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vibro - impacting systems belong to a very interesting and important class of nonsmooth and nonlinear dynamical systems dibernardo2008,luo2006,awrejcewicz2003,filippov1988  with important technological applications stronge2000,mehta1994,knudsen1992,wiercigroch2008 .
dynamics of such systems can be extremely complicated due to velocity discontinuity arising upon impacts .
a very characteristic feature of such systems is the presence of nonstandard bifurcations such as border - collisions and grazing impacts which often lead to complex chaotic motions .
the poincar map , describing evolution from an impact to the next impact , is a natural tool to study vibro - impacting systems .
the main difficulty with investigating impacting systems is in finding instant of the next impact what typically involves solving a nonlinear equation .
however , the problem can be simplified in the case of a bouncing ball dynamics assuming a special motion of the limiter . in the present paper
we investigate motion of a material point in a gravitational field colliding with a limiter moving with piecewise constant velocity .
this class of models has been extensively studied , see @xcite  and references therein . as a motivation that inspired this work ,
we mention study of physics and transport of granular matter @xcite .
a similar model has been also used to describe the motion of railway bogies @xcite .
therefore it can be expected that some of the present results may cast light on the dynamics in such systems . on the other hand , simple motion of the limiter
makes analytical explorations possible , cf . our preliminary report okninski2009 .
the paper is organized as follows . in section 2 a one dimensional dynamics of a ball moving in a gravitational field and colliding with a table is considered and poincar map is described for piecewise linear motion of the table . in section 3
transition to chaotic dynamics from periodic motion is described .
the nature of mixing leading to chaotic dynamics is described in section 4 . in sections 5 and 6 homoclinic structures responsible for mixing are determined and computed .
finally , we discuss our results in the last section .
we consider a motion of a small ball moving vertically in a gravitational field and colliding with a moving table , representing unilateral constraints .
the ball is treated as a material point while the limiter s mass is assumed so large that its motion is not affected at impacts .
a motion of the ball between impacts is described by the newton s law of motion:@xmath0where @xmath1 and motion of the limiter is:@xmath2with a known function @xmath3 .
we shall also assume that @xmath3 is a continuous function of time .
impacts are modeled as follows : @xmath4where duration of an impact is neglected with respect to time of motion between impacts . in eqs .
( [ position ] ) , ( [ velocity ] ) @xmath5 stands for time of the @xmath6-th impact while @xmath7 , @xmath8are left - sided and right - sided limits of @xmath9 for @xmath10 , respectively , and @xmath11 is the coefficient of restitution , @xmath12 @xcite .    solving eq .
( [ point motion ] ) and applying impact conditions ( position ) , ( [ velocity ] ) we derive the poincar map @xcite : @xmath13where @xmath14 . the limiter s motion has been typically assumed in form @xmath15 , cf . @xcite and references therein .
this choice leads to serious difficulties in solving the first of eqs.([yv ] ) for @xmath16 , thus making analytical investigations of dynamics hardly possible .
accordingly , we have decided to simplify the limiter s periodic motion to make ( [ yv ] ) solvable .
let us thus assume that the table moves up with a finite constant velocity @xmath17 and then goes down with a finite constant velocity @xmath18 aobr2009 .
therefore , displacement of the table is the following periodic function of time : @xmath19with @xmath20 , @xmath21 , @xmath22 , where @xmath23 is the floor function  the largest integer less than or equal to @xmath24 .
our model consists thus of equations ( [ yv ] ) , ( [ displacement ] ) with control parameters @xmath11 , @xmath25 , @xmath26 . since the period of motion of the limiter is equal to one , the map ( [ yv ] ) is invariant under the translation @xmath27 .
accordingly , all impact times @xmath28 can be reduced to the unit interval @xmath29 $ ] .
in our recent article periodic solutions of eqs .
( [ yv ] ) , ( displacement ) have been investigated @xcite .
dynamics of eqs .
( [ yv ] ) , ( [ displacement ] ) becomes complicated when some impacts occur in time interval @xmath30 , and some in @xmath31 .    in the case of two impacts per period a @xmath32 - cycle ,
@xmath33 , @xmath34 , is stable .
next , for increasing values of @xmath25 , period doubling takes place and @xmath35 - cycle with impacts @xmath36 , @xmath37 is formed .
then , upon further increase of @xmath25 , the period doubling scenario ends abruptly when @xmath38 . for @xmath39
this happens for @xmath40 .
this critical transition is investigated in sections 4 , 5 .
equations for dynamics after the first period doubling have the following form :    @xmath41    @xmath42{fig1.eps}\ ] ]    in fig .
