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Highlight: [•] New results amend some oscillation criteria for dynamic equations on isolated time scales. [•] One can amend related oscillation results for differential equations using the method provided in this paper. [•] The results obtained in this paper complement oscillation theory of dynamic equations. [Copyright &y& Elsevier]

• The paper proposes an overview of the modeling approaches for the mathematical description and analysis of processes that combine continuous and discontinuous behavior, namely impulsive differential equations, hybrid dynamical systems, and differential equations involving Dirac delta functions. • Insights are provided on the stability and attractivity analysis of hybrid behaviors, and essential differences are highlighted to the respective stability concepts for smooth dynamical systems. • Specific phenomena are discussed which are peculiar for hybrid behaviors, like beating or Zeno phenomenon, modeling of multiple impulses at a single time instance, death and splitting of solutions, etc. • With this, the paper attempts at bringing attention of the interested researchers to the methods available in other research communities and fostering the exchange of ideas and analysis techniques. We propose an overview of the modeling approaches for the mathematical description and analysis of processes that combine continuous and discontinuous behavior, namely impulsive differential equations, hybrid dynamical systems, and differential equations involving Dirac delta functions. These classes of systems are chosen due to their dominant prevalence in physics, mathematics, and control engineering research communities. A comparison of these frameworks is provided and their applicability depending on the character of the hybrid behavior is discussed. In particular, we show that special care should be taken when equations with Dirac delta function are interpreted as impulsive differential equations. We also provide insights on the stability and attractivity analysis of hybrid behaviors, highlight their essential differences to the respective stability concepts for smooth dynamical systems, and discuss specific phenomena which are peculiar for hybrid behaviors, like beating or Zeno phenomenon, modeling of multiple impulses at a single time instance, death and splitting of solutions, etc. With this, the paper attempts at bringing attention of the interested researchers to the methods available in other research communities and fostering the exchange of ideas and analysis techniques. [ABSTRACT FROM AUTHOR]

In this paper, pinning combining with impulsive control scheme is adopted to investigate the synchronization of complex-variable dynamical network. Based on the Lyapunov function method and mathematical analysis technique, sufficient conditions for achieving synchronization is first analytically derived. This result extends the condition derived for real-variable dynamical network to complex-variable network. Further, adaptive strategy is adopted to relax the restrictions on the impulsive intervals and reduce the control cost. Noticeably, the proposed adaptive pinning impulsive control scheme is universal for different dynamical networks to some extent. The impulsive instants are chosen by solving a series of maximum problems subject to the derived conditions. Several numerical simulations are performed to illustrate the effectiveness and correctness of the derived theoretical results. [ABSTRACT FROM AUTHOR]

• Stochastic noises are introduced into the model of Alzheimer's disease. • Analytic conditions for stochastic P-bifurcation are obtained. • A formula is presented for the mean switching time from a mild impairment state to a pathological state. • A disease index is proposed for the early warning of disease. • The results provide insightful suggestions to design he strategies to slow the progression of disease. Alzheimer's disease is a worldwide disease of dementia and is characterized by beta-amyloid plaques. Increasing evidences show that there is a positive feedback loop between the level of beta-amyloid and the level of calcium. In this paper, stochastic noises are incorporated into a minimal model of Alzheimer's disease which focuses upon the evolution of beta-amyloid and calcium. Mathematical analysis indicates that solutions of the model without stochastic noises converge either to a unique equilibrium or to bistable equilibria. Analytical conditions for the stochastic P-bifurcation are derived by means of technique of slow-fast dynamical systems. A formula is presented to approximate the mean switching time from a normal state to a pathological state. A disease index is also proposed to predict the risk to transit from a normal state to a disease state. Further numerical simulations reveal how the parameters influence the evolutionary outcomes of beta-amyloid and calcium. These results give new insights on the strategies to slow the development of Alzheimer's disease. [ABSTRACT FROM AUTHOR]