# Time-Dependent Circular Billiard

Authors: J. E. Howard
Journal: Mathematical Problems in Engineering
 (2009)

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## 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.


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## Body

## 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 R(t) is T-periodic and C7 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 3/2 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 radiusR=R0(1+Asin(2πt)), where A is the oscillation amplitude. In the spirit of the classical Fermi map [2] we fix R=R0 and ignore multiple collisions with the wall. For specular collisions the angular momentum is conserved and we map (vr,t) from collision to collision. Figure 1 illustrates the geometry, where the angular momentum vθ is preserved. As in [10], we neglect variations in R (which we fix at unity) but retain V(t)=Ṙ to obtain
(2.1)T:{vr′=vr-2V(t),t′=t+2vr′vr′2+vθ2(mod1),
where vr is the  particle's radial velocity and V(t) is the specified wall velocity. It has generating functionFigure 1
Geometry of the circular billiard map.(2.2)F2(t′,vr)=t′vr-ln(vr2+vθ2)-2∫t′V(t′)dt′
and is therefore symplectic. The first equation follows from Figure 1, from which the half-chord C=(1/2)v′Δt=Rsinϕ=Rvr/v′, where ϕ is the angle between v and vθ. It has unit Jacobian and when vθ↦0, 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 R= 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 letV(t)=V0sin(ωt) and introduce the phase ϕ=ωt. Then, with vr→2V0vr,vθ→2V0vθ and taking M=ω/2πV0 gives(2.3)T2:{vr′=vr-sinϕ,ϕ′=ϕ+(2πM)vr′vr′2+vθ2(mod2π).
In the limit as vθ→0 we recover the Fermi map [2], for which ϕ′=ϕ+2πM/vr. Figure 2 compares the map T2 with the Fermi map. For vθ=0 the maps are identical but for vθ=16.5 nontwist features appear at the bottom of the billiard map.Comparison of circular billiard map with Fermi map forM=2 and (a) vθ=19.94 and (b) vθ=19.997. The island centers have been shifted to ϕ=0.
(a)(b)


### 2.1. Stability of Fixed Points

The fixed points are given byϕ=(0,π) and(2.4)Mvrvr2+vθ2=k
which occur in pairs(2.5)vr±=12(Mk±(Mk)2-4vθ2).
They are linearly stable for |τ|=|Tr𝒥|<2, where 𝒥 is the Jacobian matrix of the map [2, 12]. We find, for k=1,(2.6)0<2πMcosϕvθ2-vr2(vr2+vθ2)2<1.
With the help of the equilibrium condition (2.4) this yields the stability  boundaries.Saddle-Node Bifurcation (τ = 2)
We have eithervr=0 or
(2.7)vr=vθ=M2.Period-Doubling Bifurcation (τ = −2)
There is(2.8)π2Mcosϕ(vθ2-vr2)=vr2.
For ϕ=0 we find either vr=0 or
(2.9)vr=Mπ2(π+M),vθ=vrπ+2Mπ,
and for ϕ=π, either vr=0 or
(2.10)vr=Mπ2(π-M),vθ=vrπ-2Mπ.
Note that the second case is unlikely to occur, since it requires M<(1/2)π.


### 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](2.11)H(t,vr,n)=πMln(vr2+vθ2)+cosϕδ1(n)≈πMln(vr2+vθ2)-2πvr+cosϕ,
where δ1(n)=∑m=-∞∞δ(n-m). In deriving (2.11) we Fourier analyzed δ1(n) and averaged over higher-order resonances. We also Taylor expanded H in the vertical direction.The reconnection threshold is given by equating the value ofH̅ on neighboring separatrices. With this definition, we calculate H̅lower=1 and(2.12)H̅upper=πMeln((vr+)2+vθ2(vr-)2+vθ2)-2π(vr+-vr-)-1,
where vr±=(1/2)(Me±Me2-4vθ2). Equating Hupper=Hlower then gives(2.13)1πMe=ln(1+1-ṽ2ṽ)-1-ṽ2,
where ṽ=2vθ/Me. Figure 3 shows a typical reconnection with M=20. Equation (2.13) shows that this occurs for vθ=9.355, in excellent agreement with the figure. Essentially perfect agreement occurs for larger M, where the chaotic layer is thinner.Circular Billiard map forM=20 and (a) vθ=9.4, (b) vθ=9.3, illustrating reconnection.
(a)(b)


