Periodic Motion of a Mass-Spring Systemgremaud/spring_6_01_07.pdf · Periodic Motion of a Mass-Spring System Michael Shearer∗, Pierre Gremaud † and Kristoph Kleiner Department

Periodic Motion of a Mass-Spring System

Michael Shearer∗, Pierre Gremaud† and Kristoph Kleiner

Department of Mathematics and Center for Research in Scientific Computation

North Carolina State University, Raleigh, NC 27695.

June 5, 2007

Abstract

The equations of planar motion of a mass attached to two anchored massless springs form a sym-

metric Hamiltonian system. The system has a single dimensionless parameter L, corresponding to the

spacing between the anchors. For L > 1, there is a stable equilibrium at which the springs are in tension

and lie on a line, but for L < 1, this equilibrium has both springs in compression, and is unstable. How-

ever, there are then two stable equilibria at which both springs carry no force. Oscillations are studied

in both regimes, but more systematically in the tension case, where techniques of bifurcation theory,

numerical approximation and numerical simulation are used to explore the rich variety of periodic solu-

tions.

1 Introduction

Mechanical mass-spring systems provide a rich source of physical examples of dynamical systems, and can

be used as discrete analogs or approximations of continuum descriptions of the deformations and motion

of elastic materials. In this paper, we consider a system for the planar motion of a mass attached to two

anchored springs, shown schematically in Fig.1.1. In the absence of additional effects such as frictional

damping, this system conserves energy and has a high degree of symmetry. We also assume gravity is not

involved; either it is negligible in comparison with the tensions in the springs, or the mass-spring system is

set up on a frictionless horizontal surface. We take Hooke’s law to govern the tension-elongation relation in

each spring, but the system is nonetheless nonlinear unless the mass and the two anchors are aligned during

the motion. The system is a primitive finite difference approximation to the pair of nonlinear wave equations

governing the planar motion of an elastic string1 [1, 5, 7].

After nondimensionalization, the system has a single dimensionless parameter L, corresponding to the spac-

ing between the anchors. For L > 1, there is a stable equilibrium midway between the two springs, at which

∗Partially supported by the National Science Foundation grants DMS 0244488, DMS 0604047†Partially supported by the National Science Foundation grant DMS-04105611A more complete finite difference approximation would involve a system of n ≥ 1 masses, each joined to its neighbor by a

spring, except for the end masses, each of which is joined to an anchor on one side.

1

mass

(x(t), y(t))

-L L x

y

0

Figure 1.1: The mechanical system of two springs and a mass.

the springs are in tension and lie on the line between the anchors, but for L < 1, this equilibrium has both

springs in compression, and is unstable. However, in this case, there are two stable equilibria at which both

springs carry no force. If the x - axis passes through the two anchors, located at x = ±L, then the origin is

an equilibrium, and there are one-dimensional oscillations on the coordinate axes. Moreover, the coordinate

axes are axes of symmetry for the nonlinear system, with corresponding reduction in the Hamiltonian. To

study two-dimensional oscillations, we use a combination of techniques, from bifurcation theory, numerical

approximation and numerical simulation.

In the tensile case, L > 1, we find a rich variety of patterns of periodic motion with varying degrees of

symmetry. Many of these resemble and are related to the well-known Lissajous patterns of simultaneous

decoupled oscillators with rationally related frequencies [2, 3]. Indeed, the system linearized about the

equilibrium is a pair of decoupled oscillators, the frequency of periodic motion on the y - axis depending on

the spacing 2L of the anchors, so that periodic solutions of the linearized system are realized only for discrete

values of L. Correspondingly, oscillations in the nonlinear system also depend on L, and the challenge is to

understand how the two linear oscillators are coupled through nonlinear terms in the equations.

In §2, we formulate and nondimensionalize the system of ordinary differential equations governing the

motion. We also show that longitudinal one-dimensional solutions between the anchors correspond to simple

harmonic motion, whereas transverse one-dimensional oscillations are nonlinear. In §3, we apply bifurcation

theory near the equilibrium to characterize all small amplitude solutions with 1 : 2 linear resonance. In

particular, we find a curve of values of L and amplitude a at which these solutions bifurcate from simple

harmonic motion on the x - axis with small amplitude a. In §4, we analyze the same bifurcation, but as

a period doubling bifurcation from simple harmonic motion on the x - axis. In §5, we explore a class

of nonlinear solutions away from the bifurcation points. A return map helps detect a countable family of

periodic solutions passing through the origin, of which the 1 : 2 resonance is the simplest. In this section,

we also show a more comprehensive bifurcation diagram computed numerically, that contains bifurcation

curves for a range of different patterns of periodic solution, all passing through the origin. Specific solutions

corresponding to the diagram show a range of behaviors, and we supplement this with a gallery of periodic

solutions outside this class. In particular, for L = 0.1, for which the origin is an unstable equilibrium, we

observe more exotic periodic solutions, including one that oscillates near an additional equilibrium on the y- axis.

