Search

The difference and similarity between the velocity- and position-Verlet integrators are discussed from the viewpoint of their Hamiltonian representations for both linear and nonlinear systems. For a harmonic oscillator, the exact Hamiltonians reveal that positional trajectories generated by the two integrators follow an identical second-order differential equation and thus can be matched by adjusting initial conditions. In contrast, the series expansion of the Hamiltonians for the nonlinear discrete dynamics clearly indicates that the two integrators differ fundamentally. These analytical results are confirmed by simple numerical simulations of harmonic and anharmonic oscillators. [ABSTRACT FROM AUTHOR]

Basic multimode impedance analysis, based on the availability of nonequilibrium charge carriers and their delayed return to equilibrium, is employed to assess the state of equivalent circuit representations. This analysis highlights the necessity of surpassing these representations in nonlinear systems with inherent memory, along with their associated advantages and limitations. On the basic grounds of generation and recombination (or trapping) of nonequilibrium carriers and their relaxation times, we show how seeming complexity of frequency-dependent impedance that matches a vast universe of experimental evidences can be reduced to simple combinations of basic microscopic ingredients. Counterintuitive features, such as negative capacitances or unexpected inductances, arise when the results are projected onto linear equivalent circuit representations. This indicates the presence of certain limitations and potential ambiguities in the symbolic representation of "equivalent" circuits. Our approach further provides a microscopic perspective that exposes the linkage of an apparent flux with an apparent inductance dismissing any magnetic essence. [ABSTRACT FROM AUTHOR]

Fermi's golden rule (FGR) offers an empirical framework for understanding the dynamics of spin–lattice relaxation in magnetic molecules, encompassing mechanisms like direct (one-phonon) and Raman (two-phonon) processes. These principles effectively model experimental longitudinal relaxation rates, denoted as T 1 − 1 . However, under scenarios of increased coupling strength and nonlinear spin–lattice interactions, FGR's applicability may diminish. This paper numerically evaluates the exact spin–lattice relaxation rate kernels, employing the extended dissipaton equation of motion formalism. Our calculations reveal that when quadratic spin–lattice coupling is considered, the rate kernels exhibit a free induction decay-like feature, and the damping rates depend on the interaction strength. We observe that the temperature dependence predicted by FGR significantly deviates from the exact results since FGR ignores the higher order effects and the non-Markovian nature of spin–lattice relaxation. Our methods can be easily extended to study other systems with nonlinear spin–lattice interactions and provide valuable insights into the temperature dependence of T1 in molecular qubits when the coupling is strong. [ABSTRACT FROM AUTHOR]

In this paper, a novel neural network (NN) backstepping control method is proposed for a class of uncertain nonlinear systems with unknown control direction functions. To solve the control problems caused by unknown control direction functions, a method called "virtual control coefficient" is presented using equivalence transformations. However, to improve the flexibility of the control method, a function named "dynamic gain" is introduced to design the feedback gains of the NN backstepping controllers. Moreover, instead of using σ-modification, gradient descent algorithm is applied to train the weights of NNs such that the unknown nonlinearities of the system can be well approximated by NNs. With the help of Lyapunov stability criterion, it can be proved that the system tracking error is semi-global uniformly ultimately bounded. Finally, the effectiveness of the proposed control method is verified by the experimental results on a quad-rotor hover platform. [ABSTRACT FROM AUTHOR]

Considering the complexity and universality of nonlinearity in control systems, it is of great significance to investigate the robust proportional-derivative type (PD-type) iterative learning control (ILC) design for repetitive nonlinear systems in the Takagi–Sugeno (T–S) forms, Different from existing ILC laws, the proposed ones are designed in the finite-frequency domain. The nonlinear systems are described by Roesser-type two-dimensional (2D) uncertain T–S fuzzy models with non-repetitive disturbances. The main contribution of this article is that the novel fuzzy ILC method is proposed by exploiting the 2D linear systems theory. Further to this, the proposed design guarantees the robust asymptotic stability and finite-frequency 2D H ∞ performance of the resulting fuzzy systems, and the controller gains are obtained by solving a set of linear matrix inequality (LMI) constraints. Through the simulation of a nonlinear single-link rigid robot system, our major findings are that the proposed fuzzy control algorithm can lead to steady-state tracking performance and good robustness against non-repetitive disturbances and uncertainties, and the PD-type ILC performs significantly better than the P-type ILC, the former can efficiently improve the control effect. [ABSTRACT FROM AUTHOR]

