Nonlinear Physical Systems Research Articles

Nonlinear analysis of compound pendulum model

Abstract This paper investigates the relationships among various nonlinear physi cal equations, with a particular focus on the standard form of the ϕ^4 equation. Based on the theoretical framework of the Jacobi elliptic function, an exact solution for the ϕ^4 equation is derived. A key innovation of this work is the discovery of the consistency between the ϕ^4 equation and the motion equation of the compound pendulum. By utilizing this correspondence, an exact solution for the compound pendulum equation is obtained, grounded in the Jacobi elliptic function theory. Compared to numerical methods, this solution provides higher accuracy and has the potential to be applied to
more complex nonlinear physical systems. This model can also be applied in areas such as vibration damping in building materials, mechanical system analysis, and spacecraft control.

Predefined time fuzzy adaptive tracking control for nonlinear cyber physical systems with unmeasurable states and FDI attacks

Predefined time fuzzy adaptive tracking control for nonlinear cyber physical systems with unmeasurable states and FDI attacks

Parameter-induced logical stochastic resonance in substrate potential with deformable sine-Gordon shape

Logical stochastic resonance (LSR) systems are physical nonlinear systems capable of robust Boolean logic operations under the influence of noise. While LSR systems suitable for electronic circuit implementation have been extensively studied, research on condensed matter LSR systems remains limited to constrained bistability with saturation characteristic. Considering the presence of periodic nonlinearities in many physical systems, a deformable sine-Gordon-shaped Remoissenet–Peyrard potential (RPP)-based LSR system is proposed for the first time in this work. Testing under both noise-free and noisy conditions proves the existence of parameter-induced LSR phenomena in the proposed system. Moreover, we find that a relatively wide potential well shape contributes to enhanced noise robustness in the RPP-based LSR system. Additionally, traditional polynomial nonlinearity-based LSR systems are compared with the proposed system. Regardless of whether the noise is Gaussian white noise or composite noise containing spiking pulses, the RPP-based LSR system exhibits superior robustness. The results demonstrate the exceptional advantages of the RPP-based LSR system and encourage further design of condensed matter LSR systems based on deformable periodic nonlinearities.

Mixed localized waves and their interaction structures for a spatial discrete Hirota equation

Mixed localized wave solutions and interactions are of great significance in nonlinear physical systems. This paper aims to investigating the generalized (m,N-m)-fold Darboux transformation and the mixed localized wave solutions of a spatial discrete Hirota equation. First, we construct the generalized (m,N-m)-fold Darboux transformation for the spatial discrete Hirota equation, which can produce the interactions between the breathers, degenerate breathers and rogue waves. For the Darboux transformation formula, we discuss the above order-1,2,3 localized wave solutions, as well as their dynamics by choosing the number of m = 1. We plot some specific examples such as the spatial (time)-periodic breather, second-order and third-order degenerate breathers and higher-order rogue waves with novel patterns. Furthermore, when m > 1, we give several kinds of mixed interaction solutions between the first-order rogue waves and first (second)-order (degenerate) breathers, between the first-order breather and second-order degenerate breathers, between second-order rogue waves and first-order breathers. At last, we also sum up the various mathematical features of the degenerate breathers and the mixed localized wave solutions.

Influence of Nonstationary Processes in Drill Rigs on the Durability of Structural Elements

Assessing the effects that nonstationary dynamic processes have on the durability of structural elements belongs to an important trend in modern dynamics and technical diagnostics of machines. Normally, fatigue strength calculations are performed taking into account only periodically variable stresses, as steady operating modes of machines are much longer in comparison with transient modes. However, a significant role in fatigue failure in machines and engineering structures is also played by nonstationary loads. This is explained by emerging intensive oscillations in the mechanical system during accelerating, braking, or changing the operation mode of a machine unit, which often lead to the accumulation of fatigue damages in the materials of parts in heavy loaded assemblies. The combination of stationary and nonstationary dynamic loads manifests itself, particularly in drilling rigs, where technological cycles include steady motion modes, starts, and stops. This paper represents a generalized mathematical model describing nonstationary processes in the lift system of a drill rig, which considers the relationship between electromagnetic processes in asynchronous motors and mechanical oscillatory phenomena, with the purpose of determining dynamic loads and stresses in structural elements of the rigs. Nonlinear physical systems include mechanical members with both concentrated and clearly expressed distributed parameters. The durability of structural elements is evaluated by means of a computer algorithm for analysis of crack growth rates using the NASGRO equation obtained with the presence of plastic deformation zones. An example of the crown block axis illustrates the influence of nonstationary dynamic processes in drill rigs on the durability of structural elements.

