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Computer-assisted methods for analyzing periodic orbits in

vibrating gravitational billiards

Kevin E. M. Church

Department of Mathematics and Statistics, McGill University

805 Sherbrooke St W, Montreal, Quebec, H3A 0B9, Canada

kevin.church@mcgill.ca

Cl´ement Fortin

Department of Physics, McGill University

805 Sherbrooke St W, Montreal, Quebec, H3A 0B9, Canada

clement.fortin@mail.mcgill.ca

Using rigorous numerical methods, we prove the existence of 608 isolated periodic orbits in a

gravitational billiard in a vibrating unbounded parabolic domain. We then perform pseudo-

arclength continuation in the amplitude of the parabolic surface’s oscillation to compute large,

global branches of periodic orbits. These branches are themselves proven rigorously using

computer-assisted methods. Our numerical investigations strongly suggest the existence of mul-

tiple pitchfork bifurcations in the billiard model. Based on the numerics, physical intuition and

existing results for a simpliﬁed model, we conjecture that for any pair (k, p), there is a constant

ξfor which periodic orbits consisting of kimpacts per period pcan not be sustained for am-

plitudes of oscillation below ξ. We compute a veriﬁed upper bound for the conjectured critical

amplitude for (k, p) = (2,2) using our rigorous pseudo-arclength continuation.

Keywords: gravitational billiards, time-varying domain, periodic orbit, rigorous numerics, nu-

merical continuation

1. Introduction

Adynamical billiards is a Hamiltonian dynamical system, or an abstraction of the game billiards (or pool).

A particle moves according to some equations of motion until it hits the boundary of the domain, at which

point it undergoes specular reﬂection. Contingent upon the case study, this reﬂection can either be elastic

or inelastic. Though the dynamics can seem trivial, these systems are known to be chaotic [Chernov &

Markarian, 2003], depending on the geometry of the boundary. A number of studies [Baxter & Umble,

2007; Biswas, 1997; Boshernitzan et al., 1998; Troubetzkoy, 2005] have focused on studying the dynamics

of such systems in closed polygonal domains. Some other “billiard-like” systems have been considered in

the context of switched and impacting systems [Huang & Fu, 2019; Huang & Luo, 2017; Tang et al., 2019].

In some cases, a potential ﬁeld is incorporated, allowing for periodic orbits to exist in open domains, with

parabolic, wedge-shaped or hyperbolic boundaries. When the potential corresponds to the gravitational

ﬁeld, the dynamical system is called gravitational billiards. The chaotic behavior of these systems are well-

studied both analytically and numerically [Chatterjee et al., 1996; M´aty´as & Barna, 2011; Peraza-Mues

et al., 2017; Korsch & Lang, 1991] and often categorize the stability of periodic orbits. However, only a

few papers have studied the dynamics of a particle in a spatially time-varying domain [Hartl et al., 2013;

1

2K. E. M. Church &C. Fortin

Feldt & Olafsen, 2005; Peraza-Mues et al., 2017; Costa et al., 2015; Langer & Miller, 2015] and even less

so make use of computer-assisted proofs.

In this paper, we provide a computer-assisted approach for proving existence of periodic orbits and for

studying their behavior as we vary a parameter. To showcase the strength of such approach, we study a

nonconservative boundary-particle system, with a vertically-vibrating parabolic boundary in a gravitational

ﬁeld. That is, the billiards is moving and is subject to a (gravitational) potential. We prove the existence of

at least 608 periodic orbits for speciﬁc parameters values. Moreover, strong evidence suggests the existence

of pitchfork bifurcations when doing continuation in the amplitude of oscillation of the boundary. We do not

incorporate a method for doing continuation past pitchforks but one can be found in [Lessard et al., 2017].

The code used to ﬁnd and prove results in this paper is available through the second author’s GitHub1.

An INTLAB [Rump, 1999] license is needed to complete the computer-assisted proofs.

1.1. The rigorous numerics approach to periodic orbits

Computer-assisted methods for validation of approximate numerical zeroes of nonlinear functions have

found numerous applications in nonlinear dynamics. For an overview, we refer the reader to the survey

articles [G´omez-Serrano, 2019] and [Lessard et al., 2015], where the general method for problems of ﬁnite

and inﬁnite dimensions is outlined. We describe the idea here as it applies in ﬁnite dimensions.

Let f∈C1U, RNwhere U⊂RN. We assume that zeroes of fin some way correspond to solutions

of our nonlinear problem. We refer to this as a zero-ﬁnding problem. We are therefore initially interested

in computing zeroes of f, or reﬁning an approximate zero using an interative procedure. The basic idea is,

of course, Newton’s method.

Deﬁnition 1.1. Let f∈C1U, RNwhere U⊂RNis an open set. A point ˜x∈Uis a nondegenerate zero

of fif f(˜x) = 0 and DF (˜x) is invertible. If f∈C2U, RNwhere U⊂RNis an open set and A:RN→RN

is injective, we can form the Newton operator Tby

T(x):=x−Af(x).

Reﬁning an approximate zero xof the map fcan then be accomplished by iterating the Newton operator T

on the approximate zero. The classical Newton’s method takes A:= (Df(x))−1, but this can be computa-

tionally expensive. An alternative is to use the initial guess x∈Xfor a zero of fand set A:= (Df(x))−1.

The rigorous numerics approach is as follows.

(1) Compute an approximate zero xof f, and reﬁne it with a Newton operator until the residual ||f(x)|| is

small (i.e. near machine precision).

(2) Determine a closed ball of radius r > 0 around xon which the Newton operator Twith A:= (Df (x))−1

is a contraction.

Once this is accomplished, the contraction mapping principle guarantees that fhas a unique zero in the

ball Br(x). If rcan be chosen very small, then we get a tight enclosure (i.e. our approximate zero xis

“good”). If rcan be chosen larger, then we get information about how isolated the zero is. In practice, we

compute intervals of such suitable radii rusing the radii polynomial approach: see Theorem 1 and Theorem

3. These computations are all done on a computer, and as MATLAB is used for the implementation, we

use interval arithmetic with the INTLAB [Rump, 1999] package to track roundoﬀ error and accomplish a

few other technical tasks.

We make a brief remark that stability of the periodic orbits will not studied here. Veriﬁcation of stability

of periodic orbits of arbitrary period (and number of bounces) is a fair bit more diﬃcult, especially if we

wanted to match the level of mathematical rigorous with which we prove the existence of the orbits. It

would not be diﬃcult to gain some preliminary insight into stability using numerical simulations, but since

this system has so many isolated periodic orbits – see Theorem 2 – many of them are likely to be unstable,

and those stable ones might have very small basins of attraction.

1https://github.com/ClementFortin/BilliardOscillatingParabola

Computer-assisted methods for periodic orbits in vibrating gravitational billiards 3

1.2. Overview

In Section 2, we describe the vibrating parabolic gravitational billiards, establish an equivalent zero-ﬁnding

problem for its periodic orbits, and prove some elementary results concerning such periodic orbits. The

rigorous numerics approach to periodic orbits is presented in Section 3 with our main theorems concerning

the abundance of periodic orbits for speciﬁc parameter values and the rigorous branch continuation. Section

4 ends with a conclusion.

