Dynamic Equations Of System Research Articles

Finite Elements with Switch Detection for numerical optimal control of nonsmooth dynamical systems with set-valued heaviside step functions

This paper develops high-accuracy methods for the numerical solution of optimal control problems subject to nonsmooth differential equations with set-valued Heaviside step functions. An important subclass of these systems are Filippov systems. By writing the Heaviside step function as the solution map of a linear program and using its optimality conditions, the initial nonsmooth system is rewritten into an equivalent Dynamic Complementarity System (DCS). The Finite Elements with Switch Detection (FESD) method (Nurkanović et al., 2024) was originally developed for Filippov systems transformed via Stewart’s reformulation into DCS (Stewart, 1990). This paper extends it to the above mentioned class of nonsmooth systems. The key ideas are to start with a standard Runge–Kutta method for the DCS and to let the integration step sizes to be degrees of freedom. Then, additional conditions are introduced to allow implicit but accurate switch detection and to remove possible spurious degrees of freedom when no switches occur. The theoretical properties of the FESD method are studied. The motivation for these developments is to obtain a computationally tractable formulation of nonsmooth optimal control problems. Numerical simulations and optimal control examples are used to illustrate the favorable properties of the proposed approach. All methods introduced in this paper are implemented in the open-source software package nosnoc (Nurkanović and Diehl, 2022).

Investigation of Thermo-Electrical Instabilities in a Semiconductor as 2D Dynamical Systems

A semiconducting sample placed in cryogenic media with applied electric field generates low frequency oscillations of electric current and sample temperature and known to be thermo-electrical instabilities. Although observation of current oscillations on oscilloscope is possible, change of sample temperature cannot be detected experimentally. Description of the phenomenon through mathematical equations helps to understand relationship of the two variables as well as their connection to deep trap behavior that are involved in supporting the instability. Mathematical model for thermo-electrical instabilities in an n type semiconductor based on the two deep trap level model with non-degenerate electron statistics has been introduced in order to investigate the unique relationship between the change in time of both electric current flowing through a semiconductor sample and the sample temperature. The 3D dynamical system of nonlinear inhomogeneous ordinary differential equations has been investigated as component 2D dynamical systems (n,T), (n,n<sub>t</sub>) and (n<sub>t</sub>,T) for local behavior at isolated equilibrium and at points on individual trajectories, where n, n<sub>t</sub> and T are free electron concentration at conduction band, electron concentration at deep traps and temperature of a semiconductor sample accordingly. Each of the planar systems is expressed in canonical form and investigated as a Cauchy problem with a set of appropriate initial values. This paper presents investigation results of phase trajectories of the planar systems depending on a single parameter – the temperature of cooling media T<sub>0</sub>. Based on obtained calculation results of time sequences of the three variables n, n<sub>t</sub> and T, phase differences among these variables have been determined for different values of T<sub>0</sub>. It has been established that the change in sample temperature lags behind change in current and this lag increases with T<sub>0</sub>. Clearly defined correlations among systems (n,T), (n,n<sub>t</sub>) and (n<sub>t</sub>,T) are seen, being the result of balance between field aided and thermal ionization mechanisms for charge carrier generation and recombination processes. Thermal and field assisted generation mechanisms compete with one another in achieving steady non equilibrium state in the system depending on temperature of cooling media T<sub>0</sub>.

Dynamical system analysis of Dirac-Born-Infeld scalar field cosmology in coincident f(Q) gravity* *SG acknowledges the Council of Scientific and Industrial Research (CSIR), Government of India, New Delhi, for a junior research fellowship ( 09/1026(13105)/2022-EMR-I). RS acknowledges the University Grants Commission (UGC), New Delhi, India, for awarding a Senior Research Fellowship (UGC-Ref. No.: 191620096030). PKS acknowledges the Science and Engineering Research

In this article, we present a dynamical system analysis of a Dirac-Born-Infeld scalar field in a modified gravity context. We considered a polynomial form of modified gravity, used two different types of scalar potential, polynomial and exponential, and found a closed autonomous dynamical system of equations. We analyzed the fixed points of such a system and evaluated the conditions under which deceleration to late-time acceleration occurs in this model. We note the similarity of the two models and show that our result is consistent with a previous study on Einstein's gravity. We also investigated the phenomenological implications of our models by plotting EoS (ω), energy density (Ω), and deceleration parameter (q) w.r.t. to e-fold time and comparing to the present value. We conclude the paper by observing how the dynamical system analysis differs in the modified gravity, and present the future scope of our research.

