Alexander Loskutov

Edson D. Leonel

Time-Dependent Billiards

Some dynamical properties for a bouncer model—a classical particle of mass <svg style="vertical-align:-0.1638pt;width:13.5375px;" id="M1" height="7.9499998" version="1.1" viewBox="0 0 13.5375 7.9499998" width="13.5375" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D45A" d="M766 88q-40 -45 -83.5 -72.5t-63.5 -27.5q-39 0 -16 103l49 224q15 68 -10 68q-36 0 -112.5 -78.5t-105.5 -133.5q-27 -101 -38 -165q-35 -4 -77 -18l-6 6q42 153 70 288q12 55 10 78t-19 23q-38 0 -114 -80.5t-101 -131.5q-24 -68 -41 -165q-36 -3 -76 -18l-8 6&#xA;q56 202 87 347q7 33 -3 33q-8 0 -31.5 -17.5t-41.5 -35.5l-12 28q42 45 84 72t63 27q44 0 10 -124l-20 -75h2q92 121 193 178q36 21 64 21q62 0 28 -159l-8 -37h2q50 67 103.5 112t94.5 65q39 19 62 19q60 0 23 -156l-45 -189q-9 -39 2 -39q17 0 71 49z" /></g></svg>
 falling in the
presence of a constant gravitational field <svg style="vertical-align:-3.56265pt;width:9.8000002px;" id="M2" height="12.175" version="1.1" viewBox="0 0 9.8000002 12.175" width="9.8000002" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D454" d="M546 430q-14 -33 -37.5 -140t-38.5 -209q-29 -195 -156 -289q-29 -21 -70 -37t-78 -16q-62 0 -102.5 37t-40.5 69q0 27 26 48q19 15 25 -3q13 -43 46.5 -69t87.5 -26q55 0 83 21q29 22 50 65t43 140q14 64 29 163h-2q-55 -81 -149 -152q-59 -44 -102 -44q-23 0 -43.5 30&#xA;t-20.5 85q0 74 32.5 145.5t84.5 117.5q40 35 100 58.5t117 23.5q21 0 65 -8q30 -6 44 -6zM456 386q-40 16 -81 16q-29 0 -64 -13q-42 -17 -85 -97q-19 -36 -31.5 -86t-12.5 -86q0 -64 30 -64q20 0 52.5 21t60.5 52q74 81 95 123q26 62 36 134z" /></g></svg>
 and hitting elastically a periodically moving wall—in the
presence of drag force that is assumed to be proportional to the particle's velocity are studied. The
dynamics of the model is described in terms of a two-dimensional nonlinear mapping obtained via
solution of the second Newton's law of motion. We characterize the behavior of the average velocity
of the particle as function of the control parameters as well as the time. Our results show that
the average velocity starts growing at first and then bends towards a regime of constant value, thus
confirming that the introduction of drag force is a sufficient condition to suppress Fermi acceleration
in the model.

Francys Andrews de Souza

Lucas Eduardo Azevedo Simões

Mário Roberto da Silva

Can Drag Force Suppress Fermi Acceleration in a Bouncer Model?

Cold Atoms bouncing on modulated atomic mirror exhibits acceleration and dynamical localization subject to modulated strength. We explain characteristics of acceleration and define control
parameters in terms of effective Planck's constant. We show that the effective Planck's constant plays a vital  role in limiting
classical-like overall linear growth of the variance of accelerated atoms with time. For large values
of the effective Planck's constant the atomic quantum acceleration is seized as localization window
overlaps the accelerated window.

Javed Akram

Khalid Naseer

Inam-ur Rehman

Farhan Saif

Acceleration of Material Waves in Fermi Accelerator

We investigate a time-dependent circular billiard with a two-frequency driving function and derive a new simplified form for the map, which is a symplectic nontwist map. Stability boundaries
and reconnection thresholds are derived for fixed points and period-two vortex pairs. An island
interspersal condition is derived such that neighboring island chains of the first frequency are exactly
separated by those of the second. The results show a strong enhancement of the KAM barrier to
chaotic diffusion with increasing second frequency content and overall amplitude.

J. E. Howard

Time-Dependent Circular Billiard

A phase transition from integrability to nonintegrability in two-dimensional Hamiltonian mappings is described and characterized in terms of scaling arguments. The mappings considered produce a mixed structure in the phase space in the sense that, depending on the combination of the
control parameters and initial conditions, KAM islands which are surrounded by chaotic seas that
are limited by invariant tori are observed. Some dynamical properties for the largest component of
the chaotic sea are obtained and described in terms of the control parameters. The average value
and the deviation of the average value for chaotic components of a dynamical variable are described
in terms of scaling laws, therefore critical exponents characterizing a scaling function that describes
a phase transition are obtained and then classes of universality are characterized. The three models
considered are: The Fermi-Ulam accelerator model, a periodically corrugate waveguide, and variant of
the standard nontwist map.

