Impulse Points Research Articles

Research on the dynamic characteristics of the cavitation bubble collapsing between multiple particles

The combined action of multiple particles and cavitation bubbles can severely damage hydraulic machinery. Combining the Kelvin impulse theory and the results of high-speed photography experiments, this paper researches the dynamic behaviors of a single bubble located between three equal-sized spherical particles. Non-spherical morphological evolution characteristics in the collapse stage of the bubble are described. The influence of the arrangement of the three particles on the direction and intensity of the bubble migration is quantitatively analyzed. On this basis, the spatial distribution characteristics of the zero impulse points with the Kelvin impulse equal to zero are explored. The results show that: (1) As the bubble is induced in the symmetric positions, three typical cases of collapse characteristics are summarized according to the bubble morphology, including V-shaped, T-shaped, and ginkgo leaf-shaped. (2) As the bubble incipient position is shifted on the symmetry axis, the Kelvin impulse intensity shows a non-monotonic trend with its direction varying many times. Both the impulse intensity and direction are significantly affected by the arrangement of particles. (3) There are multiple zero impulse points among the three particles, and both the spatial location and the number of the zero impulse points are affected by the arrangement of the particles.

Existenceof Heteroclinic Solutions in Nonlinear Differential Equations of the Second-Order Incorporating Generalized Impulse Effects with the Possibility of Application to Bird Population Growth

This work considers the existence of solutions of the heteroclinic type in nonlinear second-order differential equations with ϕ-Laplacians, incorporating generalized impulsive conditions on the real line. For the construction of the results, it was only imposed that ϕ be a homeomorphism, using Schauder’s fixed-point theorem, coupled with concepts of L1-Carathéodory sequences and functions along with impulsive points equiconvergence and equiconvergence at infinity. Finally, a practical part illustrates the main theorem and a possible application to bird population growth.

Impulsive Effects and Complexity Dynamics in the Anti-Predator Model with IPM Strategies

When confronted with the imminent threat of predation, the prey instinctively employ strategies to avoid being consumed. These anti-predator tactics involve individuals acting collectively to intimidate predators and reduce potential harm during an attack. In the present work, we propose a state-dependent feedback control predator-prey model that incorporates a nonmonotonic functional response, taking into account the anti-predator behavior observed in pest-natural enemy ecosystems within the agricultural context. The qualitative analysis of this model is presented utilizing the principles of impulsive semi-dynamical systems. Firstly, the stability conditions of the equilibria are derived by employing pertinent properties of planar systems. The precise domain of the impulsive set and phase set is determined by considering the phase portrait of the system. Secondly, a Poincaré map is constructed by utilizing the sequence of impulsive points within the phase set. The stability of the order-1 periodic solution at the boundary is subsequently analyzed by an analog of the Poincaré criterion. Additionally, this article presents various threshold conditions that determine both the existence and stability of an order-1 periodic solution. Furthermore, it investigates the existence of order-k (k≥2) periodic solutions. Finally, the article explores the complex dynamics of the model, encompassing multiple bifurcation phenomena and chaos, through computational simulations.

Neural Network Methods Based on Efficient Optimization Algorithms for Solving Impulsive Differential Equations

In view of the fact that impulsive differential equations have the discreteness due to the impulse phenomenon, this paper proposes a single hidden layer neural network method based extreme learning machine and a physicsinformed neural network method combined with learning rate attenuation strategy to solve linear impulsive differential equations and nonlinear impulsive differential equations respectively. For the linear impulsive differential equations, firstly, the interval is segmented according to the impulse points, and a single hidden layer neural network model is constructed, the weight parameters of hidden layer are randomly set, the optimal output parameters and solution of the first segment are obtained by the extreme learning machine algorithm, then we calculate the initial value of the second segment according to the jumping equation, and the remaining segments are solved in turn in the same way. Although the single hidden layer neural network method proposed can solve linear equations with high accuracy, it is not suitable for solving nonlinear equations. Therefore, we propose the physics-informed neural network combined with learning rate attenuation strategy to solve the nonlinear impulsive differential equations, then the Adam algorithm and L-BFGS algorithm are combined to find the optimal approximate solution of each segment. Numerical examples show that the single hidden layer neural network method with Legendre polynomials as the activation function and the physics-informed neural network method combined with learning rate attenuation strategy can solve linear and nonlinear impulsive differential equations with higher accuracy.