Design of hyperbolic quasi-zero stiffness metastructures coupled with nonlinear stiffness for low-frequency vibration isolation

Quasi-zero stiffness (QZS) metastructures with simple structures exhibit a high low-frequency vibration isolation performance, but the bearing capacity and working regions are limited due to the linear positive stiffness elements. In this study, a novel hyperbolic QZS metastructure coupled with nonlinear stiffness is designed by combining a pair of cosine-shaped beams and arc-shaped beams for low-frequency vibration isolation. Finite element models (FEM) and experimental analysis are utilized to examine the static characteristics of the metastructure, revealing a wide QZS region and significant load-bearing capacity. The metastructure can achieve effective vibration isolation properties through reasonably designing the unit cells. The influence of geometrical parameters on the QZS region is investigated to optimize and realize desirable QZS features. The dynamic equation and response of the vibration isolation system are determined using the harmonic balance method. Moreover, the corresponding displacement transmissibility is confirmed by numerical simulation. The results of the dynamic sweep frequency experiment further verify the validity of the theoretical analysis model. The hyperbolic QZS metastructure system possesses good vibration isolation performance due to the introduction and compensation of a nonlinear positive stiffness arc beam element. The proposed hyperbolic QZS metastructure provides a significant route for the stiffness compensation of metastructures.

GESO based RISE Controller for Active Flutter Suppression for Aeroelastic System

Abstract The paper presents a Robust Integral of Signum of Error (RISE) controller for active suppression of flutter of a two dimensional aerofoil and it is integrated with a General Extended State Observer (GESO). Towards this, aerofoil model in state space is first changed into a canonical form. Then, the controller is designed using RISE control technique. As equations of system dynamics are not into integral chain form, a GESO is formulated to observe system states and disturbances. The controller is implemented using estimated states and it is robustified using estimated disturbance. Stability of the proposed GESO-RISE controller is established using the Lyapunov theory. Simulations are performed to check the proposed controller's performance against variations in airspeed, uncertainties in model parameters, external disturbances, time delay and unmodeled dynamics. Comparison of the proposed GESO-RISE controller is carried out with existing controllers using two performance criteria i.e. Control Efforts and Integral of Absolute Error (IAE). It is discovered from simulations that the proposed GESO-RISE controller significantly reduces IAE and control efforts. Additionally, Monte Carlo method is utilized to check robustness of the proposed GESO-RISE controller. The proposed GESO-RISE controller significantly enhances flutter boundary of aerofoil and is completely implementable.

A unified model of particle physics and cosmology: Origin of inflation, baryogenesis, neutrino mass, CDM and dark energy

I propose a unified model of particle physic and cosmology based on both a new extension of the standard particle model and the fundamental principle of the standard cosmology. Beyond the standard model, I introduce several species of dark fields with the dark symmetry of [Formula: see text], the inflation field, cold dark matter (CDM) and dark energy (DE) are of course included among them. The model can coherently and completely describe the origin and evolution of the universe from the primordial inflation to the reheating, to the baryogenesis, to the current CDM and DE, and also the neutrino mass generation mechanism. For the energy evolution of each phase, I give its dynamical system of equations, and solve them by some special techniques. The numerical results can clearly show how each evolution is successfully implemented, furthermore, the model demonstrates that the DE genesis is purely from the CDM condensation, which is essentially a reverse process of the primordial slow-roll inflation, there is not the so-called “cosmological constant energy”. By use of fewer input parameters, the model not only exactly reproduces the measured inflationary data and the current energy density budget, but also finely predicts such important values as the tensor-to-scalar ratio [Formula: see text], the inflaton mass [Formula: see text][Formula: see text]GeV, the reheating temperature [Formula: see text][Formula: see text]GeV, the CDM mass [Formula: see text][Formula: see text]GeV, [Formula: see text] and [Formula: see text], in particular, the model smoothly clarifies and eliminates the “Hubble tension”. In short, the unified model newly establishes the deeper and closer relations between particle physics and cosmology, therefore we expect the ongoing and future experiments to test the model.

