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In this letter we present a method for control and stabilizing a pacing quadruped robot using state feedback switching. In the pacing gait, a quadruped cycles between the left and right pairs of legs to achieve locomotion. This results in two discrete stance configurations, each being unstable. The quadruped can achieve dynamic stability by switching between these stances in an appropriate manner. We model our system using two sets of non-linear dynamical equations which we have the control of switching between actively. In order to stabilize our system, we propose a state feedback switching law based on the linearized models of our system for each set of equations. Using this switching law, we are able to stabilize the robot in the roll direction. This, combined with the passive stability achieved with the design of the robot’s legs, results in a stable pacing gait. The performance of the proposed controller is evaluated on a complete dynamical model in simulation and stability is verified using a Poincare map.

In this paper, the far-distance nonplanar rapid cooperative rendezvous with multiple specific-direction thrusts is investigated. In the optimization process, the indirect method is adopted. The homotopy method is used to smooth the dynamical equations. The solutions of the initial co-state variables of the energy-optimal problem are obtained by the quantum-behaved particle swarm optimization (QPSO) algorithm. Sequential quadratic programming (SQP) algorithm is coupled into the homotopy method to overcome the difficulty in nonsmooth integration when the thrust switches on & off frequently. The solutions of the fuel-optimal problem are obtained by the homotopy-SQP coupled method. The simulation results show the feasibility of the cooperative rendezvous with multiple specific-direction thrusts. The “suitable point” of normal thrust is investigated.

There are approximately 2900 active space objects cataloged, and the number will increase continually in future. All these space objects may maneuver for some purpose occasionally, including highly elliptical orbit (HEO) transfer, station-keeping, orbit altitude adjustment, etc. Hence maneuver detection is very important for space situational awareness (SSA). It also plays an important role in the catalog maintenance by correlating the orbits before and after maneuver. In the paper, the maneuver detection methods based on orbital dynamical model are proposed. Firstly, the perturbations before and after maneuver are analyzed. Then two kinds of maneuver models are derived using the relative dynamical equations. Finally, several kinds of maneuver cases are performed to verify the models proposed. It is demonstrated that these two models are feasible and effective for the detection of longitude maneuvers.

This research is dedicated to developing an analytical physics-based model of a reciprocating compressor based on a bond graph approach. At first, a mathematical model is developed for one-cylinder reciprocating compressor based on multi-body dynamics, particularly considering planar motion of the rigid bodies using body fixed reference frames. All the dynamical equations of motions are represented and linked altogether by the bond graph that simulates a complete system model. The model has unique capability to show forces at various joints, which enhances our understanding of complex system behavior. Moreover, the consequences of changing coefficients of those forces are accentuated. The utility of the model in fault diagnosis and condition monitoring are emphasized by highlighting the effects of a crack in a joint and a dent on cylinder wall. Nonlinear differential equations extracted from bond graph model are solved efficiently using a MATLAB stiff solver. Responses of the model parts are analyzed later by force analysis along with a comparison with a previous research to examine model validity.

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 work, we show that by restricting to the subgroup of time-independent coordinate transformations, then it is possible to derive the Ricci flow from the Bianchi identities. To achieve this, we first show that the field equations of the gravitational field, the Newton’s second law of classical dynamics, and the Maxwell field equations of the electromagnetic field all share the same mathematical structure. Consequently, the Ricci flow itself may be regarded as dynamical equations used to describe physical processes associated with the gravitational field, such as the process of smoothing out irregularities of distribution of matter in space.

We study variational obstacle avoidance problems on complete Riemannian manifolds and apply the results to the construction of piecewise smooth curves interpolating a set of knot points in systems with impulse effects. We derive the dynamical equations for extrema in the variational problem, and show the existence of minimizers by using lower-continuity arguments for weak convergence on an infinite-dimensional Hilbert manifold. We then provide conditions under which it is possible to ensure that the extrema will safely avoid a given obstacle within some desired tolerance.

In this article, a new generalized exponential rational function method (GERFM) is employed to extract new solitary wave solutions for the ionic currents along microtubules dynamical equations, which is very interested in nanobiosciences. In this article, the stability of the solutions is also studied. As a result, a variety of solitary waves are obtained with free parameters such as periodic wave solution and dark and bright solitary wave solutions. The solutions are plotted and used to describe physical phenomena of the problem. The work shows the power of GERFM. We found that the proposed method is reliable and effective and gives analytical and exact solutions.

