Active dampers are on the verge of appearing in commercial machines as devices that assist the avoidance of machine tool chatter. The adjustment of control parameters in these devices is mostly guided by models that do not consider the dynamics within the control loop of active damper. Therefore, these models neglect the dynamics of actuation, measurement, and filtering, which can result in inaccurate stability predictions that hinder the efficient tuning of active dampers. To formulate a more realistic model for milling processes assisted by active damping, this paper derives a novel mathematical model that takes into account the internal dynamics of the actuator, measuring device, and discrete filtering. This study shows that accurate stability prediction requires the incorporation of actuator and filter dynamics into the model, especially at high spindle speeds and large feedback gains.

In this paper, the time-fractional modified (2 + 1)-dimensional Konopelchenko–Dubrovsky equations have been solved numerically using the Kansa method, in which the multiquadrics is used as radial basis function. To achieve this, a numerical scheme based on finite difference and Kansa method has been proposed. The stability and convergence of the proposed time-discretized scheme are theoretically proven. Also, the solitary wave solutions have been obtained by using Kudryashov technique. The computed results are compared with the exact solutions as well as with the soliton solutions obtained by Kudryashov technique to show the accuracy of the proposed method.

In this paper, we research the fractional telegraph equation with the Atangana-Baleanu-Caputo derivative. We use the Laplace method to find the exact solution of the problems. We construct the difference schemes for the implicit finite method. We prove the stability of difference schemes for the problems by the matrix method. We demonstrate the accuracy of the method by some numerical experiments. The obtained results confirm the accuracy and effectiveness of the proposed method. Additionally, the numerical results demonstrate that the expected physical properties of the model are also observed.

In this paper, we introduce three versions of fractional-order chaotic (or hyperchaotic) complex Duffing-van der Pol models. The dynamics of these models including their fixed points and their stability are investigated. Using the predictor-corrector method and Lyapunov exponents we calculate numerically the intervals of their parameters at which chaotic, hyperchaotic solutions and solutions that approach fixed points to exist. These models appear in several applications in physics and engineering, e.g., viscoelastic beam and electronic circuits. The electronic circuits of these models with different fractional-order are proposed. We determine the approximate transfer functions for novel values of fractional-order and find the equivalent tree shape model (TSM). This TSM is used to build circuits simulations of our models. A good agreement is found between both numerical and simulations results. Other circuits diagrams can be similarly designed for other fractional-order models.

Dynamical behaviors of the time-fractional nonlinear model of the coupled spring-mass system with damping have been explored here. Fractional derivatives with singular and non-singular kernels are used to assess the suggested model. The fractional Adams-Bashforth numerical method based on Lagrange polynomial interpolation is applied to solve the system with non-local operators. Existence, Ulam-Hyers stability, and uniqueness of the solution are established by using fixed-point theory and nonlinear analysis. Further, the error analysis of the present method has also been included. Finally, the behavior of the solution is explained by graphical representations through numerical simulations.

Nonlinear vibrations of axially moving plates partially immersed in fluid are investigated in this paper. The system has time dependency in velocity and tension in axial direction. The Galerkin method is used to solve the nonlinear vibration differential equation. The method of multiple scales and Runge–Kutta method are applied to solve the nonlinear vibration response of the system. Additionally, the stability conditions of trivial and nontrivial solutions are analyzed using the Routh–Hurwitz criterion. The effects of mean velocity, amplitude of pulsating velocity, mean tension, amplitude of pulsating tension, and pulsating frequency on the complex dynamics of the system are obtained. The study results reveal rich dynamic behaviors of fluid–structure coupling system.

In this paper, the dynamical properties of multiple van der Pol-Duffing oscillators with time delays are studied. The amplitude death and bifurcation curves in the parameter plane are determined by using the space decomposition method. Different patterns of bifurcated solutions are given on the basis of the symmetric bifurcation theory. The properties of bifurcated solutions are shown by using the norm forms on the center manifold. The interactions of bifurcations are discussed and their dynamical behaviors are shown. An electronic circuit platform is implemented by means of nonlinear circuit and time delay circuit. The revealed behaviors of the circuit reach an agreement with the obtained results. It is shown that the nonlinearity and time delays have great effects on the system performance and can induce interesting and abundant dynamic features.

