Abstract
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.
1. Introduction
There is much current interest in the construction of billiard maps for systems with moving walls. For an introduction to the physics of particle confinement with moving walls, see Loskutov et al. [1]. For single frequency motion, it is well known that Fermi acceleration does not occur in the Fermi map with smooth wall motion, but does occur for a sawtooth profile [2]. In the case of the circular billiard Kamphorst and de Carvalho [3] (KdC) have proven that if the wall motion is -periodic and then Fermi acceleration does not occur, which suggests a need for numerical experiments. Two-frequency techniques were first used as a means of enhancing plasma wave heating [4]. Indeed, the two-frequency Fermi map was studied as a simplified model for plasma wave heating [4]. Each frequency carries its own series of resonances, and parameters are chosen such that the resonant islands are interspersed at a desired location. For a nearly integrable system each island chain possesses a surrounding chaotic layer, and it is these layers that “overlap", destroying local invariant circles and thereby enhancing local chaoticity. Later on, similar techniques were applied to the important physical problem of microwave ionization of Rydberg atoms [5]. Here the system is degrees of freedom and one must make several canonical transformations in order to identify local wave-particle resonances. An experiment was designed and successfully run [6, 7], which verified a predicted enhancement in the ionization curve.
For billiards the motivating question is the existence or nonexistence of Fermi acceleration, that is, unbounded energy gain. KdC [3] showed that, based on an unpublished theorem of Douady [8], space-spanning invariant circles exist in a 2D map for harmonic oscillations of a circular billiard. On the other hand it has recently been shown that the four-dimensional time-dependent elliptic billiard exhibits Fermi accleration [9]. Here we modulate a simplified version of the 2D KdC map with a second frequency in such a way that the total energy of the two components is preserved, a direct generalization of earlier work [10]. An island interspersal condition is derived such that the second frequencies' islands fall exactly halfway in-between those of the first. The results show significant raising of the KAM barrier to global diffusion with increasing harmonic content. An important difference from previous examples is the presence of nontwist islands [11], which occur in pairs and can complicate the island overlap criterion.
2. The Circular Billiard Map
Following KdC we consider a breathing billiard of radius , where is the oscillation amplitude. In the spirit of the classical Fermi map [2] we fix and ignore multiple collisions with the wall. For specular collisions the angular momentum is conserved and we map from collision to collision. Figure 1 illustrates the geometry, where the angular momentum is preserved. As in [10], we neglect variations in (which we fix at unity) but retain to obtain where is the particle's radial velocity and is the specified wall velocity. It has generating function
and is therefore symplectic. The first equation follows from Figure 1, from which the half-chord , where is the angle between and . It has unit Jacobian and when , reduces to the area preserving Fermi map. The factors of two arise from the fact that there is no “there and back" for the billiard map. The same map can also be obtained by setting constant in KdC's somewhat more complex version. It agrees well with KdC's full map.
Before analyzing this map we first transform it so that direct comparison with the Fermi map is possible. First let and introduce the phase . Then, with and taking gives
In the limit as we recover the Fermi map [2], for which . Figure 2 compares the map with the Fermi map. For the maps are identical but for nontwist features appear at the bottom of the billiard map.
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2.1. Stability of Fixed Points
The fixed points are given by and
which occur in pairs
They are linearly stable for , where is the Jacobian matrix of the map [2, 12]. We find, for ,
With the help of the equilibrium condition (2.4) this yields the stability boundaries.
Saddle-Node Bifurcation (τ = 2)
We have either or
Period-Doubling Bifurcation (τ = −2)
There is
For we find either or
and for , either or
Note that the second case is unlikely to occur, since it requires .
2.2. Reconnection
Reconnection is a global bifurcation wherein fixed points do not change type, but the separatrix layers surrounding them undergo a topological rearrangement [11]. As the effect is generic and described in detail elsewhere [11], we will be succinct. Near a resonance we have the approximate Hamiltonian [2]
where . In deriving (2.11) we Fourier analyzed and averaged over higher-order resonances. We also Taylor expanded in the vertical direction.
The reconnection threshold is given by equating the value of on neighboring separatrices. With this definition, we calculate and
where . Equating then gives
where . Figure 3 shows a typical reconnection with . Equation (2.13) shows that this occurs for , in excellent agreement with the figure. Essentially perfect agreement occurs for larger , where the chaotic layer is thinner.
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2.3. Vortex Pairs
Vortex pairs also exist for this map. Figure 4 shows such a pair for and and . To understand this structure, consider a general period-two orbit, for which , so that either or . The former orbits do not exist for all parameters, being born in a pitchfork bifurcation. The latter orbits turn out to have , with . They are stable for , where . It follows that and , so that
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Thus, vortex pairs annihilate when , in agreement with Figure 4. They are formed when -points of adjacent island chains merge. Note that vortex pair configuration is extremely ephemeral and therefore probably of little importance in particle transport. For details see [11].
3. A Two-Frequency Map
Following the program of [10] we employ the bichromatic wall function
where and are the amplitudes, and and are the frequencies of the two components. The condition
maintains constant energy in the two waves, while the frequencies are chosen such that
where and are coprime integers, in order to maintain periodicity.
Defining the phase , the map can be written
where and
with . Thus (3.1) becomes
where .
3.1. Island Interspersal
For the map we may distinguish three distinct families: common resonances, which are shared by both -and -fold frequencies and defined by ,
with effective or
with solution
The angle is given by
In addition, when there exists a set of fixed points given by , which includes the common set (3.7):
These fixed points have period , where is reduced to the lowest terms. Similarly, when , setting yields period- fixed points, given by
which also includes (3.7) as a subset. At intermediate values of the - and -fold islands at and are unperturbed, while the other fixed points move according to (3.10), which locates .
Figure 5 compares the two-frequency Billiard map for for and . The general effect follows closely that for the Fermi map [10]. For one can see the groups of three a-fold islands interspersed with groups of two b-fold islands. One can also see chains of five islands, due to nonlinear mixing of the 2 and 3-fold chains. Increasing the admixture causes the amplitude of the b-fold islands to increase to the point where two neighboring a-chains “overlap." The net effect is to raise the KAM barrier; maximal chaos is attained for . Figure 6 shows the result of iterating a single orbit with intial condition . For the orbit hangs up on the twistless torus near , but for it breaks through and reaches a new KAM barrier near . Interesting complications can arise due to pairs of islands and their bifurcations. Using the “two-thirds rule," one can be quite quantitative about overlap, [10] but that is not our present goal, which is to demonstrate that the KAM barrier can indeed be raised substantially.
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4. Discussion
We have derived a simplified version of the circular billiard map, which promises to be very useful in parametric studies. As an example we have studied the effects of two frequencies and demonstrated the raising of the KAM barrier to diffusion. The map can be “straightened" by adding appropriate terms to the map. It was surprising to encounter a nontwist map and one wonders how this fact would manifest itself in the 4D elliptic billiard. We emphasize that the chaos enhancement effect of using two frequencies is not Fermi acceleration; the motion is still firmly bounded by the existence of space-spanning invariant circles.
Acknowledgments
It is a pleasure to acknowledge the contributions of Professor J. D. Meiss, A. Loskutov, S. M. Soskin and S. Kamphorst. A preliminary version of this paper was given at the Conference Billiards09, in Aquas de Lindóia, Brazil. The author is grateful to Professor E. Leonel for inviting him to this conference.