How to model non-smooth dynamical systems and study its non-smooth global dynamics by developing analytical methods is now an important topic. In this paper, a new bistable nonlinear oscillator with bilateral elastic constraints and a three-piecewise nonlinear restoring force is established to study the perturbation of viscous damping and an external harmonic excitation. A five-piecewise linear approach to the restoring force can transform the original dimensionless dynamical equations into a two-dimensional piecewise-defined Hamiltonian systems, separated by four switching manifolds with symmetrical characteristic and subjected to damping dissipation and a periodic excitation. The geometrical structure of the unperturbed system has a pair of symmetrical piecewise-defined homoclinic orbits and each of them transversally and continuously crosses two switching manifolds. The analytical Melnikov method is extended here to fit the theoretical framework for analyzing homoclinic bifurcations and chaotic dynamics for non-smooth oscillators with multiple switching manifolds. A global perturbation technique is presented to calculate the distance between the stable and unstable manifolds and derive the Melnikov function in the form of piecewise-integration. A major innovation here is no need to extend the vector filed near the switching manifolds and carry out rigorous perturbation analysis with geometrical intuition to derive the Hamilton energy differences of trajectories involving infinite time. The developed Melnikov function can be directly used to detect the parameter threshold of homoclinic chaos in the bistable nonlinear oscillator under bilateral elastic collision. Finally, the effectiveness of the Melnikov analysis for homoclinic chaos is verified by simulations.

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 suggest a universal map capable to recover a behavior of a wide range of dynamical systems given by ODEs. The map is built as an artificial neural network whose weights encode a modeled system. We assume that ODEs are known and prepare training datasets using the equations directly without computing numerical time series. Parameter variations are taken into account in the course of training so that the network model captures bifurcation scenarios of the modeled system. Theoretical benefit from this approach is that the universal model admits using common mathematical methods without needing to develop a unique theory for each particular dynamical equations. Form the practical point of view the developed method can be considered as an alternative numerical method for solving dynamical ODEs suitable for running on contemporary neural network specific hardware. We consider the Lorenz system, the Roessler system and also Hindmarch-Rose neuron. For these three examples the network model is created and its dynamics is compared with ordinary numerical solutions. High similarity is observed for visual images of attractors, power spectra, bifurcation diagrams and Lyapunov exponents., Comment: 17 pages, 10 figures

We establish a generalization of the thermoelasticity wave equation to the porous case, including the Lord–Shulman (LS) and Green–Lindsay (GL) theories that involve a set of relaxation times ( $$\tau _i, \ i = 1, \ldots , 4$$ ). The dynamical equations predict four propagation modes, namely, a fast P wave, a Biot slow wave, a thermal wave, and a shear wave. The plane-wave analysis shows that the GL theory predicts a higher attenuation of the fast P wave, and consequently a higher velocity dispersion than the LS theory if $$\tau _1 = \tau _2 > \tau _3$$ , whereas both models predict the same anelasticity for $$\tau _1 = \tau _2 = \tau _3$$ . We also propose a generalization of the LS theory by applying two different Maxwell–Vernotte–Cattaneo relaxation times related to the temperature increment ( $$\tau _3$$ ) and solid/fluid strain components ( $$\tau _4$$ ), respectively. The generalization predicts positive quality factors when $$\tau _4 \ge \tau _3$$ , and increasing $$\tau _4$$ further enhances the attenuation. The wavefields are computed with a direct meshing algorithm using the Fourier pseudospectral method to calculate the spatial derivatives and a first-order explicit Crank–Nicolson time-stepping method. The propagation illustrated with snapshots and waveforms at low and high frequencies is in agreement with the dispersion analysis. The study can be useful for a comprehensive understanding of wave propagation in high-temperature high-pressure fields.

