Stability Of Dynamical Systems Research Articles

New insight into bifurcation of fractional-order 4D neural networks incorporating two different time delays

Delay has a vital influence on the dynamics of neural networks. Exploring the effect of time delay on the dynamics of neural networks has become a hot issue in mathematics and engineering fields. In this current manuscript, on the basis of the earlier publications, we put forward a new fractional-order 4D neural networks incorporating two different time delays. First of all, the existence and uniqueness, boundedness of the solution of the fractional-order 4D neural networks incorporating two different time delays. are analyzed by applying contraction mapping principle, construct of an adaptive function, respectively. Next, the stability and the emergence of Hopf bifurcation are explored by making use of the stability and bifurcation theory of fractional-order dynamical system. A series of novel stability criteria and bifurcation conditions guaranteeing the stability and the emergence of Hopf bifurcation of the considered fractional-order 4D neural networks under the different delay cases are built. What,s more, the impact of delay on stabilizing neural networks and controlling the emergence of Hopf bifurcation of neural networks is adequately uncovered. At last, Matlab simulation figures are presented to confirm scientificness of the derived prime conclusions. The derived prime conclusions of this manuscript are perfectly innovative and own momentous theoretical reference value in the control issue and design aspect of neural networks.

Eigenvalue spectra and stability of directed complex networks.

Quantifying the eigenvalue spectra of large random matrices allows one to understand the factors that contribute to the stability of dynamical systems with many interacting components. This work explores the effect that the interaction network between components has on the eigenvalue spectrum. We build on previous results, which usually only take into account the mean degree of the network, by allowing for nontrivial network degree heterogeneity. We derive closed-form expressions for the eigenvalue spectrum of the adjacency matrix of a general weighted and directed network. Using these results, which are valid for any large well-connected complex network, we then derive compact formulas for the corrections (due to nonzero network heterogeneity) to well-known results in random matrix theory. Specifically, we derive modified versions of the Wigner semicircle law, the Girko circle law, and the elliptic law and any outlier eigenvalues. We also derive a surprisingly neat analytical expression for the eigenvalue density of a directed Barabási-Albert network. We are thus able to deduce that network heterogeneity is mostly a destabilizing influence in complex dynamical systems.

Generalized Markov–Bernstein Inequalities and Stability of Dynamical Systems

Generalized Markov–Bernstein Inequalities and Stability of Dynamical Systems

Non-Euclidean Contraction Theory for Robust Nonlinear Stability

In this article, we study necessary and sufficient conditions for contraction and incremental stability of dynamical systems with respect to non-Euclidean norms. First, we introduce weak pairings as a framework to study contractivity with respect to arbitrary norms, and characterize their properties. We introduce and study the sign and max pairings for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{1}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _\infty$</tex-math></inline-formula> norms, respectively. Using weak pairings, we establish five equivalent characterizations for contraction, including the one-sided Lipschitz condition for the vector field as well as logarithmic norm and Demidovich conditions for the corresponding Jacobian. Third, we extend our contraction framework in two directions: we prove equivalences for contraction of continuous vector fields, and we formalize the weaker notion of equilibrium contraction, which ensures exponential convergence to an equilibrium. Finally, as an application, we provide incremental input-to-state stability and finite input-state gain properties for contracting systems, and a general theorem about the Lipschitz interconnection of contracting systems, whereby the Hurwitzness of a gain matrix implies the contractivity of the interconnected system.

Disturbance Observer-Based Sliding Mode Controller for Underwater Electro-Hydrostatic Actuator Affected by Seawater Pressure

This paper presents a disturbance-observer-based sliding mode control strategy for an underwater electro-hydrostatic actuator, particularly considering that electro-hydrostatic actuators (EHAs) significantly suffer from sea pressure disturbance, which makes it hard to achieve high-precision position control. Therefore, a nonlinear disturbance observer was designed to aim at the matched and mismatched disturbance caused by sea pressure disturbance. Then, a nonlinearities model for an underwater EHA was established, and a related non-singular fast terminal sliding mode (NFTSM) controller was designed by changing the conventional sliding mode surface to further improve the control accuracy. In addition, the backstepping tool was used to guarantee the robust stability of the entire three-order hydraulic dynamic system. Finally, a comparative simulation was conducted with different load forces in AMESim and Simulink, which effectively verified the high tracking performance of the proposed control strategy.

