaning or having to modify the Einstein's Field Equations., 15 pages, 5 figures, IJMPD Accepted

The present study introduces a Collective Variable (CV) approach to investigate an important type of Schrodinger equation called the Lakshmanan-Porsezian-Daniel (LPD) model which has great deals in optical materials. A state of numerical simulation via the application of the fourth-order Runge-Kutta method is further employed for the numerical treatment of the resultant system of dynamical equations of motion. The CV method gives the dynamics of the pulse parameters. Graphical representations of the pulse width, amplitude, temporal position, chirp, frequency, and phase of the pulse verses the propagation coordinate are displayed, correspondingly. More so, a significant periodicity is observed in the soliton’s amplitude, width and chirp. Other vital features with regards to the present study are also deduced.

The dynamical variables $\psi$ of a classical system, undergoing stochastic stirring forces, satisfy equations of motion with noise terms. Hence, these dynamical variables show a stochastic evolution themselves. The probability of each possible realization of $\psi$ within a given time interval, arises from the interplay between the deterministic parts of dynamics and the statistics of noise terms. In this work, we discuss the construction of the stochastic Lagrangian out of the dynamical equations, that is a tool to calculate the realization probabilities of the dynamical variables as path integrals. In this formulation, the study of classical statistical dynamics can benefit from all the techniques developed in Quantum Mechanics of path integrals; moreover, as the path integral is expressed in terms of a Lagrangian, the invariance properties of the system become transparent. After a coincise review of the stochastic Lagrangian formalism, some applications of it to physically relevant cases are illustrated. Then, the advantages and maturity of this approach, and its expected future developments, are outlined.

The main focus of this manuscript is to explore the effectiveness of Rastall theory of gravity in describing very early epoch of the universe, called cosmic inflation. In this framework, we develop the basic inflationary dynamical equations for flat FRW metric. To find out the exact solutions of inflaton field and corresponding potentials, we start with a basic technique of choosing a known solution of the scale factor, i.e., intermediate and logamediate forms and then extracted the effective potential necessary to support the inflation under slow-roll condition. Via these solutions, we calculate the inflationary observable such as the slow-roll parameters, e-folding number, scalar/tensor power spectra, scalar spectral index and tensor–scalar ratio in both strong and weak dissipative regimes. The recent Planck 2018 data sets bounds for scalar power spectrum, scalar spectral index, n s = 0 . 966 and tensor–scalar ratio, R 0 . 16 . We have plotted graphical trajectories of R − n s by constraining the model parameters, which show that Planck bounds are fully satisfied by Rastall theory of gravity. Hence, this modified gravity theory has ability to describe the early as well as the late-time cosmic eras elegantly.

We provide new insights into the contact Hamiltonian and Lagrangian formulations of dissipative mechanical systems. In particular, we state a new form of the contact dynamical equations, and we review two recently presented Lagrangian formalisms, studying their equivalence. We define several kinds of symmetries for contact dynamical systems, as well as the notion of dissipation laws, prove a dissipation theorem and give a way to construct conserved quantities. Some well-known examples of dissipative systems are discussed., Comment: 24 pp. This is the final version of the article. Some minor misprints are corrected. Some bibliographycal references are uploaded

In what follows we shall deal a lot with fields, hence it is important to formulate Lagrange equations for relativistic fields on a regular basis. After doing this, we go on to construct the action and derive the dynamical equations for electromagnetic potential in relativistic form.

In this paper we present a new extended complex nonlinear Klein–Gordon Lagrangian density, which bears a single non-topological soliton solution with a specific rest frequency ω s in 1 + 1 dimensions. There is a proper term in the new Lagrangian density, which behaves like a massless spook that surrounds the single soliton solution and opposes any internal changes. In other words, any arbitrary variation in the single soliton solution leads to an increase in the total energy. Moreover, just for the single soliton solution, the general dynamical equations are reduced to those versions of a special type of the standard well-known complex nonlinear Klein–Gordon systems, as its dominant dynamical equations.

The lightcone gauge is a set of what are called the observational coordinates adapted to our past lightcone. We develop this gauge by producing a perturbed spacetime metric that describes the geometry of our past lightcone where observations are usually obtained. We connect the produced observational metric to the perturbed Friedmann-Lema&icirc, tre-Robertson-Walker (FLRW) metric in the standard general gauge or what is the so-called 1+3 gauge. We derive the relations between these perturbations of spacetime in the observational coordinates and those perturbations in the standard metric approach, as well as the dynamical equations for the perturbations in observational coordinates. We also calculate the observables in the lightcone gauge and re-derive them in terms of Bardeen potentials to first order. A verification is made of the observables in the perturbed lightcone gauge with those in the standard gauge. The advantage of the method developed is that the observable relations are simpler than in the standard formalism, and they are expressed in terms of the metric components which in principle are measurable. We use the perturbed lightcone gauge in galaxy surveys and the calculations of galaxy number density contrast. The significance of the new gauge is that by considering the null-like light propagations the calculations are much simpler due to the non-consideration of the angular deviations.

