This book presents the recently introduced and already widely referred semi-discretization method for the stability analysis of delayed dynamical systems. Delay differential equations often come up ...in different fields of engineering, like feedback control systems, machine tool vibrations, balancing/stabilization with reflex delay. The behavior of such systems is often counter-intuitive and closed form analytical formulas can rarely be given even for the linear stability conditions. If parametric excitation is coupled with the delay effect, then the governing equation is a delay differential equation with time periodic coefficients, and the stability properties are even more intriguing. The semi-discretization method is a simple but efficient method that is based on the discretization with respect to the delayed term and the periodic coefficients only. The method can effectively be used to construct stability diagrams in the space of system parameters.
Piecewise-smooth Dynamical Systems di Bernardo, Mario; Budd, Christopher J; Champneys, Alan R ...
01/2008, Letnik:
163
eBook, Book Chapter
Recenzirano
Many dynamical systems that occur naturally in the description of physical processes are piecewise-smooth. That is, their motion is characterized by periods of smooth evolutions interrupted by ...instantaneous events. Traditional analysis of dynamical systems has restricted its attention to smooth problems, thus preventing the investigation of non-smooth processes such as impact, switching, sliding and other discrete state transitions. These phenomena arise, for example, in any application involving friction, collision, intermittently constrained systems or processes with switching components.
Consider a fast–slow system of ordinary differential equations of the form x˙=a(x,y)+ε−1b(x,y), y˙=ε−2g(y), where it is assumed that b averages to zero under the fast flow generated by g. We give ...conditions under which solutions x to the slow equations converge weakly to an Itô diffusion X as ε→0. The drift and diffusion coefficients of the limiting stochastic differential equation satisfied by X are given explicitly.
Our theory applies when the fast flow is Anosov or Axiom A, as well as to a large class of nonuniformly hyperbolic fast flows (including the one defined by the well-known Lorenz equations), and our main results do not require any mixing assumptions on the fast flow.
This book examines the main theorems in bifurcation theory. It covers both the local and global theory of one-parameter bifurcations for operators acting in infinite-dimensional Banach spaces and ...shows how to apply the theory.
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