A general theory for infectious disease dynamics

We present a general theory of infection spreading. It directly follows from conservation laws once known the probability density functions of latent times. The theory can deal with any distribution of compartments latent times. Real probability density function can be then employed, thus overcoming the limitations of standard SIR, SEIR and other similar models that implicitly make use of exponential or exponential-related distributions. SIR and SEIR-type models are, in fact, a subclass of the theory here presented. We show that beside the infection rate, the probability density functions of latent times in the exposed and infectious compartments govern the dynamics of infection spreading. We study the stability of such dynamical system and provide the general solution of the linearized equations in terms of the characteristic functions of latent times probability density functions. We exploit the theory to simulate the spreading of COVID-19 infection in Italy during the first 120 days.

Finite-time stabilization of state dependent impulsive dynamical linear systems

The problem of finite-time stabilizing control design for state-dependent impulsive dynamical linear systems (SD-IDLS) is tackled in this paper. Such systems are characterized by continuous-time, linear, possibly time-varying, dynamics coupled with discrete-time, linear, possibly time-varying, dynamics. The continuous-time part determines the system evolution in any time interval between two consecutive resetting events, while the discrete-time part governs its instantaneous state jump whenever the system trajectory intersects a resetting set, i.e. a region of the state space assumed to be time-independent. By making use of a quadratic control Lyapunov function, the finite-time stabilization of SD-IDLS through a static output feedback control design is specifically discussed in this paper. A sufficient and constructive result is provided based on the conical hulls of the resetting set subregions and on some cone copositivity properties of the chosen control Lyapunov function. Such a result is based on the solution of a feasibility problem that involves a set of coupled Difference/Differential Linear Matrix Inequalities (D/DLMI), which is shown to be less conservative and more numerically amenable with respect to other results available in the literature. An example illustrates the effectiveness of the proposed approach.

Stochastic Power System Dynamic Simulation and Stability Assessment Considering Dynamics From Correlated Loads and PVs

The integration of uncertain photovoltaics (PVs) and flexible loads leads to uncertainties in the power system dynamic simulation results. Furthermore, geographically close PV farms are correlated and may exhibit nonlinear correlations. This article proposes a copula-based sparse polynomial chaos expansion (PCE) framework for quantifying the impacts of uncertain dynamic PVs and loads on power system dynamic simulations and stability. The dynamics include both PV and load stochasticity and those governed by differential and algebraic equations. The copula statistics are utilized to accurately characterize the dependence structure of PVs and further used to develop the copula-PCE for quantifying the impacts of uncertain PVs and loads. A probabilistic TSI is also developed to assess the uncertainties from PVs and loads on the system stability. To address the cases, where both stable and unstable conditions coexist, a preprocessing step via sample classification is proposed. The effects of different dependence structures of PVs and different numbers of uncertain sources are investigated. Comparison results with other methods on the modified IEEE 39- and 118-bus systems, including the Monte Carlo method, Latin hypercube sampling, and traditional PCE without consideration of uncertain input correlations show that the proposed method is able to accurately quantify the uncertain dynamic simulations and transient system stability while being computationally efficient.

Discrete Dynamics of Balls in Cageless Ball Bearings

Cageless ball bearings are often preferred as a back-up bearing for active magnetic bearings to support a falling rotor, but the contact between the balls of the cageless ball bearing may lead to the deterioration of the bearing performance and affect the dynamic stability of the rotor system. Thus, we studied the discrete dynamics of cageless ball bearings. First, a model is proposed to change the groove curvature center of the local outer raceway to control the ball velocity to achieve dispersion. Combined with the spatial geometry theory, the mathematical model of the discrete raceway is established, the collision between the balls is considered as an abruptly added constraint, and the non-smooth dynamics equation of the cageless ball bearing with a local discrete raceway is established. Then, the fourth-order Adams prediction correction algorithm is used to numerically solve the dynamic discretization of the ball, and the structural parameters of the discrete raceway are preferably selected, according to the phase diagram of the ball and the change in the angular spacing. The results show that the structure of the discrete raceway has a strong influence on the discrete dynamics of the ball.