### 2.3. Vortex Pairs

Vortex pairs also exist for this map. Figure4 shows such a pair for M=20 and vθ=19.94 and 19.997. To understand this structure, consider a general period-two orbit, for which sinϕ+sinϕ′=0, so that either ϕ′=-ϕ or ϕ′=ϕ-π. The former orbits do not exist for all parameters, being born in a pitchfork bifurcation. The latter orbits turn out to have ϕ=π,ϕ′=0, with 2Mvr/(vr2+vθ2)=1. They are stable for πMe|f′(vr)|<1, where f(vr)=vr/(vr2+vθ2). It follows that vr=vr′ and vr2-2Mvr+vθ2=0, so thatVortex pair formation for circular billiard map forM=20 and (a) vθ=19.99, (b) vθ=19.997.
(a)(b)(2.14)vr=M±M2-vθ2.
Thus, vortex pairs annihilate when vθ=M, in agreement with Figure 4. They are formed when x-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].


## 2.1. Stability of Fixed Points

The fixed points are given byϕ=(0,π) and(2.4)Mvrvr2+vθ2=k
which occur in pairs(2.5)vr±=12(Mk±(Mk)2-4vθ2).
They are linearly stable for |τ|=|Tr𝒥|<2, where 𝒥 is the Jacobian matrix of the map [2, 12]. We find, for k=1,(2.6)0<2πMcosϕvθ2-vr2(vr2+vθ2)2<1.
With the help of the equilibrium condition (2.4) this yields the stability  boundaries.Saddle-Node Bifurcation (τ = 2)
We have eithervr=0 or
(2.7)vr=vθ=M2.Period-Doubling Bifurcation (τ = −2)
There is(2.8)π2Mcosϕ(vθ2-vr2)=vr2.
For ϕ=0 we find either vr=0 or
(2.9)vr=Mπ2(π+M),vθ=vrπ+2Mπ,
and for ϕ=π, either vr=0 or
(2.10)vr=Mπ2(π-M),vθ=vrπ-2Mπ.
Note that the second case is unlikely to occur, since it requires M<(1/2)π.


## 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](2.11)H(t,vr,n)=πMln(vr2+vθ2)+cosϕδ1(n)≈πMln(vr2+vθ2)-2πvr+cosϕ,
where δ1(n)=∑m=-∞∞δ(n-m). In deriving (2.11) we Fourier analyzed δ1(n) and averaged over higher-order resonances. We also Taylor expanded H in the vertical direction.The reconnection threshold is given by equating the value ofH̅ on neighboring separatrices. With this definition, we calculate H̅lower=1 and(2.12)H̅upper=πMeln((vr+)2+vθ2(vr-)2+vθ2)-2π(vr+-vr-)-1,
where vr±=(1/2)(Me±Me2-4vθ2). Equating Hupper=Hlower then gives(2.13)1πMe=ln(1+1-ṽ2ṽ)-1-ṽ2,
where ṽ=2vθ/Me. Figure 3 shows a typical reconnection with M=20. Equation (2.13) shows that this occurs for vθ=9.355, in excellent agreement with the figure. Essentially perfect agreement occurs for larger M, where the chaotic layer is thinner.Circular Billiard map forM=20 and (a) vθ=9.4, (b) vθ=9.3, illustrating reconnection.
(a)(b)


## 2.3. Vortex Pairs

Vortex pairs also exist for this map. Figure4 shows such a pair for M=20 and vθ=19.94 and 19.997. To understand this structure, consider a general period-two orbit, for which sinϕ+sinϕ′=0, so that either ϕ′=-ϕ or ϕ′=ϕ-π. The former orbits do not exist for all parameters, being born in a pitchfork bifurcation. The latter orbits turn out to have ϕ=π,ϕ′=0, with 2Mvr/(vr2+vθ2)=1. They are stable for πMe|f′(vr)|<1, where f(vr)=vr/(vr2+vθ2). It follows that vr=vr′ and vr2-2Mvr+vθ2=0, so thatVortex pair formation for circular billiard map forM=20 and (a) vθ=19.99, (b) vθ=19.997.
(a)(b)(2.14)vr=M±M2-vθ2.
Thus, vortex pairs annihilate when vθ=M, in agreement with Figure 4. They are formed when x-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(3.1)V(ϕ)=Vasin(ωat)+Vbsin(ωbt)Va2+Vb2,
where Va and Vb are the amplitudes, and ωa and ωb are the frequencies of the two components. The condition(3.2)Va2+Vb2=const
maintains constant energy in the two waves, while the frequencies are chosen such that(3.3)ωaωb=ab
where a and b are coprime integers, in order to maintain periodicity.Defining the phaseϕ=ωat/a, the map T2 can be written(3.4)T3:{vr′=vr-V(ϕ),ϕ′=ϕ+(4πMa+b)vr′vr′2+vθ2(mod2π),
where M=ω̅/π and(3.5)ωaa=ωbb=ω̅a+b
with ω̅=(1/2)(ωa+ωb). Thus (3.1) becomes(3.6)V(ϕ)=[sin(aϕ)+μsin(bϕ)]1+μ2,
where μ=Vb/Va.


