Integral Manifolds for Impulsive Differential Problems with Applications

1. Basic theory<br>1.1 Introduction<br>1.2 Impulsive differential equations<br>1.2.1 Impulsive ordinary differential equations with variable impulsive perturbations<br>1.2.2 Impulsive ordinary differential equations with fixed moments of impulsive perturbations<br>1.3 Impulsive functional differential equations<br>1.4 Impulsive fractional differential equations<br>1.5 Impulsive conformable differential equations<br>1.6 Integral manifolds<br>1.7 Lyapunov method and impulsive differential equations<br>1.7.1 Piecewise continuous Lyapunov functions<br>1.7.2 Lyapunov–Razumikhin method<br>1.7.3 Fractional Lyapunov function method<br>1.7.4 Conformable Lyapunov function method<br>1.8 Comparison results<br>1.9 Notes and comments<br><br>2. Impulsive differential equations and existence of integral manifolds<br>2.1 Integral manifolds for impulsive differential equations<br>2.1.1 Integral manifolds for impulsive functional differential equations<br>2.1.2 Integral manifolds for impulsive uncertain functional differential equations<br>2.1.3 Integral manifolds for impulsive fractional functional differential equations<br>2.2 Impulsive differential equations and (ρ, η)-integral manifolds<br>2.2.1 Integral manifolds of (ρ, η)-type and perturbations of the linear part of impulsive differential equations<br>2.2.2 (ρ, η)-integral manifolds for singularly perturbed impulsive differential equations<br>2.3 Affinity integral manifolds for linear singularly perturbed systems of impulsive differential equations<br>2.4 Integral manifolds of impulsive differential equations defined on a torus<br>2.5 Notes and comments<br><br>3. Impulsive differential equations and stability of integral manifolds<br>3.1 Lyapunov method and stability of integral manifolds<br>3.2 Stability of moving integral manifolds<br>3.2.1 Stability of moving integral manifolds for impulsive ordinary differential equations<br>3.2.2 Stability of conditionally moving integral manifolds for impulsive ordinary differential equations<br>3.2.3 Stability of moving integral manifolds for impulsive functional differential equations<br>3.3 Stability with respect to h-manifolds<br>3.3.1 Practical stability with respect to h-manifolds for impulsive functional differential equations with variable impulsive perturbations<br>3.3.2 Stability with respect to h-manifolds for impulsive functional differential systems of fractional order<br>3.3.3 Practical stability with respect to h-manifolds for impulsive conformable differential equations<br>3.4 Reduction principle and stability of integral manifolds<br>3.4.1 Integral manifolds and the reduction principle for impulsive differential equations<br>3.4.2 Integral manifolds and the reduction principle for singularly perturbed impulsive differential equations<br>3.5 Notes and comments<br><br>4. Applications: integral manifolds and impulsive differential models<br>4.1 Impulsive neural networks and integral manifolds<br>4.1.1 Integral manifolds for impulsive cellular neural networks<br>4.1.2 Stability with respect to h-manifolds of impulsive Cohen–Grossberg neural networks<br>4.1.3 Integral manifolds for impulsive reaction-diffusion neural networks<br>4.2 Integral manifolds for impulsive models in biology and medicine<br>4.2.1 Stable manifolds for impulsive Lotka–Volterra models<br>4.2.2 Integral manifolds for impulsive Lasota–Wazewska models<br>4.2.3 Integral manifolds for impulsive epidemic and virus dynamic models<br>4.2.4 Integral manifolds for impulsive Kolmogorov models<br>4.3 Integral manifolds for impulsive models in finance<br>4.4 Notes and comments<br>References<br>Index