Analytical spectral design of mechanical metamaterials with inertia amplification

Functional metamaterials offering superior dynamic performances can be conceived by introducing local mechanisms of inertia amplification in the periodic microstructure of cellular composite media. The minimal physical realization of an inertially amplified metamaterial is represented by a one-dimensional crystal lattice, characterized by an intracellular pantograph mechanism. The microstructural dynamics of the periodic tetra-atomic cell is synthetically described by a low-dimension lagrangian model. The lagrangian coordinates account for the longitudinal motion of a pair of elastically coupled principal atoms, rigidly connected by the pantograph arms to a pair of transversely-oscillating secondary atoms, serving as inertial amplifiers. Within the range of small-oscillations, the free propagation of undamped harmonic waves is described by a linear difference equation, governed by a symplectic transfer matrix. First, the band structure of the complex-valued dispersion spectrum is determined analytically, by properly exploiting the formal (time-to-space) analogy with the Floquet Theory for the stability of non-autonomous dynamic systems (direct problem). Second, a spectral design problem is stated and solved by inverting analytically the functions expressing parametrically the boundaries separating attenuation (stop) and propagation (pass) bands in the frequency spectrum (inverse problem). Specifically, the inverse problem solution has the merit of providing simple formulas for parametric design. Such formulas explicitly and exactly identify the mechanical parameters of the inertially amplified metamaterial possessing a desired pass-stop-pass band structure, assigned according to functional design requirements. The existence and uniqueness of the solution in the admissible range of mechanical parameters is discussed. The discussion provides (i) mathematical demonstration for the existence of feasible design solutions or multi-solutions, (ii) complete definition of the physically realizable band structures in the frequency domain, and (iii) alternative criteria to design iso-band structured metamaterials within the admissible domain of mechanical parameters. The extra-customization possibilities offered by iso-band structured metamaterials are analyzed in terms of static and dynamic performances. As feasible limit cases of the analytical spectral design, extreme mechanical metamaterials working as phononic superfilters and/or superpropagators are realizable.

Uniform global asymptotic stability for nonautonomous nonlinear dynamical systems

Research on uniform global asymptotic stability has the advantage of being able to predict the asymptotic velocity to the origin for all solutions of equations describing nonlinear phenomena. This study elucidates the sufficient conditions under which the zero solution of a nonautonomous nonlinear dynamical system of second order is uniformly globally asymptotically stable. From the results obtained, a certain first-order nonlinear differential equation associated with the second-order dynamical system plays a vital role in uniform global asymptotic stability. More precisely, under several reasonable assumptions, if the integral from σ to t+σ of a particular solution of the first-order differential equation diverges uniformly with respect to σ, then the zero solution of the dynamical system is guaranteed to be uniformly globally asymptotically stable. For the proof, the behavior of the solutions of the dynamical system is carefully tracked. An example that also includes a linear case is provided to illustrate the main result. Simulations are also presented to facilitate understanding of the example, and a local theorem corresponding to the main result and its application to an ecosystem model are discussed.

Stability of Logical Dynamic Systems With a Class of Constrained Switching

In this paper, a novel constrained switching rule, called “time-triggered logical switching” (TTLS) is considered for switched logical dynamic systems (SLDSs). Compared with the general time-triggered switching, the activation mode of TTLS cannot be arbitrary and is pre-allocated according to the logic operation, which is more practical. The TTLS is described as an LDS, and according to the characteristics of LDS, the stability analysis of SLDSs with TTLS is converted into the stability analysis of SLDSs under the logical switching cycle sequences. Firstly, based on the equivalent algebraic form of SLDSs with TTLS, combining the Lyapunov theory of LDS with average dwell-time method, several sufficient conditions are put forward for ensuring the point stability of the considered SLDSs. Then, by defining the switching cycle invariant subset and constructing a new system, the set stability analysis of the original system is transformed into the point stability analysis of the new system, and further, the obtained results for the point stability analysis are applied to the set stability analysis. At last, the validity of obtained results is illustrated by simulation on gene and protein signaling activity patterns.

Research on Frequency Transmission Scheme of Wind Farm in the Scenario of DC Islanding

With the wide application of new energy technologies, wind power has become one of the technologies with the largest development scale. Voltage Source Converter based High Voltage Direct Current Transmission system has unique advantages in its active and reactive power decoupling control, etc., and has become the main solution to solve the wind power transmission of large-scale wind farms. This paper studies the control characteristics and basic principles of new energy generation in the passive transmission system of DC island power grids, and proposes a frequency signal transmission scheme in the external transmission scenario. The simulation results verify the feasibility of the frequency signal transmission scheme between the sending-end wind farm and the receiving-end power grid in the DC islanding scenario, and that the VSC-HVDC system can quickly provide active power support and improve the dynamic frequency stability of the interconnected system.