Stability analysis of a stochastic port-Hamiltonian car-following model

Port-Hamiltonian systems are pertinent representations of many nonlinear physical systems. In this study, we formulate and analyse a general class of stochastic car-following models with a systematic port-Hamiltonian structure. The model class is a generalisation of classical car-following approaches, including the optimal velocity model of Bando et al (1995 Phys. Rev. E 51 1035), the full velocity difference model of Jiang et al (2001 Phys. Rev. E 64 017101), and recent stochastic following models based on the Ornstein–Uhlenbeck process. In contrast to traditional models where the interaction is totally asymmetric (i.e. depending only on the speed and distance to the predecessor), the port-Hamiltonian car-following model also depends on the distance to the follower. We determine the exact stability condition of the finite system with N vehicles and periodic boundaries. The stable system is ergodic with a unique Gaussian invariant measure. Other properties of the model are studied using numerical simulation. It turns out that the Hamiltonian component improves the flow stability and reduces the total energy in the system. Furthermore, it prevents the problematic formation of stop-and-go waves with oscillatory dynamics, even in the presence of stochastic perturbations.

Monitoring and Reconstruction of Actuator and Sensor Attacks for Lipschitz Nonlinear Dynamic Systems Using Two Types of Augmented Descriptor Observers

Fault data injection attacks may lead to a decrease in system performance and even a malfunction in system operation for an automatic feedback control system, which has motive to develop an effective method for rapidly detecting such attacks so that appropriate measures can be taken correspondingly. In this study, a secure descriptor estimation technique is proposed for continuous-time Lipschitz nonlinear cyber physical systems affected by actuator attacks, sensor attacks, and unknown process uncertainties. Specifically, by forming a new state vector composed of original system states and sensor faults, an equivalent descriptor dynamic system is built. A proportional and derivate sliding-mode observer is presented so that the system states, sensor attack, and actuator attack can be reconstructed successfully. The observer gains are obtained by using linear matrix inequality to secure robustly stable estimation error dynamics. Moreover, a robust descriptor fast adaptive observer estimator is presented as a complement. Finally, the efficacy levels of the proposed design approaches are validated using a vertical take-off and landing aircraft system. Comparison studies are also carried out to assess the tracking performances of the proposed algorithms.

Lie symmetry analysis and solitary wave solution of biofilm model Allen-Cahn

The investigation presented in this study delves into the analysis of Lie symmetries for the bistable Allen-Cahn (BAC) equation with a quartic potential, specifically applied to the biofilm model. By employing the Lie symmetry method, we have acquired the Lie infinitesimal generators for the considered model. Using a transformation method, the nonlinear partial differential equations (NPDEs) are converted into various nonlinear ordinary differential equations (NLODEs), providing the numerous closed-form solitary wave solutions. The obtained solutions manifest in various forms including dark, bright, kink, anti-kink, and periodic types using diverse strategies. To enhance the physical interpretation, the study presents 3D, 2D, and contour plots of the acquired solutions. Every graph’s wave-like structure contains information about the structural behaviour of the bacteria that build biofilms on surfaces where rectangles have different densities. This analysis enhances comprehension of the complex dynamics present in areas like fluid dynamics, fiber optics, biology, ocean physics, coastal engineering, and nonlinear complex physical systems.

Modeling Nonlinear Physical Systems in a Real-Time Operating System Environment for Control and Cyber Threat Mitigation

Nonlinearity control and interconnection between physical systems are essential for many contemporary systems. Cyber Physical systems came about because of the growing number of security concerns with network control systems. Nonlinear physical systems are the primary subject of this study's investigation into control and cyber threat mitigation. Nonlinear physical systems are the focus of this study, which also delves into their control, modelling, and real-time implementation. Here, physical systems are defined as two kinds of benchmark nonlinear robotic systems: the Segway transporter and the cruise control system. Many see the Segway as an unstable and nonlinear physical system. By factoring in kinetic and potential energy, Lagrangian dynamics has allowed for the derivation of equations of motion. Next, a state-space model of the system is obtained by linearizing the nonlinear differential equations using Taylor series expansion of functions. This model is then transformed into two transfer functions, one for the tilt angle and one for the yaw angle. The Model Predictive Controller (MPC), GA Tuned PID, and PID are the building blocks of MATLAB's effective simulation study of nonlinear control.

H ∞-based truncated predictive control for switched nonlinear cyber physical systems subjected to actuator faults and deception attacks

ABSTRACT This paper emphasises the study of switched nonlinear cyber-physical systems (CPSs) subjected to input delay, actuator faults and deception attacks by a H ∞ -based truncated predictive control design. Particularly, the predictive controller uses delayed state information in addition to the traditional feedback control to estimate the present state for the feedback. Furthermore, the proposed control design is modelled with a truncated predictive approach which is also used to compensate the effects of actuator faults and attacks that also satisfy a predefined H ∞ index simultaneously. With the assistance of the Lyapunov stability theory, a novel set of adequate criteria are acquired in terms of LMIs which assure the asymptotical stability of the switched nonlinear CPS under prescribed predefined H ∞ performance levels. Eventually, the proposed theoretical results are validated through two numerical examples.

