Integral Manifolds for Impulsive Differential Problems with Applications

Dr. Ivanka Stamova is a Professor of Mathematics at the University of Texas, San Antonio, TX, USA. She has authored numerous articles on nonlinear analysis, stability, and control of nonlinear systems, as well as three books, Functional and Impulsive Differential Equations of Fractional Order: Qualitative Analysis and Applications (2017), Applied Impulsive Mathematical Models (2016), and Stability Analysis of Impulsive Functional Differential Equations (2009). Her current research interests include qualitative analysis of nonlinear dynamical systems, fractional-order systems and models, impulsive control, and applications. Dr. Gani Stamov is a Professor of Instruction at the University of Texas, San Antonio, TX, USA. His current research interests include nonlinear analysis, applied mathematics, control theory, and uncertain nonlinear systems. He has authored over 150 articles, as well as three books, Functional and Impulsive Differential Equations of Fractional Order: Qualitative Analysis and Applications (2017), Applied Impulsive Mathematical Models (2016), and Almost Periodic Solutions of Impulsive Differential Equations (2012).

• 1. Basic theory1.1 Introduction1.2 Impulsive differential equations1.2.1 Impulsive ordinary differential equations with variable impulsive perturbations1.2.2 Impulsive ordinary differential equations with fixed moments of impulsive perturbations1.3 Impulsive functional differential equations1.4 Impulsive fractional differential equations1.5 Impulsive conformable differential equations1.6 Integral manifolds1.7 Lyapunov method and impulsive differential equations1.7.1 Piecewise continuous Lyapunov functions1.7.2 Lyapunov–Razumikhin method1.7.3 Fractional Lyapunov function method1.7.4 Conformable Lyapunov function method1.8 Comparison results1.9 Notes and comments2. Impulsive differential equations and existence of integral manifolds2.1 Integral manifolds for impulsive differential equations2.1.1 Integral manifolds for impulsive functional differential equations2.1.2 Integral manifolds for impulsive uncertain functional differential equations2.1.3 Integral manifolds for impulsive fractional functional differential equations2.2 Impulsive differential equations and (ρ, η)-integral manifolds2.2.1 Integral manifolds of (ρ, η)-type and perturbations of the linear part of impulsive differential equations2.2.2 (ρ, η)-integral manifolds for singularly perturbed impulsive differential equations2.3 Affinity integral manifolds for linear singularly perturbed systems of impulsive differential equations2.4 Integral manifolds of impulsive differential equations defined on a torus2.5 Notes and comments3. Impulsive differential equations and stability of integral manifolds3.1 Lyapunov method and stability of integral manifolds3.2 Stability of moving integral manifolds3.2.1 Stability of moving integral manifolds for impulsive ordinary differential equations3.2.2 Stability of conditionally moving integral manifolds for impulsive ordinary differential equations3.2.3 Stability of moving integral manifolds for impulsive functional differential equations3.3 Stability with respect to h-manifolds3.3.1 Practical stability with respect to h-manifolds for impulsive functional differential equations with variable impulsive perturbations3.3.2 Stability with respect to h-manifolds for impulsive functional differential systems of fractional order3.3.3 Practical stability with respect to h-manifolds for impulsive conformable differential equations3.4 Reduction principle and stability of integral manifolds3.4.1 Integral manifolds and the reduction principle for impulsive differential equations3.4.2 Integral manifolds and the reduction principle for singularly perturbed impulsive differential equations3.5 Notes and comments4. Applications: integral manifolds and impulsive differential models4.1 Impulsive neural networks and integral manifolds4.1.1 Integral manifolds for impulsive cellular neural networks4.1.2 Stability with respect to h-manifolds of impulsive Cohen–Grossberg neural networks4.1.3 Integral manifolds for impulsive reaction-diffusion neural networks4.2 Integral manifolds for impulsive models in biology and medicine4.2.1 Stable manifolds for impulsive Lotka–Volterra models4.2.2 Integral manifolds for impulsive Lasota–Wazewska models4.2.3 Integral manifolds for impulsive epidemic and virus dynamic models4.2.4 Integral manifolds for impulsive Kolmogorov models4.3 Integral manifolds for impulsive models in finance4.4 Notes and commentsReferencesIndex