Integral Manifolds for Impulsive Differential Problems with Applications
- 1st Edition - May 28, 2025
- Authors: Ivanka Stamova, Gani Stamov
- Language: English
- Paperback ISBN:9 7 8 - 0 - 4 4 3 - 3 0 1 3 4 - 6
- eBook ISBN:9 7 8 - 0 - 4 4 3 - 3 0 1 3 5 - 3
Integral Manifolds for Impulsive Differential Problems with Applications offers readers a comprehensive resource on integral manifolds for different classes of differential equati… Read more
Integral Manifolds for Impulsive Differential Problems with Applications offers readers a comprehensive resource on integral manifolds for different classes of differential equations which will be of prime importance to researchers in applied mathematics, engineering, and physics. The book offers a highly application-oriented approach, reviewing the qualitative properties of integral manifolds which have significant practical applications in emerging areas such as optimal control, biology, mechanics, medicine, biotechnologies, electronics, and economics. For applied scientists, this will be an important introduction to the qualitative theory of impulsive and fractional equations which will be key in their initial steps towards adopting results and methods in their research.
- Offers a comprehensive resource of qualitative results for integral manifolds related to different classes of impulsive differential equations, delayed differential equations and fractional differential equations
- Presents the manifestations of different constructive methods, by demonstrating how these effective techniques can be applied to investigate qualitative properties of integral manifolds
- Discusses applications to neural networks, fractional biological models, models in population dynamics, and models in economics of diverse fields
Graduate students, researchers, and professionals in applied mathematics, engineering, and physics for whom integral manifolds approaches are important tools to study the qualitative properties of different classes of dynamical systems, including impulsive differential equations and equations that follow fractional-order dynamics
1. Basic theory
1.1 Introduction
1.2 Impulsive differential equations
1.2.1 Impulsive ordinary differential equations with variable impulsive perturbations
1.2.2 Impulsive ordinary differential equations with fixed moments of impulsive perturbations
1.3 Impulsive functional differential equations
1.4 Impulsive fractional differential equations
1.5 Impulsive conformable differential equations
1.6 Integral manifolds
1.7 Lyapunov method and impulsive differential equations
1.7.1 Piecewise continuous Lyapunov functions
1.7.2 Lyapunov–Razumikhin method
1.7.3 Fractional Lyapunov function method
1.7.4 Conformable Lyapunov function method
1.8 Comparison results
1.9 Notes and comments
2. Impulsive differential equations and existence of integral manifolds
2.1 Integral manifolds for impulsive differential equations
2.1.1 Integral manifolds for impulsive functional differential equations
2.1.2 Integral manifolds for impulsive uncertain functional differential equations
2.1.3 Integral manifolds for impulsive fractional functional differential equations
2.2 Impulsive differential equations and (ρ, η)-integral manifolds
2.2.1 Integral manifolds of (ρ, η)-type and perturbations of the linear part of impulsive differential equations
2.2.2 (ρ, η)-integral manifolds for singularly perturbed impulsive differential equations
2.3 Affinity integral manifolds for linear singularly perturbed systems of impulsive differential equations
2.4 Integral manifolds of impulsive differential equations defined on a torus
2.5 Notes and comments
3. Impulsive differential equations and stability of integral manifolds
3.1 Lyapunov method and stability of integral manifolds
3.2 Stability of moving integral manifolds
3.2.1 Stability of moving integral manifolds for impulsive ordinary differential equations
3.2.2 Stability of conditionally moving integral manifolds for impulsive ordinary differential equations
3.2.3 Stability of moving integral manifolds for impulsive functional differential equations
3.3 Stability with respect to h-manifolds
3.3.1 Practical stability with respect to h-manifolds for impulsive functional differential equations with variable impulsive perturbations
3.3.2 Stability with respect to h-manifolds for impulsive functional differential systems of fractional order
3.3.3 Practical stability with respect to h-manifolds for impulsive conformable differential equations
3.4 Reduction principle and stability of integral manifolds
3.4.1 Integral manifolds and the reduction principle for impulsive differential equations
3.4.2 Integral manifolds and the reduction principle for singularly perturbed impulsive differential equations