Impulsive Destabilization Effect on Novel Existence of Solution and Global μ-Stability for MNNs in Quaternion Field

In this paper, a novel memristor-based non-delay Hopfield neural network with impulsive effects is designed in a quaternion field. Some special inequalities, differential inclusion, Hamilton rules and impulsive system theories are utilized in this manuscript to investigate potential solutions and obtain some sufficient criteria. In addition, through choosing proper μ(t) and impulsive points, the global μ-stability of the solution is discussed and some sufficient criteria are presented by special technologies. Then, from the obtained sufficient criteria of global μ-stability, other stability criteria including exponential stability and power stability can be easily derived. Finally, one numerical example is given to illustrate the feasibility and validity of the derived conclusions.

STABILITY OF QUATERNION-VALUED IMPULSIVE DIFFERENTIAL EQUATIONS

We consider a class of quaternion-valued impulsive differential equations (QIDEs). We first use impulsive Cauchy matrices which are essential to explore the stability of QIDEs and whose exponential structure are analyzed in light of eigenvalues of matrices via the distance between impulsive points. Then, a number of sufficient criteria are acquired for the asymptotic stability of linear QIDEs and linear QIDEs with perturbation. In addition, the existence, uniqueness and Ulam–Hyers–Rassias stability of solutions of nonlinear QIDEs are investigated. Most importantly, the results which we obtain are presented in the sense of quaternion-valued and complex-valued, respectively. Finally, examples are given to indicate the utility of our theoretical results.

Existence, stability and controllability results for a class of switched evolution system with impulses over arbitrary time domain

We establish several qualitative properties of a neutral switched impulsive evolution system on an arbitrary time domain by using the theory of time scales. This is the first attempt for switched evolution systems with impulses in abstract spaces. First, we investigate the existence of a unique solution and Ulam’s type stability results. After that, we establish the total controllability results, i.e., controllability not only with respect to the endpoint of the interval but also on the impulsive points. We transform the controllability problem into a solvability problem of an operator equation. We used the Banach fixed point theory and evolution operator theory to establish these results. To illustrate the effectiveness and implications of the developed results, we provide theoretical and simulated numerical examples for different time domains.

Exponential Stability of Fractional-Order Switched Systems With Mode-Dependent Impulses and Its Application.

Most exiting results for impulsive switched systems (ISSs) are mainly built on the synchronous switching and impulses case; however, the impulses can not only occur in switched interval including switched instants but also the switched signals may exist between two impulsive points in practical instants. Under asynchronous impulses and switching signals, the main objective of this article is to study the exponential stability of fractional-order hybrid systems. In order to better characterize stability, some novel criteria are presented by adopting the mode-dependent average impulsive interval and induction method. The obtained impulsive switched criteria lead to a tradeoff between fractional-order α and impulsive strength. Especially, the impulsive effects (positive or negative) with the order α are also discussed in detail, which extends the previous integer order results. Moreover, numerical examples are given to interpret and verify the effectiveness of the obtained criteria.

Controllability of switched Hilfer neutral fractional dynamic systems with impulses

Abstract The aim of this work is to investigate the controllability of a class of switched Hilfer neutral fractional systems with non-instantaneous impulses in the finite-dimensional spaces. We construct a new class of control function that controls the system at the final time of the time-interval and controls the system at each of the impulsive points i.e. we give the so-called total controllability results. Also, we extend these results to the corresponding integro-system. We mainly use the fixed point theorem, Laplace transformation, Mittag-Leffler function, Gramian type matrices and fractional calculus to establish these results. In the end, we provide a simulated example to verify the obtained analytical results.

Extremal mild solutions to fractional delay integro-differential equations with non-instantaneous impulses

We study the existence of extremal mild solutions to fractional delay integro-differential equations with non-instantaneous impulses by employing the monotone iterative technique coupled with the method of lower and upper solutions. Our existence results are established by appealing to two different theories. In our first result, the semigroup generated by the linear operator associated with the problem is assumed to be compact. On the other hand, in our second result, the semigroup need not be compact and our conclusions are obtained using the theory of measures of noncompactness. In order to illustrate our results, we also discuss a concrete problem regarding heat conduction in a metal bar with predefined impulsive points.

