Unperturbed System Research Articles

Difference Equation With Minima-Based Discrete-Time Sliding Mode Control

This note proposes a reaching law based on a difference equation with minima, for discrete-time sliding mode control, providing a solution that has properties of modified Gao’s reaching law when far away from the origin and Utkin’s equivalent control law when close to the origin. On one hand, the proposed approach eliminates chattering and makes the system states stay on the sliding hyperplane perfectly, while on the other hand, it restricts the rate of change of the sliding variable by tuning a design parameter. As a result, the control law derived from this reaching law is not as aggressive as it is for Gao’s reaching law, for very large initial conditions. The proposed reaching law based discrete-time sliding mode control is designed for both unperturbed and perturbed systems. For the unperturbed system, the sliding variable is made zero in finite time, and for the perturbed system, the sliding variable remains in the neighborhood of the sliding hyperplane. An example is simulated to show the effectiveness of the proposed reaching law.

Mean Square Exponential Stability of Semi-Linear Stochastic Perturbed Differential Equation Via Lyapunov Function Approach

In this work, a class of stochastically perturbed differential systems with standard Brownian motion of ordinary unperturbed differential system is considered and studied. The necessary conditions for the existence of a unique solution of the stochastic perturbed semi-linear system of differential equations are suggested and supported by concluding remarks. Some theoretical results concerning the mean square exponential stability of the nominal unperturbed deterministic differential system and its equivalent stochastically perturbed system with the deterministic and stochastic process as a random noise have been stated and proved. The proofs of the obtained results are based on using the stochastic quadratic Lyapunov function method. Form an application point of view of the proposed approach, an illustrative example is considered and implemented.

A Sliding Mode Control-Based Guidance Law for a Two-Dimensional Orbit Transfer with Bounded Disturbances

The aim of this paper is to analyze the performance of a state-feedback guidance law, which is obtained through a classical sliding mode control approach, in a two-dimensional circle-to-circle orbit transfer of a spacecraft equipped with a continuous-thrust propulsion system. The paper shows that such an inherently robust control technique can be effectively used to obtain possible transfer trajectories even when the spacecraft equations of motion are affected by perturbations. The problem of the guidance law design is first addressed in the simplified case of an unperturbed system, where it is shown how the state-feedback control may be effectively used to obtain simple mathematical relationships and graphs that allow the designer to determine possible transfer trajectories that depend on a few control parameters. It is also shown that a suitable combination of the controller parameters may be exploited to obtain trade-off solutions between the flight time and the transfer velocity change. The simplified control strategy is then used to investigate a typical heliocentric orbit raising/lowering in the presence of bounded disturbances and measurement errors.

Analysis of bifurcation and chaos in the traveling wave solution in optical fibers using the Radhakrishnan–Kundu–Lakshmanan equation

The study examines the propagation of solitary waves in optical fibers controlled by the Radhakrishnan–Kundu–Lakshmanan model (RKL) equation, which includes a cubic–quintic non-Kerr refractive index. The first step in this study involves utilizing the extended algebraic method to derive solitonic solutions for the given equation. These solutions are subsequently graphically represented for a variety of parameter values. The proposed strategy allows for the discovery of a range of solutions.The controlling model is incorporated into a planar dynamical system via the Galilean transformation, resulting in the transformation of the governing model. The effects of both the quintic nonlinearity and self-phase modulation on the perturbed and unperturbed systems are explored using the effective potential and related phase portrait. The governing model is thus transformed into a planar dynamical system. Furthermore, this study harnessed a variety of numerical techniques. These methods encompassed time series analysis, phase portraits examination, and meticulous sensitivity inspections. Additionally, we employed a Poincaré section to assess the equation’s responsiveness to external perturbations. The outcomes reveal a fascinating observation: the perturbed system’s behavior shifts from a quasi-periodic state to a chaotic one, contingent upon the amplitude and frequency of the external perturbation. Moreover, we conducted multistability analysis for specific sets of physical parameters, uncovering the equation’s tendency to exhibit multistability in such scenarios.