2. Vibrating Parabolic Domain in Two Dimensions

Let (x, y) be the position coordinates in R2of a particle and assume without loss of generality that its

mass is 1. Denote (v, w) such that

˙x=v,

˙y=w,

˙v= 0,

˙w=−g. (1)

Let a parabolic oscillating domain P(t) be deﬁned by

P(t) = {(x, y)∈R2:y≥αx2+sin(2πωt)}(2)

where α, , ω ∈R+describe the steepness of the curve, the amplitude of its oscillation and its frequency

of oscillation, respectively. We eliminate the frequency parameter by deﬁning the rescaled time ˆ

t=ωt,

gravity ˆg=g/ω and velocities ˆv=ωv and ˆw=ωw. Completing the change of variables and dropping the

hats, the parameterization of the boundary of P(t) becomes

y=αx2+sin(2πt),(3)

which is referred to as the oscillating surface.

From an initial condition (x0, y0, v0, w0) at time t0>0, the particle evolves according to (1). At some

time t∗≥t0and position (x(t∗), y(t∗)), the particle comes into contact with the surface, thereby changing

velocity. In other words, we have that

y(t∗) = αx(t∗)2+sin(2πt∗).

The particle’s new velocity is then a function of its velocity before the impact, the position at which the

impact happened as well as the time at which it happened.

When the surface is not vibrating – that is = 0 – the global attractor is always a periodic orbit or the

union of periodic orbits [Korsch & Lang, 1991]. We are interested in studying periodic orbits when > 0.

To this end, in what follows the symbol k∈Nwill denote the number of impacts of a periodic orbit. The

symbol ϕwill typically be used to refer to a periodic orbit (in terms of a coordinate system we will later

specify), and p > 0 its period: that is the time it takes the particle to come back to its original position

(x0, y0) with velocity (v0, w0). Because of equation (3), the period pof any periodic orbit is necessarily a

positive integer. In the following section, we build a zero-ﬁnding problem whose zeroes will encode periodic

orbits, and characterize diﬀerent types of its solutions.

2.1. Zero-Finding Problem

Let us deﬁne a coeﬃcient of restitution e∈[0,1] which relates the pre-impact velocity of the particle to its

post-impact velocity. For e6= 1 the collision is not perfectly elastic and a portion of the particle’s kinetic

energy is lost. The velocity component aﬀected by this collision is the one parallel to the surface’s normal

force. Let B ∈ R2×2be the linear operator that changes the usual x-y coordinate system to the surface’s

perpendicular and parallel coordinate system, such that

B=1

p1 + (2αx)21 2αx

−2αx 1

4K. E. M. Church &C. Fortin

The reset law R:R4→R2is obtained by applying B−1EB with

E=1 0

0−e

to the particle’s velocity in the surface frame of reference using a Galilean transformation, and moving

back to the particle’s reference frame so that

R=R1

R2=B−1EB v

w−2π cos(2πt)+0

2π cos(2πt).

More explicitly,

R(t, x, v, w) = 1

1 + (2αx)21−e(2αx)22αx(1 + e)

2αx(1 + e) (2αx)2−ev

w+2π cos(2πt)(e+ 1)

1 + (2αx)2−2αx

1(4)

where (α, e, ) are ﬁxed parameters. For presentation purposes, we denote ˙

x+= (v+, w+) = R(t, x, ˙

x)>,

the particle’s velocity vector immediately after an impact occurs.

Deﬁne tnthe time at which the nth impact occurs, xn= (xn, yn) the position of the nth impact and

˙

xn= (vn, wn) the velocity immediately before the nth impact. The periodic orbits ϕare speciﬁed by the

coordinates (tn,xn,˙

xn). We will ﬁrst derive an equation that characterizes the impact times tn. Since each

impact happens on the surface, the n+ 1th impact has to satisfy

yn+1(t) = αx2

n+1 +sin(2πtn+1),(5)

for any n∈ {0,1, ..., k −1}, where the particle’s post-impact velocity vector can be found using the reset

law. From there, the particle will evolve according to (1) and Newton’s equations of motion are used to

ﬁnd (xn+1,˙

xn+1) i.e. the coordinates of the next impact:

xn+1 =xn+v+

n(tn+1 −tn),

yn+1 =yn+w+

n(tn+1 −tn)−g(tn+1 −tn)2

2.

Substituting both of these in (5) we obtain a zero-ﬁnding problem that tn+1 must satisfy;

0 = α(xn+v+

n(tn+1 −tn))2+sin(2πtn+1)−yn−w+

n(tn+1 −tn) + g(tn+1 −tn)2

2.

Since every impact happens on the surface, the height of the particle must be the same as the surface’s

such that yn=αx2

n+sin(2πtn), for any n∈ {0,1, ..., k −1}. Henceforth, let T:R5→Rbe deﬁned by

T(τ, t, x, R(t, x, ˙

x)) = α[(x+R1(t, x, ˙

x)(τ−t))2−x2] + [sin(2πτ )−sin(2πt)]

− R2(t, x, ˙

x)(τ−t) + g(τ−t)2

2,(6)

where ˙

x= (v, w) and where τand tcorrespond to the time values of the next and current impact

respectively. The impact coordinates must therefore satisfy

T(t1, t0, x0,R(t0, x0,˙

x0)) = 0,

T(t2, t1, x1,R(t1, x1,˙

x1)) = 0,

.

.

.

T(tk, tk−1, xk−1,R(tk−1, xk−1,˙

xk−1)) = 0.

(7)

and hence for kimpacts of total period pthere is a sequence of time values

t0< t1< ... < tk−1< tk=p+t0

Computer-assisted methods for periodic orbits in vibrating gravitational billiards 5

representing the time at which each impact (xn,˙

xn) occurs (for any n∈ {0,1, ..., k −1}). In some cases,

there can be multiple diﬀerent values for tn+1 that solve T(tn+1, tn, . . . ) = 0. However, only one value

satisﬁes

tn+1 = min{t>tn:T(t, tn, xn,R(tn, xn,˙

xn)) = 0}.(8)

Physically, this condition makes sure that the particle never crosses the surface in the interval [tn, tn+1].

Once a candidate for a periodic orbit has been computed, we verify that it satisﬁes (8) using the function

verifynlssall of INTLAB [Rump, 1999; Hargreaves, 2002] which allows for rigorous enclosure of all zeros

of elementary functions within a speciﬁed interval. For a validated periodic orbit, the particle is hence

restricted to the domain P(t).

To obtain the position-velocity components of the particle for the n+ 1th impact, deﬁne the map

S:R6→R4with

S(τ, t, x,R(t, x, ˙

x)) =

x+R1(t, x, ˙

x)(τ−t)

y+R2(t, x, ˙

x)(τ−t)−g(τ−t)2

2

R1(t, x, ˙

x)

R2(t, x, ˙

x)−g(τ−t)

,(9)

such that

[xn+1,˙

xn+1] = S(tn+1 , tn,xn,R(tn, xn,˙

xn)).