This work considers the numerical approximation of linear and nonlinear singularly perturbed initial value coupled systems of first-order, for which the diffusion parameters at each equation of the system are distinct and also they can have a different order of magnitude. To do that, we use two efficient discretization methods, which combine the backward differences and an appropriate splitting by components. Both a priori and a posteriori error estimates are proved for the proposed discretization methods. The developed numerical methods are more computationally efficient than those classical methods used to solve the same type of coupled systems. Extensive numerical experiments strongly confirm in practice the theoretical results and corroborate the superior performance of the current approach compared with previous existing approaches. [ABSTRACT FROM AUTHOR]

In this paper, we explore the dynamical properties of a class of nonlinear systems governed by delay differential equations with multitime periodic switching. The systems incorporate piecewise-smooth birth and death functions to capture complex population dynamics under seasonal variations. Assuming monotonicity for both birth and death functions, we obtain a novel equivalence result: when the delay is a positive integer multiple of the switching period, the existence and stability of periodic solutions for the systems are equivalent to those in the nondelay case. To illustrate and validate the theoretical findings, a logistic model with seasonal switching is presented. Numerical simulations further confirm that the system exhibits consistent dynamical behaviors across varying delay values. [ABSTRACT FROM AUTHOR]

This work presents a mathematical framework based on uncertain numbers to address the inherent uncertainty in nonlinear systems, a challenge that traditional mathematical frameworks often struggle to fully capture. By establishing five axioms, a formal system of uncertain numbers is developed and embedded within set theory, providing a comprehensive characterization of uncertainty. This framework allows phenomena such as infinity and singularities to be treated as uncertain numbers, offering a mathematically rigorous analytical approach. Subsequently, an algebraic structure for uncertain numbers is constructed, defining fundamental operations such as addition, subtraction, multiplication, and division. The framework is compatible with existing mathematical paradigms, including complex numbers, fuzzy numbers, and probability theory, thereby forming a unified theoretical structure for quantifying and analyzing uncertainty. This advancement not only provides new avenues for research in mathematics and physics but also holds significant practical value, particularly in improving numerical methods to address singularity problems and optimizing nonconvex optimization algorithms. Additionally, the anti-integral-saturation technique, widely applied in control science, is rigorously derived within this framework. These applications highlight the utility and reliability of the uncertain number framework in both theoretical and practical domains. [ABSTRACT FROM AUTHOR]

This paper studies the synchronization problem of multi-agent systems (MASs) with completely unknown nonlinear dynamics. Firstly, the recurrent neural network is adopted to identify the completely unknown systems, and after sufficient training, the exact nonlinear MASs with model identification errors are obtained. Secondly, model identification error for each system is seen as the bounded disturbance, and then the robust leaderless consensus and leader-follower consensus problems with disturbances are considered, respectively. Then, to achieve the cooperative synchronization of MASs with disturbances, based on triggering mechanisms, two adaptive consensus protocols are constructed. The former solves the consensus issue without leader, and does not need continuous information transmission between neighbors. The latter achieves the consensus problem with one leader; the design of consensus protocol together with triggering mechanism has nothing to do with any global information of topology. Finally, the simulation results verify rationality of the obtained theoretical results. [ABSTRACT FROM AUTHOR]

We present some new results concerning Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA) in uniformly smooth Banach spaces X . First, we generalize Temlyakov's theorem (Temlyakov in Forum Math Sigma 2(12):26, 2014) to cover situations in which the modulus of smoothness and the A 3 parameter are not necessarily power functions. Secondly, we apply this new theorem to the Zygmund spaces X = L p (log L) α , with 1 < p < ∞ and α ∈ R , and show that, when the Haar system is used, then exact recovery of N-sparse signals occurs when the number of iterations is ϕ (N) = O (N max { 1 , 2 / p ′ } (log N) | α | p ′ ) . Moreover, this quantity is sharp when p ≤ 2 . Finally, an expression for ϕ (N) in the case of the trigonometric system is also given. [ABSTRACT FROM AUTHOR]