The present study deals with the spherically symmetric distributions of dust dissipative fluids with a central vacuum cavity, evolving under the condition of vanishing of expansion scalar. The work is based on two aspects—on the one hand the description of model in terms of structures scalars emerging in the orthogonal splitting of Riemann tensor. Secondly, to obtain the solutions of dynamical equations under some special cases and assumption on implicit function of variables. The solutions here discussed, model the evolution of the vacuum cavity of fluid distribution, thereby showing the potential of expansion free motion.

In this note we perform the $n+1$ decomposition, or Arnowitt Deser Misner (ADM) formulation of $f($Lovelock$)$ gravity theory. The hamiltonian form of Lovelock gravity was known since the work of C. Teitelboim and J. Zanelli in 1987, but this result had not yet been extended to $f($Lovelock$)$ gravity. Besides, field equations of $f($Lovelock$)$ have been recently be computed by P. Bueno et al., though without ADM decomposition. We focus on the non-degenerate case, ie. when the Hessian of $f$ is invertible. Using the same Legendre transform as for $f(\mathrm{R})$ theories, we can identify the partial derivatives of $f$ as scalar fields, and consider the theory as a generalised scalar-tensor theory. We then derive the field equations, and project them along a $n+1$ decomposition. We obtain an original system of constraint equations for $f($Lovelock$)$ gravity, as well as dynamical equations. We give explicit formulas for the $f(\mathrm{R},$ Gauss-Bonnet$)$ case., Comment: 23 pages

According to a framework based on Clifford algebra $$C\ell (1,3)$$ , this paper gives a classification for elementary fields, and then derives their dynamical equations and transformation laws in detail. These results provide an outline on elementary fields and some new insights into their unusual properties. All elementary fields exist in pairs, and one part of the pair is a complex field. Some intrinsic symmetries and constraints such as Lorentz gauge condition are automatically included in the canonical equation. Clifford algebra $$C\ell (1,3)$$ is a natural language to describe the world. In this language, the representation formalism of dynamical equation is symmetrical and elegant with no more or less contents. This paper is also a summary of some previous problem-oriented researches. Solutions to some simple equations are given.

Constraints work in the simplest way among a variety of methods of modeling the mechanical behaviors on the interface between bodies. The constraints within a persistent point contact usually fall into the following three categories: geometry-dependent constraints due to non-penetration limitation between the two rigid bodies; velocity- or force-dependent constraints due to the vanishing of tangential velocity or sliding friction engaged in tangential interaction. Though those constraints may be intuitively obtained for some simple problems, they are essentially associated with the evolution of location parameters denoting the temporal position of the contact point. Focusing on a multibody system subject to a persistent point contact, we propose a uniform and programmable procedure to formulate the constraint equations. Kinematic analysis along the procedure can clearly expose the dependence of the constraint equations on the location parameters, unveil the reason why the velocity-dependent constraints may become nonholonomic, and exhibit the fulfillment of the Appell–Chetaev’s rule naturally. Furthermore, we employ d’Alembert–Lagrangian principle to yield the dynamical equations of the system via the method of Lagrange’s multipliers. The dynamical equations so obtained are then compared with those derived from a quite different method that characterizes the contact interplay as a pair of contact force vectors. Accordingly, the correlations between the Lagrange multipliers and the components of the real contact force can be clarified. The clarification enables us to correctly embed the force-dependent constraints into the dynamical equations. A classical example of a thin disk contacting a horizontal rough surface is provided to demonstrate the validation of the proposed theory and method.

Advanced structural models, based on variable one-, two-, and three-dimensional kinematics, are proposed in this paper and applied to the analysis of the free vibration of reinforced aircraft shell structures. The used models go beyond classical structural theories, that is, Euler–Bernoulli (for one-dimensional beams) and Kirchhoff (for two-dimensional plates) type assumptions. The order of the expansion of the displacement fields over the cross section (one-dimensional case) and along the plate thickness (two-dimensional case) is, in fact, a free parameter of the problem. In this paper, Lagrange polynomials are used to build such expansions, and as a consequence, only displacements are used as the problem unknowns (no rotations or derivatives of displacements, which are typical of one-dimensional/two-dimensional classical theories, are introduced). The finite-element method is used to provide numerical solutions. The related arrays and the governing dynamical equations are written in terms of a few funda...

This paper concerns with the full coupled linear theory of elasticity for triple porosity materials. The system of the governing equations based on the equations of motion, conservation of fluid mass, the constitutive equations and Darcy’s law for materials with triple porosity. The cross-coupled terms are included in the equations of conservation of mass for the fluids of the three levels of porosity and in the Darcy’s law for materials with triple porosity. The system of general governing equations of motion is expressed in terms of the displacement vector field and the pressures in the three pore systems. Five spatial cases of the dynamical equations are considered: equations of steady vibrations, equations in Laplace transform space, equations of quasi-static, equations of equilibrium and equations of steady vibrations for rigid body with triple porosity. The fundamental solutions of the systems of these PDEs are constructed explicitly by means of elementary functions and finally, the basic properties of the fundamental solutions are established.