THIRD-ORDER GENERALIZED DISCONTINUOUS IMPULSIVE PROBLEMS ON THE HALF-LINE

In this paper, we improve the existing results in the literature by presenting weaker sufficient conditions for the solvability of a third-order impulsive problem on the half-line, having generalized impulse effects. More precisely, our nonlinearities do not need to be positive nor sublinear and the monotone assumptions are local ones. Our method makes use of some truncation and perturbed techniques and on the equiconvergence at infinity and the impulsive points. The last section contains an application to a boundary layer flow problem over a stretching sheet with and without heat transfer.

Exponential stability of non-linear neutral stochastic delay differential system with generalized delay-dependent impulsive points

This work aims to analyze the exponential stability of a non-linear impulsive neutral stochastic delay differential system. In this study, impulse perturbation is considered a delay-dependent state variable. The solution of the delay-dependent impulsive neutral stochastic delay differential system is associated with the solution of the system without impulses. First, we developed a relation connecting the solution of the neutral stochastic delay differential system without impulses and the solution of the corresponding system with impulses. Then, the conditions of the exponential stability of the proposed impulsive system are derived by determining the stability analysis of the respective system without impulse. The numerical approach for the neutral stochastic delay system without impulses is generated using the Euler-Maruyama method and adopted for the corresponding impulsive system. Finally, the achieved theoretical results are illustrated for applying the Malthusian single species neutral stochastic delay population model with immigration impulses.

An Improved Adaptive Time Difference Estimation Method Based on Median Filter in Impulse Environment

Boiler tube burst is the main reason for the unplanned shutdown of thermal power plants. The time difference can be estimated and analyzed by the acoustic signal generated when the boiler tube leaks, and the time difference can be converted into a distance difference to locate the leakage point. The accuracy of positioning depends on the accuracy of the time difference estimation, and the various background noises produced by the complex production environment of the boiler and the impulse noise generated by the uncertain factors in the signal acquisition circuit have a non-negligible effect on the accuracy of the time difference estimation. Aiming at the problem that the signal does not have obvious second-order statistics in the environment of impulse noise, a Wiener weighted adaptive time difference estimation method based on median filtering is proposed. First, the median filter is used to remove the impulse points in the noise to make it obey the normal distribution and have second-order statistics; Next, select the generalized correlation method based on the Wiener weighting function of linear minimum mean square error to eliminate the influence of noise; Finally, according to the characteristics of the generalized correlation method that relies on the prior knowledge of the signal but the ability to suppress noise and the characteristics of the adaptive method that the ability to suppress noise is weak but does not rely on the prior knowledge of the signal, the two methods are combined to form a Wiener-weighted generalized correlation and its adaptive method. Simulation experiments prove that the Wiener weighted adaptive time difference estimation method based on median filtering has better estimation performance under impulsive noise than the adaptive minimum average p-norm method based on fractional low-order statistics.

Optimal estimation of unknown data of periodic boundary value problems for first order linear impulsive systems of ordinary differential equations from indirect noisy observations of their solutions

We consider boundary value problems with periodic boundary conditions for first-order linear systems of impulsive ordinary differential equations with unknown right-hand sides and jumps of solutions at the impulse points entering into the statement of these problems which are assumed to be subjected to some quadratic restrictions. From indirect noisy observations of their solutions on a finite system of intervals, we obtain the optimal, in certain sense, estimates of images of their right-hand sides under linear continuous operators. Under the condition that the unknown correlation functions of noises in observations belong to some special sets, it is established that such estimates and estimation errors are expressed explicitly via solutions of special periodic boundary value problems for linear impulsive systems of ordinary differential equations.

Exponential stability of non-linear stochastic delay differential system with generalized delay-dependent impulsive points

Abstract This paper is concerned with a non-linear stochastic delay differential system with delay-dependent impulsive perturbations. In this work, the size of the jump is defined as a general non-linear delay-dependent state variable and the solution of the impulsive stochastic delay differential system corresponding to the system without impulsive perturbations is given. This work is based on the relation between the solution of the equivalent model of stochastic delay differential system without impulses corresponding to the solution of the system with impulses. Then the conditions of the exponential stability of the proposed impulsive system are obtained by deriving stability criteria of the corresponding system without impulses. The numerical approximation for the stochastic delay system without impulses is developed using the Runge-Kutta-Maruyama method and it is suitably applied for the corresponding impulsive system. Finally, the obtained theoretical results are illustrated graphically for a stochastic delay system with impulses.