Trolling mode atomic force microscopy (TR-AFM) can considerably reduce the liquid-resonator interaction forces, and hence, has overcome many imaging problems in liquid environments. This mode increases the quality factor (QF) significantly compared with the conventional AFM operation in liquid; therefore, the duration to reach the steady-state periodic motion of the oscillating probe is relatively high. As a result, utilizing conventional imaging techniques, which are based on measuring the amplitude and phase, are significantly slower when compared to our proposed method. This research presents a high-speed scanning technique based on an estimation law to obtain the topography of various samples utilizing a two-degree-of-freedom model of TR-AFM. The effect of the nanoneedle tip horizontal displacement on the estimation process is investigated, and a solution to compensate for its undesirable effect is also presented.

During epidemic outbreaks, there are various types of information about epidemic prevention disseminated simultaneously among the population. Meanwhile, the mass media also scrambles to report the information related to the epidemic. Inspired by these phenomena, we devise a model to discuss the dynamical characteristics of the co-evolution spreading of multiple information and epidemic under the influence of mass media. We construct the co-evolution model under the framework of two-layered networks and gain the dynamical equations and epidemic critical point with the help of the micro-Markov chain approach. The expression of epidemic critical point show that the positive and negative information have a direct impact on the epidemic critical point. Moreover, the mass media can indirectly affect the epidemic size and epidemic critical point through their interference with the dissemination of epidemic-relevant information. Though extensive numerical experiments, we examine the accuracy of the dynamical equations and expression of the epidemic critical point, showing that the dynamical characteristics of co-evolution spreading can be well described by the dynamic equations and the epidemic critical point is able to be accurately calculated by the derived expression. The experimental results demonstrate that accelerating positive information dissemination and enhancing the propaganda intensity of mass media can efficaciously restrain the epidemic spreading. Interestingly, the way to accelerate the dissemination of negative information can also alleviate the epidemic to a certain extent when the positive information hardly spreads. Current results can provide some useful clues for epidemic prevention and control on the basis of epidemic-relevant information dissemination.

We investigate a braneworld model generated by a global monopole in the context of Brans-Dicke gravity. After solving the dynamical equations we found a model capable to alleviate the so-called hierarchy problem. The obtained framework is described by a hybrid compactification scheme endowed with a seven-dimensional spacetime, in which the brane has four non-compact dimensions and two curled extra dimensions. The relevant aspects of the resulting model are studied and the requirements to avoid the well known seesaw-like behavior are discussed. We show that under certain conditions it is possible to circumvent such a pathological behavior that characterizes most of the models that exhibit hybrid compactification. Lastly, we deepen our analysis by considering possible extensions of this model to a setup with multiple branes and orbifold-like extra dimension. For this, we compute the consistency conditions to be obeyed by this more general configuration as predicted by the braneworld sum rules formalism. This study indicates the possibility of exclusively positive brane tensions in the model., Comment: 10 pages, 1 figure. Improved version with more discussion added to the Introduction, section II and conclusion. A new plot was added as well as new references. Matches version accepted in EPJC

Most numerical solvers used to determine the free variables of dynamical systems rely on firstorder derivatives of the state of the system with respect to the free variables. The number of free variables can be fairly large. One of the approaches to obtaining these derivatives is the integration of the derivatives simultaneously with the dynamical equations, which is best done with automatic differentiation techniques. Even though there exist many automatic differentiation tools, none has been found to be scalable and usable for practical purposes of modeling dynamical systems. Landau is a Turing incomplete statically typed domain-specific language aimed to fill this gap. The Turing incompleteness allows for a sophisticated source code analysis and, as a result, a highly optimized compiled code. Among other things, the language syntax supports functions, compile-time ranged for loops, if/else branching constructions, real variables and arrays, and allows for manually discarding calculations where the automatic derivative values are expected to be negligibly small. In spite of reasonable restrictions, the language is rich enough to express and differentiate any cumbersome equation with practically no effort. Bibliography: 11 titles.

Turbo-machineries, as key components, have a wide utilization in fields of civil, aerospace, and mechanical engineering. By calculating natural frequencies and dynamical deformations, we have explained the rationality of the series form for the aerodynamic force of the blade under the subsonic flow in our earlier studies. In this paper, the subsonic aerodynamic force obtained numerically is applied to the low pressure compressor blade with a low constant rotating speed. The blade is established as a pre-twist and presetting cantilever plate with a rectangular section under combined excitations, including the centrifugal force and the aerodynamic force. In view of the first-order shear deformation theory and von-Kármán nonlinear geometric relationship, the nonlinear partial differential dynamical equations for the warping cantilever blade are derived by Hamilton’s principle. The second-order ordinary differential equations are acquired by the Galerkin approach. With consideration of 1:3 internal resonance and 1/2 sub-harmonic resonance, the averaged equation is derived by the asymptotic perturbation methodology. Bifurcation diagrams, phase portraits, waveforms, and power spectrums are numerically obtained to analyze the effects of the first harmonic of the aerodynamic force on nonlinear dynamical responses of the structure.

