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Dynamic behaviors of a particle (or a bouncing ball) in a generalized Fermi-acceleration oscillator are investigated. The motion switching of a particle in the Fermi-oscillator causes the complexity and unpredictability of motion. Thus, the mechanism of motion switching of a particle in such a generalized Fermi-oscillator is studied through the theory of discontinuous dynamical systems, and the corresponding analytical conditions for the motion switching are developed. From solutions of linear systems in subdomains, four generic mappings are introduced, and mapping structures for periodic motions can be constructed. Thus, periodic motions in the Fermi-acceleration oscillator are predicted analytically, and the corresponding local stability and bifurcations are also discussed. From the analytical prediction, parameter maps of periodic and chaotic motions are achieved for a global view of motion behaviors in the Fermi-acceleration oscillator. Numerical simulations are carried out for illustrations of periodic and chaotic motions in such an oscillator. In existing results, motion switching in the Fermi-acceleration oscillator is not considered. The motion switching for many motion states of the Fermi-acceleration oscillator is presented for the first time. This methodology will provide a useful way to determine dynamical behaviors in the Fermi-acceleration oscillator.

The Fermi-acceleration oscillator was first presented [

The similar studies of impacting systems have also
been carried out in mechanical engineering because the impact is an important
phenomenon in mechanical engineering. For instance, in 1982, Holmes [

In 2005, Luo [

This paper will investigate dynamics of a generalized Fermi-acceleration oscillator. The domains and boundaries for such a problem will be introduced because of the discontinuity, and the analytical conditions of stick and grazing motions will be developed at the boundaries. The mapping technique will be used for the analytical prediction of periodic motions of the extended Fermi-acceleration oscillator. The local stability and bifurcation of periodic motions will be discussed using the eigenvalue analysis. Bifurcation scenario and analytical prediction of motions will be presented, and numerical simulations of periodic and chaotic motions will be carried out. In addition, the Poincare mapping sections will be presented for illustrations of chaotic motions, and the parameter maps will be presented as well.

A generalized Fermi-acceleration oscillator consists of a particle moving vertically between a
fixed wall and the moving piston in a vibrating oscillator. The piston of mass

Mechanical model.

If the particle does not stay on
the piston, the corresponding motion of such a system is called the

If the particle stays on the piston and they move
together, such a motion is called

For nonstick motion, the impact relation between the particle
and the fixed wall is

In this section, the domains and boundaries of the extended Fermi-oscillator will be introduced in the absolute and relative coordinates. From such domains and boundaries, the analytical conditions of stick and grazing motions to each switching boundary will be developed from the theory of discontinuous dynamical systems.

To analyze the motion discontinuity in the Fermi-acceleration oscillator, the origin of the absolute coordinates is set at the equilibrium of piston. The domains for the particle and piston without stick in the absolute coordinates are defined as

Absolute domains and boundaries without stick: (a) particle and (b) piston.

For this system, there is a stick motion of the particle and piston. Thus, the stick motion will appear and vanish under certain conditions. Such
onset and disappearance of the sticking motion will generate new boundaries and
domains. The domains

Absolute domains and boundaries with stick: (a) particle and (b) piston.

From the aforedefined domains, the vectors for absolute
motions are defined as

Because the switching boundary varies with time, it is very
difficult to develop the switching conditions. Thus, the relative coordinates
for such a Fermi-oscillator are adopted herein. The relative displacement,
relative velocity, and relative acceleration between the particle and the piston are

Relative domains and boundaries definition: (a)

The vectors for relative motions are

The analytical conditions of stick and grazing motion will
be developed from the theory of non-smooth dynamical systems in Luo [

Based on the

In a similar fashion, the grazing
conditions for the boundary of the stick motion are

To geometrically explain the above
analytical conditions, consider a flow of the motion approaching the stick
boundary

(a) Passable motion and (b) grazing motion.

In this section, the switching sets and mapping structures will be introduced to symbolically describe motions in the generalized Fermi-acceleration oscillator. The switching sets will be defined from the switching boundaries. From the switching sets, the generic mappings in domains will be introduced. The mapping structure for periodic and chaotic motions will be constructed from such generic mappings. The stability and bifurcation conditions will also be discussed via the eigenvalue analysis.

Based on the switching boundaries in (

Switching sets and generic mappings for nonstick motion (in absolute coordinates): (a) particle and (b) piston.

Similarly, from the switching boundaries in (

Switching sets and generic mappings for stick motion: (a) particle and (b) piston.

From the above definitions, a mapping will
map a switching set into another switching set (or itself) through the corresponding
dynamical system in a specified domain. In such a domain, the dynamical system
given in (

The notation for mapping action is introduced as

Once the
grazing bifurcation of the periodic motion of

Without stick,
there are two types of motions: (i) impact only at the boundary

Mapping structure for motions: (a)

Consider a map

The
mapping structure of a simple periodic motion of

In this section, bifurcation scenario for the generalized Fermi-acceleration oscillator will be presented first, and analytical predictions of periodic motions will be completed through the mapping structures. Periodic and chaotic motions will be presented for illustration of the analytical conditions. Poincare mapping sections will be given for illustrations of chaotic motions in the Fermi-acceleration oscillator. The parameter maps for certain parameters will also be presented for an overview of dynamical behaviors of particle and piston in the generalized Fermi-acceleration oscillator.

