Silvio L.T. de Souza

Calculation of Lyapunov exponents in systems with impacts

We apply a model based algorithm for the calculation of the spectrum of the Lyapunov exponents of attractors of mechanical systems with impacts. For that, we introduce the transcendental maps that describe solutions of integrable differential equations, between impacts, supplemented by transition conditions at the instants of impacts. We apply this procedure to an impact oscillator and to an impact-pair system (with periodic and chaotic driving). In order to show the method precision, for large parameters range, we calculate Lyapunov exponents to classify attractors observed in bifurcation diagrams. In addition, we characterize the system dynamics by the largest Lyapunov exponent diagram in the parameter space.

Controlling chaotic orbits in mechanical systems with impacts

Abstract We stabilize desired unstable periodic orbits, embedded in the chaotic invariant sets of mechanical systems with impacts, by applying a small and precise perturbation on an available control parameter. To obtain such perturbation numerically, we introduce a transcendental map (impact map) for the dynamical variables computed just after the impacts. To show how to implement the method, we apply it to an impact oscillator and to an impact-pair system.

Basins of Attraction and Transient Chaos in a Gear-Rattling Model

The authors numerically investigate basins of attraction of coexisting periodic and chaotic attrac tors in a gear-rattling impact model. These attractors are strongly dependent on small changes of the initial conditions. Gradually varying a control parameter, the size of these basins of attraction is modified by global bifurcations of their boundaries. Moreover, the topology of these basins is also modified by appearance or disappearance of coexisting attractors. Furthermore, for the considered control parameter range, the frac tal basin boundaries are so interleaved that trajectories are practically unpredictable in some regions of phase space. The authors also examine an example of a crisis on which a chaotic attractor is converted into a chaotic transient that goes to a periodic attractor. For this crisis, the authors show the evolution of transient lifetime dependence of the initial conditions as the control parameter is varied.

Microcantilever chaotic motion suppression in tapping mode atomic force microscope

The tapping mode is one of the mostly employed techniques in atomic force microscopy due to its accurate imaging quality for a wide variety of surfaces. However, chaotic microcantilever motion impairs the obtention of accurate images from the sample surfaces. In order to investigate the problem the tapping mode atomic force microscope is modeled and chaotic motion is identified for a wide range of the parameters values. Additionally, attempting to prevent the chaotic motion, two control techniques are implemented: the optimal linear feedback control and the time-delayed feedback control. The simulation results show the feasibility of the techniques for chaos control in the atomic force microscopy.

Multistability and Self-Similarity in the Parameter-Space of a Vibro-Impact System

The dynamics of a dissipative vibro-impact system called impact-pair is investigated. This system is similar to Fermi-Ulam accelerator model and consists of an oscillating one-dimensional box containing a point mass moving freely between successive inelastic collisions with the rigid walls of the box. In our numerical simulations, we observed multistable regimes, for which the corresponding basins of attraction present a quite complicated structure with smooth boundary. In addition, we characterize the system in a two-dimensional parameter space by using the largest Lyapunov exponents, identifying self-similar periodic sets.