Phase Transition in Dynamical Systems: Defining Classes of Universality for Two-Dimensional Hamiltonian Mappings via Critical Exponents

Ac-driven asymmetric pulses 
                        can be used to control the Fermi 
                        acceleration between three different 
motions, <svg style="vertical-align:-0.0pt;width:11.875px;" id="M1" height="11.425" version="1.1" viewBox="0 0 11.875 11.425" width="11.875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.363)"><path id="x41" d="M673 0h-245v28q47 6 58 16t1 39l-55 154h-213q-31 -87 -36 -129q-16 -47 -5 -61t68 -19v-28h-226v28q50 7 68.5 23.5t43.5 80.5l214 524l24 9l193 -536q23 -63 41.5 -79.5t68.5 -21.5v-28zM418 280l-81 255h-2q-74 -197 -101 -255h184z" /></g></svg>: the <i>accelerated</i> mode, <svg style="vertical-align:-0.0pt;width:12.6125px;" id="M2" height="11.175" version="1.1" viewBox="0 0 12.6125 11.175" width="12.6125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x44" d="M43 650h280q194 0 282 -85q85 -82 85 -215q0 -173 -123 -267q-107 -83 -297 -83h-236v27q65 5 79.5 19.5t14.5 77.5v402q0 45 -6 62.5t-21.5 24t-57.5 9.5v28zM213 547v-416q0 -55 24.5 -75t81.5 -20q133 0 203 86t70 214q0 80 -30 143t-88 97q-66 41 -170 41&#xA;q-62 0 -78 -13q-13 -9 -13 -57z" /></g></svg>: the <i>decelerated</i> mode, and <svg style="vertical-align:-0.0pt;width:13.15px;" id="M3" height="11.175" version="1.1" viewBox="0 0 13.15 11.175" width="13.15" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x48" d="M729 650v-28q-62 -5 -76 -19.5t-14 -75.5v-404q0 -62 13.5 -76t75.5 -19v-28h-268v28q65 5 79.5 19t14.5 76v199h-341v-199q0 -62 14 -76t76 -19v-28h-263v28q61 5 74.5 19t13.5 76v404q0 62 -14 76t-76 19v28h259v-28q-58 -5 -71 -19t-13 -76v-162h341v162&#xA;q0 61 -13.5 75.5t-72.5 19.5v28h261z" /></g></svg>: the <i>hyperaccelerated</i> mode. In this paper we show that dissipation strongly affects the particles velocity, reducing the possibility for an accurate control of the dynamics. The saturation time, where the mean velocity starts to be constant due to dissipation, decays with a power law <svg style="vertical-align:-3.56265pt;width:37.862499px;" id="M4" height="21.1875" version="1.1" viewBox="0 0 37.862499 21.1875" width="37.862499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,16.7)"><path id="x223C" d="M548 311q-21 -57 -59 -94t-89 -37q-53 0 -128 52q-55 38 -84 38q-39 0 -70.5 -23t-45.5 -60l-32 12q21 57 59 94t89 37q53 0 128 -52q55 -38 84 -38q39 0 70.5 23t45.5 60z" /></g><g transform="matrix(.017,-0,0,-.017,14.749,16.7)"><path id="x1D6FE" d="M478 372q0 -39 -31 -97t-61 -98t-78 -98q-45 -55 -73 -102q-13 -79 -13 -197q-11 -11 -43.5 -25.5t-53.5 -15.5l-15 17q5 35 26 101.5t47 123.5q8 72 -1.5 174.5t-37.5 178.5q-14 37 -29 37q-20 0 -67 -65l-25 21q37 60 73 90.5t63 30.5q47 0 72 -112q13 -56 17.5 -141.5&#xA;t0.5 -143.5h2q155 193 155 297q0 26 -12 47q-5 8 -5 15q0 16 12.5 27t29.5 11q21 0 34 -21t13 -55z" /></g><g transform="matrix(.012,-0,0,-.012,23.275,8.537)"><path id="x2212" d="M535 230h-483v50h483v-50z" /></g><g transform="matrix(.012,-0,0,-.012,30.26,8.537)"><path id="x1D6FD" d="M558 587q0 -32 -14 -61t-40 -53.5t-48.5 -41t-54.5 -36.5q144 -51 144 -174q0 -55 -43.5 -108t-104.5 -87q-77 -42 -131 -42q-31 0 -54 20t-31 47l11 18q48 -29 108 -29q79 0 119.5 43t40.5 109t-44.5 107.5t-119.5 50.5l22 47q34 1 65 21q96 61 96 157q0 42 -24 67.5&#xA;t-62 25.5q-24 0 -43.5 -9t-35 -29.5t-27 -44t-22.5 -63t-19.5 -75.5t-18.5 -91q-57 -294 -68 -380q-26 -190 -35 -200q-26 -31 -97 -37l-4 26q19 9 48 170l77 413q23 121 52.5 187.5t83.5 114.5q70 62 148 62q51 0 88.5 -34t37.5 -91z" /></g></svg>, where <svg style="vertical-align:-3.56265pt;width:8.6374998px;" id="M5" height="12.175" version="1.1" viewBox="0 0 8.6374998 12.175" width="8.6374998" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D6FE" /></g></svg> is the dissipation parameter and <svg style="vertical-align:-3.56265pt;width:10px;" id="M6" height="16.6625" version="1.1" viewBox="0 0 10 16.6625" width="10" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D6FD" /></g></svg>
 is close to 1. The value of the saturated mean velocity also decays with a power law with exponent <svg style="vertical-align:-3.56265pt;width:24.700001px;" id="M7" height="16.6625" version="1.1" viewBox="0 0 24.700001 16.6625" width="24.700001" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D6FD" /></g><g transform="matrix(.017,-0,0,-.017,14.647,12.162)"><use xlink:href="#x223C" /></g></svg>
 0.6 for case <svg style="vertical-align:-0.0pt;width:13.15px;" id="M8" height="11.175" version="1.1" viewBox="0 0 13.15 11.175" width="13.15" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x48" /></g></svg>, and <svg style="vertical-align:-3.56265pt;width:24.700001px;" id="M9" height="16.6625" version="1.1" viewBox="0 0 24.700001 16.6625" width="24.700001" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D6FD" /></g><g transform="matrix(.017,-0,0,-.017,14.647,12.162)"><use xlink:href="#x223C" /></g></svg> 0.3 for case <svg style="vertical-align:-0.0pt;width:11.875px;" id="M10" height="11.425" version="1.1" viewBox="0 0 11.875 11.425" width="11.875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.363)"><use xlink:href="#x41" /></g></svg>. In case <svg style="vertical-align:-0.0pt;width:12.6125px;" id="M11" height="11.175" version="1.1" viewBox="0 0 12.6125 11.175" width="12.6125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x44" /></g></svg> this velocity is almost constant for small dissipations.