Many practical systems have inherent time delays that cannot be ignored; thus, their dynamics are described using delay differential equations (DDEs). The Galerkin approximation method is one strategy for studying the stability of time-delay systems (TDS). In this work, we consider delays that are time-varying and, specifically, time-periodic. The Galerkin method can be used to obtain a system of ordinary differential equations (ODEs) from a second-order time-periodic DDE in two ways: either by converting the DDE into a second-order time-periodic partial differential equation (PDE) and then into a system of second-order ODEs, or by first expressing the original DDE as two first-order time-periodic DDEs, then converting into a system of first-order time-periodic PDEs, and finally converting into a first-order time-periodic ODE system. The difference between these two formulations in the context of control is presented in this paper. Specifically, we show that the former produces spurious Floquet multipliers at a spectral radius of 1. We also propose an optimization-based framework to obtain feedback gains that stabilize closed-loop control systems with time-periodic delays. The proposed optimization-based framework employs the Galerkin method and Floquet theory and is shown to be capable of stabilizing systems considered in the literature. Finally, we present experimental validation of our theoretical results using a rotary inverted pendulum apparatus with inherent sensing delays as well as additional time-periodic state-feedback delays that are introduced deliberately.

The tracking control of underactuated systems is a challenging problem due to the structural differences compared to fully actuated systems. Contrarily to fully actuated systems, resolving the inverse kinematics problem of underactuated systems is not possible independently from the dynamic equations. Instead, the inverse dynamics must be addressed. It is common to extend the computed torque control (CTC) technique with servoconstraints. Besides the CTC's clearness, the stability of the system cannot be always guaranteed. A novel predictive controller (PC) is presented in this paper. Our PC applies the variational principle to design the motion of the system in order to achieve a stable motion with the lowest possible tracking error. To demonstrate the applicability and the performance of the PC method, a numerical study is presented for a planar manipulator resulting in about 20% RMS error compared to the CTC method from the literature.

This paper presents Jacobi stability analysis of 23 simple chaotic systems with only one Lyapunov stable equilibrium by Kosambi–Cartan–Chern theory, and analyzes the chaotic behavior of these systems from the geometric viewpoint. Different from Lyapunov stability, the unique equilibrium for each system is always Jacobi unstable. Moreover, the dynamical behaviors of deviation vector near equilibrium are discussed to reveal the onset of chaos for these 23 systems and show geometrically the coexistence of unique Lyapunov stable equilibrium and chaotic attractor for each system. The obtaining results show that these chaotic systems are not robust to small perturbations of the equilibrium, indicating that the systems are extremely sensitive to the internal environment. This reveals that the chaotic flows generated by these systems may be related to Jacobi instability of the equilibrium.

A new technique for the analysis of dynamical equations with quasi-periodic coefficients (so-called quasi-periodic systems) is presented. The technique utilizes Lyapunov–Perron (L–P) transformation to reduce the linear part of a quasi-periodic system into the time-invariant form. A general approach for the construction of L–P transformations in the approximate form is suggested. First, the linear part of a quasi-periodic system is replaced by a periodic system with a “suitable” large principal period. Then, the state transition matrix (STM) of the periodic system is computed in the symbolic form using Floquet theory. Finally, Lyapunov–Floquet theorem is used to compute approximate L–P transformations. A two-frequency quasi-periodic system is studied and transformations are generated for stable, unstable, and critical cases. The effectiveness of these transformations is demonstrated by investigating three distinct quasi-periodic systems. They are applied to a forced linear quasi-periodic system to generate analytical solutions. It is found that the closeness of the analytical solutions to the exact solutions depends on the principal period of the periodic system. A general approach to obtain the stability bounds on linear quasi-periodic systems with stochastic perturbations is also discussed. Finally, the usefulness of approximate L–P transformations is presented by analyzing a nonlinear quasi-periodic system with cubic nonlinearity using time-dependent normal form (TDNF) theory. The closed-form solution generated is found to be in good agreement with the exact solution.