Since ynis uniquely deﬁned in terms of xnand tnaccording to (5), we need not consider ynas an explicit

variable. Therefore, periodic orbits are characterized by kimpacts, each determined by the four associated

coordinates (tn, xn, vn, wn). We hence build a map with f:ϕ∈R4k→R4ksuch that

f(ϕ):=

T(t1, t0, x0,R(t0, x0,˙

x0))

x0+R1(t0, x0,˙

x0)(t1−t0)

R1(t0, x0,˙

x0)

R2(t0, x0,˙

x0)−g(t1−t0)

−

x1

v1

w1

.

.

.

T(tn+1, tn, xn,R(tn, xn,˙

xn))

xn+R1(tn, xn,˙

xn)(tn+1 −tn)

R1(tn, xn,˙

xn)

R2(tn, xn,˙

xn)−g(tn+1 −tn)

−

xn+1

vn+1

wn+1

.

.

.

T(t0+p, tk−1, xk−1,R(tk−1, xk−1,˙

xk−1))

xk−1+R1(tk−1, xk−1,˙

xk−1)(t0+p−tk−1)

R1(tk−1, xk−1,˙

xk−1)

R2(tk−1, xk−1,˙

xk−1)−g(t0+p−tk−1)

−

x0

v0

w0

(10)

where ϕ= (t0, x0, v0, w0, . . . , tk−1, xk−1, vk−1, wk−1). Zeros of fmust satisfy (4), (7) and (9) and thus deﬁne

periodic orbits ϕof kimpacts.

Remark 2.1. is treated here as a constant but will later be considered a variable. The domain of fwill

thus change from R4kto R4k+1.

For a periodic orbit ϕto exist with e6= 1, there must be a balance between the loss in the particle’s

kinetic energy due to the coeﬃcient of restitution and the gain that the surface induces upon impact.

Hence, it should be suspected that no periodic orbits can exist if , the amplitude of the oscillating surface,

is too small.

6K. E. M. Church &C. Fortin

2.2. Symmetric periodic orbits and their properties

The following deﬁnitions specify types of periodic orbits.

Deﬁnition 2.1. We call periodic solutions with k > 1symmetric and denote them ϕsym if they have

opposite horizontal coordinates between consecutive impacts and equal vertical coordinates for all impacts.

That is,

xn+1 =−xn, yn+1 =yn,

vn+1 =−vn, wn+1 =wn,

for all n∈ {0,1, . . . , k −1}.

Deﬁnition 2.2. Periodic orbits that have xn=vn= 0 for all n∈ {0,1, . . . , k −1}are called trivial as they

describe a one-dimensional motion.

Intuitively, it might seem like nontrivial symmetric solutions should arise for any value of α, given ﬁxed

values of k, p, e, and . However, as the following proposition illustrates, there exists only one speciﬁc α

for which this is possible.

Proposition 1. For any pair (k, p)∈N2and (e, )∈R2, nontrivial symmetric periodic orbits can only

exist if αgp2= 2k2.

Proof. Since symmetric solutions must satisfy f(ϕsym) = 0, we have that v+

n−vn+1 = 0 and hence

v+

n=vn+1 =−vn. Therefore,

xn+v+

n(tn+1 −tn)−xn+1 =−xn+1 −v+

n+1(tn+2 −tn+1) + xn+2 = 0,

vn+1(tn+1 −tn) = vn+1 (tn+2 −tn+1),

tn+1 −tn=tn+2 −tn+1,

where we have used the fact that xn=xn+2. By transitivity, we infer that the time between two impacts

is the same for any two impacts. Let us now determine this value. Remark that t2−t1=t1−t0⇐⇒

t2= 2(t1−t0) + t0. This establishes the base case in our inductive argument. Suppose ti=i(t1−t0) + t0

for some i > 0, then ti+1 −ti=ti−ti−1=· · · =t1−t0by the above argument. Therefore, ti+1 =

i(t1−t0) + t0+t1−t0= (i+ 1)(t1−t0) + t0and by induction, tn=n(t1−t0) + t0for all n∈N∗.

Recall that for kimpacts, one has that tk=t0+p, where pis the total period between the ﬁrst impact

and the last. Thus,

tk=t0+p=k(t1−t0) + t0,

t1−t0=p

k.

We now show that if xn6= 0, ϕsym can only exist for a speciﬁc value of α. From

xn+v+

n(tn+1 −tn)−xn+1 = 0,

yn+w+

n(tn+1 −tn)−g

2(tn+1 −tn)2=yn=yn+1,

we get that the post-impact velocity components are

v+

n=−2xnk

p=−vn, w+

n=gp

2k.

Moreover, 0 = w+

n−g(tn+1 −tn)−wn+1 =gp/2k−gp/k −wn+1 ⇐⇒ wn+1 =wn=−gp/2k. Denote

Computer-assisted methods for periodic orbits in vibrating gravitational billiards 7

βn:= 2αxn=αvnp/k and consider the ﬁrst output of the Rmap for the nth impact,

v+

n=(1 −eβ2

n)vn+βn(1 + e)wn−2πβncos(2πtn)(1 + e)

1 + β2

n

,

−2xnk

p(1 + β2

n) = (1 −eβ2

n)2xnk

p−(1 + e)βngp

2k−2πβncos(2πtn)(1 + e),

0 = 2βnk

αp +β3

nk

αp (1 −e)−βngp

2k(1 + e)−2πβncos(2πtn)(1 + e),

0 = βnβ2

nk

αp (1 −e) + 2k

αp −(1 + e)gp

2k−2π cos(2πtn)(1 + e).

Since βn= 2αxnwith xn6= 0, then βn6= 0. Thus,

2π cos(2πtn)(1 + e) = β2

nk

αp (1 −e) + 2k

αp −(1 + e)gp

2k.(11)

We now consider the second output of the Rmap,

w+

n=βn(1 + e)vn+ (β2

n−e)wn+ 2π cos(2πtn)(1 + e)

1 + β2

n

,

gp

2k(1 + β2

n) = (1 + e)β2

nk

αp + (e−β2

n)gp

2k+ 2π cos(2πtn)(1 + e),

0 = β2

n

αp(1 + e) + gp

2k(e−1) −β2

ngp

k+ 2π cos(2πtn)(1 + e)

and

2π cos(2πtn)(1 + e) = β2

ngp

k−k

αp(1 + e)−gp

2k(e−1).(12)

Equating (11) and (12),

β2

nk

αp (1 −e) + 2k

αp −(1 + e)gp

2k=β2

ngp

k−k

αp(1 + e)−gp

2k(e−1),

β2

nk

αp(1 −e)−gp

k+k

αp(1 + e)= (1 + e)gp

2k−2k

αp −gp

2k(e−1),

β2

n2k

αp −gp

k=−2k

αp −gp

k.