Folding State within a Hysteresis Loop: Hidden Multistability in Nonlinear Physical Systems.

Identifying hidden states in nonlinear physical systems that evade direct experimental detection is important as disturbances and noises can place the system in a hidden state with detrimental consequences. We study a cavity magnonic system whose main physics is photon and magnon Kerr effects. Sweeping a bifurcation parameter in numerical experiments (as would be done in actual experiments) leads to a hysteresis loop with two distinct stable steady states, but analytic calculation gives a third folded steady state "hidden" in the loop, which gives rise to the phenomenon of hidden multistability. We propose an experimentally feasible control method to drive the system into the folded hidden state. We demonstrate, through a ternary cavity magnonic system and a gene regulatory network, that such hidden multistability is in fact quite common. Our findings shed light on hidden dynamical states in nonlinear physical systems which are not directly observable but can present challenges and opportunities in applications.

Numerical analysis of the kinetic equation describing isotropic 4-wave interactions in non-linear physical systems

We develop a numerical method for solving kinetic equations (KEs) that describe out-of-equilibrium isotropic nonlinear four-wave interactions in optics, deep-water wave theory, physics of superfluids and Bose gases, and in other applications. High complexity of studying numerically the wave kinetics in these applications is related with the multi-scale nature of turbulence and with power-law behaviour of turbulent spectra in the Fourier space. When solving the Cauchy problem for KE, this leads to emergence of spectra with extremely steep gradients and to occurrence of singular points in the collision integral standing in the right-hand side. To solve these problems, we develop special fast-convergent cubature formulas, highly-accurate rational approximations of the KE solution, and a new stable method for time marching. We apply the developed methods for solving the test problems of integration arisen from applications, for studying the wave kinetics in random fiber lasers and for analysing the Bose–Einstein condensation. In these applications we used KEs obtained from the Ginzburg–Landau and from the Gross–Pitaevskii equations.

Event-triggered finite-time command-filtered tracking control for nonlinear time-delay cyber physical systems against cyber attacks

Event-triggered finite-time command-filtered tracking control for nonlinear time-delay cyber physical systems against cyber attacks

Event-triggered prescribed performance adaptive secure control for nonlinear cyber physical systems under denial-of-service attacks

For a class of nonlinear cyber physical systems (CPSs) under intermittent denial-of-service attacks (DoS), this paper investigates a novel adaptive security control scheme. A switched fuzzy observer is designed to estimate unmeasurable states. In addition, the relative threshold-based event-triggered approach is introduced to reduce the released data, and the dynamic surface control technique is used to remove the ‘explosion of terms’ problem. Meanwhile, an observer-based adaptive event-triggered prescribed performance controller is constructed, which ensures that all closed-loop signals remain bounded and system output approaches a designed performance bound within a predefined finite time. Simultaneously, the Zeno behavior can be thoroughly eliminated. Finally, a numerical simulation is used to demonstrate the validity of the results.

New chirp soliton solutions for the space-time fractional perturbed Gerdjikov–Ivanov equation with conformable derivative

The space–time perturbed fractional Gerdjikov–Ivanov equation is the main topic of this work, together with quintic nonlinearity and self-steepening, as it involves several intricate physical phenomena including nonlinearity, self-steepening and fractional calculus, where the fractional derivative is described by employing a conformable derivative. In addition, the governing equation is transformed into an integer-order ordinary differential equation by using an appropriate fractional complex transformation. Under certain restrictions, a direct algebraic method is employed to investigate the structures of chirp soliton solutions enfolding hyperbolic functional terms. The dynamic behaviour and bifurcation of equilibria of the system are thoroughly examined; chirp soliton solutions under specified constraints are investigated and the evolving profiles of the obtained solutions are visualized. Moreover, this research offers valuable perspectives on the behaviour of chirp solitons under specific conditions, which have practical applications in nonlinear physical systems and optical communication systems. The significant contribution of this work is the investigation and obtaining of novel chirp soliton solutions with hyperbolic functional terms under particular limitations using a novel approach. It further extends the prior approaches by treating difficult fractional differential equations from a fresh angle, offering new tools, and closely examining soliton solutions.