Periodic states and their implications in gas hydrate systems

Gas hydrates are one of the largest marine carbon reservoirs on Earth. The conventional understanding of hydrate dynamics assumes that the system, in the absence of external triggers, converges to a steady-state over geological time-scales, achieving fixed concentrations of gas hydrate and free gas phase. However, using a high-fidelity numerical model and consistently resolving phase states across multiple fluid-fluid and fluid-solid phase boundaries, we have identified well-defined periodic states embedded within hydrate system dynamics. These states lead to cyclic formation and dissolution of massive hydrate layers that is self-sustaining for the majority of natural marine settings. This previously unresolved characteristic could manifest as spontaneous gas migration and pressure release in, supposedly, unperturbed systems. Our findings show that the gas hydrate systems are not bound to have unique steady-state solutions. Instead, existence of periodic states introduces an irreducible, but, quantifiable uncertainty in gas hydrate dynamics which adds significant error bars to global gas hydrate inventory estimates.

Dynamical features and sensitivity visualization of thin-film Polarisation equation

The present investigation describes the dynamical behavior, multi-stability, and traveling wave solutions of thin-film polarisation equations (TFPE) which describes the propagation of waves in thin-film ferroelectric materials. The extended direct algebraic technique is used to construct the traveling wave patterns. Visual representations of a few randomly selected solutions are provided for physical comprehension. The ordinary differential equation can be expressed in the planar dynamical system using the Galilean transformation. Using various initial conditions for the unperturbed dynamical system, phase portraits with various sorts of trajectories are created. Additionally, the Runge-Kutta method is used to plot nonlinear periodic waves and super nonlinear waves. Additionally, the Hamiltonian function for this undisturbed dynamical system is computed and shown. It also included the source term with amplitude and frequency parameters for the chaotic and quasi-periodic behaviors, and the system is also stated in the non-autonomous form. For the dynamical system under investigation, multi-stability is also thoroughly described. Furthermore, a full inspection of the sensitivity of the perturbed dynamical structure under various initial conditions has been conducted.

Quasistatic deformation of yield stress materials: Homogeneous or localized?

We analyze a mesoscopic model of a shear stress material with a three-dimensional slab geometry, under an external quasistatic deformation of a simple shear type. Relaxation is introduced in the model as a mechanism by which an unperturbed system achieves progressively mechanically more stable configurations. Although in all cases deformation occurs via localized plastic events (avalanches), we find qualitatively different behavior depending on the degree of relaxation in the model. For no or low relaxation, yielding is homogeneous in the sample, and even the largest avalanches become negligible in size compared with the system size (measured as the thickness of the slab L_{z}) when this is increased. On the contrary, for high relaxation, the deformation localizes in an almost two-dimensional region where all avalanches occur. Scaling analysis of the numerical results indicates that in this case, the linear size of the largest avalanches is comparable with L_{z}, even when this becomes very large. We correlate the two scenarios with a qualitative difference in the flow curve of the system in the two cases, which is monotonous in the first case and velocity weakening in the second case.

Melnikov-type method for chaos in a class of hybrid piecewise-smooth systems with impact and noise excitation under unilateral rigid constraint

The Melnikov method for chaos is developed in a class of hybrid piecewise-smooth systems with impact and noise excitation under unilateral rigid constraint, the corresponding unperturbed Hamiltonian system has the three-piecewise homoclinic orbit. This homoclinic orbit crosses the first and second switching manifolds continuously, then jumps symmetrically with x-axis on the third switching manifold through the rigid impact. It will not cross the third switching manifold and enter the next zone under unilateral rigid constraint. Especially, the perturbed trajectory crosses the first switching manifold through the elastic impact rule expressed by the reset map. Afterwards, the random non-smooth Melnikov function is obtained, the chaotic criteria with and without noise excitation are derived. Furthermore, the theoretical results obtained are applied to analyse the chaos in an actual non-smooth mechanical system with impacts, dry friction and noise excitation.

Stochastic averaging for a completely integrable Hamiltonian system with fractional Brownian motion

This paper proposes an effective approximation result for the behavior of a small transversal perturbation to a completely integrable stochastic Hamiltonian system on a symplectic manifold. We derive an averaged stochastic differential equations (SDEs) in the action space for the action component of the perturbed system, where the averaged drift coefficient is characterized by the averages of that of the action component with respect to the invariant measure of the unperturbed system on the corresponding invariant manifolds. Then, the averaging principle is shown to be valid such that the action component of the perturbed system converges to the solution of averaged SDEs in the mean square sense.