2

2 Equations and one-dimensional solutions

Consider a mass joined by two identical massless springs to points fixed at (±L, 0) (Fig. 1.1) Assuming the

motion of the mass is in the plane, in the absence of gravity, and assuming Hooke’s law for the springs, we

obtain the system of second order ODE for the position (x(t), y(t)) of the mass at time t :

(

x′′

y′′

)

(t) = −ξ− − 1

ξ−

(

x − Ly

)

− ξ+ − 1

ξ+

(

x + Ly

)

, ξ± =√

(L ± x)2 + y2, (2.1)

These equations have been nondimensionalized as follows. Let ℓ > 0 denote the natural length of the

springs, the boundary between tension and compression, for which there is no force in the spring. Let

m > 0 be the mass of the object with location (x(t), y(t)), and let k > 0 be Hooke’s constant, the elastic

constant. If the springs are fixed at x = ±L, then L = L/ℓ is the dimensionless distance of these fixed

anchor points from the origin. The time scale set by k and m gives the dimensionless time variable√

mk

t.Finally, x and y are measured in multiples of ℓ, and hence are dimensionless. Note that motion with the

springs in extension, ξ > 1, is of most interest, but we also consider motions in which compressive strains

0 < ξ < 1 appear. In particular, for L < 1, compressive strains can be expected.

The equations may be written in a more compact form as

U ′′ = −∇V (U), (2.2)

in which U(t) =

(

xy

)

(t), and the potential V is given by

V (x, y) = f(ξ+) + f(ξ−), f(ξ) =1

2ξ2 − ξ,

Since it is autonomous, the system is invariant under time translation, and it is also time-reversible. Fur-

thermore, the system is symmetric with respect to the coordinate axes: it is invariant under transformations

x → −x and y → −y. Consequently, the system has the form

x′′ = xF (x2, y2)

y′′ = yG(x2, y2),(2.3)

in which F (x2, 0) = −2, G(0, 0) = −2(1 − 1L). In particular, the x - axis and the y - axis are invariant;

motion on the x - axis is simple harmonic motion, with equation x′′ + 2x = 0.

Being a conservative mechanical system, it is Hamiltonian, with Hamiltonian given by the sum of kinetic

and potential energy:

H(x, y, p, q) = 12(p2 + q2) + V (x, y), p = x′, q = y′. (2.4)

The Hamiltonian is constant on trajectories, thereby constraining orbits (x(t), y(t), p(t), q(t)) to lie on the

three-dimensional manifold H = const. Such systems with symmetries have been studied theoretically and

numerically [4, 6].

3

Expanding the potential V (x, y) about the rest state x = y = 0, reveals a little of the structure of the

equations:

V (x, y) = −2L + L2 + x2 +

(

1 − 1

L

)

y2 − 1

L3x2y2 +

1

4L3y4 + ...

In particular,

√

1 − 1L

is the ratio of vibrational frequencies for the system linearized about the equilibrium

at the origin, and the x4 term is missing (as are all higher powers of x alone, since the system is linear when

y = 0).

2.1 One-dimensional oscillations.

Oscillations on the x axis are linear, governed by the equation

x′′ = −2x. (2.5)

Solutions with y = 0 therefore correspond to simple harmonic motion:

x = a sin(√

2t + c), y = 0.

On the y axis, motion is nonlinear:

y′′ = −2y(1 − 1/ξ), ξ =√

L2 + y2, x = 0. (2.6)

For each L > 1, the origin is a center, and there is a one-parameter family of periodic orbits, lying on the

curves H(0, y, 0, y′) = H(0, y0, 0, 0) parameterized by y0 > 0 (see Fig. 2.1):

12y′2 +

(

√

L2 + y2 − 1)2

=

(

√

L2 + y20 − 1

)2

. (2.7)

The period T = T (y0) of these orbits is easily determined from (2.7):

T (y0) = 2√

2

∫ y0

0

dy√

(

√

L2 + y20 − 1

)2−(

√

L2 + y2 − 1)2

. (2.8)

For 0 < L < 1, the motion is rather different, as there are additional equilibria on the y axis, both

centers, in addition to the equilibrium at the origin, which is now a saddle point. The new equilibria y =±√

1 − L2, x = 0 correspond to zero force in the springs, while the springs are compressed when the mass

is at the origin. The phase portrait is shown in Fig. 2.1. Note that time-dependent solutions on the y -

axis with L < 1 involve both compression and tension. Moreover, the origin has two homoclinic orbits;

consequently, the period of the motion around each of the stable equilibria is unbounded, even though the

amplitude of the oscillations is bounded.

