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Chaos Workshop

Instituto de Fısica - IFUSP - Universidade de Sao Paulo - USP

Global Ballistic Acceleration in a Bouncing-Ball Model

Andre Luıs Prando Livorati

Universidade Estadual Paulista - UNESP - Rio Claro

July 30th 2015.

ALP Livorati (IFUSP) Global ballistic acceleration 07/30/2015 1 / 24

Collaborators

Tiago Kroetz (UTFPR - Pato Branco -Brazil)

Edson Denis Leonel (UNESP - Rio Claro - Brazil)

Ibere Luiz Caldas (IFUSP - Sao Paulo - Brazil)


Outline

I. IntroductionII. The bouncing ball dynamics

III. Fermi acceleration and accelerator modesIV. Deaccelerator modes and manifoldsV. Final Remarks


Introduction

Introduction

Hamiltonian systems are typical non-ergodic and non-integrable [1]. Their phase space aredivided into regions with regular and chaotic dynamics, where we can observe KAM islandsand invariant tori surrounded by chaotic seas.

Such a division leads to the stickiness phenomenon, which is manifested through the factthat a phase trajectory in a chaotic region passing near enough a Kolmogorov-Arnold-Moser(KAM) island, evolves there almost regularly during a time that may be very long [2].

Also, in these systems, transport and diffusion properties can present anomalous behaviour,depending on the initial conditions and control parameters. This implies in the occurence ofseveral and complex nonlinear and chaotic phenomenon [3,4].


Introduction

The bouncing ball model

The bouncing ball model: A mass m under gravity, collides elastically with a vibrating heavyplate.

Figure 1: The Bouncer model and its parameters.


Introduction

Fermi acceleration

Fermi acceleration (FA) was[6] as an attempt to explain the possible origin of the high ener-gies of the cosmic rays.

Fermi claimed that the charged cosmic particles could acquire energy from the movingmagnetic fields present in the cosmos. His original idea generated a prototype model whichexhibits unlimited energy growth and is called the bouncer model.

The model consists of a free particle (making allusions to the cosmic particles) which isfalling under influence of a constant gravitational field g (a mechanism to inject the particleback to the collision zone) and suffering collisions with a heavy and time-periodic movingwall (denoting the magnetic fields).

When dissipative dynamics is considered, invariant curves are destroyed, and the stableelliptical points become attracting fixed points (sinks) and chaotic attractors are observed[3,4,5]. Also, Fermi Acceleration is no longer observed [7,8].


The bouncing ball dynamics

Mapping description

The vibrating plate equation is xw (t) = A [cos(ωt + ϕ)− 1], and the for the particle we havexp(t) = h0 + vt − gt2/2. A collision happens when xw (t) = xp(t).

The complete version takes into account the real movement of the vibrating plate. Weevaluate the time spent for the particle to go up, reach null velocity and then go down tosuffer a collision. Thus, we obtain a discrete mapping considering the recurrence of thevelocity and the time between the collisions.

Since the the control parameters A, ω and g are not independent, the mapping should begiven in terms of a dimensionless variables as Vn = ωvn/(πg), φn = ωtn. The only controlparameter now is K = ω2A/(πg), which is interpreted as a ratio between accelerations.

{K [cos(ϕn)− cos(ϕn+1)] + Vn(ϕn+1 − ϕn)− (ϕn+1 − ϕn)2/2π = 0Vn+1 = −Vn + (ϕn+1 − ϕn)/π − 2K sin(ϕn+1)

. (1)

The first expression above is a transcendental equation, and should be solved numerically.


The bouncing ball dynamics

Simplified approach

The moving wall is said to be fixed now, so the time to go up is equally to the time for theparticle goes down, but when the particle collides with the wall, they exchange momentumand energy as if the wall was moving normally.

{ϕn+1 = ϕn + 2πVnVn+1 = |Vn − 2K sinϕn+1|

. (2)

Such approximation retains the nonlinearity of the system, none transcendental equationneed to be solved, and it useful in analytical approximations.

Also, for the complete version Det J = Vn+K sin(ϕn)Vn+1+K sin(ϕn+1)

. On the other hand, for the simplifiedapproach Det J = 1.

So, the complete version is not symplectic, and the simplified approach is symplectic. [9,10]


Fermi acceleration and accelerator modes

Phase spaceFor K = 0, the system is integrable, and when we increase K , there is a transition fromlocal chaos, to global chaos.

Such transition is crucial for the FA phenomenon to occur. Here, we have the destruction ofthe invariant spanning curves, allowing the union of the local chaotic seas [9].

So a chaotic orbit has a “free path” to diffuse along the velocity axis [9].

Figure 2: Phase Space for some values of K . In (a) K = 0.0318, (b) K = 0.06366, (c)

K = 0.07957, and (d) K = 0.09549.ALP Livorati (IFUSP) Global ballistic acceleration 07/30/2015 9 / 24


Average velocityThe average velocity over an ensemble of M = 5000 initial conditions has a tendency togrow as function of the number of collisions, which is an evidence of the influence of FA.

Vi (n,K ) =1n

n∑j=1

Vj,i ,V =1M

M∑i=1

Vi (n,K ) . (3)

Two growing regimes as characterized. A power law growth, as V ∝ nβ1 (FA), and a lineargrowth one V ∝ nβ2 accelerator mode (AM) [10].

Figure 3: Average velocity curves for different values of K . Two growing regimes are set.



Growing regimesThe power law growing regime (FA) is characterized by the chaotic properties of the dy-namics. However the linear growing regime is caused by resonances (AM) in the phasespace.

Figure 4: Difference between FA and the AM.



Accelerator modes

One can see a fine structure on the quantity 〈V 〉104 . This feature indicates some meta-morphosis suffered by the AM as we vary K .

