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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.

In order to study cosmic ray accelerated to high energy, Enrico Fermi proposed an accelerator model as a dynamical system [

In the engineering context, mathematical models describing mechanical impacts similar to those considered in the Fermi-Ulam model have been intensively studied, like gearbox model [

In this paper, we consider a prototypical vibro-impact system, known as impact-pair. This system is comprised of a ball moving between two oscillatory walls. Numerical studies have shown a rich dynamical behavior with several nonlinear phenomena observed, such as bifurcations, chaotic regimes [

This paper is organized as follows. In Section

In this section we present the basic equations of the impact-pair system [

Schematic view of an impact-pair system.

The impact-pair system is composed of a point mass

In an absolute coordinate systems, equation of motion of the point mass

Integrating (

An impact occurs wherever

Therefore, the dynamics of the impact-pair system is obtained from (

Since there is an analytical solution for the motion between impacts, we can obtain an impact map. First, we define the discrete variables

As mentioned before, this map, obtained by considering the analytical solutions and the sequence of impact times, is only used to evaluate the Lyapunov exponents [

The dynamics was investigated using bifurcation diagrams, phase portraits, Lyapunov exponents, basins of attraction, uncertainty exponent, and parameter space diagrams. We fix the control parameters at

In order to obtain a representative example of the kind of dynamics generated by the impact-pair system, we use a bifurcation diagram for the velocity,

In Figure

(a) Bifurcation diagram showing coexisting attractors. Velocity

In Figures

Phase portrait of velocity versus displacement of two coexisting periodic attractors (a) and (b). Time histories of two equilibrium points (c) and (d). For the control parameters

The corresponding basins of attraction of the four possible solutions are depicted in Figure

Basins of attraction and successive magnifications for the parameters

As can be seen, the structure of the basins is quite complex and the basin boundary between periodic attractors is convoluted and apparently fractal (Figures

As a consequence of the complex basins of attraction observed, uncertainty in initial conditions leads to uncertainty in the final state. To evaluate the final state sensitivity we can calculate the uncertainty exponent,

To obtain the uncertainty exponent, we choose randomly a large number of set initial conditions

For the basins of attraction shown in Figure

Uncertain fraction versus uncertainty radius

In order to obtain a further insight into the influence of the restitution coefficient

(a) Parameter space diagram with periodic structures. (b)–(d) Successive magnifications of parameter space diagram.

To clarify how the parameter space diagram was constructed, we fix the restitution coefficient at

Bifurcation diagram of velocity

On examining the parameter space diagram (Figure

To finalize, in Figures

(a) Parameter space diagram with periodic structures. (b) Magnification of a portion of the previous figure.

In this paper, we considered the impact-pair system studying its dynamics by a means of numerical simulations. Initially, we discussed the coexistence of attractors with smooth basin boundary, but with complicated and evoluted basins structure. According to the uncertainty exponent evaluated for a certain region in phase space, this type basins of attraction is associated with effect of obstruction on predictability of time-asymptotic final state (attractor). In other words, the uncertainty in initial conditions leads to uncertainty in the final state.

In the end, we explored the dynamics in the two-dimensional parameter space by using the largest Lyapunov exponents. We identified several periodic sets embedded in chaotic region. Some of them, known as shrimps, present self-similar structures and organize themselves along a very specific direction in parameter space with a series of accumulations. In addition, the periodic sets cross each other indicating a large quantity of coexisting periodic attractors.

This work was made possible by partial financial support from the following Brazilian government agencies: FAPESP and CNPq.