Billiard Physics: Motion of a Cue Ball after Hit by a Cue Stick Horizontally

Billiards and pool games provide an excellent medium to discuss several physics principles including Newton’s laws of motion, friction, impulse-momentum theorem, work-energy theorem, kinematics and dynamics of rotational motion, and rolling motion. This paper provides a complete analysis of the motion of a cue ball that is given a sharp impulse, which is parallel to the horizontal surface. The horizontal impulse, in this work, is restricted to be in the plane passing through the ball’s center of mass and the contact point such that it sets the ball in horizontal motion with no sidespin. The motion of the cue ball until it eventually starts to roll without slipping is analyzed for all different impulse points on the surface of the ball. These analyses provide physics instructors a set of exciting examples on the rolling motion.

Stability of a uniform attractor for a second-order evolution equation with discontinuous trajectories

The work is devoted to the study of the qualitative behavior of solutions of the wave equation, whose trajectories have impulsive perturbations in moments when they reach a fixed (impulsive) subset of the phase space. Using the general constructing scheme of the infinite-dimensional impulsive dynamical system and using the concept of a uniform attractor — a minimal compact uniformly attracting set, we obtained the result about the existence and the explicit form of a uniform attractor for the corresponding impulsive dynamical system. Trajectories of such system can have an infinite number of impulsive points when they encounter an impulsive subset of the phase space. Thus, a uniform attractor may have a non-empty intersection with the impulsive set and, as a result, may not have the stability property. However, due to the additional conditions on the impulsive parameters of the problem, we managed to prove the stability property for the non-impulsive part of the uniform attractor.

Fuel-Optimal Multi-Impulse Orbit Transfer Using a Hybrid Optimization Method

For the $N$ -impulse transfer between two earth orbits, this paper introduces $N - 1$ intermediate orbits to describe the orbit transfer scheme. The eccentricity vector of the intermediate orbits and the true anomalies corresponding to the impulsive points are chosen as the optimization variables. Based on the patched conic theory, candidate solution can be analytically derived, constraints are removed from the optimization model, and the original problem is converted to a parameter optimization problem. The only difficulty lies in the initialization because the number of optimization variables increases linearly with $N$ , which can be very large. This is settled by a hybrid optimization algorithm that comprises two searching methods. The problem is solved first by an improved particle swarm optimization method and, then, by an adaptive conjugate gradient method. The proposed method is adaptive to problems with any finite $N$ and can calculate the optimal $N$ in any transfer scenarios. The simulation validates the proposed method with some well-known cases and demonstrates its adaptation to $N$ .

Projective Synchronization of Delayed Neural Networks With Mismatched Parameters and Impulsive Effects.

In this paper, the impulsive effects on projective synchronization between the parameter mismatched neural networks with mixed time-varying delays have been analyzed. Since complete synchronization is not possible due to the existence of parameter mismatch and projective factor, a drive has been taken to achieve the weak projective synchronization of different neural networks under impulsive control strategies. Through the use of matrix measure technique and the extended comparison principle based on the formula of variation of parameters for mixed time-varying delayed impulsive systems, sufficient criteria have been derived for exponential convergence of the networks under the effects of extensive range of impulse. Instead of upper or lower bound of the impulsive interval, the concept of the average impulsive interval is applied to estimate the number of impulsive points in an interval. The concept of calculus is applied for optimizing the synchronization error bounds which are obtained because of different ranges of impulse. Finally, the numerical simulations for various impulsive ranges for different cases are presented graphically to validate the efficiency of the theoretical results.

Continuous Dependence of Solutions of Integer and Fractional Order Non-Instantaneous Impulsive Equations with Random Impulsive and Junction Points

This paper gives continuous dependence results for solutions of integer and fractional order, non-instantaneous impulsive differential equations with random impulse and junction points. The notion of the continuous dependence of solutions of these equations on the initial point is introduced. We prove some sufficient conditions that ensure the solutions to perturbed problems have a continuous dependence. Finally, we use numerical examples to demonstrate the obtained theoretical results.