Part One: One-Dimensional Dynamics Examples of Dynamical Systems Preliminaries from Calculus Elementary Definitions Hyperbolicity An example: the quadratic family An Example: the Quadratic Family Symbolic Dynamics Topological Conjugacy Chaos Structural Stability Sarlovskiis Theorem The Schwarzian Derivative Bifurcation Theory Another View of Period Three Maps of the Circle Morse-Smale Diffeomorphisms Homoclinic Points and Bifurcations The Period-Doubling Route to Chaos The Kneeding Theory Geneaology of Periodic Units Part Two: Higher Dimensional Dynamics Preliminaries from Linear Algebra and Advanced Calculus The Dynamics of Linear Maps: Two and Three Dimensions The Horseshoe Map Hyperbolic Toral Automorphisms Hyperbolicm Toral Automorphisms Attractors The Stable and Unstable Manifold Theorem Global Results and Hyperbolic Sets The Hopf Bifurcation The Hnon Map Part Three: Complex Analytic Dynamics Preliminaries from Complex Analysis Quadratic Maps Revisited Normal Families and Exceptional Points Periodic Points The Julia Set The Geometry of Julia Sets Neutral Periodic Points The Mandelbrot Set An Example: the Exponential Function

An introduction to chaotic dynamical systems (2nd edition), by Robert L. Devaney. Pp 336. £34·95. 1989. ISBN 0-201-13046-7 (Addison-Wesley)

Bifurcation analysis of an automatic dynamic balancing mechanism for eccentric rotors

A periodically forced linear oscillator with impact attachment has been studied. An asymptotical analytical method has been developed to obtain the fixed points and to analyze the transient 1:1 resonance (two impacts per cycle) of the modulated response. The influence of parameters on dynamics has been analyzed around the slow invariant manifold (SIM). Five different response regimes have been observed from theoretical and numerical results. It is demonstrated that they are closely related to the topological structure and relative position of fixed points. The bifurcation, route to chaos and the efficiency of targeted energy transfer (TET) with the variation of different parameters (i.e., amplitude and frequency of excitation, clearance, damping, mass ratio and restitution coefficient) have been investigated and well explained around SIM. Experimental results validate the existence of different regimes and different routes to chaos by the variation of the return map of time difference between consecutive impact moments. TET phenomenon has been analyzed for a strongly modulated response, and different cases of TET have been observed and analyzed. It is clearly observed that TET depends not only on whether there exists 1:1 resonance, but also on impulse strength during the transient resonance capture.

https://hal.insa-toulouse.fr/hal-01820034/document

On the dynamics around targeted energy transfer for vibro-impact nonlinear energy sink

An experimental investigation of an impacting system is presented. The dynamical behaviour of the experimental system is compared with that of a simple model system using the common notion of an in ...

Experimental investigation of some consequences of low velocity impacts in the chaotic dynamics of a mechanical system

This paper studies resonance phenomena in a piecewise-smooth dynamical system with external periodic action and examines transitions to chaos via border-collision bifurcations of cycles on a two-dimensional torus. As an example we consider a control system with pulse-width modulation described by a three-dimensional set of piecewise-linear non-autonomous equations. It is shown that the domains of synchronization of quasiperiodic oscillations for piecewise-smooth dynamical systems differ in an essential way from the classical Arnol'd tongues. The difference lies in the inner structure and bifurcational transitions. There are two different kinds of synchronization domains, one of which contains regions of bistability. The structure of border-collision bifurcation boundaries of synchronization tongues and transitions to chaos via border-collision bifurcations of cycles on a two-dimensional torus are described in detail.

C. K. Sung

Papers

The application of ball-type balancers for radial vibration reduction of high-speed optic disk drives

Dynamics of a harmonically excited impact damper: Bifurcations and chaotic motion