Since βn∈R,then β2

n≥0 and we must have that 2k2=αgp2.

Remark 2.2. Suppose αgp2= 2k2, so the conclusion of the previous proposition holds. We can explicitly

compute the candidates for horizontal impact positions xnbased on the associated impact times. From

(11),

2π cos(2πtn)(1 + e) = β2

nk

αp (1 −e) + 2k

αp −(1 + e)gp

2kwhere α=2k2

gp2, βn= 2αxn,

=8k3x2

n

gp3(1 −e) + gp

2k(1 −e).(13)

Solving for xnwe obtain

xn=±s2π cos(2πtn)(1 + e) + gp

2k(e−1)gp3

8k3(1 −e).(14)

Now suppose for example that k= 2. Since x0=−x1for a nontrivial symmetric solution and t1=t0+p

for some positive integer p, it suﬃces to choose some t0such that the radical of the above equation (for

n= 0) is positive. The particular situation where (e, ) = (1,0) is treated in [Korsch & Lang, 1991].

8K. E. M. Church &C. Fortin

Remark 2.3. The time diﬀerence of two subsequent impacts p/k has to be a multiple of the surface’s period

of oscillation, which is given by 1. This comes from the deﬁnition of a symmetric solution as well as the

injective (bijective) behavior of the reset law when considering one side of the y-axis. Physically, the particle

has to bounce back from a given position (x, y) with a speciﬁc velocity vector induced by the surface’s

normal force. Hence, p/k ∈N∗.

Referring to Remark 2.2, suppose we restrict to the case k= 2. There is a continuous interval of time

values for which the radical of (14) is nonnegative. Since these time values uniquely determine symmetric

solutions with k= 2 impacts per period, such solutions are parameterized by a continuous parameter,

namely t0. As such, these symmetric solutions are not isolated and hence do not qualify as nondegenerate

zeros of f, see Deﬁnition 1.1. This hinders the process of identifying such solutions as the injective linear

operator Agiven by the inverse of the Jacobian will suﬀer from numerical instabilities. A similar argument

applies for nontrivial symmetric solutions with a generally even number of impacts per period. For this

reason, we will not be searching for nontrivial symmetric periodic orbits.

We now characterize the existence of trivial ϕsym using an adapted result of [Luo & Han, 1996] from

the Period-1 Motion section.

Proposition 2. Trivial (symmetric) solutions either exist in pair or do not exist. Additionally, if the

periodic orbit’s impacts are not perfectly elastic, such solutions do not exist for

∈0,gp

4πk

1−e

1 + e.(15)

Proof. By deﬁnition, a trivial solution satisﬁes xn= 0, vn= 0, wn=−gp/2ksuch that

R2t0,0,0,−gp

2k=gp

2k=egp

2k+ 2π cos(2πt0)(1 + e),

cos(2πt0) = gp

4πk

1−e

1 + e.(16)

for all n∈ {0,1, . . . , k −1}. Since the coeﬃcient of restitution is smaller or equal to one, the last equation’s

right-hand side is always nonnegative. Therefore, whenever (k, p) are such that the latter is smaller than

one, exactly two diﬀerent values of t0will satisfy this equation:

t0=1

2πarccos gp

4πk

1−e

1 + e, t0= 1 −1

2πarccos gp

4πk

1−e

1 + e.

We now consider the second claim. If = 0, then necessarily e= 1 and the motion is perfectly elastic. If

instead > 0, we use (16) to get

2π(1 + e)≥2π cos(2πt0)(1 + e) = gp

2k(1 −e),

from which we conclude that must satisfy (15).

3. Computer-assisted proof of solutions and branches

With the equivalence between the zero-ﬁnding problem (10) and periodic orbits established, we will now

show how to prove the existence of periodic orbits. We start with isolated zeroes in Section 3.1 before

moving to branches in Section 3.2.

3.1. Isolated periodic solutions

To ﬁnd a solution, we use Newton’s method in higher dimensions. We start with an initial guess ϕ(0) for a

solution of the fmap. Newton’s method deﬁnes a sequence of vectors {ϕ(0), ϕ(1) ,...ϕ(n), ϕ(n+1), . . . }such

that

ϕ(n+1) =ϕ(n)−(Df (ϕ(n)))−1f(ϕ(n)).

Computer-assisted methods for periodic orbits in vibrating gravitational billiards 9

If the zero-ﬁnding problem evaluated at ϕ(n+1) is within a certain tolerance of zero such that kf(ϕ(n+1))k<

tol, we stop the process and denote ¯ϕ=ϕ(n+1) a numerical solution of the system. The nonzero partial

derivatives of the Jacobian matrix Df are shown in Appendix A.

The following theorem will be used to construct an enclosure of an approximate zero ϕof the map f

of (10) – that is, a closed ball Br(ϕ) in which fhas a unique zero.

Theorem 1. Let Xbe a Banach space and T:X→Xa Frchet diﬀerentiable mapping. Let x0∈X, and

suppose that r∗, Y, Z > 0have

kT(x0)−x0kX≤Y, (17)

kDT (z)kB(X)≤Z, where z∈Br∗(x0).(18)

Deﬁne the linear radii polynomial

p(r)=(Z−1)r+Y.

If there exists r0such that r∗≥r0>0and p(r0)<0,then there is a unique ﬁxed point ˜x∈Br0(x0)of

the contraction T.

Proof. Assume that r∗≥r0>0 has that p(r0)<0. Then,

Zr0+Y < r0,

and since r06= 0 we have that

Z+Y

r0

<1.

Since Y, r0are positive we have that Z < 1. For x, y ∈Br0(x0) we have that

kT(x)−T(y)kX≤sup

z∈Br0(x0)

kDT (z)kB(X)kx−ykX≤Zkx−ykX.

Since Z < 1, Tis a contraction on Br0(x0). To see that Tmaps the closed ball into itself, choose x∈Br0(x0).

Then,

kT(x)−x0kX≤ kT(x)−T(x0)kX+kT(x0)−x0kX

≤sup

z∈Br0(x0)

kDT (z)kB(X)kx−x0kX+Y

≤Zr0+Y

< r0.

It follows from the contraction mapping theorem that there exists a unique ﬁxed point ˜x∈Br0(x0) of T.

In this case, the contraction mapping T:R4k→R4kis deﬁned as T(ϕ):=ϕ−Af(ϕ), where R4k

equipped with the sup-norm is a Banach space. Remark that f∈C∞, which allows us to set A:=

(Df ( ¯ϕ))−1and Y:=kDf( ¯ϕ)−1f( ¯ϕ)kwhere we compute Anumerically for each ¯ϕ∈R4k. In particular,

the invertibility of the linear operator A yields that D T ( ¯ϕ) is well-deﬁned over all of R4k. With this in

mind, deﬁne Br∗( ¯ϕ)⊂R4ka ball of radius r∗centered at ¯ϕwith r∗>0. Since r∗is ﬁnite, Br∗( ¯ϕ)⊂R4kis

a compact subset and DT is bounded over Br∗( ¯ϕ) by the extreme value theorem. We can now set

Z:= sup

z∈Br∗( ¯ϕ)

kDT (z)k∞, Y :=kT( ¯ϕ)−¯ϕk∞(19)

with the radii polynomial given by

p(r)=(Z−1)r+Y.