To
obtain a bifurcation scenario in the generalized Fermi-acceleration oscillator,
the closed-form solutions of particle and piston in the appendix are used for numerical
simulations. To achieve a motion with stick, the stick conditions in (

Consider a set of
parameters (

Summary of driving frequency for periodic motions (

Mapping | Excitation | Mapping | Excitation | ||||
---|---|---|---|---|---|---|---|

Structure | Frequency | Structure | |||||

1 | P(2 | (6.33, 6.56) | 16 | P(4 | (11.01, 11.05) (11.35, 11.40) | ||

2 | P( | (6.56, 7.01) | 17 | P(2 | (11.05, 11.16) (11.40, 11.52) | ||

3 | P(2 | (7.41, 7.43) | 18 | P(8 | (11.75, 11.76) | ||

4 | P(3 | (7.60, 7.80) (7.91, 7.93) | 19 | P(4 | (11.76, 11.77) | ||

5 | P(6 | (7.87, 7.89) | 20 | P(4 | (11.77, 11.79) | ||

6 | P(3 | (8.05, 8.07) (8.18, 8.20) | 21 | P(8 | (11.79, 11.80) | ||

7 | P(6 | (8.27, 8.29) | 22 | P(4 | (19.15, 19.40) | ||

8 | P(6 | (8.62, 8.64) | 23 | P(2 | (19.40, 21.25) | ||

9 | P(8 | (8.76, 8.78) | 24 | P( | (21.25, 35.20) | ||

10 | P(4 | (8.94, 8.96) | 25 | P(8 | (56.42, 56.92) | ||

11 | P(8 | (9.06, 9.07) | 26 | P(4 | (56.92, 59.35) | ||

12 | P(4 | (9.07, 9.09) (9.81, 9.83) | 27 | P(2 | (59.35, 70.23) | ||

14 | P(6 | (10.42, 10.44) | 28 | Chaos & Complex Motion | — | (7.01, 19.15) (19.15, 35.20) (70.23, 80.00) | |

15 | P(8 | (11.00, 11.01) (11.34, 11.35) |

Bifurcation scenario varying with driving
frequency: (a) switching displacement
and (b) switching velocity of the particle; (c) switching displacement and (d)
switching velocity of the piston; (e) Switching phase (

The analytical prediction of
periodic motions is based on the corresponding mapping structures presented in
Section

Consider

Analytical prediction of periodic motion for

Eigenvalues varying with excitation frequency for

The methodology presented in Section

Analytical prediction of periodic motion relative to mapping structure

Eigenvalues varying with excitation frequency for

For a better
understanding of the switching dynamics of the Fermi-acceleration oscillator, a
parameter map for periodic motions and chaos should be developed from the
analytical prediction, and the corresponding local stability and bifurcation
conditions with the grazing and stick conditions will be adopted. The parameter map about excitation amplitude
and frequency is presented in Figure

Parameter maps: (a) overall view, (b)
zoomed view, (

From the previous section, periodic motions were analytically predicted. It is significant to illustrate motions for a better understanding of the dynamics of the Fermi-acceleration oscillator. To illustrate motions in the generalized Fermi-acceleration oscillator, the closed-form solutions of particle and piston in the appendix are used again for numerical simulations. As in the section of bifurcation scenario, the corresponding conditions for motion switching should be embedded in the computer program. The time histories of displacement and velocity of both the particle and piston will be presented for periodic motions. In addition, phase planes of both particle and piston will be illustrated. For chaotic motions, the Poincare mapping sections consisting of the switching points on the switching boundaries will be presented. In numerical illustrations, the switching points are labeled by circular symbols. The large and small circular symbols are for the particle and piston in the Fermi-acceleration oscillator, respectively.

Consider
a set of parameters (

Periodic motion of

As discussed before, there are two
kinds of motions for motion switching: (i) stick motion and nonstick motion. In other words, the nonstick motion means that the particle does not stay on
the piston, and the stick motion means that the particle stays on the piston
and moves together. In Figure

Initial conditions for periodic motions (

Mapping | Initial time | |||
---|---|---|---|---|

Periodic
motion of

Periodic
motion of

Phase
planes. Solid and dashed curves represent motions of particle and piston, respectively,
(

Finally, a chaotic motion relative
to

Chaotic motion (

Poincare mapping sections of chaotic motion (

In this paper, the mechanism of motion switching of a particle in such a generalized Fermi-oscillator was studied through the theory of discontinuous dynamical systems, and the corresponding analytical conditions for the motion switching were developed. From the solutions of linear systems in each domain, the generic mappings are introduced. Further, the mapping structures for periodic motions were developed, and such periodic motions in the Fermi-acceleration oscillator were predicted analytically. The corresponding local stability and bifurcation are carried out. From the analytical prediction, parameter maps of regular and chaotic motions were achieved for a global view of motions in the Fermi-acceleration oscillator. Illustrations of periodic and chaotic motions in such an oscillator were done. This methodology will provide a useful way to determine dynamical behaviors in the Fermi-acceleration oscillator.

Equation of motion for the particle in the generalized
Fermi-oscillator is

The equation of motion for piston and stick motion in the
extended Fermi-oscillator is