As a first attempt, the vibration and stability analysis of magnetically embedded spinning axially functionally graded (AFG) nanotubes conveying fluid under axial loads is performed based on the nonlocal strain gradient theory (NSGT). A detailed parametric investigation is conducted to elucidate the influence of key factors such as material distribution type and size-dependent parameters on the divergence and flutter instability borders. Also, a comparative study is conducted to evaluate the available theories in the modeling of nanofluidic systems. The material characteristics of the system are graded along the longitudinal direction based on the power-law and exponential distribution functions. To accurate model and formulate the system, the no-slip boundary condition is considered. Adopting the Laplace transform and Galerkin discretization technique, the governing size-dependent dynamical equations of the system are solved. The backward and forward natural frequencies, as well as critical fluid and spin velocities of the system, are extracted. Besides, an analytical approach is applied to identify the instability thresholds of the system. Dynamical configurations, Campbell diagrams, and stability maps are analyzed. Meanwhile, it is concluded that, in contrast to the influence of nonlocal and density gradient parameters, the increment of strain gradient and elastic modulus gradient parameters expands the stability regions and alleviate the destabilizing effect of the axial compressive load.

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

This paper investigates the dynamical characteristics for meminductor and memcapacitor via fractal–fractional-order domain of Caputo–Fabrizio. A chaos circuit is modeled for the highly nonlinear and non-fractional governing differential equations of meminductor and meminductor for knowing the hyperchaos, abrupt chaos and coexisting attractors. The time-scale transformation on dynamical equations is invoked within non-classical approach through newly presented fractal–fractional differential operator of Caputo–Fabrizio. The nonlinear fractionalized governing differential equations of meminductor and meminductor have been simulated by means of Adams–Bashforth–Moulton method. In order to disclose the functionalities of capacitive and inductive elements so-called meminductor and memcapacitor, we specified the fractal–fractional differential operator of Caputo–Fabrizio in three categories as (i) variation in both fractional and fractal parameters, (ii) variation in fractional parameter keeping fractal parameters equal to one, and (iii) variation in fractal parameter keeping fractional parameters equal to one. At the end, our numerically simulated results elaborate that chaotic behavior and unpinched hysteresis loops obtained via fractal–fractional approach are more efficient than ordinary approach.

We introduce an individual-based model with dynamical equations for susceptible-infected-susceptible (SIS) epidemics on clustered networks. Linking the mean-field and quenched mean-field models, a general method for deriving a cluster approximation for three-node loops in complex networks is proposed. The underlying epidemic threshold condition is derived by using the quasi-static approximation. Our method thus extends the pair quenched mean-field (pQMF) approach for SIS disease spreading in unclustered networks to the scenario of epidemic outbreaks in clustered systems with abundant transitive relationships.We found that clustering can significantly alter the epidemic threshold, depending nontrivially on topological details of the underlying population structure. The validity of our method is verified through the existence of bounded solutions to the clustered pQMF model equations, and is further attested via stochastic simulations on homogeneous small-world artificial networks and growing scale-free synthetic networks with tunable clustering, as well as on real-world complex networked systems. Our method has vital implications for the future policy development and implementation of intervention measures in highly clustered networks, especially in the early stages of an epidemic in which clustering can decisively alter the growth of a contagious outbreak.

The aim of this outline is to investigate the consequences of viscous dissipative self-gravitational collapse in non-static spherically symmetric geometry by adopting a Misner–Sharp approach in the context of f(R) metric formalism. In this outline, we propose a viable f(R) model for the physical significance of a self-gravitating system. We determine the junction conditions between interior and exterior space–times. Furthermore, we extend our work for the dissipative dark source case in both forms of heat transformation and free radiation streaming. Finally, the dissipative variables such as heat flux, bulk, and shear viscosity are evaluated in the dynamical equation and then coupled with full causal transport equations in the background of the Israel–Stewart notion. The present analysis discusses the whole study of self-gravitational collapse and recognizes the form of a reduced collapse due to decreasing gravitational mass density, which depends on thermodynamical collapsing variables.