Risk Analysis and Simulation of Large Bridge Construction Based on System Dynamics

To reduce risk incidents in large bridge construction, it is necessary to study the interaction mechanisms and dynamic changes among various risk factors during the construction of large bridges. First, the evolution mechanism of construction risks for large bridges is analyzed, and a risk factor structure system is established. Then, based on system dynamics theory, a causal loop diagram and flow–stock diagram are constructed, and system dynamics equations for each variable in the flow diagram are established to build a system dynamics model. Finally, taking a large bridge construction project as an example, the variable equations are assigned actual values, and the construction risk level is simulated and analyzed. The results show that, ➀ continuous investment in safety funds can effectively reduce the overall risk level of the system, ➁ changes in the management risk subsystem have a significant impact on the overall risk level of large bridge construction, and ➂ increasing the safety investment ratio in both the personnel risk subsystem and the management risk subsystem can effectively reduce the total risk level of the system.

Modeling of Cooperative Robotic Systems and Predictive Control Applied to Biped Robots and UAV-UGV Docking with Task Prioritization.

This paper studies a cooperative modeling framework to reduce the complexity in deriving the governing dynamical equations of complex systems composed of multiple bodies such as biped robots and unmanned aerial and ground vehicles. The approach also allows for an optimization-based trajectory generation for the complex system. This work also studies a fast-slow model predictive control strategy with task prioritization to perform docking maneuvers on cooperative systems. The method allows agents and a single agent to perform a docking maneuver. In addition, agents give different priorities to a specific subset of shared states. In this way, overall degrees of freedom to achieve the docking task are distributed among various subsets of the task space. The fast-slow model predictive control strategy uses non-linear and linear model predictive control formulations such that docking is handled as a non-linear problem until agents are close enough, where direct transcription is calculated using the Euler discretization method. During this phase, the trajectory generated is tracked with a linear model predictive controller and addresses the close proximity motion to complete docking. The trajectory generation and modeling is demonstrated on a biped robot, and the proposed MPC framework is illustrated in a case study, where a quadcopter docks on a non-holonomic rover using a leader-follower topology.

Bifurcation analysis and new waveforms to the fractional KFG equation

This study presents novel waveforms and bifurcation analysis for the fractional Klein–Fock–Gordon (KFG) structure, which is widely used in particle and condensed matter physics. To examine the bifurcation, chaos, and sensitivity of the model, we derive the dynamical system from the Galilean transformation. We present and explore a diverse range of waveforms, including periodic waves, quasi-periodic waves, bright and dark solitons, kink waves, and anti-kink waves. Graphic diagrams from simulations illustrate the diverse features and existence of these solutions. Nonlinear models can benefit from the potency, brevity, and effectiveness of integration procedures used in contemporary scientific and engineering contexts.

A nonlinear low frequency quasi zero stiffness vibration isolator using double-arc flexible beams

There is an increasing demand for integrated small vibration isolators in precision sensing instruments and medical equipment. Therefore, a quasi-zero stiffness (QZS) vibration isolator with double-arc flexible beams (D-AFB) is proposed. The QZS isolator consists of D-AFB units, the base, platform, fixture, guide rail, and linear bearing. The static performance of D-AFB is analyzed using the FEA method, and the QZS condition is obtained. The harmonic balance method (HBM) and the pseudo arc length method (PALM) are used to solve the dynamic equations of the nonlinear system, and the analytical solutions of displacement transmissibility and Force transmissibility are derived. In theory, the initial frequency is 77.6 % lower than that of the equivalent linear system. The QZS characteristic of D-AFB is verified by static experiment, and the low-frequency isolation performance of the QZS isolator is verified by dynamic experiment under constant amplitude and constant acceleration excitation. The D-AFB has a simple structure and can easily satisfy QZS conditions. So the D-AFB can be customized according to load requirements. Multiple vibration isolation loads can be obtained by adjusting the assembly of the D-AFB. The QZS isolator can adapt to load changes within 10 % and has an isolation effect on vibrations above 4 Hz. The idea of designing a QZS isolator with D-AFB as an array unit has a good research prospect and great application value in meeting the requirements of miniaturization and integration.

