^{1}

^{2}

^{2}

^{1}

^{2}

Some dynamical properties for a one-dimensional hybrid Fermi-Ulam-bouncer model are studied under the framework of scaling description. The model is described by using a two-dimensional nonlinear area preserving mapping. Our results show that the chaotic regime below the lowest energy invariant spanning curve is scaling invariant and the obtained critical exponents are used to find a universal plot for the second momenta of the average velocity.

The investigation of nonlinear dynamical systems has
awaken special interest along last decades since they can explain and/or
predict some phenomena until then incomprehensible. A special class of systems
that present nonlinear phenomena and that can be described via recursive
equations is the so-called classical nonlinear billiard problems [

It is well known in the literature that the phase
space of billiard problems highly depends on the shape of the boundary. The
dynamics of the particle might generate phase spaces of different kinds that
can be settled in three different classes of universality including (i)
integrable, (ii) ergodic, and (iii) mixed. A typical example of case (i) is the
circle billiard, as the integrability of such a
case resembles the angular momentum conservation. Two examples of case (ii) are
the Bunimovich stadium [

In this paper, we will consider a 1D model that is described using the formalism
of discrete mappings, the so-called hybrid Fermi-Ulam-bouncer model [

Thus, the hybrid Fermi-Ulam-bouncer model consists of
a classical particle which is confined in and bouncing elastically between two
rigid walls in the presence of a constant gravitational field. Thus, properties
that are individually observed in the FUM and bouncer model come together and
coalesce in the hybrid Fermi-Ulam-bouncer model [

This paper is organized as follows. In Section

We discuss in this section all the details needed for
the mapping construction. We also present the phase space and obtain the
positive Lyapunov exponent for the low-energy chaotic sea. The one-dimensional
hybrid Fermi-Ulam-bouncer model thus consists of a classical particle confined
to bounce elastically between two rigid walls. One of the walls is assumed to
be fixed at the position

Finally, for the case (iii), where

Figure

(a) Phase space for the complete hybrid
Fermi-Ulam-bouncer model. (b) Behavior of the average energy,

Figure

In this section, we discuss a simplification used in
the model. For the complete model, the instant of the impact of the particle
with the moving wall is obtained via a solution of a transcendental equation,
which yields the simulations to be long-time consuming. However, instead of
considering solving transcendental equations, we will use a simplification in
the model which is commonly used in the literature [

collision without reflection in the upper
wall. Such a condition is verified if

Collision with reflection in the upper wall.
For such a collision, the condition that must be observed is

The phase space for the mapping (

(a) Phase space generated from iteration of mapping (

The main goal of this section is to describe a scaling
present in the low-energy regime. We discuss in full detail the investigation
for the simplified and then, at the end of section, we present the
corresponding results for the complete version. Thus, we now discuss the
procedures used to obtain the average velocity on the chaotic low-energy
region. The average velocity (average along an orbit) is defined
as

(a) Behavior of the deviation of the average velocity
for different values of the control parameter

when

as the iteration number increases,

the cross-over iteration number that marks the
change from growth to the saturation is written as

After considering these three initial suppositions, we
are now able to describe the deviation of the average velocity in terms of a
scaling function of the type

We begin considering that

Choosing now

Note that the scaling exponents are determined if the
critical exponents

(a) Plot of

(a) Different curves of the

Let us now discuss our numerical results for the
complete version of the model. Once the equations of the mapping now be solved
numerically, we have considered an ensemble of less different initial
conditions. Such a consideration is mainly to produce a simulation not so
longer. However, it is still relevant to characterize statistical properties of
the model. For the complete version of the model, we have considered an
ensemble of

(a) Different
curves of the

As a final
remark of the present paper, we have studied a simplified and the complete
version of the hybrid Fermi-Ulam-bouncer model considering elastic collisions
with the walls. We show that the average energy as well as the deviation around
the average velocity for chaotic orbits for both the complete and simplified
versions of the model exhibit scaling properties with the same critical
exponents. Moreover, we have shown that there is an analytical relation between
the critical exponents

D. F. M. Oliveira and R. A. Bizão are grateful to CNPq; E. D. Leonel thanks FAPESP, FUNDUNESP, and CNPq for financial support.