Oscillation of a class of second-order nonlinear ODE with impulses

Xiu-li, WU; Chen, Siyang; Ji, Hong

doi:10.1016/s0096-3003(02)00073-5

Cited by 18 publications

(4 citation statements)

References 3 publications

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“…The behavior of all solutions is an important issue of impulsive differential equations. Therefore, much effort has been made to investigate the oscillation criteria and the asymptotic behavior of all solutions of impulsive differential equations [8][9][10][11][12][13][14][15][16]. For instance, in [9], the authors investigated the asymptotic behavior of solutions of second order nonlinear impulsive differential equations.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Behavior of nth Order Impulsive Differential Equations

Zheng,

Pan

2024

SJAMS

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This paper devotes to the asymptotic behavior of all solutions of nth order impulsive differential equations. Based on impulsive differential inequality, boundedness and zero tendency of every solution for nth order impulsive differential equations are obtained. In addition, we derive globally uniformly exponential stability of every solution under Lyapunov function and impulsivetechnique, and these results are extend to nth order differential equations with periodic coefficient and periodic impulse. Meanwhile, an example with simulations are provided to verify the conclusion.

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Section: Introductionmentioning

confidence: 99%

Asymptotic Behavior of nth Order Impulsive Differential Equations

Zheng,

Pan

2024

SJAMS

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“…Ordinary Differential Equations (ODEs) have been widely used to model dynamic systems in various scientific fields such as physics (Ferrell Jr et al, 2011;Xiu-Li et al, 2003;Massatt, 1983), biology (Hemker, 1972;Huang et al, 2006;Lu et al, 2011), and economics (Tu, 2012;Weber and Kryazhimskiy, 2011;Norton and Scandizzo, 1981). In recent years, ODEs are also attracting increasing attention in the machine learning community.…”

Section: Introductionmentioning

confidence: 99%

Identifiability and Asymptotics in Learning Homogeneous Linear ODE Systems from Discrete Observations

Wang¹,

Huang²,

Gong³

et al. 2022

Preprint

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Ordinary Differential Equations (ODEs) have recently gained a lot of attention in machine learning. However, the theoretical aspects, e.g., identifiability and asymptotic properties of statistical estimation are still obscure. This paper derives a sufficient condition for the identifiability of homogeneous linear ODE systems from a sequence of equally-spaced error-free observations sampled from a single trajectory. When observations are disturbed by measurement noise, we prove that under mild conditions, the parameter estimator based on the Nonlinear Least Squares (NLS) method is consistent and asymptotic normal with n −1/2 convergence rate. Based on the asymptotic normality property, we construct confidence sets for the unknown system parameters and propose a new method to infer the causal structure of the ODE system, i.e., inferring whether there is a causal link between system variables. Furthermore, we extend the results to degraded observations, including aggregated and time-scaled ones. To the best of our knowledge, our work is the first systematic study of the identifiability and asymptotic properties in learning linear ODE systems. We also construct simulations with various system dimensions to illustrate the established theoretical results.

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“…We may note that this type of work is very rare in the literature signifying that the impulse of the differential equation follows a difference equation of same type. In this direction, we refer the reader to some of the related works [3,4,5,6,7,8,9,11,12,15,16,17,19,26,27,32,33,34] and the references cited there in. Definition 1.1.…”

Section: Introductionmentioning

confidence: 99%