Origami-inspire quasi-zero stiffness structure for flexible low-frequency vibration isolation

As an emerging technology, origami exhibits rich nonlinear static and dynamic properties. However, simulating creases remains a key challenge hindering practice application, especially for volumetric (deformable) origami. Inspired by the bistable behavior and axial-rotational coupling mechanism of triangulated cylindrical origami (TCO), this paper proposes a novel combined truss-spring structure with additional inertial mechanism (TCTS-RIM) for low-frequency vibration isolation. Unlike pure TCO, the TCTS-RIM uses elastic linkages to simulate all creases and modifies crease positions, enhancing physical realization in engineering applications and offering tunable stiffness and inertia on demand. Static analysis shows the modified truss-spring structure in TCTS-RIM has superior quasi-linear negative stiffness compared to pure TCO, which enables a very wide quasi-zero stiffness (QZS) range via linear spring compensation. The wide QZS range for different loads can be realized by setting appropriate structural parameters. From system dynamic equations, nonlinear inertia, nonlinear quadratic force, and nonlinear damping due to RIM are identified and discussed. The Alternating frequency–time harmonic balance method is used to obtain the displacement transmissibility, and its parametric influencing investigations verify that the proposed TCTS-RIM has excellent low-frequency vibration isolation performance and rich tunability. Comparative discussions show the wide QZS range significantly reduces the resonance frequency, broadens the isolation frequency band, and yields superior dynamic performance versus pure TCO isolators. Moreover, equivalent mass, anti-resonance and softening behavior induced by the rotational inertia mechanism further enhance low-frequency vibration isolation performance. An experimental prototype is built, and experimental results validate the correctness of the theoretical analyses. This work presents a simple mechanical implementation of volumetric origami, which may progress the engineering applications of origami and provide a viable approach to low-frequency vibration control.

Application of nonlinear energy sink for vibration suppression of mine boring machines

The propulsion system of mine boring machines often experiences severe vibration during mining. To suppress the vibration of the main beam system of roadheader and prevent the excessive fatigue of propulsion system, a magnetic nonlinear energy sink (MNES) is designed to suppress vibration. In this paper, firstly, the structure and design principle of MNES are introduced, and the dynamic equations of the main beam-MNES system are established. On this basis, the genetic algorithm is applied to optimize the vibration suppression of MNES, and the optimal parameters are determined. In results, the accumulative energy dissipation rate of optimal MNES in transient vibration can reach 90.2% and the vibration reduction effect of the optimal MNES in steady-state vibration is about 82.2%.

Physics-constrained robust learning of open-form partial differential equations from limited and noisy data

Unveiling the underlying governing equations of nonlinear dynamic systems remains a significant challenge. Insufficient prior knowledge hinders the determination of an accurate candidate library, while noisy observations lead to imprecise evaluations, which in turn result in redundant function terms or erroneous equations. This study proposes a framework to robustly uncover open-form partial differential equations (PDEs) from limited and noisy data. The framework operates through two alternating update processes: discovering and embedding. The discovering phase employs symbolic representation and a novel reinforcement learning (RL)-guided hybrid PDE generator to efficiently produce diverse open-form PDEs with tree structures. A neural network-based predictive model fits the system response and serves as the reward evaluator for the generated PDEs. PDEs with higher rewards are utilized to iteratively optimize the generator via the RL strategy and the best-performing PDE is selected by a parameter-free stability metric. The embedding phase integrates the initially identified PDE from the discovering process as a physical constraint into the predictive model for robust training. The traversal of PDE trees automates the construction of the computational graph and the embedding process without human intervention. Numerical experiments demonstrate our framework's capability to uncover governing equations from nonlinear dynamic systems with limited and highly noisy data and outperform other physics-informed neural network-based discovery methods. This work opens new potential for exploring real-world systems with limited understanding.