Cesar Manchein

Marcus W. Beims

Dissipation Effects in the Ratchetlike Fermi Acceleration

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.

Silvio L. T. de Souza

Iberê L. Caldas

Ricardo L. Viana

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

Let <svg     style="vertical-align:-3.27605pt;width:21.112499px;" id="M1" height="15.25" version="1.1" viewBox="0 0 21.112499 15.25" width="21.112499"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
    
        
            <g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path id="x1D448" d="M751 650l-6 -28q-53 -1 -69 -12.5t-24 -38.5q-23 -85 -43 -192l-13 -69q-34 -181 -130 -262q-79 -65 -185 -65t-171 57q-87 77 -54 258l42 232q12 58 0.5 73t-75.5 19l8 28h263l-6 -28q-63 -5 -78 -18t-26 -74l-41 -226q-25 -135 22 -201t139 -66q99 0 161.5 68t91.5 217&#xA;l11 57q28 149 28 193q0 28 -19 38t-79 12l6 28h247z" /></g>

            <g transform="matrix(.012,-0,0,-.012,13.038,15.187)"><path id="x1D439" d="M599 650l-18 -160l-30 -2q-2 70 -11 94q-5 16 -26 24t-75 8h-94q-27 0 -33.5 -6.5t-12.5 -32.5l-43 -226h108q52 0 72.5 5t31 18t28.5 52h28l-40 -200h-28q-1 42 -6 56.5t-24.5 21t-70.5 6.5h-108l-32 -177q-13 -65 1 -80.5t88 -22.5l-8 -28h-279l6 28q61 5 77.5 20.5&#xA;t29.5 81.5l72 387l7.5 40.5t1 26.5t-4 18t-15.5 10t-24 6t-37 4l8 28h461z" /></g>

        
    
</svg> be the Floquet operator of a time periodic
Hamiltonian <svg     style="vertical-align:-2.3205pt;width:32.587502px;" id="M2" height="15.0875" version="1.1" viewBox="0 0 32.587502 15.0875" width="32.587502"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
    
