This book is devoted to bifurcations of periodic, subharmonic and chaotic oscillations, and travelling waves in nonlinear differential equations and discrete dynamical systems by using the topological degree theory both for single-valued and multi-valued mappings in Banach spaces. Original bifurcation results are proved with applications to a broad variety of nonlinear problems ranging from non-smooth and discontinuous mechanical systems, weakly coupled oscillators, systems with relay hysteresis, through infinite chains of differential equations on lattices involving also spatially discretized partial differential equations, and to string and beam partial differential equations. Next, the chaotic behaviour is also investigated for maps possessing topologically transversally intersecting invariant manifolds. Moreover, periodic orbits with arbitrarily high periods, the so-called blue sky catastrophe, are shown for reversible differential systems and maps. Finally, bifurcations of large amplitude oscillations for discontinuous undamped wave partial differential equations are given as well.
This book is mainly intended for post-graduate students and researchers in mathematics with an interest in applications of topological bifurcation methods to dynamical systems and nonlinear analysis, in particular to differential equations and inclusions, and maps. But, among others, it could also be used either by physicists studying oscillations of nonlinear mechanical systems or by engineers investigating vibrations of strings and beams, and electrical circuits.