Nonlinear & Complex Systems

This book aims to provide an overview of the recent research devoted to nonlinear and chaotic systems by discussing the modeling of different phenomena. It will encompass methods and techniques from the analysis of nonlinear ordinary differential equations, perturbation theory, bifurcation analysis and chaos synchronization. It contains 19 chapters, organized into 3 sections:

Section 1 - Resonance Phenomena, Nonlinear Vibrations and Waves.
Section 2 - Perturbation Theory and Homotopy Perturbation Methods.
Section 3 - Bifurcation Analysis and Chaos Synchronization.



Section 1 is centered on the analysis of nonlinear ordinary differential equations. It devoted to the study of resonance, oscillations and waves phenomena of from the point of view of dynamical systems close to resonances or bifurcation points. This will include review approximate methods for strongly nonlinear differential equations with oscillations (e.g. Lindstedt-Poincare method and Krylov-Bogoliubov first approximate method) and introduce the normal forms theory for simplifying nonlinear dynamical systems. It will consider examples of coupled nonlinear oscillators, including Drinfeld-Sokolov system and Haken-Kelso-Bunz model (rhythmic movement patterns in human bimanual coordination). Section 2 is focused on the perturbation methods, singular perturbation problem, homotopy perturbation method, variational method, and successive complementary expansion method (a combination of the homotopy and singular perturbation methods). It will include examples of these methods applied to boundary problems, nonlinear large-amplitude oscillations, and nonlinear integral-differential equations. Section 3 is first focused bifurcation analysis, in particular, the Hopf bifurcation of nonlinear dynamic systems. It will include the examples of the Hopf bifurcation as applied to neural networks and Boussinesq systems. Next, it will investigate a higher-order nonlinear Schroedinger equation with non-Kerr term by using the bifurcation theory method as well as the dynamics of a ring of seven unidirectionally coupled nonlinear Duffing oscillators using Fast Fourier Transform analysis presented in the form of a bifurcation graph. The second subject covered in this section is the chaos synchronization, which occurs when two or more dissipative chaotic systems are coupled. The first example will be an adaptive control scheme developed to analyze the generalized adaptive chaos synchronization with uncertain chaotic behavior parameters between two identical chaotic dynamic systems. The second example is the synchronization of a four-dimensional Lorenz-Stenflo system modeled by an electronic circuit. The last example investigates the chaos synchronization of a three-dimensional autonomous nonlinear system (called the general T-system).