An approach to constraint nonlinear optimisation is proposed, where the variables of the objective function and of the constraint function belong to the state of a nonlinear dynamical system. A nonlinear closed-loop feedback system is designed, whose solutions converge to a target set consisting of the optimal points of the underlying optimisation problem. The feedback law for the control input of the nonlinear dynamical system is designed by exact linearisation of a dynamical formulation of a first-order optimality condition. For the problem formulation and the stability analysis of the nonlinear closed-loop feedback system, the notion of V-detectability is considered. An application to energy-optimal feed-forward torque control for different types of synchronous machines is presented, which demonstrates the usefulness of the method. In this regard, anisotropic permanent magnet synchronous machines and electrically excited synchronous machines are covered. [ABSTRACT FROM AUTHOR]

The component-by-component construction Kalman filter (CBCKF) is developed from the lattice rules in quasi-Monte Carlo (QMC) by incorporating the component-by-component (CBC) construction method to deterministically generate uniformly distributed sample points. The CBC construction systematically optimizes QMC points for specific integration dimensions and accuracy requirement to enhance the precision of multidimensional integrals in the Gaussian environment, leading to the improvement of state estimation accuracy. To further enhance the numerical stability of CBCKF, the square-root CBCKF (SCBCKF) is proposed accordingly. Meanwhile, to address the parameter selection issue existing in both CBCKF and SCBCKF, the number of sampling points is optimized through the maximum likelihood estimation method. Additionally, the Cramér-Rao lower bound (CRLB) of the nonlinear system model is introduced as a performance metric, and the detailed discussions on error propagation and computational complexities of the proposed filters are also provided. The performance superiorities of the proposed CBCKF and SCBCKF are verified through the simulations with two nonlinear examples. [ABSTRACT FROM AUTHOR]

Our paper is devoted to studying the existence results of distributional solutions in the anisotropic Sobolev space with variable exponents and zero boundary $ \mathring {W}^{1,\overrightarrow {p}(\cdot)}(\Omega,\mathbb {R}^m) $ W ˚ 1 , p → (⋅) (Ω , R m) for a family of boundary value problems, which is a variable exponent anisotropic nonlinear elliptic system with $ L^{\overrightarrow {p'}(\cdot)} $ L p ′ → (⋅) data for the datum f (i.e. $ \in L^{\overrightarrow {p'}(\cdot)}(\Omega,\mathbb {R}^m) $ ∈ L p ′ → (⋅) (Ω , R m)), and has a sum of two Carathéodory nonlinearities, one of which includes the solution u and the other its partial derivatives $ \partial _iu,\,i=1,\ldots,N $ ∂ i u , i = 1 , ... , N. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to solve the problem of adaptive predefined-time control for spacecraft rendezvous maneuver with input quantization and unknown parameters. Design/methodology/approach: A significant error-shifting function is established to ensure that the limitation of the initial condition can be ignored. Then, a combination of neural networks method and minimum-learning-parameter method is used to mitigate the adverse effects caused by nonlinear system dynamics. An input quantization method is applied to improved efficiency of data communication between controller and actuator in rendezvous maneuver control system. Findings: The simulation results verify the correctness and effectiveness of the designed control strategy which can effectively overcome the disadvantages caused by coupling nonlinear system dynamics. Originality/value: This paper designs an adaptive predefined-time control method for spacecraft rendezvous maneuver control with input quantization based on backstepping control method. [ABSTRACT FROM AUTHOR]

The paper investigates the finite-time adaptive consensus control of nonlinear multi-agent systems (MASs) by output-feedback. During the process of control design, the fuzzy logic system (FLS) is employed to approximate nonlinear function, as well as constructing a fuzzy state observer to estimate unmeasured states. Combining the dynamic surface control (DSC) with adaptive backstepping method conquers the complexity explosion issue. Then, semi-global practical finite-time stability (SGPFS) for closed-loop system is obtained under arbitrary switching communication topologies. All the signals of the overall system are uniformly ultimately bounded (UUB). Finally, the availability of developed control approach will be demonstrated through a simulation. [ABSTRACT FROM AUTHOR]