Predicting the Stability of Hierarchical Triple Systems with Convolutional Neural Networks

The dynamical stability of hierarchical triple systems is a long-standing question in celestial mechanics and dynamical astronomy. Assessing the long-term stability of triples is challenging because it requires computationally expensive simulations. Here we propose a convolutional neural network model to predict the stability of equal-mass hierarchical triples by looking at their evolution during the first 5 × 105 inner binary orbits. We employ the regularized few-body code tsunami to simulate 5 × 106 hierarchical triples, from which we generate a large training and test data set. We develop 12 different network configurations that use different combinations of the triples’ orbital elements and compare their performances. Our best model uses six time series, namely, the semimajor axes ratio, the inner and outer eccentricities, the mutual inclination, and the arguments of pericenter. This model achieves excellent performance, with an area under the ROC curve score of over 95% and informs of the relevant parameters to study triple systems stability. All trained models are made publicly available, which allows predicting the stability of hierarchical triple systems 200 times faster than pure N-body methods.

Stability of a Bomb with a Wind-Stabilised-Seeker

The primary purpose of this article is to consider the main factors affecting the stability of a bomb with a Wind-Stabilised-Seeker (WSS). The article contains a full mathematical description of the bomb-WSS system motion. It allows you to calculate the spatial motion of the bomb and the motion of WSS in relation to the bomb. The method of calculating the aerodynamic forces is also described. The sample simulation results were presented and discussed. The equations of motion of the system were determined. They constituted the basis for the development of a computer program to simulate the motion of the system. Aerodynamic forces and moments were calculated using the Prodas software and based on the results of the bomb tests in the wind tunnel. The mass and geometrical data of the system relate to the tested prototype of an aerial bomb. The result of the research is a comprehensive assessment of the influence of the geometric parameters of the mobile tracking system on the dynamic stability of the bomb-WSS system. This knowledge is necessary for proper design of control laws based on the signals registered by the WSS. The main parameter influencing the WSS-bomb stability is the aerodynamic focus position of the tracking system. It should be behind the WSS attachment point. The position of the WSS centre of mass in relation to the attachment point also has an adverse effect on its stability. The length of the stick connecting the WSS with the bomb is irrelevant.

Consistency optimal coordination control of underground heavy-load robot in nonstructural environment

In order to guarantee the dynamic stability of robots in nonstructural environment, this paper proposes a new consistency optimal coordination (COC) control strategy. Firstly, by analyzing the topological structure, motion correlation and control coupling relationship among subsystems, the general expression of multi-body coupling system (MCS) for underground heavy-load robot is realized. Then, the transient spatial output deviations are proposed as the characterization of dynamic stability for the underground heavy-load robot. Afterwards, in order to reduce the strong coupling relationship among subsystems, the underground heavy-load robot is decoupled into fixed-point and non-fixed-point operation modes, the Nyquist stability and Lyapunov stability are applied to discriminate the dynamic stability of two working modes respectively. Finally, based on the idea of minimum loop gain compensation of mechanism, the COC control strategy is proposed, which can coordinate the subsystems to be consistent, so as to realize the dynamic stability of overall system in nonstructural environment. Both the experimental and situational results verify that the COC control strategy proposed in this paper not only realizes stable output of robot, but also effectively reduces the lag caused by instability. The work of this paper improves the dynamic stability analysis theory for robots, and promotes adaptability and dynamic response capability of moving robots as well.