A new approach to the entropy of a transitive BE-algebra with countable partitions

The concept of entropy and information gain of BE-algebras in scientific disciplines such as information theory, data science, supply chain and machine learning assists us to calculate the uncertanity of the scientific processes of phenomena. In this respect the notion of filter entropy for a transitive BE-algebra is introduced and its properties are investigated. The notion of a dynamical system on a transitive BE-algebra is introduced. The concept of the entropy for a transitive BE-algebra dynamical system is developed and, its characteristics are considered. The notion of equivalent transitive BE-algebra dynamical systems is defined, and it is proved the fact that two equivalent BE-algebra dynamical systems have the same entropy. Theorems to help calculate the entropy are given. Specifically, a new version of Kolmogorov– Sinai Theorem has been proved. The study introduces the concept of information gain of a transitive BE-algebra with respect to its filters and investigates its properties. This study proposes the use of filter entropy to approximate the level of risk introduced by a BE-algebra dynamical system. This aim is reached by defining the information gain with respect to the filters of a BE-algebra. This methodology is well developed for use in engineering, especially in industrial networks. This paper proposes a novel approach to assess the quantity of uncertainty, and the impact of information gain of a BE-algebra dynamical system.

Parameter optimization of vibration control system for adjacent building structures based on negative stiffness inerter damper

Building structures are subjected to strong earthquakes, which result in lateral collisions between them. Such collisions often cause severe structural damage and exacerbate the seismic hazard risk of building structures during earthquake events. This paper discusses the application of vibration control devices based on negative stiffness inerter damper in single-story adjacent building structures. The dynamic equations of the vibration control system containing different types of negative stiffness inerter damper under seismic excitation are established as a unified model. The H2 norm theory and Monte Carlo pattern search method are used to optimize the design parameters to improve the vibration control performance of the system, and the dynamic characteristics of the system are investigated. The results demonstrate that attaching negative stiffness inerter damper to adjacent building structures can effectively improve the overall seismic capacity reserve of the building and reduce the risk of collision of adjacent building structures; improve the robustness and stability of the system, and better reduce the displacement response of the building structure under seismic excitation. In addition, the potential of NSID-based vibration control devices to convert seismic energy into usable electricity has been investigated.

Site Quality Models and Fuel Load Dynamic Equation Systems Disaggregated by Size Fractions and Vegetative States in Gorse and High Heath Shrublands in Galicia (NW Spain)

Compatible model systems were developed for estimating fuel load dynamics in Ulex europaeus (gorse) and in Erica australis (Spanish heath) dominated shrub communities at stand level. The models were based on intensive, detailed destructive field sampling and were fitted simultaneously to fulfill the additivity principle. The models enable, for the first time, estimation of the biomass dynamics of the total shrub layer, size fractions and vegetative stage, with reasonably good accuracy. The approach used addresses the high variability in shrub biomass estimates by using a site index (SI) based on biomass levels at a reference age of 10 years. Analysis of the effect of climatic variables on site index confirmed the preference of gorse for mild temperatures and the ability of high heath communities to tolerate a wider range of temperatures. In the gorse communities, SI tended to increase as summer rainfall and the mean temperature of the coldest month increased. However, in the heath communities, no relationships were observed between SI and any of the climatic variables analyzed. The study findings may be useful for assessing and monitoring fuel hazards, updating fuel mapping, planning and implementing fuel reduction treatments and predicting fire behavior, among other important ecological and biomass use-related applications.

Dynamic Analysis and Optimization on Passive/Active Vibration Reduction of a Beam Structure with Distributed Smart Foams

A smart foam is a vibration absorption structure integrated with piezoelectric material and shows promise for a highly efficient structural vibration reduction both passively and actively. The arrangement of the smart foams on the primary structure is a key issue in practical engineering. In this work, the vibration reduction performances of distributed smart foams attached to a simply supported beam are investigated, and the distribution scheme is optimized. The dynamic equations of the vibration system are established for passive and active vibration reduction, respectively. The frequency responses are analyzed based on a Newmark-[Formula: see text] method, and the accuracy is verified by a finite element analysis. An optimization method based on the genetic algorithm is proposed for the smart foams’ quantity and locations. The study reveals that, with the same total attached mass (1% of the beam mass), a limited distribution of smart foams achieves a larger vibration reduction ratio than a single smart foam or uniformly distributed smart foams. For the passive reduction mode, the optimal scheme is to arrange four smart foams at the middle of the beam, and a reduction ratio of 38% is achieved. It is related to both the first-order modal shape of the beam and the mass distribution of smart foams. For the active reduction mode, the optimal scheme is to arrange two smart foams separately, and 98% of the vibration is reduced. The optimal location is no longer at the middle of the beam, because the active moment of the piezoelectric beam makes the main contribution, rather than the passive vibration absorption. This research provides an optimization method and instructions for both passive and active vibration reductions of smart foams.