        
            <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x1D43B" d="M865 650q-1 -4 -4 -14t-4 -14q-62 -5 -77 -19.5t-29 -82.5l-74 -394q-12 -61 -0.5 -77t75.5 -21l-6 -28h-273l8 28q64 5 82 21t29 76l36 198h-380l-37 -197q-11 -64 0.5 -78.5t79.5 -19.5l-6 -28h-268l6 28q60 6 75.5 21.5t26.5 76.5l75 394q13 66 2 81.5t-77 20.5l8 28&#xA;h263l-6 -28q-58 -5 -75.5 -21t-30.5 -81l-26 -153h377l29 153q12 67 2 81t-74 21l5 28h268z" /></g><g transform="matrix(.017,-0,0,-.017,14.97,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,20.852,12.138)"><path id="x1D461" d="M324 430l-26 -36l-112 -4l-55 -265q-13 -66 7 -66q13 0 44.5 20t50.5 40l17 -24q-38 -40 -85.5 -73.5t-87.5 -33.5q-50 0 -21 138l55 262h-80l-2 8l25 34h66l25 99l78 63l10 -9l-37 -153h128z" /></g><g transform="matrix(.017,-0,0,-.017,26.649,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g>

        
    
</svg>. 
 For each positive  and discrete observable <svg     style="vertical-align:-0.0pt;width:12.5875px;" id="M3" height="11.4625" version="1.1" viewBox="0 0 12.5875 11.4625" width="12.5875"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
    
        
            <g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path id="x1D434" d="M716 28l-8 -28h-254l8 28q55 5 66 17.5t5 47.5l-30 145h-232l-74 -138q-21 -43 -11 -54.5t72 -17.5l-6 -28h-235l8 28q53 7 73.5 21t53.5 70l319 540l33 8q6 -43 33 -176l80 -381q10 -49 26.5 -63t72.5 -19zM495 281l-49 264h-2l-149 -264h200z" /></g>

        
    
</svg> (which we call a <i>probe energy</i>), we derive  a formula for the Laplace time average
of its expectation value up to time <svg     style="vertical-align:-0.0pt;width:11.45px;" id="M4" height="11.6125" version="1.1" viewBox="0 0 11.45 11.6125" width="11.45"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
    
        
            <g transform="matrix(.017,-0,0,-.017,.062,11.55)"><path id="x1D447" d="M649 676l-22 -187l-33 -2q3 56 -12 94q-8 20 -31.5 26.5t-86.5 6.5h-74l-90 -491q-4 -23 -6 -36.5t1 -25t7 -16.5t18 -9t27 -5t41 -3l-6 -28h-286l4 28q68 5 84 18.5t28 76.5l94 491h-55q-74 0 -100.5 -6t-41.5 -23q-24 -29 -54 -98l-32 1q32 98 53 188h22q7 -18 15 -22&#xA;t37 -4h417q23 0 33.5 5t25.5 21h23z" /></g>

        
    
</svg> in terms of its eigenvalues and Green functions at the circle of radius <svg     style="vertical-align:-0.1638pt;width:26.325001px;" id="M5" height="16.85" version="1.1" viewBox="0 0 26.325001 16.85" width="26.325001"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
    
        
            <g transform="matrix(.017,-0,0,-.017,.062,16.588)"><path id="x1D452" d="M391 364q0 -28 -20.5 -53.5t-50 -43.5t-70 -34t-73.5 -25t-68 -17v-29q0 -48 20 -81t65 -33q73 0 157 76l16 -23q-107 -113 -221 -113q-21 0 -40.5 6.5t-39 22t-31.5 47t-12 75.5q0 69 33.5 139.5t95.5 114.5q77 55 143 55q42 0 69 -24.5t27 -59.5zM313 350&#xA;q0 26 -14.5 40.5t-37.5 14.5q-32 0 -44 -7q-82 -46 -104 -175q200 49 200 127z" /></g>

            <g transform="matrix(.012,-0,0,-.012,7.1,8.425)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g><g transform="matrix(.012,-0,0,-.012,12.811,8.425)"><path id="x2F" d="M368 703l-264 -866h-60l265 866h59z" /></g><g transform="matrix(.012,-0,0,-.012,17.714,8.425)"><use xlink:href="#x1D447"/></g>

        
    
</svg>. Some simple applications are provided which support its usefulness.

César R. de Oliveira

Mariza S. Simsen

Quantum Energy Expectation in Periodic Time-Dependent Hamiltonians via Green Functions

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.

A. C. J. Luo

Y. Guo

Motion Switching and Chaos of a Particle in a Generalized Fermi-Acceleration Oscillator

We present results from computational experiments of one-dimensional inhomogeneous granular
column. With characteristic properties lying between those of liquids and solids, this class of material
consists of assemblies of solid particles which interact via mechanical forces (contact and friction)
and are maintained together by a gravitational field. The system is composed of a vertical column
of spherical beads driven by a sinusoidally vibrating plate. Results for 10 bead experiments show
the fluidization and condensation phenomena, both dependent on the amplitude and frequency of
the driving plate. Some scaling properties arising from the dynamics of the beads are also shown.

Marcus V. Carneiro

Joaquim J. Barroso

Elbert E. N. Macau

Simulation of Inhomogeneous Columns of  Beads under Vertical Vibration

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Mathematical Problems in Engineering

Time-Dependent Billiards

Time-Dependent Billiards

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