Asynchronous event‐triggered adaptive robust switching filtering for nonlinear industrial cyber physical systems under data injection attacks

AbstractBy fully taking the effects of data injection attacks and nonlinear disturbances into consideration, based on dynamic asynchronous event‐triggered approach, an adaptive robust H‐Infinity switching filtering (ARSF) method is proposed for the stability of industrial cyber physical systems. An adaptive feedback controller utilizing anticipated states and error‐dependent parameters can proficiently counteract the deleterious impacts of data injection attacks originating from the measurement channel. Furthermore, a filter, which includes the value of the measurement signal, is designed to monitor the system's state changes in real‐time and ensure the stable operation of the system under attacks. Based on the design of an adaptive feedback controller and a filter eliminating the effects of data injection attacks originating from the measurement channel, a switching scheme is further proposed for compensating the effect of data injection attacks in the control channel. Finally, simulation tests are conducted by using MATLAB to control the robustness of a practical quadrotor UAV system's angles by adopting ARSF, whose results verify that the correctness and effectiveness of the proposed ARSF.

A unified methodology for the power efficiency analysis of physical systems

In this paper, the problem of power efficiency evaluation for n-ports physical systems is investigated. The efficiency analysis that we perform highlights the necessary and sufficient conditions for the system to be passive, and outlines the guidelines for the efficiency maps computation. After addressing the problem from a formal point of view, the analysis is deepened for the case of two-ports linear and nonlinear physical systems, and for the cases of three and four-ports linear systems. The efficiency analysis and the computation of the efficiency maps are addressed as a function of the power variables characterizing all the energetic ports of the considered systems. Furthermore, the salient properties of the efficiency are highlighted and discussed. The theoretical analysis which is developed is then applied to some physical systems of interest for industries and engineers working in the electromechanical, hydraulic and automotive fields: a DC electric motor driving an hydraulic pump for the two-ports systems class, a single-stage planetary gear set for the three-ports systems class, and a Ravigneaux planetary gear set for the four-ports systems class.

Twisted curve geometry underlying topological invariants

Topological invariants such as winding numbers and linking numbers appear as charges of topological solitons in diverse nonlinear physical systems described by a unit vector field defined on two and three dimensional manifolds. While the Gauss-Bonnet theorem shows that the Euler characteristic (a topological invariant) can be written as the integral of the Gaussian curvature (an intrinsic geometric quantity), the intriguing question of whether winding and linking numbers can also be expressed as integrals of some other kinds of intrinsic geometric quantities has not been addressed in the literature. In this paper we provide the answer by showing that for the winding number in two dimensions, these geometric quantities involve torsions of the two evolving space curves describing the manifold. On the other hand, in three dimensions we find that in addition to torsions, intrinsic twists of the space curves are necessary to obtain a nontrivial winding number and linking number. They arise from the hitherto unknown connections that we establish between these topological invariants and the corresponding appropriately normalized global space curve anholonomies (i.e., geometric phases) that can be associated with the unit vector fields on the respective manifolds. An application of our results to a 3D Heisenberg ferromagnetic model supporting a topological soliton is also presented.

Multi-objective optimization strategy for parallel tuning of multiple fractional order controllers in a PMSG based WECS

In this paper, a novel multi-objective optimization strategy is proposed for the parallel tuning of six fractional order controllers used in regulation loops of a PMSG based wind energy conversion system connected to the electric grid. The nonlinear nature of the WECS components has made controllers design challenging. To enhance the transient response of system variables, Fractional Order Proportional Integral (FOPI) controllers are considered as they are more suitable for such nonlinear physical systems. The additional parameters introduced by FOPI are normally tuned using a single objective, multi-dimensional particle swarm optimization algorithm. However, the inter-dependence of regulation loops introduces additional complexity that is solved in this work using a multi-objective optimization strategy based on a succession of Single-Objective PSO and Multi-Objective PSO. A simulation study has been conducted in order to demonstrate the higher performance and superior tracking accuracy of the proposed multi-objective optimization strategy in variable wind speed operating conditions.

Reinforcement learning-based fixed-time resilient control of nonlinear cyber physical systems under false data injection attacks and mismatch disturbances

This paper presents a fixed-time adaptive resilient control framework based on reinforcement learning (RL), mismatch disturbance observer and nonsingular fast terminal sliding mode scheme for nonlinear cyber-physical systems (CPS) to enhance cyber-security under False Data Injection (FDI) attacks, as well as match and mismatch disturbances. The RL algorithm contains an actor-critic neural network (NN) in which the actor NN is employed to estimate the uncertainty while the critic NN is applied to evaluate the performance cost function. Next, to compensate the detrimental effects of attacks and mismatch disturbances, a new sliding mode-based adaptive disturbance observer is designed. To achieve high precision trajectory tracking within a fixed-time interval, a nonsingular fast terminal sliding mode scheme is designed. This scheme ensures fixed-time convergence of the tracking error and facilitates disturbance attenuation and attack mitigation which are the key features of the proposed fixed-time secure control strategy. Finally, the fixed-time stability analysis of is performed through Lyapunov theory. Results reveal the effectiveness of the proposed resilient control for wind turbine system as the nonlinear CPS.