This paper reviews the three-pattern decomposition of global atmospheric circulation (3P-DGAC) developed in recent years, including the decomposition model and the dynamical equations of global horizontal, meridional, and zonal circulations. Compared with the traditional two-dimensional (2D) circulation decomposition method, the 3P-DGAC can effectively decompose the actual vertical vorticity into two components that are caused by the horizontal circulation and convergent/divergent movement (associated with the meridional and zonal circulations). It also decomposes the vertical velocity into the components of the meridional vertical circulation and the zonal vertical circulation, thus providing a new method to study the dynamical influences of convergent/divergent motions on the evolution of actual vertical vorticity and an accurate description of local vertical circulations. The 3P-DGAC is a three-dimensional (3D) circulation decomposition method based on the main characteristics of the actual atmospheric movements. The horizontal, meridional, and zonal circulations after the 3P-DGAC are the global generalization of Rossby waves in the middle-high latitudes and Hadley and Walker circulations in low latitudes. Therefore, the three-pattern decomposition model and its dynamical equations provide novel theoretical tools for studying complex interactions between middle-high and low latitude circulations as well as the physical mechanisms of the abnormal evolution of large-scale atmospheric circulations under the background of global warming.

This paper focuses on the derivation of the aerodynamic force for the cantilever plate in subsonic flow. For the first time, a new analytical expression of the quasi-steady aerodynamic force related to the velocity and the deformation for the high-aspect-ratio cantilever plate in subsonic flow is derived by utilizing the subsonic thin airfoil theory and Kutta-Joukowski theory. Results show that aerodynamic force distribution obtained theoretically is consistent with that calculated by ANSYS FLUENT. Based on the first-order shear deformation and von Karman nonlinear geometric relationship, nonlinear partial differential dynamical equations of the high-aspect-ratio plate subjected to the aerodynamic force are established by using Hamilton’s principle. Galerkin approach is applied to discretize the governing equations to ordinary differential equations. Numerical simulation is utilized to investigate the relation between the critical flutter velocity and some parameters of the system. Results show that when the inflow velocity reaches the critical value, limit cycle oscillation occurs. The aspect ratio, the thickness, and the air damping have significant impact on the critical flutter velocity of the thin plate.

It is well known that there is a deep connection between the symmetric and traveling wave solutions. It has been shown that all symmetric waves are traveling waves. In this paper, we establish new analytic solution collections of nonlinear conformable time-fractional water wave dynamical equation in a complex domain. For this purpose, we construct a new definition of a symmetric conformable differential operator (SCDO). The operator has a symmetric representation in the open unit disk. By using SCDO, we generalize a class of water wave dynamical equation type time-space fractional complex Ginzburg–Landau equation. The results show that the obtainable approaches are powerful, dependable and prepared to apply to all classes of complex differential equations.

We present the concept of an adiabatic limit of Ginzburg—Landau dynamical equations on ℝ1+2 and Seiberg—Witten equations on four-dimensional symplectic manifolds. We show that the Seiberg—Witten equations can be regarded as a complex version of the Ginzburg—Landau equations.

This paper proposes a systematic methodology to obtain a closed-form formulation for dynamics analysis of a new design of a fully spherical robot that is called a 3(RSS)-S parallel manipulator with real co-axial actuated shafts. The proposed robot can completely rotate about a vertical axis and can be used in celestial orientation and rehabilitation applications. After describing the robot and its inverse position, velocity and acceleration analysis is performed. Next, based on Kane’s method, a methodology for deriving the dynamical equations of motion is developed. The elaborated approach shows that the inverse dynamics of the manipulator can be reduced to solving a system of three linear equations in three unknowns. Finally, a computational algorithm to solve the inverse dynamics of the manipulator is advised and several trajectories of the moving platform are simulated.