Several industrial chemical processes exhibit severe nonlinearity. This paper addresses the computational and nonlinear issues occurring in many typical industrial problems in aspects of its stability, strength of nonlinearity, and input output dynamics. In this article, initially, a prospective investigation is conducted on various nonlinear processes through phase portrait analysis to understand their stability status at different initial conditions about the vicinity of the operating point of the process. To estimate the degree of nonlinearity, for input perturbations from its nominal value, a novel nonlinear measure Δ 0 is put forward that anticipates on the converging area between the nonlinear and their linearized responses. The nonlinearity strength is fixed between 0 and 1 to classify processes to be mild, medium, or highly nonlinear. The most suitable operating point, for which the system remains asymptotically stable, is clearly identified from the phase portrait. The metric Δ 0 can be contemplated as a promising tool to measure the nonlinearity of Industrial case studies at different linear approximations. Numerical simulations are executed in matlab to compute Δ 0 , which conveys that the nonlinear dynamics of each industrial example is very sensitive to input perturbations at different linear approximations. In addition to the identified metric, nonlinear lemmas are framed to select appropriate control schemes for the processes based on its numerical value of nonlinearity.

The piezoelectric and electromagnetic hybrid vibration energy harvester (HVEH) has proven to be a favorable option to deal with the low power generation issue and overcome the drawbacks of each individual transduction mechanism. Besides, colored Lévy noise consisting of small perturbations, large jumps, and correlation time turns out to be a relatively suitable tool for describing the complex environments. For the purpose of enhancing the harvesting performance of HVEH, the stochastic dynamics induced by colored Lévy noise in a tri-stable HVEH is mainly investigated in this paper. The stationary probability density, the largest Lyapunov exponent, the signal-to-noise ratio, and the mean harvested power are calculated to explore the stochastic dynamics of system, such as the stochastic response, the stochastic stability, the stochastic resonance (SR), and the energy harvesting performance. The results show that the colored Lévy noise can induce stochastic P-bifurcation, D-bifurcation, and SR phenomenon. In particular, the comparisons between colored Lévy noise and colored Brownian noise in dynamics and harvesting performance are also discussed in detail. It is found that the colored Lévy noise can make a greater contribution than colored Brownian noise in the effective voltage and help to improve the mean harvested power through the SR effect.

The post-buckling and nonlinear dynamic response of shallow spherical caps subjected to external pressure is analyzed. The Novozhilov's nonlinear thin shell theory is used to express the strain–displacement relations. Following the Rayleigh-Ritz method, the displacement fields are expanded using a mixed series: Legendre polynomials in the meridional direction, harmonic functions in the circumferential direction. Once the linear analysis is completed, the displacement fields are re-expanded and the nonlinear dynamic model is obtained by using the Lagrange equations. The response of clamped caps, made of isotropic and homogeneous material, is investigated. The bifurcation analyses of equilibrium points and periodic orbits are presented by using continuation techniques. Benchmark results are provided in terms of natural frequencies and critical buckling loads. The dynamic effects due to the interaction between static and dynamic pressure are investigated. Numerical results pointed out that, under particular load conditions, dynamic bifurcation results in nonnegligible asymmetric states activation in the response of the structure.

In this work, the authors draw comparisons between the Floquet theory and Normal Forms technique and apply them towards the investigation of stability bounds for linear time periodic systems. Though the Normal Forms technique has been predominantly used for the analysis of nonlinear equations, in this work, the authors utilize it to transform a linear time periodic system to a time-invariant system, similar to the Lyapunov–Floquet (L–F) transformation. The authors employ an intuitive state augmentation technique, modal transformation, and near identity transformations to facilitate the application of time-independent Normal Forms. This method provides a closed form analytical expression for the state transition matrix (STM). Additionally, stability analysis is performed on the transformed system and the comparative results of dynamical characteristics and temporal variations of a simple linear Mathieu equation are also presented in this work.