1 transition to chaos is shown .
the initial dynamical state with two impacts per period bifurcates at @xmath43 ( this value can be computed analytically , see @xcite ) . for @xmath44 time of the second impact tends to @xmath45 and this mode of dynamics is impossible for @xmath46 .
it turns out that for @xmath47 there are two attractors : a noisy , probably chaotic , attractor coexisting with a @xmath48cycle which appears just before the transition . at @xmath49 the noisy
attractor disappears and is substituted by a more irregular attractor , see figs .
full circles indicate positions of small clouds of points . we have studied these potentially chaotic attractors in detail .
first of all , we have checked numerically that the attractors are non  periodic .
indeed , computations show that after @xmath50 iterations the points generated by the map ( [ yv ] ) , ( [ displacement ] ) stay on the corresponding attractor and do not repeat .
we have also computed lyapunov exponents for both attractors . in the case of the attractor shown in fig .
2 the lyapunov exponent is @xmath51 while for the attractor in fig . 3 @xmath52 .
it follows that in both cases dynamics is chaotic and is more mixing in the second case .
the mechanism of mixing is explained in the next section .
@xmath53{fig2.eps}\ ] ]    @xmath53{fig3.eps}\ ] ]
mixing can arise due to corner events @xcite when impacts occur at points where motion of the limiter loses smoothness at time instances @xmath54 , @xmath55 .
let us investigate the second possibility more closely . in fig .
4 the stable @xmath56 - cycle with four impacts per two periods : @xmath36 , @xmath37 and unstable @xmath32 - cycle are shown schematically .
@xmath53{fig4.eps}\ ] ]    for increasing value of the control parameter @xmath25 we have @xmath57 , see fig .
the map ( [ yv ] ) , ( [ displacement ] ) is invariant under translation @xmath58 and thus the phase space is topologically equivalent to the cylinder and hence we have to glue the end points of the time interval @xmath59   $ ] obtaining thus a circle .
therefore , a small neighborhood of @xmath60 is a union of two sets , @xmath61 , where @xmath62 are small and positive , see fig .
@xmath53{fig5.eps}\ ] ]    now , let @xmath63 .
it follows from eqs .
( [ yv ] ) , ( [ displacement ] ) that time of the next impact , @xmath16 , as well as the corresponding post  impact velocity , @xmath64 , depend discontinuously on @xmath28 . in other words
, we get different solutions , @xmath65 , depending on whether @xmath66 or @xmath67 .
it follows that mixing will necessarily be present if a trajectory recurrently visits the interval @xmath68 .
we shall study this possibility in the next section .
let us consider critical case : @xmath69 , @xmath70 .
this is described by eq .
( equations ) with four impacts per two periods such that @xmath71 , @xmath72:@xmath73where we have substituted @xmath74 and the periodicity conditions : @xmath75 , @xmath76 .
solutions of eq .
( [ hom1 ] ) depend on roots of the following algebraic equation:@xmath77where @xmath78 with coefficients @xmath79 depending on @xmath11 , @xmath26 in a complicated way .
the coefficients are listed below:@xmath80@xmath81:@xmath82@xmath83:@xmath84@xmath85:@xmath86@xmath87:@xmath88@xmath89:@xmath90@xmath91:@xmath92    let us stress here that acceptable solution for the time of the first impact must fulfill consistency condition @xmath93 .
now it follows that necessary conditions for existence of this solution can be formulated .
indeed , the condition @xmath94 guarantees existence of solution @xmath95 . furthermore ,
after change of variable @xmath96 the equation ( [ poly ] ) is written as @xmath97 and the condition @xmath98 guarantees existence of the solution @xmath99 and hence existence of solution @xmath100 .
region of acceptable values of parameters @xmath11 , @xmath101 is shown in fig .
6 - it is placed between thin solid lines ( which correspond to the condition @xmath102 ) and below medium solid line ( the condition @xmath103 ) .
@xmath53{fig6.eps}\ ] ]    the solution of eq .