10 K. E. M. Church &C. Fortin

Applying Theorem 1 and denoting r∗≥r0>0 such that p(r0)<0, we conclude that there exists a unique

equilibrium solution ˜ϕin the ball Br( ¯ϕ) for any r∈I= [r0, r∗], where Iis called the existence interval.

Lastly, ˜ϕsatisﬁes f( ˜ϕ) = 0 with Df ( ˜ϕ) invertible. In practice, Zis computed with INTLAB by computing

DT (z) for a thick interval zcorresponding to the ball Br∗(ϕ).

Since a periodic orbit ϕis equivalent (by the zero-ﬁnding problem) to a set of vectors, one can rep-

resent it in diﬀerent ways that are in some sense equivalent. The following proposition characterizes this

equivalence.

Proposition 3. An asymmetric periodic orbit ϕwith kimpacts has kequivalent ways of being represented

at the dynamics level, in the sense that these orbits trace the same path in state space. If ϕis a nontrivial

periodic orbit, it is equivalent to a family of 2knontrivial periodic orbits by way of horizontal reﬂections

and cyclic permutations of coordinates in terms of the variables of the fmap.

Proof. Let ϕ∈R4ka periodic orbit representing the sequence of vectors {φ0, φ1, . . . , φk−1}where φn=

(tn, xn,˙

xn). Since there are kmany vectors, there are kcyclic permutations of the sequence, namely

{φ0, φ1, . . . , φk−2, φk−1},

{φ1, φ2, . . . , φk−1, φ0},

.

.

.

{φk−1, φ0, . . . , φk−3, φk−2},

where the time values are modiﬁed such that t0∈[0,1). The ordered sequence of impact coordinates shown

above can be found by applying the maps R,Tand Siteratively. Since the path drawn by the particle

throughout the periodic orbit is the same for each of the above sequences, the latter all have equivalent

dynamics. In the case where ϕis not a trivial periodic orbit, each sequence has a reﬂection with respect to

the y axis, see Figure 1 for a visual representation. However, a sequence and its reﬂection do not draw the

same path in parameter space. Thus they do not have equivalent dynamics and are only equivalent at the

fmap’s level.

Fig. 1. Simpliﬁed visual representation of two nontrivial periodic orbits that are the reﬂection of one another. Three permu-

tations are possible as there are k= 3 impacts. The surface’s oscillation is not shown for presentation purposes.

Remark 3.1. The Zbound depends on the size of r∗, thus the choice of r∗will be case dependent so as to

maximize the existence interval of the unique equilibrium solution ˜ϕ. In this case, r0provides tight bounds

on the location of the real unique solution ˜ϕ, whereas r∗corresponds to the latter’s domain of isolation.

With the method outlined above, we rigorously compute solutions of the fmap for diﬀerent combinations

of the parameters (α, e, ) and (k, p). Since periodic orbits of one impact only bounce vertically at x= 0,

we consider cases with k > 1. To ﬁnd solutions, we generate random guesses that live in 4kdimensions.

We will restrict the search to two impacts, as it requires less computation and facilitates presentation.

Computer-assisted methods for periodic orbits in vibrating gravitational billiards 11

However, our code is general, and using it together with Theorem 1, we are able to prove solutions for any

(k, p).

3.1.1. Results for parameters g= 9.81 and (α, e, ) = (2,0.778,1)

Let g= 9.81 and (α, e, ) = (2,0.778,1) such that nontrivial ϕsym do not exist. Since t0= 0 and t2=p+t0,

the zero-ﬁnding problem is of eight dimensions. Indeed, the variables are ϕ= (t0, x0, v0, w0, t1, x1, v1, w1)

with y0=αx2

0+sin(2πt0) and y1=αx2

1+sin(2πt1). The zero-ﬁnding problem (10) then becomes

f=

T(t0, x0,R(t0, x0,˙

x0))

x0+R1(t0, x0,˙

x0)(t1−t0)

R1(t0, x0,˙

x0)

R2(t0, x0,˙

x0)−g(t1−t0)

−

x1

v1

w1

T(t1, x1,R(t1, x1,˙

x1))

x1+R1(t1, x1,˙

x1)(p+t0−t1)

R1(t1, x1,˙

x1)

R2(t1, x1,˙

x1)−g(p+t0−t1)

−

x0

v0

w0

.(20)

To ﬁnd zeroes, we apply Newton’s method numerically until fattains a certain tolerance. In this case,

we set kf( ¯ϕ)k∞<tol such that tol = 10−13. We search for solutions that have periods between p= 1 and

p= 10, as there is an abundance of orbits for these values. Furthermore, for a ﬁxed period, we focus on

p= 2 as it is the least value that allows for (trivial) symmetric solutions. For these cases, we give a lower

bound on the total number of periodic orbits that exist. The plots in Figure 2 correspond to one complete

period of solutions that have (k, p) = (2,2).

Fig. 2. Trajectory of periodic orbits (red) bouncing on the oscillating surface and its corresponding impact coordinates. Left:

nontrivial (asymmetric) orbits. Right: trivial orbits. The uniqueness intervals I= [Imin, Imax] all have Imin ≤1.9·10−10 and

Imax ≥8.5×10−6. Their coordinates are provided in Table 3.1.1.

With Proposition 3 in mind and the computer-assisted proof complete, we are able to give a more

accurate lower bound on the number of periodic orbits there are for the parameter set in question.

Theorem 2. For g= 9.81,(α, e, ) = (2,0.778,1) and k= 2, there are at least 608 periodic orbits with

12 K. E. M. Church &C. Fortin

Table 1. Coordinates (in impact time and state-space location) of the periodic orbits from Figure 2.

Figure t0x0v0w0t1x1v1w1

Top Left 0.76171 -0.10155 -0.25742 -9.64597 0.84303 0.39236 6.07384 5.14225

Top Right 0.23446 0 0 -4.905 1.23446 0 0 -4.905

Bottom Left 0.73258 1.65511 3.31680 4.23915 2.21578 -0.05902 -1.15570 -9.63543

Bottom Right 0.09841 0 0 2.60147 1.73768 0 0 -9.00244

their ﬁrst bounce t0occurring in the interval [0,1), with

8periodic orbits for p= 1,40 periodic orbits for p= 2,

42 periodic orbits for p= 3,46 periodic orbits for p= 4,

52 periodic orbits for p= 5,70 periodic orbits for p= 6,

68 periodic orbits for p= 7,92 periodic orbits for p= 8,

92 periodic orbits for p= 9,98 periodic orbits for p= 10.

In a given class of (k, p), these periodic orbits are isolated.