The aim of this paper is to develop a fractional order mathematical model for describing the spread of hepatitis B virus (HBV). We also provide a rigorous mathematical analysis of the stability of the disease-free equilibrium (DFE) and the endemic equilibrium of the system based on the basic reproduction number. Here, the infectious disease HBV model is described mathematically in a nonlinear system of differential equations in a caputo sense, and hence, Jacobi collocation method is used to reduce into a system of nonlinear equations. Finally, Newton Raphson method is used for the systems of nonlinear equations to arrive at an approximate solution and matlab 2018 has helped us to simulate the nature of each compartment and effects of the possible control strategies (i.e., vaccination and isolation).

Being able to identify instability regions is an important task for the designers of rotating machines. It allows discarding, since the early design stages, those configurations which may lead to catastrophic failures. Instability can be induced by different occurrences such as an unbalanced disk, torsional, and axial forces on the shaft or periodic variation of system parameters known as “parametric excitation.” In this paper, the stability of a Jeffcott rotor, parametrically excited by the time-varying stiffness of the rolling bearings, is studied. The harmonic balance method (HBM) is here applied as an approximate procedure to obtain the well-known “transition curves (TCs)” which separate the stable from the unstable regions of the design parameter space. One major challenge in the HBM application is identifying an adequate harmonic support (i.e., number of harmonics in the Fourier formulation), necessary to produce trustworthy results. A procedure to overcome this issue is here proposed and termed “trained HBM” (THBM). The results obtained by THBM are compared to those computed by Floquet theory, here used as a reference. The THBM proves to be able to produce reliable TCs in a timely manner, compatible with the design process.

In this paper, a high-dimensional system of nearest-neighbor coupled neural networks with multiple delays is proposed. Nowadays, most present researches about neural networks have studied the connection between adjacent nodes. However, in practical applications, neural networks are extremely complicated. This paper further considers that there are still connection relationships between nonadjacent nodes, which reflect the intrinsic characteristics of neural networks more accurately because of the complexity of its topology. The influences of multiple delays on the local stability and Hopf bifurcation of the system are explored by selecting the sum of delays as bifurcation parameter and discussing the related characteristic equations. It is found that the dynamic behaviors of the system depend on the critical value of bifurcation. In addition, the conditions that ensure the stability of the system and the criteria of Hopf bifurcation are given. Finally, the correctness of the theoretical analyses is verified by numerical simulation.

Noise-induced motions are a significant source of uncertainty in the response of micro-electromechanical systems (MEMS). This is particularly the case for electrostatic MEMS where electrical and mechanical sources contribute to noise and can result in sudden and drastic loss of stability. This paper investigates the effects of noise processes on the stability of electrostatic MEMS via a lumped-mass model that accounts for uncertainty in mass, mechanical restoring force, bias voltage, and AC voltage amplitude. We evaluated the stationary probability density function (PDF) of the resonator response and its basins of attraction in the presence noise and compared them to that those obtained under deterministic excitations only. We found that the presence of noise was most significant in the vicinity of resonance. Even low noise intensity levels caused stochastic jumps between co-existing orbits away from bifurcation points. Moderate noise intensity levels were found to destroy the basins of attraction of the larger orbits. Higher noise intensity levels were found to destroy the basins of attraction of smaller orbits, dominate the dynamic response, and occasionally lead to pull-in. The probabilities of pull-in of the resonator under different noise intensity level are calculated, which are sensitive to the initial conditions.

In this study, we explore the event-triggered control of a flexible three-dimensional (3D) Euler–Bernoulli beam modeled by partial differential equations (PDEs). A novel event-triggered control strategy is developed, which can minimize the communication burden in the process of signal transmission. A larger threshold value can be selected when the control input signal value is large to limit the communication burden; a smaller threshold value can be selected when the control input signal value is close to zero in order to improve the control accuracy. The threshold is time-varying, which further preserves the system resources. The proposed control laws can realize both vibration and event-triggered control. All the system signals can be guaranteed as uniform ultimate bounds. The effectiveness of this approach is verified in a series of simulations.