Stability Under Perturbations for the Attractor of a Dissipative PDF-ODF-Type System

We consider a dissipative evolutionary system formed by a parabolic system of reaction-diffusion type and a system of ordinary differential equations perturbed by bounded external signals. We prove that the global attractor of the unperturbed system is stable in the input-to-state stability sense with respect to the value of perturbations.

Piecewise Smooth Perturbations to a Class of Planar Cubic Centers

This paper studies the limit cycle bifurcations of a class of planar cubic isochronous centers. For different values of two key parameters, we give an estimate of the maximum number of limit cycles bifurcating from the period annulus of the unperturbed systems under arbitrarily small piecewise smooth polynomial perturbation. The main method and technique are based on the first order averaging theory for discontinuous systems and the Argument Principle in complex analysis.

Thermophoretic motion of a charged single colloidal particle.

We calculate the thermophoretic drift of a charged single colloidal particle with hydrodynamically slipping surface immersed in an electrolyte solution in response to a small temperature gradient. Here we rely on a linearized hydrodynamic approach for the fluid flow and the motion of the electrolyte ions while keeping the full nonlinearity of the Poisson-Boltzmann equationof the unperturbed system to account for possible large surface charging. The partial differential equationsare transformed into a coupled set of ordinary differential equationsin linear response. Numerical solutions are elaborated for parameter regimes of small and large Debye shielding and different hydrodynamic boundary conditions encoded in a varying slip length. Our results are in good agreement with predictions from recent theoretical work and successfully describe experimental observations on thermophoresis of DNA. We also compare our numerical results with experimental data on polystyrene beads.

The classical Aharonov–Bohm interaction as a relativity paradox

The situation of a charged particle passing down the symmetry axis through a magnetic toroid presents a relativity paradox; different inertial frames suggest different forces on the charge and on the toroid due to the unperturbed systems. We review the charge-toroid interaction and suggest that the magnetic Aharonov–Bohm situation is misunderstood because of unfamiliarity with the acceleration fields following from the Darwin Lagrangian, which go unmentioned in recent textbooks of classical electromagnetism.

On limit measures and their supports for stochastic ordinary differential equations

This paper studies limit measures of stationary measures of stochastic ordinary differential equations on the Euclidean space and tries to determine which invariant measures of an unperturbed system will survive. Under the assumption for SODEs to admit the Freidlin–Wentzell or Dembo–Zeitouni large deviations principle with weaker compactness condition, we prove that limit measures are concentrated away from repellers which are topologically transitive, or equivalent classes, or admit Lebesgue measure zero. We also preclude concentrations of limit measures on acyclic saddle or trap chains. This illustrates that limit measures are concentrated on Liapunov stable compact invariant sets. Applications are made to the Morse–Smale systems, the Axiom A systems including structural stability systems and separated star systems, the gradient or gradient-like systems, those systems possessing the Poincaré–Bendixson property with a finite number of limit sets to obtain that limit measures live on Liapunov stable critical elements, Liapunov stable basic sets, Liapunov stable equilibria and Liapunov stable limit sets including equilibria, limit cycles and saddle or trap cycles, respectively. Four nontrivial examples admitting a unique limit measure are provided, two of which possess infinite equivalent classes and their supports are Liapunov stable periodic orbits, one is supported on saddle-nodes.

Uniform bifurcation of comet-type periodic orbits in the restricted (n + 1)-body problem with non-Newtonian homogeneous potential

We treat the restricted (n+1)-body problem with a non-Newtonian homogeneous potential where the n primaries move on an arbitrary 2π-periodic orbit. We prove that the satellite equation has infinitely many periodic solutions that emerge from the infinity, asymptotically homothetic to the circular solutions of a central force problem. These solutions are obtained as critical solutions of a family of time-dependent perturbed Lagrangian systems, bifurcating uniformly from a compact set of periodic solutions of the unperturbed Lagrangian system.