The data (e.g. coordinates, radii of isolation) associated to these periodic orbits can be found at the

second author’s GitHub2. Note that there exist periodic orbits for periods larger than p= 10 as well. For

each period p, equation (16) is used to verify the existence of a pair of trivial symmetric orbits. As per

Proposition 3, some of the asymmetric solutions of the theorem trace out the same path in the phase space.

However, in the extended phase space consisting of spatial and time coordinates (with initial impact times

normalized to t0∈[0,1)), they are distinct.

3.2. Branches of periodic orbits: rigorous parameter continuation

After rigorously identifying periodic orbits of the fmap, one wonders how these solutions behave as we vary

a given parameter. For that, we use a predictor-corrector algorithm, meaning that we produce an initial

guess at a point lying on the branch of solutions and correct it using Newton’s method until it converges

within a prescribed tolerance. In particular, we use a method called pseudo-arclength continuation. This

method considers the parameter as a variable and parameterizes the branch of solutions by pseudo-arclength

using the parameter ∆s > 0. This allows for continuation past saddle-node bifurcations. The method

outlined in this section allows one to rigorously prove the existence of a smooth solution curve between

two points lying on the curve. Furthermore, we show that the union of connecting smooth curves yields

a smooth curve and we provide a strategy for detecting secondary bifurcations. For proofs of Theorem 3,

Theorem 4, Theorem 5 and Corollary 3.1 please refer to [Breden et al., 2013].

We focus on doing continuation in the surface’s amplitude of oscillation and consider the other

parameters as constants. As previously stated in Remark 2.1, becomes a variable and the new unknown

variable which describes the periodic orbit is Φ = (ϕ, ). We redeﬁne the fmap such that f:R4k+1 →R4k

where the problem becomes f(Φ) = 0. We build the predictor by computing a unit tangent vector ˙

Φ0∈

R4k+1 to the branch at the solution point Φ0using the fact that

DΦf(Φ0)·˙

Φ0= 0 ∈R4k.(21)

In MATLAB, the null space is computed using singular value decomposition. The predictor is thus given

by

ˆ

Φ1:= Φ0+ ∆s˙

Φ0∈R4k+1.

To correct the predictor, we consider the hyperplane perpendicular to the tangent vector ˙

Φ0at the

predictor:

E(Φ) := (Φ −ˆ

Φ1)·˙

Φ0

2https://github.com/ClementFortin/BilliardOscillatingParabola

Computer-assisted methods for periodic orbits in vibrating gravitational billiards 13

and we apply Newton’s method to the new zero-ﬁnding problem F:R4k+1 →R4k+1 deﬁned by

F(Φ) :=E(Φ)

f(Φ)= 0,(22)

where the initial condition is given by the predictor ˆ

Φ1. Denote the new solution by Φ1where kF(Φ1)k<tol.

To prove the existence of this numerical solution, we apply Theorem 1 just as previously.

We now show that there exists a smooth curve between Φ0and Φ1parameterized by Φ = Φswith

s∈[0,1] such that F(Φs)≈0. We ﬁrst compute a unit tangent vector to the curve ˙

Φ1at the point Φ1by

making use of equation (21) and we denote

Φs= Φ0+s∆Φ,

˙

Φs=˙

Φ0+s∆˙

Φ,

where ∆Φ = Φ1−Φ0and ∆ ˙

Φ = ˙

Φ1−˙

Φ0. Moreover, let Es(Φ) := (Φ −Φs)·˙

Φssuch that

Fs(Φ) :=Es(Φ)

f(Φ) .

The following theorem proves the existence of a solution curve between Φ0and Φ1.

Theorem 3. Consider f∈Ck(Rn+1,Rn)with k∈ {2, ..., ∞}. For any s∈[0,1] ,consider the pre-

dictors Φsand the tangent vectors ˙

Φswith Fsdeﬁned as in (22). Let A∈Mn+1(R)be such that

A≈DΦF(Φ0)−1. Let r∗>0be such that Br∗(Φs)⊂Rn+1 is a cylinder. Consider nonnegative bounds

Y0,ˆ

Y0, Z0,ˆ

Z0and Z2: (0, r∗]→[0,∞)satisfying

kAF0(Φ0)k ≤ Y0,(23)

kAFs(Φs)−F0(Φ0)k ≤ ˆ

Y0,for all s∈[0,1] (24)

kId −ADΦF0(Φ0)k ≤ Z0,(25)

kADΦFs(Φs)−DΦF0(Φ0)k ≤ ˆ

Z0,for all s∈[0,1] (26)

kADΦFs(C)−DΦFs(Φs)k ≤ Z2(r)r, for all C∈Br(Φs)with r∈(0, r∗].(27)

Deﬁne the radii polynomial

p(r):=Z2(r)r2−(1 −Z0−ˆ

Z0)r+Y0+ˆ

Y0.

If there exists r0∈(0, r∗]such that p(r0)<0, then there exists a function

˜

Φ: [0,1] →[

s∈[0,1]

Br0(Φs)

with ˜

Φ∈Ck(0,1),Rn+1, and such that

f(˜

Φ(s)) = 0,for all s∈[0,1] .

In this case, the solution curve is parameterized by ˜

Φ and its existence interval is given by a cylinder. The

existence interval of this solution curve is given by I= [rmin, rmax]∩(0, r∗] such that any r∈Iyields

p(r)<0. The function p(r) is called a radii polynomial.

Remark 3.2. It is often hard to set the Z2bound of (27) such that r > 0 with p(r)<0. As shown in

[Calleja et al., 2019], a general constant bound Z2=Z2(r∗) satisfying (27) and obtained via the mean

value inequality is provided in the case of the ∞-norm:

Z2(r∗) = sup

b∈Br∗(Φs)

max

1≤i≤n+1 X

1≤k,m≤n+1 X

1≤j≤n+1

Aij

∂2(Fs)j

∂Φm∂Φk

(b)

.(28)

14 K. E. M. Church &C. Fortin

Note that we do not employ any Z1bounds in this paper, as these are only necessary when an approximate

inverse is used in the context of inﬁnite-dimensional zero-ﬁnding problems [Calleja et al., 2019; Lessard

et al., 2017].

Theorem 4. Suppose the hypotheses of Theorem 3 are satisﬁed with the norm k·k∞, yielding a branch of

solution curve S0:=∪s∈[0,1] ˜

Φ(s). If

−∆Φ ·˙

Φ0+r0k∆˙

Φk∞+|∆Φ ·∆˙

Φ|<0,

then S0is a smooth curve, that is, for all s∈[0,1] , ds˜

Φ(s)6= 0.

Assume we have computed two smooth solution curves, S0and S1using the above results. We want

to prove that the curves connect smoothly and hence produce a single smooth curve.

Theorem 5. Let S0and S1be two smooth curves, such that the hypotheses of Theorem 3 and Theorem 4

are satisﬁed between Φ0and Φ1, and Φ1and Φ2, respectively. Then, S0∪ S1is a smooth curve.