In this paper, a simulation model for frequency modulation atomic force microscopy (FM-AFM) operating in constant amplitude dynamic mode is presented. The model is based on the slow time varying function theory. The mathematical principles to derive the dynamical equations for the amplitude and phase of the FM-AFM cantilever-tip motion are explained and the stability and performance of its closed-loop controller to keep the amplitude at constant value and phase at 90° is analysed. Then, the performance of the theoretical model is supported by comparison of numerical simulations and experiments. Furthermore, the transient behaviour of amplitude, phase and frequency shift of FM-AFM is investigated and the effect of controller gains on the transient motion is analysed. Finally, the derived FM-AFM model is used to simulate the single molecule/nanoscale force spectroscopy and study the effect of sample viscosity, stiffness and Hamaker constant on the response of FM-AFM.

In this study, the output tracking of delayed logical control networks (DLCNs) with state and control constraints is further investigated. Compared with other delays, state-dependent delay updates its value depending on the current state values and a pseudo-logical function. Multiple constraints mean that state values are constrained in a nonempty set and the design of the controller is conditioned. Using the semi-tensor product of matrices, dynamical equations of DLCNs are converted into an algebraic description, and an equivalent augmented system is constructed. Based on the augmented system, the output tracking problem is transformed into a set stabilization problem. A deformation of the state transition matrix is computed, and a necessary and sufficient condition is derived for the output tracking of a DLCN with multi-constraint. This condition is easily verified by mathematical software. In addition, the admissible state-feedback controller is designed to enable the outputs of the DLCN to track the reference signal. Finally, theoretical results are illustrated by an example.

This brief studies robust output regulation in discrete-time singular systems with actuator saturation and matched uncertainties. The singularity of dynamical equations makes the design procedure more complex due to the appearance of both algebraic and difference equations in such systems. In addition to this, the actuator saturation constraint along with uncertain terms complicates the design problem. The idea for robust output regulation is to add an additional control term to the composite nonlinear feedback (CNF) controller which guarantees the output regulation of the uncertain closed-loop singular system with an ultimately bounded tracking error. In this regard, a theorem is given which guarantees the output regulation even if the saturation takes place. The accuracy of the theoretical results is tested through simulations for a practical example.

In this paper we analytically derive the exact closed dynamical equations for a liquid with short-ranged interactions in large spatial dimensions using the same statistical mechanics tools employed to analyze Brownian motion. Our derivation greatly simplifies the original path-integral-based route to these equations and provides insight into the physical features associated with high-dimensional liquids and glass formation. Most importantly, our construction provides a route to the exact dynamical analysis of important related dynamical problems, as well as a means to devise cluster generalizations of the exact solution in infinite dimensions. This latter fact opens the door to the construction of increasingly accurate theories of vitrification in three-dimensional liquids.

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.

Dissipative interaction facilitates the global phase locking of a network of lasers with generally complex graph topologies. Recently, we showed that the equilibrium states of such laser networks are optimal solutions of governing cost functions that are of non-convex quadratic form. This realization allows for mapping a large class of non-convex optimization problems onto laser networks, which provides a promising route for optically solving such computationally-hard problems. Here, by numerical simulation of the underlying dynamical equations, we investigate the accuracy of the solutions obtained for well-known non-convex problems and discuss the role of parameter tuning for escaping the local minima.

We formulate a class of stochastic partial differential equations based on Kelvin’s circulation theorem for ideal fluids. In these models, the velocity field is randomly transported by white-noise vector fields, as well as by its own average over realizations of this noise. We call these systems the Lagrangian averaged stochastic advection by Lie transport (LA SALT) equations. These equations are nonlinear and non-local, in both physical and probability space. Before taking this average, the equations recover the Stochastic Advection by Lie Transport (SALT) fluid equations introduced by Holm (Proc R Soc A 471(2176):20140963, 2015). Remarkably, the introduction of the non-locality in probability space in the form of momentum transported by its own mean velocity gives rise to a closed equation for the expectation field which comprises Navier–Stokes equations with Lie–Laplacian ‘dissipation’. As such, this form of non-locality provides a regularization mechanism. The formalism we develop is closely connected to the stochastic Weber velocity framework of Constantin and Iyer (Commun Pure Appl Math 61(3):330–345, 2008) in the case when the noise correlates are taken to be the constant basis vectors in $$\mathbb {R}^3$$ R 3 and, thus, the Lie–Laplacian reduces to the usual Laplacian. We extend this class of equations to allow for advected quantities to be present and affect the flow through exchange of kinetic and potential energies. The statistics of the solutions for the LA SALT fluid equations are found to be changing dynamically due to an array of intricate correlations among the physical variables. The statistical properties of the LA SALT physical variables propagate as local evolutionary equations which when spatially integrated become dynamical equations for the variances of the fluctuations. Essentially, the LA SALT theory is a non-equilibrium stochastic linear response theory for fluctuations in SALT fluids with advected quantities.

