Nonlinear Hybrid Continuous/Discrete-Time Models

This book examines the author's recently-introduced theory of equations with piece-wise constant argument of generalized type, demonstrating its use in the modeling of such real world problems as neural networks, blood pressure distribution and others.

The book is mainly about hybrid systems with continuous/discrete-time dynamics. The major part of the book consists of the theory of equations with piece-wise constant argument of generalized type. The systems as well as technique of investigation were introduced by the author very recently. They both generalized known theory about differential equations with piece-wise constant argument, introduced by K. Cook and J. Wiener in the 1980s. Moreover, differential equations with fixed and variable moments of impulses are used to model real world problems. We consider models of neural networks, blood pressure distribution and a generalized model of the cardiac pacemaker. All the results of the manuscript have not been published in any book, yet. They are very recent and united with the presence of the continuous/discrete dynamics of time. It is of big interest for specialists in biology, medicine, engineering sciences, electronics. Theoretical aspects of the book meet very strong expectations of mathematicians who investigate differential equations with discontinuities of any type.

1. Introduction.- 2. Linear and quasi-linear systems with piecewise constant argument.- 3. The reduction principle for systems with piecewise constant argument.- 4. The small parameter and differential equations with piecewise constant argument.- 5. Stability.- 6. The state-dependent piecewise constant argument.- 7. Almost periodic solutions.- 8. Stability of neural networks.- 9. The blood pressure distribution.- 10. Integrate-and-fire biological oscillators.