Theorem 3, Theorem 4 and Theorem 5 allow one to prove the existence of a global smooth curve

S:=

j−1

[

i=0

Si

of f(Φ) = 0 for jiterations, starting at the numerical solution Φ0which was found using Newton’s method

together with Theorem 1. As the shape of the smooth curve Sis often unknown, it is beneﬁcial to be able

to recognize any secondary bifurcations that the solution curve might be undergoing. Hence, the following

corollary provides a condition for verifying the path’s regularity.

Corollary 3.1. Assume that the hypotheses of Theorem 3 and Theorem 4 hold. Then, for every s∈

[0,1] ,dim Ker DΦf˜

Φ(s)= 1,that is ˜

Φ(s)is a regular path.

As is shown in Section 3.2.1, strong evidence suggest that branch solutions in the billiard always appear

from pitchfork bifurcations.

3.2.1. Results for parameters g= 9.81 and (α, e) = (2,0.778)

Applying Theorem 3 and verifying both Theorem 4 and Corollary 3.1 at every step, we do rigorous pseudo-

arclength continuation in the parameter for solutions of (α, e, ) = (2,0.778,1) and (k, p) = (2,2). We

continued from all of the (k, p) = (2,2) periodic orbits of Theorem 2 until the radii polynomial method

failed. The result is we were able to prove the following.

Theorem 6. The curves appearing in Figure 3 and Figure 6 correspond uniquely to curves of periodic

orbits of type (k, p) = (2,2) of the gravitational billiards with parameters g= 9.81 and (α, e) = (2,0.778)

for varying amplitude . These branches are located in the half-space ≥0.195, in the sense that at least

one periodic orbit (corresponding to a point on the branch) exists at = 0.195.

The branches seem to meet at several points in Figure 3. However, rigorously determining existence

of a smooth curve passing through either of these points proves diﬃcult as the Z2bound in (27) blows

up, even when using (28). Moreover, the kernel of the numerical approximation of DΦfat these points is

two-dimensional, and hence the curve does not deﬁne a regular path, see Corollary 3.1. By plotting the

radius of the domain of uniqueness obtained from Theorem 3 vs the amplitude of oscillation, we observe

sharp drops at particular amplitudes; see Figure 5. This is readily explained from the fact that branches

converge towards these values and thus get closer together. All of this strongly indicates the existence of

pitchfork bifurcations at the points represented by black dots in Figure 3. See Figure 4 for a more global

view of the (likely) bifurcation diagram.

In the left pane of Figure 3, two sets of horizontally opposite periodic orbit come into existence at

≈4.45 and at ≈1.545 before connecting in saddle-nodes at ≈0.511. In the right pane, pitchfork

Computer-assisted methods for periodic orbits in vibrating gravitational billiards 15

Fig. 3. Plots of connected branch solutions at (α, e) = (2,0.778) and (k, p) = (2,2). Left: 5 solution branches in the (t0, v0)

projection. Right: 8 diﬀerent branches in the (t1, x0) projection. Branches meet at black dots at ≈1.545 and ≈4.45 on the

top left and at ≈1.667 and twice at ≈0.195 on the top right.

0 1 2 3 4 5 6

6

7

8

9

10

11

12

13

14

15

Fig. 4. Bifurcation diagram in the the 2-norm of periodic orbit representatives ϕ(i.e. ﬁxed points of the map (10)). The pale

blue curve corresponds to the branches appearing in the left pane of Figure 3, and the black curve to the right pane. We have

windowed to solutions to 6 ≤ ||ϕ||2≤15, since this region contains all likely pitchfork bifurcations and most of the branch

structure. Intersections of the black and blue curves are not bifurcation points, since these two curves are spatially isolated

from each other.

bifurcations are hypothesized to exist at amplitudes of ≈0.195 and ≈1.667. For (k, p) = (2,2), we

make the following distinctions between branch solutions.

(i) Child branches: branches that appear out of pitchfork bifurcations;

(ii) Parent branches: branches that beget child branches;

(iii) Wild branches: branches that are not connected to pitchfork bifurcations.

Since branches of the ﬁrst family are born through pitchforks, they always come in pairs from branches of

the second family. The yellow and red curves in the left pane of Figure 3 and the (left) dark blue, yellow,

orange and purple curves in the right pane are examples of child branches. The green (purple) and blue

curves in the left pane and the dark red, (right) dark blue, light blue and green curves in the right pane

are examples of parent branches. Using this classiﬁcation, it is possible for a branch to be both a child and

parent branch.

16 K. E. M. Church &C. Fortin

1.5 2 2.5 3 3.5

0

0.5

1

1.5

2

2.5

310-4

1234567

0.5

1

1.5

2

2.5 10-5

Fig. 5. Plots of the radius of the domain of uniqueness as a function of the amplitude of oscillation. The left plot corresponds

to the green curve in the top left pane of Figure 3, and the right plot corresponds to the red curve lying on the x0= 0 line in

the top right pane. Sharp drops in rmax are observed at ≈1.545 and ≈1.667, respectively.

Near pitchforks, the step size ∆shas to be carefully chosen so that it is not too large that it crosses

the bifurcation, in which case the Z2bound blows up, but not too small that it takes too long to advance

near the pitchfork. Therefore, it might happen that wild branches do not actually exist but are rather

a consequence of a poorly chosen step size and bifurcate oﬀ of a parent branch. In any case, potential

candidates of this branch family are shown in Figure 6.

0 2 4 6 8 10 12

12

13

14

15

16

17

18

19

20

21

Fig. 6. Wild branch candidates for (α, e) = (2,0.778) and (k, p) = (2,2). Left: 6 branches in the (v0, w0) projection. Right:

Two-norm of the solution branches of the left plot.

Remark 3.3. The code provided through Github3uses an adaptive step size, as it is dependent upon i,

the number of times the Newton operator has been applied to reach convergence in the previous proof of

existence of solution branch segment. The step size of the (n+ 1)th iteration is given by

∆s(n+1) = 24−i

3∆s(n),

with an initial (and maximum) step size of ∆s(0) = 0.1. Nonetheless, one might notice that convergence is

sometimes hard to achieve even with a small step size. Indeed, there are unstable (in the sense of iteration

of the Newton operator) eight-dimensional zones near branch solutions. In some cases, a small step size is

enough to yield a diverging sequence. One can usually solve this problem by taking a larger step size.

3https://github.com/ClementFortin/BilliardOscillatingParabola

REFERENCES 17

In Figure 3, branches always stop before the surface’s amplitude becomes zero. Indeed, child branches

connect back to other child branches through saddle-nodes and parent branches stop at speciﬁc points. For

collisions that are not perfectly elastic i.e. e < 1, existence of periodic orbits is simply not possible past a

certain value of . In the particle-surface system, there is a loss in energy that the surface is no longer able

to balance out, preventing the particle from describing a periodic trajectory. The following conjecture is a

generalization of Proposition 2.

Conjecture 1. For ﬁxed numbers of impacts k, period pand parameters (g, α, e), there is a critical amplitude

ξ > 0 such that no periodic orbit exist for ∈[0, ξ).