In this paper, a simplified dynamic model of a dual-rotor system coupled with blade disk is built, and the effects of blade parameters of an aircraft engine on the dynamic characteristics of a dual-rotor system are studied. In the methodology, the blade is simplified as a cantilever structure, and the dynamical equations are obtained by the means of a finite element method. The amplitude-frequency response curves and orbits of shaft centre-vibration shape diagram are used to analyze the effects of blade parameters on dynamic characteristics of a dual-rotor system. The results indicate that the properties of the blades have huge impacts on the critical speed and other dynamic characteristics of the system. With an increase of the length of the blade, the second-order critical speed decreases obviously, but the first-order critical speed is almost invariant; this means that the blades attached on the low-pressure compressor do not affect the first-order critical speed of the dual-rotor system. Meanwhile, note that the high-pressure rotor and low-pressure turbine rotor can excite the first-order resonance of the dual-rotor system, while the low-pressure compressor rotor can only excite the second-order resonance, and then the dynamic model of this six-point support dual-rotor system can further be simplified as a relatively independent single-rotor system with one disk and a four-support dual-rotor system with dual disks.

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.

One of the problems arising in modern celestial mechanics is the need of precise numerical integration of dynamical equations of motion of the Moon. The action of tidal forces is modeled with a time delay and the motion of the Moon is therefore described by a functional differential equation (FDE) called delay differential equation (DDE). Numerical integration of the orbit is normally being performed in both directions (forwards and backwards in time) starting from some epoch (moment in time). While the theory of normal forwards-in-time numerical integration of DDEs is developed and well-known, integrating a DDE backwards in time is equivalent to solving a different kind of FDE called advanced differential equation, where the derivative of the function depends on not yet known future states of the function. We examine a modification of Adams--Bashforth--Moulton method allowing to perform integration of the Moon's DDE forwards and backwards in time and the results of such integration.

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.

This paper focuses on indirect adaptive fuzzy control of nonlinear descriptor systems described by both uncertain algebraic and differential equations aiming to guarantee asymptotic tracking of a regular and impulse-free descriptor reference model. The proposed controller exploits the universal approximation capability of Takagi–Sugeno–Kang (TSK) fuzzy models for the identification of the unknown system dynamics. More specifically, it is assumed that only the system order is known while all the dynamical equations of the system are completely unknown. In the proposed method, the asymptotic tracking of the reference model is guaranteed by suitable adaptation laws for the parameters of the TSK fuzzy model. Simulation results are presented to demonstrate the effectiveness of the proposed method.

It is shown that dynamical equations for quantum tomograms retain the normalizations of their solutions during time evolution only if the solutions satisfy a set of special conditions. These conditions are found explicitly. On the contrary, it is also illustrated that the classical Liouville equation, Moyal equation for Wigner function, and evolution equation for Husimi function retain normalizations of any initially normalized and quickly decaying at infinity functions on the phase space. Necessary and sufficient conditions for optical and symplectic tomogram-like functions to be tomograms of physical states are also discussed.

By using the reductive perturbation technique, the nonlinear dust ion-acoustic solitary wave models of the damped Korteweg–de Vries (D-KdV) and modified damped Korteweg–de Vries (D-mKdV) equations are formulated. We constructed the more general and new solitary wave solutions of nonlinear damped KdV and damped mKdV equations by using the modified mathematical technique. These obtained solutions are more useful in the development of quantum plasma, dynamics of solitons, dynamics of adiabatic parameters, dynamics of fluid and problems of biomedical, industrial phenomena. The new solutions are obtained in the shape of dark solitons, bright solitons, traveling wave and kink and anti-kink wave solitons. We show the physical structure of new solutions by two and three-dimensions graphical to know the physical interpretation of different structure of dust ion-acoustic solitary wave. The calculations show that this new technique is more powerful, effective, straightforward, and fruitfulness to study analytically other nonlinear complex physical models in plasma physics, mathematical physics, fluid mechanics, hydrodynamics, mathematical biology and many other physical sciences.