Response solutions of 2-dimensional degenerate systems under quasi-periodic perturbations

This paper considers quasi-periodic perturbations of 2-dimensional degenerate systems. It is proved that if the equilibrium point of the unperturbed system is hyperbolic-type degenerate, then the perturbed system has a small response solution. The proof is based on the topological degree theory and some KAM techniques developed in [3].

Quantum first detection of a quantum walker on a perturbed ring

The problem of quantum first detection time has been extensively investigated in recent years. Employing the stroboscopic measurement protocol, we consider such a monitored quantum walk on sequentially or periodically perturbed rings and focus on the statistics of first detected return time, namely, the time it takes a particle to return to the initial state for the first time. Using time-independent perturbation theory, we obtain the general form of the eigenvalues and eigenvectors of the Hamiltonian. For the case of a sequentially perturbed ring system, we find steplike behaviors of ${\ensuremath{\sum}}_{n=1}^{\mathcal{N}}n{F}_{n}$ ($\ensuremath{\rightarrow}\ensuremath{\langle}n\ensuremath{\rangle}$ as $\mathcal{N}\ensuremath{\rightarrow}\ensuremath{\infty}$) when $\mathcal{N}$ increases, with two plateaus corresponding to integers, where ${F}_{n}$ is the first detected return probability at the $n\mathrm{th}$ detection attempt. Meanwhile, if the initial condition preserves the reflection symmetry, the mean return time is the same as the unperturbed system. For the periodically perturbed system, similar results can also appear in the case where the symmetry is preserved; however, the size of the ring, the interval between adjacent perturbations, and the initial position may change the mean return time in most cases. In addition, we find that the decay rate of the first detection probability ${F}_{n}$ decreases with the increase in perturbation amplitude. More profoundly, the symmetry-preserving setup of the initial conditions leads to the coincidence of ${F}_{n}$. The symmetry of the physical systems under investigation is deeply reflected in the quantum detection time statistics.

Taylor-Hood Like Finite Elements for Nearly Incompressible Strain Gradient Elasticity Problems

We propose a family of mixed finite elements that are robust for the nearly incompressible strain gradient model, which is a fourth-order singular perturbed elliptic system. The element is similar to [C. Taylor and P. Hood, Comput. & Fluids, 1(1973), 73–100] in the Stokes flow. Using a uniform discrete B-B inequality for the mixed finite element pairs, we show the optimal rate of convergence that is robust in the incompressible limit. By a new regularity result that is uniform in both the materials parameter and the incompressibility, we prove the method converges with 1/2 order to the solution with strong boundary layer effects. Moreover, we estimate the convergence rate of the numerical solution to the unperturbed second-order elliptic system. Numerical results for both smooth solutions and the solutions with sharp layers confirm the theoretical prediction.

On the Solvability of a Periodic Problem for a System of Ordinary Differential Equations with the Main Positive Homogeneous Nonlinearity

We study the solvability of a periodic problem for a system of ordinary differential equations in which we separate the main nonlinear part that is positive homogeneous mapping (of order greater than unity), with the rest called a perturbation. It is proved that if the unperturbed system of equations has no nonzero bounded solutions, then the periodic problem is solvable under any perturbation if and only if the degree of the positive homogeneous mapping on the unit sphere is nonzero. The result obtained is of interest from the point of view of the application and development of methods of nonlinear analysis in the theory of differential and integral equations.

Stability tests and solution estimates for non-linear differential equations

This article deals with certain systems of delay differential equations (DDEs) and a system of ordinary differential equations (ODEs). Here, five new theorems are proved on the fundamental properties of solutions of these systems. The results on the properties of solutions consist of sufficient conditions and they dealt with uniformly asymptotically stability (UAS), instability and integrability of solutions of unperturbed systems of DDEs, boundedness of solutions of a perturbed system of DDEs at infinity and exponentially stability (ES) of solutions of a system of nonlinear ODEs. Here, the techniques of proofs depend upon the Lyapunov- Krasovski? functional (LKF) method and Lyapunov function (LF) method. For illustrations, in particular cases, four examples are constructed as applications. Some results of this paper are given at first time in the literature, and the other results generalize and improve some related ones in the literature.