( [ hom1 ] ) is unstable and leads to a homoclinic  type orbit and thus can be referred to as the homoclinic cycle ( see devaney2003 for the definition of a homoclinic point and a homoclinic orbit ) .
indeed , for the initial condition @xmath104 the orbit is attracted by the @xmath35 - cycle @xmath105 , while for @xmath106 the fixed point @xmath107 is repelling , but the orbit returns eventually to the @xmath35 - cycle ( provided that a coexisting attractor does not capture the trajectory ) .
we shall compute the repelling branch in the next section .
to analyse structure of the chaotic attractor shown in fig .
2 we have solved eq .
( [ hom1 ] ) for @xmath108 , @xmath109 , computing thus critical value of the parameter @xmath25 and the homoclinic cycle where @xmath110 is the homoclinic point:@xmath111    this solution is attracting for initial condition @xmath112 and @xmath113 .
the sequence @xmath114 starting from such initial condition belongs to the attracting branch of the homoclinic orbit .
the repelling branch is obtained in the following way .
we start from @xmath115 , @xmath116 .
@xmath117 and @xmath118 are computed from the following equations:@xmath119    the solution of the first equation is of course @xmath74 .
we assume now in ( [ homrb ] ) that at the impact the table is just about going up with velocity @xmath120 rather than it has just finished going down with velocity @xmath121 ( therefore this equation differs from the second of equations in ( [ hom1 ] ) ) .
we thus compute from eq .
( [ homrb ] ) , for @xmath108 , @xmath109 and @xmath122 , that @xmath123 ( let us stress again that using the second of eqs .
( [ hom1 ] ) we get @xmath124)@xmath125 . due to symmetry of the dynamics @xmath27 the first point of the repelling branch of the homoclinic orbit can be assumed as @xmath126 , @xmath127 .
we have thus computed numerically the repelling branch of the homoclinic orbit starting from the initial condition @xmath128 , @xmath129 ( @xmath108 , @xmath109 , @xmath130 ) .
@xmath53{fig7.eps}\ ] ]    we have shown in fig . 6 first twenty six points ( full circles ) of the repelling branch of the homoclinic orbit starting from the point @xmath131 - the outermost full circle in the figure , lying on the vertical axis .
these points agree very well with positions of twenty six clouds of points belonging to the chaotic attractor shown in fig .
2 , computed for @xmath132 .
the next points ( dots ) of the repelling branch of the homoclinic trajectory enter four connected parts placed as in fig . 2 and tend , as an attracting branch , to the homoclinic cycle ( larger open circles ) containing the homoclinic point @xmath133 - the outermost open circle .
this homoclinic structure is preserved in the interval @xmath134 , @xmath108 , @xmath109  until this attractor is substituted by a new one due to crisis ( the unstable cycle collides with one of clouds of points belonging to the attractor ) , see the bifurcation diagram , fig . 1 , and figs . 2,3 .
we have found a generic scenario of transition to chaos for dynamics of a ball moving vertically in gravitational field and colliding with a table moving vertically with piecewise constant velocity .    according to this scenario
a periodic and stable solution is destroyed via a corner bifurcation @xcite in a corner event , @xmath135 or @xmath136 . in the present paper
the solution , defined analytically by @xmath38 in eq .
( [ equations ] ) , is a homoclinic  type orbit and leads to mixing and hence to chaotic dynamics .
this homoclinic  type orbit is untypical in the sense that it is not a saddle structure but its origin is related to discontinuous dynamics in the neighborhood of @xmath137 .
a. c. j. luo , y. guo , motion switching and chaos of a particle in a generalized fermi - acceleration oscillator , mathematical problems in engineering , vol . *
2009 * , article i d 298906 , 40 pages , 2009 .
doi:10.1155/2009/298906 .
a. okninski , b. radziszewski , chaotic dynamics in a simple bouncing ball model , proceedings of the 10th conference on dynamical systems : theory and applications , december 7 - 10 , 2009 .
d , poland , j. awrejcewicz , m. kazmierczak , p. olejnik , j. mrozowski ( eds . ) , pp .
651 - 656 .
we study dynamics of a ball moving in gravitational field and colliding with a moving table . 
 the motion of the limiter is assumed as periodic with piecewise constant velocity  
 it is assumed that the table moves up with a constant velocity and then moves down with another constant velocity . 
 the poincar map , describing evolution from an impact to the next impact , is derived and scenarios of transition to chaotic dynamics are investigated analytically .