3.5 Notes and comments
4. Applications: integral manifolds and impulsive differential models
4.1 Impulsive neural networks and integral manifolds
4.1.1 Integral manifolds for impulsive cellular neural networks
4.1.2 Stability with respect to h-manifolds of impulsive Cohen–Grossberg neural networks
4.1.3 Integral manifolds for impulsive reaction-diffusion neural networks
4.2 Integral manifolds for impulsive models in biology and medicine
4.2.1 Stable manifolds for impulsive Lotka–Volterra models
4.2.2 Integral manifolds for impulsive Lasota–Wazewska models
4.2.3 Integral manifolds for impulsive epidemic and virus dynamic models
4.2.4 Integral manifolds for impulsive Kolmogorov models
4.3 Integral manifolds for impulsive models in finance
4.4 Notes and comments
References
Index
1.1 Introduction
1.2 Impulsive differential equations
1.2.1 Impulsive ordinary differential equations with variable impulsive perturbations
1.2.2 Impulsive ordinary differential equations with fixed moments of impulsive perturbations
1.3 Impulsive functional differential equations
1.4 Impulsive fractional differential equations
1.5 Impulsive conformable differential equations
1.6 Integral manifolds
1.7 Lyapunov method and impulsive differential equations
1.7.1 Piecewise continuous Lyapunov functions
1.7.2 Lyapunov–Razumikhin method
1.7.3 Fractional Lyapunov function method
1.7.4 Conformable Lyapunov function method
1.8 Comparison results
1.9 Notes and comments
2. Impulsive differential equations and existence of integral manifolds
2.1 Integral manifolds for impulsive differential equations
2.1.1 Integral manifolds for impulsive functional differential equations
2.1.2 Integral manifolds for impulsive uncertain functional differential equations
2.1.3 Integral manifolds for impulsive fractional functional differential equations
2.2 Impulsive differential equations and (ρ, η)-integral manifolds
2.2.1 Integral manifolds of (ρ, η)-type and perturbations of the linear part of impulsive differential equations
2.2.2 (ρ, η)-integral manifolds for singularly perturbed impulsive differential equations
2.3 Affinity integral manifolds for linear singularly perturbed systems of impulsive differential equations
2.4 Integral manifolds of impulsive differential equations defined on a torus
2.5 Notes and comments
3. Impulsive differential equations and stability of integral manifolds
3.1 Lyapunov method and stability of integral manifolds
3.2 Stability of moving integral manifolds
3.2.1 Stability of moving integral manifolds for impulsive ordinary differential equations
3.2.2 Stability of conditionally moving integral manifolds for impulsive ordinary differential equations
3.2.3 Stability of moving integral manifolds for impulsive functional differential equations
3.3 Stability with respect to h-manifolds
3.3.1 Practical stability with respect to h-manifolds for impulsive functional differential equations with variable impulsive perturbations
3.3.2 Stability with respect to h-manifolds for impulsive functional differential systems of fractional order
3.3.3 Practical stability with respect to h-manifolds for impulsive conformable differential equations
3.4 Reduction principle and stability of integral manifolds
3.4.1 Integral manifolds and the reduction principle for impulsive differential equations
3.4.2 Integral manifolds and the reduction principle for singularly perturbed impulsive differential equations
3.5 Notes and comments
4. Applications: integral manifolds and impulsive differential models
4.1 Impulsive neural networks and integral manifolds
4.1.1 Integral manifolds for impulsive cellular neural networks
4.1.2 Stability with respect to h-manifolds of impulsive Cohen–Grossberg neural networks
4.1.3 Integral manifolds for impulsive reaction-diffusion neural networks
4.2 Integral manifolds for impulsive models in biology and medicine
4.2.1 Stable manifolds for impulsive Lotka–Volterra models
4.2.2 Integral manifolds for impulsive Lasota–Wazewska models
4.2.3 Integral manifolds for impulsive epidemic and virus dynamic models
4.2.4 Integral manifolds for impulsive Kolmogorov models
4.3 Integral manifolds for impulsive models in finance
4.4 Notes and comments
References
Index
- Edition: 1
- Published: May 28, 2025
- Language: English
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Ivanka Stamova
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.
Affiliations and expertise
University of Texas at San Antonio, Department of Mathematics, San Antonio, TX, United StatesGS
Gani Stamov
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).
Affiliations and expertise
University of Texas at San Antonio, San Antonio, USARead Integral Manifolds for Impulsive Differential Problems with Applications on ScienceDirect