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.

Point symmetries allowed by plasticity equations in the dynamical case are used to construct solutions for the dynamical equations of ideal plasticity. These symmetries make it possible to convert the exact solutions of stationary dynamical equations to nonstationary solutions. The so-constructed solutions include arbitrary functions of time. The solutions allow us to describe the plastic flow between the plates changing their shape under the action of dynamical loads. Some new spatial self-similar solution is also presented.

This thesis is devoted to study the thermal vibration of graphene sheet embedded in viscoelastic medium based on nonlocal shear deformation theory. Some of basic concepts of the elasticity theory and plate theory are presented. The fundamental principles of elasticity theory such as the stresses, strains and the constitutive relations are given. Also, the equilibrium equations and Hooke's law for different elastic materials are given. Some of theories such as the classical thin plate theory and first-order shear deformation plate theory are presented. The nonlocal theory of elasticity is used to present thermal vibration of a single-layered graphene sheet (SLGS) embedded in viscoelastic medium. The viscous damping term is added to the elastic foundation medium to get a three-parameter visco-Pasternak medium. The nonlocal shear deformation theory is applied to obtain the governing dynamical equations of simply-supported SLGSs. Effects of nonlocal parameter as well as the length of SLGS, mode numbers, the three-parameter of foundation and the thermal parameter are discussed carefully for the vibration problem. The validation of the present frequencies is discussed and additional results for thermal vibration of SLGSs are investigated to take into account the effect of nonlocal, thermal, and visco-Pasternak’s parameters for future comparisons.

The aim of this article is to argue that a temporal asymmetry may be established within the framework of quantum field theory, independently of any violation of CP, and thereby T, in weak interactions, and independently of the property of time reversal invariance that its dynamical equations instantiate. Particularly, I shall argue that the temporal asymmetry can be stemmed from assessing the links between the proper group of symmetries of the theory and the ontology of the theory: arguments applied to establish which elements and magnitudes remain invariants under group transformations can also be used to establish a temporal asymmetry.

In this paper, a voltage-based sliding mode control (SMC) is presented to control the position of n rigid-link flexible-joint (RLFJ) serial robot manipulator in the presence of structured and unstructured uncertainties. In addition to good performance, the proposed method has three attracting properties: First, it has been developed for a class of RLFJ robot manipulators with a n degree of freedom (DOF). Second, the design process includes all the manipulator dynamical equations, the mechanical and the electrical part of the actuator. Third, the simple structure and low volume of computing make practical implementation possible. Using the idea of the independent joint control strategy, the system dynamic equations are decomposed into n independent subsystems. For each subsystem, three sliding surfaces are defined. Then using these sliding surfaces, control input laws are designed for all subsystems simultaneously. It is proved that the proposed method can guarantee the asymptotic stability of each subsystems and the global asymptotic stability of the closed-loop system in the presence of uncertainties. The results of simulations, as well as experimental results produced using MATLAB/Simulink external mode control on a flexible-joint electrically driven robot manipulator, illustrate high efficiency of the proposed control schemes.

In this paper, we have discussed the dynamics of non-adiabatic gravitating compact object in f(R, T) theory using the Misner–Sharp approach. Here, we have extended the work of Herrera and Santos, to the modified f(R, T) theory of gravity. In order to discuss the effects of heat flux on the dynamics of compact object, we have taken the non-adiabatic source inside spherically symmetric compact object and modified field equations have been evaluated in f(R, T) gravity. The Misner–Sharp mass has been used to discuss the variation of total energy inside the gravitating compact object. The dynamical equations have been formulated by using the Bianchi identities in f(R, T) gravity. Further, the dynamical equations have been coupled with the heat transportation equation resulting from the framework of M u ¨ l l e r –Israel–Stewart formalism. To analyze the results deeper, we have used the functional form of f(R, T) model, for such model the matter-curvature coupling constant λ has some direct effects on inertial mass density and gravitational forces. It makes the gravitational force less active thereby reducing the rate of collapse.