### 3.1. Island Interspersal

For the mapT3 we may distinguish three distinct families: common resonances, which are shared by both a-and b-fold frequencies and defined by ϕ↦ϕ+2πk,(3.7)vrvr2+vθ2=kMe
with effective Me=2M/(a+b) or(3.8)vr2-Mekvr+vθ2=0
with solution(3.9)vrk±=12(Mek±(Mek)2-4vθ2).
The angle ϕ0 is given by(3.10)sinaϕ+μsinbϕ=0.In addition, whenμ=0 there exists a set of fixed points given by aϕ↦aϕ+2πk, which includes the common set (3.7):(3.11)avrkvrk2+vθ2=kMe.
These fixed points have period P, where P/Q is a/b reduced to the lowest terms. Similarly, when μ↦∞, setting bϕ↦bϕ+2πk yields period-b fixed points, given by(3.12)bvrkvrk2+vθ2=kMe,
which also includes (3.7) as a subset. At intermediate values of μ the a- and b-fold islands at ϕ0=0 and π are unperturbed, while the other fixed points move according to (3.10), which locates ϕ0(μ).Figure5 compares the two-frequency Billiard map for a=3,b=2,vθ=16 for μ=0.0 and μ=0.25. The general effect follows closely that for the Fermi map [10]. For μ=0.25 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 μ 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 μ≈0.5. Figure 6 shows the result of iterating a single orbit with intial condition ϕ=0,vr=10. For μ=0.0 the orbit hangs up on the twistless torus near vr=20, but for μ=0.5 it breaks through and reaches a new KAM barrier near vr=70. 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.A Two-frequency Fermi map forM=600,a=3,b=2,vθ=16 for (a) μ=0.0,  (b) μ=0.25.
(a)(b)Single orbits;106 iterations for M=600,a=3,b=2,vθ=16 (a) with μ=0,  (b) μ=0.5.
(a)(b)


## 3.1. Island Interspersal

For the mapT3 we may distinguish three distinct families: common resonances, which are shared by both a-and b-fold frequencies and defined by ϕ↦ϕ+2πk,(3.7)vrvr2+vθ2=kMe
with effective Me=2M/(a+b) or(3.8)vr2-Mekvr+vθ2=0
with solution(3.9)vrk±=12(Mek±(Mek)2-4vθ2).
The angle ϕ0 is given by(3.10)sinaϕ+μsinbϕ=0.In addition, whenμ=0 there exists a set of fixed points given by aϕ↦aϕ+2πk, which includes the common set (3.7):(3.11)avrkvrk2+vθ2=kMe.
These fixed points have period P, where P/Q is a/b reduced to the lowest terms. Similarly, when μ↦∞, setting bϕ↦bϕ+2πk yields period-b fixed points, given by(3.12)bvrkvrk2+vθ2=kMe,
which also includes (3.7) as a subset. At intermediate values of μ the a- and b-fold islands at ϕ0=0 and π are unperturbed, while the other fixed points move according to (3.10), which locates ϕ0(μ).Figure5 compares the two-frequency Billiard map for a=3,b=2,vθ=16 for μ=0.0 and μ=0.25. The general effect follows closely that for the Fermi map [10]. For μ=0.25 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 μ 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 μ≈0.5. Figure 6 shows the result of iterating a single orbit with intial condition ϕ=0,vr=10. For μ=0.0 the orbit hangs up on the twistless torus near vr=20, but for μ=0.5 it breaks through and reaches a new KAM barrier near vr=70. 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.A Two-frequency Fermi map forM=600,a=3,b=2,vθ=16 for (a) μ=0.0,  (b) μ=0.25.
(a)(b)Single orbits;106 iterations for M=600,a=3,b=2,vθ=16 (a) with μ=0,  (b) μ=0.5.
(a)(b)


## 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 isnot Fermi acceleration; the motion is still firmly bounded by the existence of space-spanning invariant circles.



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