Figure 5: Peaks of intensity for the accelerator modes.



Fixed Points and AMThe period-1 fixed points for the bouncing ball can be obtained considering the repetitionstructure for the velocity in the phase space. So, we have

V ac = l where; l = 1, 2, 3, ... ϕac = arcsin(−V ac/2K

). (4)

To determine the stability we linearize the system around (V ac , ϕac) via the Jacobian matrix.The calculation of eigenvalues leads to

P(λ) = λ2 − λ(2− 4πK cos(ϕac)) + 1 = 0 . (5)

The stability condition will be satisfied since the eigenvalues are complex (elliptical fixedpoints).

|2− 4πK cos(ϕac)| < 2 → 0 < 4πK cos(ϕac) < 4. (6)

So, the stability condition of the period-1 AM as a function of parameter K is [9,10]

0.5 < K <√

1/π2 + 1/4. (7)



Period-1 AM evolution and bifurcations

Figure 6: Evolution of period-1 AM. (a) K = 0.5055, (b) K = 0.5252, (c) K = 0.5285, (d)

K = 0.54, (e) K = 0.5491, (f) K = 0.5530, (g) K = 0.5773, (h) K = 0.5958 and (i) K = 0.6012.



Period-3 AM evolution and bifurcations

Figure 7: Evolution of period-3 AM. (a) K = 0.2354, (b) K = 0.2374, (c) K = 0.2380, (d)

K = 0.2389, (e) K = 0.2409, (f) K = 0.2411, (g) K = 0.2418, (h) K = 0.2465 and (i)

K = 0.2480.ALP Livorati (IFUSP) Global ballistic acceleration 07/30/2015 15 / 24

Deaccelerator modes and manifolds

Phase space according det(J)Since the complete mapping is not symplectic, the det(J) = Vn+K sin(ϕn)

Vn+1+K sin(ϕn+1), present va-

lues larger or smaller than the unity. So, the phase space presents contract (red) regionsand expansion (blue) regions [10].

In these regions, we found deaccelerator modes (DM) and accelerator modes (AM) in theanti-symmetric position of each other.

Figure 8: Phase space for: (a) K = 0.54, (b) K = 0.322 and (c) K = 0.237.



Evolution for DM and AM

Figure 9: Evolution for a single initial condition for period-3: (a) DM and (b) AM.ALP Livorati (IFUSP) Global ballistic acceleration 07/30/2015 17 / 24


Repelling and attracting nature

Considering the modulated phase space V → V mod(1), we can see the repelling naturefor the DM, and the attracting nature of the AM [10].

Figure 10: In (a) basin of repelling for the DM and in (b) the “attracting” nature for the AM.



Manifolds and escape basinsSince we have the nature of attracting and repelling for respective AM and DM in the mo-dulated phase space, we may consider escape (attracting) basins for the AM, and repellingbasins for the DM.

We drawn the stable and unstable manifold for the central saddle point, and confirm thatthese manifolds are respectively drawing the boundary of the escape and repelling basins,for AM and DM [10].

Figure 11: In (a) we have the escape and repelling basins for respective AM and DM, in (b)

stable and unstable manifolds drawn the boundaries of the escape and repelling basins.ALP Livorati (IFUSP) Global ballistic acceleration 07/30/2015 19 / 24

Conclusions

Conclusions

The dynamics of the bouncing model was investigated using a two dimensional mapping.Chaotic properties where characterized, and transition from local to global chaos allows thephenomenon of Fermi Acceleration (FA) to occur.

Two growing regimes were characterized for FA. A power law one (β1 ≈ 1/2), set bynormal diffusion from the chaotic sea; and a linear growth (β2 ≈= 1), related with featuredresonances known as accelerator modes (AM).

The period-1 AM stability were characterized according the range of the control parameterK . The bifurcation process that it suffers, is related with the intensity of the peaks for 〈V 〉.

Considering the complete version (not symplectic), we analysed the modulated phase space,and characterized an attractive nature of the AM. Also, deaccelerator modes(DM), whichhas repelling nature, we set in the anti-symmetric position of the AM [10].

Since we have the attracting and repelling nature for AM and DM, we observed basinsof acceleration and deacceleration, where their boundaries are drawn by the stable andunstable manifolds respectively [10].


Conclusions

Published paper: Physical Review E, 92, 012905,(2015).


Acknowledgements

Acknowledgements


References

References

[1] L. Markus, K. R. Meyer, “Mem. Am. Math. Soc.”144, 1 (1978).[2] G. M. Zaslasvsky, “Hamiltonian Chaos and Fractional Dynamics”, Oxford University Press,New York (2008).[3] A. J. Lichtenberg, M.A. Lieberman, Regular and Chaotic Dynamics. Appl. Math. Sci. 38,Springer Verlag, New York, 1992.[4] K. T. Alligood, T. D. Sauer, and J. A. Yorke. Chaos. Springer, 1996.[5] R. C. Hilborn, Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers.Oxford University Press, New York, 1994.[6] E. Fermi, Phys. Rev., 75, 1169 (1949).[7] E. D. Leonel and A. L. P. Livorati, Physica. A, 387, 1155, (2008).[8] A. L. P. Livorati, D. G. Ladeira and E. D. Leonel, Phys. Rev. E, 78, 056205, (2008).[9] A. L. P. Livorati, T. Kroetz, C. P. Dettmann, I. L. Caldas and E. D. Leonel, Phys. Rev. E, 86,036203, (2012).[10] T. Kroetz, A. L. P. Livorati, E. D. Leonel and I. L. Caldas, Phys. Rev. E, 92, 012905, (2015).


References

Thank you for your attention!!