A consequence of Theorem 6 is that if the above conjecture is true, the critical amplitude for (g, α, e) =

(9.81,2,0.778) and (k, p) = (2,2) satisﬁes ξ < 0.195. While physically this conjecture seems perfectly

reasonable, we make no attempt to prove it here.

4. Concluding Remarks

The motion of a particle is studied in a domain bounded by a vertically-vibrating parabola. The particle is

subject to a constant gravitational ﬁeld. We build a nonlinear map in (10) for which the zeros correspond

to periodic orbits of the particle-boundary system. We then classify periodic orbits as symmetric when

the path drawn by the particle is symmetric with respect to the vertical axis, and prove some elementary

results concerning when such orbits can exist. Using a computer-assisted approach, we rigorously prove

the existence of at least 608 asymmetric or trivial (i.e. vertical motion only) periodic orbits for speciﬁc

test parameter values. We then do rigorous pseudo-arclength continuation in the boundary’s amplitude of

oscillation. Again, using computer-assisted proofs, we provide strong evidence for the existence of pitchfork

bifurcations while proving the existence of some global smooth branches of periodic orbits. Lastly, we

conjecture that no periodic orbit exist for a small enough value of the boundary’s amplitude of oscillation,

and rigorously compute an upper bound for this critical amplitude for the test parameters. It remains

to rigorously study the likely pitchfork bifurcations, characterize wild orbits, and study the behavior of

(nontrivial) symmetric periodic orbits as the vibration amplitude varies. We have proposed that periodic

orbits can not persist below a critical amplitude of surface oscillation, and have computed an upper bound

for this critical amplitude in a particular case. It would be interesting to explore this conjecture in more

detail even for the particular test parameters and (k, p) = (2,2) periodic orbits, as we have done here in a

preliminary way.

Acknowledgments

Thank you to the anonymous reviewers, whose comments led to several improvements to the paper. We

thank Olivier H´enot for providing the proof of Theorem 1. Kevin E. M. Church acknowledges the support of

NSERC (Natural Sciences and Engineering Research Council of Canada) through the NSERC Postdoctoral

Fellowships Program.

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Appendix A Partial Derivatives

For (tn, xn, vn, wn, tn+1, xn+1 , vn+1, wn+1 ) and f= [f1, f2, . . . , f4k]|, we present the entries of the Jacobian

Df of (10). The nonzero derivatives with respect to the function’s n+ 1 row i.e. Tn, are given by

∂f4n+1

∂tn

= 2αxn+v+

n(tn+1 −tn) ∂v+

n

∂tn

(tn+1 −tn)−v+

n!−2π cos(2πtn) + w+

n

− ∂w+

n

∂tn

+g!(tn+1 −tn),

∂f4n+1

∂xn

= 2α(tn+1 −tn) v+

n+xn

∂v+

n

∂xn!+ 2αv+

n

∂v+

n

∂xn

(tn+1 −tn)2

−

∂w+

n

∂xn

(tn+1 −tn),

∂f4n+1

∂vn

= 2α∂v+

n

∂vn

(tn+1 −tn)[xn+v+

n(tn+1 −tn)] −

∂w+

n

∂vn

(tn+1 −tn),

∂f4n+1

∂wn

= 2α∂v+

n

∂wn

(tn+1 −tn)[xn+v+

n(tn+1 −tn)] −

∂w+

n

∂wn

(tn+1 −tn),

∂f4n+1

∂tn+1

= 2αv+

nxn+v+

n(tn+1 −tn)+ 2π cos(2πtn+1 )−w+

n+g(tn+1 −tn).

REFERENCES 19

The derivatives of the n+ 2, n + 3 and n+ 4 rows are

∂f4n+2

∂tn

=∂v+

n

∂tn

(tn+1 −tn)−v+

n,∂f4n+2

∂xn

= 1 + ∂ v+

n

∂xn

(tn+1 −tn),∂f4n+2

∂vn

=∂v+

n

∂vn

(tn+1 −tn),∂f4n+2

∂wn

=∂v+

n

∂wn

(tn+1 −tn),

∂f4n+2

∂tn+1

=v+

n,∂f4n+2

∂xn+1

=−1,∂f4n+2

∂vn+1

= 0,∂f4n+2

∂wn+1

= 0.

∂f4n+3

∂tn

=∂v+

n

∂tn

,∂f4n+3

∂xn

=∂v+

n

∂xn

,∂f4n+3

∂vn

=∂v+

n

∂vn

,∂f4n+3

∂wn

=∂v+

n

∂wn

,

∂f4n+3

∂tn+1

= 0,∂f4n+3

∂xn+1

= 0,∂f4n+3

∂vn+1

=−1,∂F4n+3

∂wn+1

= 0,

∂f4n+4

∂tn

=∂w+

n

∂tn

+g, ∂f4n+4

∂xn

=∂w+

n

∂xn

,∂F4n+4

∂vn

=∂w+

n

∂vn

,∂F4n+4

∂wn

=∂w+

n

∂wn

,

∂f4n+4

∂tn+1

=−g, ∂f4n+4

∂xn+1

= 0,∂f4n+4

∂vn+1

= 0,∂f4n+4

∂wn+1

=−1,

where the partial derivatives of v+

nand w+

nare taken directly from the reset law,

∂v+

n

∂tn

=8απ2xnsin(2πtn)(e+ 1)

1 + (2αxn)2,∂w+

n

∂tn

=

−4π2sin(2πtn)(e+ 1)

1 + (2αxn)2,

∂v+

n

∂xn

=

−2α(e+ 1) 4α2wnx2

n

−wn+ 2π cos(2πtn)+4αvnxn

−8α2x2

nπcos(2πtn)

(1 + (2αxn)2)2,

∂w+

n

∂xn

=2α(e+ 1) vn

−4α2vnx2

n+ 4αwnxn

−8αxnπcos(2πtn)

(1 + (2αxn)2)2,

∂v+

n

∂vn

=1−4eα2x2

n

1 + (2αxn)2,∂w+

n

∂vn

=2αxn(e+ 1)

1 + (2αxn)2,∂v+

n

∂wn

=2αxn(e+ 1)

1 + (2αxn)2,∂w+

n

∂wn

=4α2x2

n

−e

1 + (2αxn)2.

The partial derivatives with respect to the parameter are given by

∂f4n+1

∂ = 2α∂ v+

n

∂ (tn+1 −tn)xn +v+

n(tn+1 −tn)+ sin (2πtn)−

∂w+

n

∂ (tn+1 −tn),

∂f4n+2

∂ =∂ v+

n

∂ (tn+1 −tn),∂ f4n+3

∂ =∂ v+

n

∂ ,∂ f4n+4

∂ =∂ w+

n

∂

and the derivatives with respect to the parameter yield

∂v+

n

∂ =2πcos (2πtn) (e+ 1) (−2αxn)

1 + (2αxn)2,∂w+

n

∂ =2πcos (2πtn) (e+ 1)

1 + (2αxn)2

where n∈ {0,1, ..., k −1}.