Numerical Construction of Lyapunov Functions Using Homotopy Continuation Method

Lyapunov functions are frequently used for investigating the stability of linear and nonlinear dynamical systems. Though there is no general method of constructing these functions, many authors use polynomials in $ p-forms $ as candidates in constructing Lyapunov functions while others restrict the construction to quadratic forms. By focussing on the positive and negative definiteness of the Lyapunov candidate and the Hessian of its derivative, and using the sum of square decomposition, we developed a method for constructing polynomial Lyapunov functions that are not necessarily in a form. The idea of Newton polytope was used to transform the problem into a system of algebraic equations that were solved using the polynomial homotopy continuation method. Our method can produce several possibilities of Lyapunov functions for a given candidate. The sample test conducted demonstrates that the method developed is promising

Dynamic power-based temporary frequency support scheme for a wind farm

Doubly fed induction generators can participate in frequency support following a disturbance by releasing their rotational energies. However, when regaining the rotor speed, a secondary frequency dip (SFD) tends to produce a sudden output drop. This study suggests a dynamic power-based stepwise inertial control (IC) scheme of a wind power plant for minimizing the SFD while reducing the maximum frequency deviation (MFD), considering the non-negligible wake. To this end, the reference of the output power increases to the torque limit. Afterward, the power reference smoothly decays with the dynamically decreasing incremental power and then automatically switches to the maximum power point tracking operation. The performance of the temporary frequency support with the suggested stepwise IC strategy is investigated with various penetration levels of wind power. Test results demonstrate that the suggested stepwise IC strategy can minimize the SFD and reduce mechanical stresses on the wind turbine during the recovery of the rotor speed while reducing the MFD. Therefore, the suggested stepwise IC strategy secures the dynamic frequency stability of an electric power system dominated by wind power generations.

Increasing the stability of hydrogen production in the wind energy-hydrogen system through the use of synthetic inertia of the wind turbine

One of the trends in the development of the electric power industry in the world is the use of renewable resources. In recent years, the development of hydrogen energy has been widely discussed, which, when combined with renewable energy sources, makes it possible to obtain “green” hydrogen. One of the promising solutions for obtaining green hydrogen is the development of wind energy-hydrogen systems. However, one of the limiting factors, in addition to economic aspects, are technical limitations. Sustainable hydrogen production is complicated by the variability of wind energy. In addition, energy systems with a predominant share of wind power units have a low total inertia value. Consequently, such systems become more sensitive to changes in load modes in the grid and become less stable in emergency modes. Together, these facts reduce the reliability of wind energy-hydrogen system. The paper proposes to increase the stability of wind energy-hydrogen system and power system as a whole to apply synthetic inertia for wind power plants. It is shown that by adjusting the parameters of synthetic inertia can provide not only the required inertial response, but also increase the dynamic stability of the power system and as a consequence wind energy-hydrogen system. The influence of the measurement window time and calculation of the frequency change rate, which is important in certain modes, is also investigated.

Comprehensive stiffness analysis of the cylindrical roller bearing for aircraft engines considering the radial clearance

PurposeThe rotor system supported by the cylindrical roller bearings is widely used in various fields such as aviation, space and machinery due to its importance. In the study of the dynamic characteristics of the cylindrical roller bearings, it is important to accurately calculate the stiffness of the cylindrical roller bearings. The stiffness of the cylindrical roller bearings is very important in the analysis of the vibration characteristics of the rotor system. Therefore, in this paper, the method of creating a comprehensive stiffness model of the cylindrical roller bearing is mentioned. The purpose of this study is to improve the dynamic stability of the rotor system supported by the cylindrical roller bearing by accurately establishing the comprehensive stiffness calculation model of the cylindrical roller bearings.Design/methodology/approachIn consideration of the radial clearance of the cylindrical roller bearing, the radial load acting on the cylindrical roller bearing was derived, and based on this, a model for calculating the Hertz contact stiffness of the cylindrical roller bearing was created. Based on the load considering the radial clearance, an oil film stiffness model of the cylindrical roller bearing was created under the elastohydrodynamic lubrication (EHL) theory. Then, the comprehensive stiffness was calculated by combining Hertz contact stiffness and the oil film stiffness of the cylindrical roller bearing, and the dynamic parameters are calculated by using the MATLAB program.FindingsWhen the radial clearance of the cylindrical roller bearing is considered, the comprehensive stiffness is larger than when the radial clearance is not taken into account, and the radial clearance of the cylindrical roller bearing is an important factor that directly affects the comprehensive stiffness of the cylindrical roller bearing.Originality/valueIn this paper, based on Hertz contact theory and the EHL theory, the authors investigated the method of creating a comprehensive stiffness model of the cylindrical roller bearing considering the radial clearance. These results will contribute to the theoretical basis for studying the mechanics of cylindrical roller bearings and optimizing their structures, and they will provide an important theoretical basis for analyzing the dynamic characteristics of the rotor system supported by the cylindrical roller bearing.