Nonlinear dynamics study of freestanding triboelectric nanogenerator system

As a nonlinear dynamic system, the dynamic equation of triboelectric nanogenerator system is not only difficult to accurately establish, but also difficult to solve. The output signals of triboelectric nanogenerator system depends on the initial conditions of the system, which reflects the dynamic behavior of the system. Currently, there is little research on using the output signals to study the dynamic behavior of triboelectric nanogenerator system. Therefore, a dynamic simulation model of a freestanding triboelectric nanogenerator (F-TENG) system was established using COMSOL software. The mathematical model of the F-TENG system was fitted. Using chaos theory and its phase-space reconstruction technology, the phase-space trajectories of the F-TENG system were reconstructed through voltage signals under external factors, and the dynamic behavior of the F-TENG system was discussed and studied. Finally, validation experiments were carried out. The research shows that with the variation of internal factors and operating frequency, the mathematical model of the F-TENG system remains basically unchanged, and the variation law of output characteristics also remains basically unchanged. The F-TENG system is in periodic motion; with the variation of external factors such as resistance and friction layer spacing, the mathematical model of the F-TENG system will evolve in a complex direction, and the output characteristics will gradually distort. The F-TENG system will evolve from periodic motion to chaotic motion. The research results not only reveal the dynamic behavior and enrich the nonlinear dynamic theory of triboelectric nanogenerator system, but also provide theoretical guidance for increasing the working bandwidth and stability of triboelectric nanogenerator system.

Sparse Estimation for Hamiltonian Mechanics

Estimating governing equations from observed time-series data is crucial for understanding dynamical systems. From the perspective of system comprehension, the demand for accurate estimation and interpretable results has been particularly emphasized. Herein, we propose a novel data-driven method for estimating the governing equations of dynamical systems based on machine learning with high accuracy and interpretability. The proposed method enhances the estimation accuracy for dynamical systems using sparse modeling by incorporating physical constraints derived from Hamiltonian mechanics. Unlike conventional approaches used for estimating governing equations for dynamical systems, we employ a sparse representation of Hamiltonian, allowing for the estimation. Using noisy observational data, the proposed method demonstrates a capability to achieve accurate parameter estimation and extraction of essential nonlinear terms. In addition, it is shown that estimations based on energy conservation principles exhibit superior accuracy in long-term predictions. These results collectively indicate that the proposed method accurately estimates dynamical systems while maintaining interpretability.

Phase trajectories and Chaos theory for dynamical demonstration and explicit propagating wave formation

This paper is subjected to study the nonlinear integrable model which is the (3+1)-dimensional Boussinesq equation which has a lot of applications in engineering and modern sciences. To find and examine the analytical exact solitary wave solutions of (3+1)-dimensional Boussinesq equation, a modified generalized exponential rational functional method is exerted. As a result, waves, singular periodic, hyperbolic, and trigonometric type solutions are obtained. These acquired solutions are more innovative and encouraging to researchers in their endeavor to study physical marvels. To illustrate how some selected exact solutions propagate, the graphical representation in 2D, Contour, and 3D of those solutions is provided with various parametric values. The considered equation is additionally transformed into the planar dynamical structure by applying the Galilean transformation. All potential phase portraits of the dynamical system are investigated using the theory of bifurcation. The Hamiltonian function of the dynamical system of differential equations is established to see that, the system is conservative over time. The presentation of energy levels through graphics provides valuable insights, and it demonstrates that the model has solutions that can be expressed in closed form. The periodic, quasi-periodic, and chaotic behaviors of the 2D, 3D, and time series are also observable once the dynamical system is subjected to an external force. Meanwhile, the sensitivity of the derived solutions is carefully examined for a range of initial conditions.