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Chaos Workshop

Instituto de Fı́sica - IFUSP - Universidade de São Paulo - USP

Global Ballistic Acceleration in a Bouncing-Ball Model

André Luı́s 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)

Iberê Luiz Caldas (IFUSP - São Paulo - Brazil)

ALP Livorati (IFUSP) Global ballistic acceleration 07/30/2015 2 / 24

Outline

I. Introduction II. The bouncing ball dynamics

III. Fermi acceleration and accelerator modes IV. Deaccelerator modes and manifolds V. Final Remarks

ALP Livorati (IFUSP) Global ballistic acceleration 07/30/2015 3 / 24

Introduction

Introduction

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

Such a division leads to the stickiness phenomenon, which is manifested through the fact that 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 of several and complex nonlinear and chaotic phenomenon [3,4].

ALP Livorati (IFUSP) Global ballistic acceleration 07/30/2015 4 / 24

Introduction

The bouncing ball model

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

Figure 1: The Bouncer model and its parameters.

ALP Livorati (IFUSP) Global ballistic acceleration 07/30/2015 5 / 24

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 moving magnetic fields present in the cosmos. His original idea generated a prototype model which exhibits unlimited energy growth and is called the bouncer model.

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

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

ALP Livorati (IFUSP) Global ballistic acceleration 07/30/2015 6 / 24

The bouncing ball dynamics

Mapping description

The vibrating plate equation is xw (t) = A [cos(ωt + ϕ)− 1], and the for the particle we have xp(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. We evaluate the time spent for the particle to go up, reach null velocity and then go down to suffer a collision. Thus, we obtain a discrete mapping considering the recurrence of the velocity and the time between the collisions.

Since the the control parameters A, ω and g are not independent, the mapping should be given in terms of a dimensionless variables as Vn = ωvn/(πg), φn = ωtn. The only control parameter 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 = 0 Vn+1 = −Vn + (ϕn+1 − ϕn)/π − 2K sin(ϕn+1)

. (1)

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

ALP Livorati (IFUSP) Global ballistic acceleration 07/30/2015 7 / 24

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 the particle goes down, but when the particle collides with the wall, they exchange momentum and energy as if the wall was moving normally.

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

. (2)

Such approximation retains the nonlinearity of the system, none transcendental equation need 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 simplified approach Det J = 1.

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

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Fermi acceleration and accelerator modes

Phase space For K = 0, the system is integrable, and when we increase K , there is a transition from local chaos, to global chaos.

Such transition is crucial for the FA phenomenon to occur. Here, we have the destruction of the 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

Fermi acceleration and accelerator modes

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

Vi (n,K ) = 1 n

n∑ j=1

Vj,i ,V = 1 M

M∑ i=1

Vi (n,K ) . (3)

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

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

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Fermi acceleration and accelerator modes

Growing regimes The 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 phase space.

Figure 4: Difference between FA and the AM.

ALP Livorati (IFUSP) Global ballistic acceleration 07/30/2015 11 / 24

Fermi acceleration and accelerator modes

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.

ALP Livorati (IFUSP) Global ballistic acceleration 07/30/2015 12 / 24

Fermi acceleration and accelerator modes

Fixed Points and AM The period-1 fixed points for the bouncing ball can be obtained considering the repetition structure 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 fixed points).

|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)

ALP Livorati (IFUSP) Global ballistic acceleration 07/30/2015 13 / 24

Fermi acceleration and accelerator modes

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.

ALP Livorati (IFUSP) Global ballistic acceleration 07/30/2015 14 / 24

Fermi acceleration and accelerator modes

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) regions and expansion (blue) regions [10].

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

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

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Deaccelerator modes and manifolds

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 a