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The book covers nonlinear physical problems and mathematical modeling, including molecular biology, genetics, neurosciences, artificial intelligence with classical problems in mechanics and astronomy and physics. The chapters present nonlinear mathematical modeling in life science and physics through nonlinear differential equations, nonlinear discrete equations and hybrid equations. Such modeling can be effectively applied to the wide spectrum of nonlinear physical problems, including the KAM (Kolmogorov-Arnold-Moser (KAM)) theory, singular differential equations, impulsive dichotomous linear systems, analytical bifurcation trees of periodic motions, and almost or pseudo- almost periodic solutions in nonlinear dynamical systems.



Chapter 1. The Solution of the Second Peskin Conjecture and Developments

The integrate-and-fire cardiac pacemaker model of pulse-coupled oscillators was introduced by C. Peskin. Because of the pacemaker’s function, two famous synchronization conjectures for identical and nonidentical oscillators were formulated. The first of Peskin’s conjectures was solved in the paper (J. Phys. A 21:L699–L705, 1988) by S. Strogatz and R. Mirollo. The second conjecture was solved in the paper by Akhmet (Nonlinear Stud. 18:313–327, 2011). There are still many issues related to the nature and types of couplings. The couplings may be impulsive, continuous, delayed, or advanced, and oscillators may be locally or globally connected. Consequently, it is reasonable to consider various ways of synchronization if one wants the biological and mathematical analyses to interact productively. We investigate the integrate-and-fire model in both cases—one with identical and another with not-quite-identical oscillators. A combination of continuous and pulse couplings that sustain the firing in unison is carefully constructed. Moreover, we obtain conditions on the parameters of continuous couplings that make possible a rigorous mathematical investigation of the problem. The technique developed for differential equations with discontinuities at nonfixed moments (Akhmet, Principles of Discontinuous Dynamical Systems, Springer, New York, 2010) and a special continuous map form the basis of the analysis. We consider Peskin’s model of the cardiac pacemaker with delayed pulse couplings as well as with continuous couplings. Sufficient conditions for the synchronization of identical and nonidentical oscillators are obtained. The bifurcation of periodic motion is observed. The results are demonstrated with numerical simulations.
M. U. Akhmet

Chapter 2. On Periodic Motions in a Time-Delayed, Quadratic Nonlinear Oscillator with Excitation

Analytical solutions of periodic motions in a time-delayed, quadratic nonlinear oscillator with periodic excitation are obtained through the finite Fourier series, and the stability and bifurcation analysis for periodic motions are discussed. The bifurcation trees of period-1 motion to chaos can be presented. Numerical illustration of periodic motion is given to verify the analytical solutions.
Albert C. J. Luo, Hanxiang Jin

Chapter 3. Mathematical Analysis of a Delayed Hematopoietic Stem Cell Model with Wazewska–Lasota Functional Production Type

In this chapter, we consider a more general model describing the dynamics of a hematopoietic stem cell (HSC) model with Wazewska–Lasota functional production type describing the cycle of proliferating and quiescent phases. The model is governed by a system of two ordinary differential equations with discrete delay. Its dynamics are studied in terms of local stability and Hopf bifurcation. We prove the existence of the possible steady state and their stability with respect to the time delay and the apoptosis rate of proliferating cells. We show that a sequence of Hopf bifurcations occurs at the positive steady state as the delay crosses some critical values. We illustrate our results with some numerical simulations.
Radouane Yafia, M. A. Aziz Alaoui, Abdessamad Tridane, Ali Moussaoui

Chapter 4. Random Noninstantaneous Impulsive Models for Studying Periodic Evolution Processes in Pharmacotherapy

In this chapter we offer a new class of impulsive models for studying the dynamics of periodic evolution processes in pharmacotherapy, which is given by random, noninstantaneous, impulsive, nonautonomous periodic evolution equations. This type of impulsive equation can describe the injection of drugs in the bloodstream, and the consequent absorption of them in the body is a random, periodic, gradual, and continuous process. Sufficient conditions on the existence of periodic and subharmonic solutions are established, as are other related results such as their globally asymptotic stability. The dynamical properties are also derived for the whole system, leading to the theory of fractals. Finally, examples are given to illustrate our theoretical results.
JinRong Wang, Michal Fečkan, Yong Zhou

Chapter 5. Boundedness of Solutions to a Certain System of Differential Equations with Multiple Delays

In this chapter, we consider a system of differential equations of second order with multiple delays. Based on the Lyapunov–Krasovskii functional approach, we investigate the boundedness of solutions. The obtained results essentially complement and improve some known results in the literature.
Cemil Tunç

Chapter 6. Delay Effects on the Dynamics of the Lengyel–Epstein Reaction-Diffusion Model

We investigate bifurcations of the Lengyel–Epstein reaction-diffusion model involving time delay under the Neumann boundary conditions. We first give stability and Hopf bifurcation analysis of the ordinary differential equation (ODE) models, including delay associated with this model. Later, we extend this analysis to the partial differential equation (PDE) model. We determine conditions on parameters of both models to have Hopf bifurcations. Bifurcation analysis for both models show that Hopf bifurcations occur by regarding the delay parameter as a bifurcation parameter. Using the normal form theory and the center manifold reduction for partial functional differential equations, we also determine the direction of the Hopf bifurcations and the stability of bifurcating periodic solutions for the PDE model. Finally, we perform some numerical simulations to support analytical results obtained for the ODE models.
Hüseyin Merdan, Şeyma Kayan

Chapter 7. Almost Periodic Solutions of Evolution Differential Equations with Impulsive Action

In an abstract Banach space we study conditions for the existence of piecewise continuous, almost periodic solutions for semilinear impulsive differential equations with fixed and nonfixed moments of impulsive action.
Viktor Tkachenko
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