4

y

y'

1 0 1 22

1

0

1

2

--

-

0.6

-0.6 0.6-1.2 1.2

0.4

0.2

0.0

0.0

-0.2

-0.4

-0.6

y

y

'

Figure 2.1: Periodic orbits with x = 0. L = 1.9 (left) L = 0.7 (right)

3 Bifurcation near equilibrium

Periodic solutions near the equilibrium (0, 0) can be studied using bifurcation theory. The method we adopt

is Lyapunov-Schmidt reduction to a finite-dimensional problem that can be analyzed simply. The simpler

equations are related to the full problem using the implicit function theorem.

3.1 Linearized equations and resonance.

The origin (x, y) = (0, 0) is an equilibrium for system (2.3). Linearizing the equations about this point, we

obtain the decoupled linear system

x′′ = −2x (a)

y′′ = −2(1 − 1L)y. (b)

(3.9)

Equation (a) has the general solution

x(t) = a sin(√

2t + c), (3.10)

while equation (b) has the general solution

y(t) = b sin

(

√

2(

1 − 1

L

)

t + d

)

. (3.11)

Note that for L ≤ 1, there are no oscillations in y of this form, consistent with Fig. 2.1(right). We refer to

a, b as amplitudes, and c, d as phases. By time translation, we can take one of the phases to be zero, say

d = 0.

5

For the pair (x(t), y(t)) to be periodic, with both amplitudes nonzero, the frequencies√

2,√

2(1 − 1L) have

to be rationally related. Since L > 1, this implies 1 − 1L

= m2

n2 for some positive integers m < n. Then

L =n2

n2 − m2. (3.12)

We refer to this as (m,n) resonance. Trajectories for (3.10), (3.11) are known as Lissajous figures [3], first

studied by Nathaniel Bowditch [2], and later by Jules Antoine Lissajous, who invented an optical device

coupled to tuning forks to visualize the figures. In classifying these figures, the values of m,n are of

primary importance, while the amplitudes and phases have a lesser role, serving to distort the pattern of the

trajectory.

3.2 Lyapunov-Schmidt reduction

In this subsection, we restrict attention to solutions of system (2.3) near the (1, 2) resonance, i.e., (3.10),

(3.11) with L near L = 43 . We also restrict to the case of zero phase: c = d = 0. Thus, the solutions of the

linear problem with L = 43 are

x(t) = a sin(√

2t), y(t) = b sin

(

t√2

)

. (3.13)

We seek periodic solutions with period near that of this solution. It is convenient to scale time so that the

frequency appears explicitly in the equation, and the function space does not change with the frequency.

Accordingly, let

t =√

2ρt, x(t) = x(t), y(t) = y(t)

Thend x(t)

dt=

1√2ρ

dx

dt.

Equations (2.3) become (dropping the tildes),

ρx′′ = xf(x2, y2)

ρy′′ = yg(x2, y2),(3.14)

in which 2f = F, 2g = G; thus, f(x2, 0) = −1, g(0, 0) = −(1− 1L). In fact, from the Taylor expansion of

the right hand sides of (3.14), we have

ρx′′ = −x +1

L3xy2 + h.o.t.

ρy′′ = −(1 − 1L)y +

1

L3x2y − 1

2L3y3 + h.o.t.,

(3.15)

where h.o.t. denotes higher order terms, of order five and higher.

We wish to set these equations up in Hilbert space in order to use the Lyapunov-Schmidt reduction. Ac-

cordingly, we define spaces based on the space L2(−2π, 2π) of functions that are square-integrable on

6

(−2π, 2π), and on the Sobolev space H2(−2π, 2π) of L2(−2π, 2π) functions whose first and second weak

derivatives are also in L2(−2π, 2π).

X = {x(t) : x ∈ L2(−2π, 2π) is 4π periodic, and odd on R, and x(2π − t) = −x(t).}

Y = {y(t) : y ∈ L2(−2π, 2π) is 4π periodic, and odd, and y(2π − t) = y(t).}

X2 = X ∩ H2(−2π, 2π);Y2 = Y ∩ H2(−2π, 2π)

Then X2 × Y2 and X × Y are Hilbert spaces with corresponding H2 and L2 inner products, respectively.

Note that X and Y are separable, so that x ∈ X, y ∈ Y have Fourier series expansions:

x ∈ X implies x(t) =∑∞

n=1 an sin nt,

y ∈ Y implies y(t) =∑∞

n=0 bn sin(n + 12)t,

with∑∞

n=1 a2n < ∞,

∑∞n=0 b2

n < ∞.

We define an operator F : X2 × Y2 × R2 → X × Y by:

F(x, y, ρ, L) = (ρx′′ − xf(x2, y2), ρy′′ − yg(x2, y2)).

The linearization of F about x = y = 0 is contained in (3.15):

DF(0, 0, ρ, L) = (ρx′′ − x, ρy′′ − (1 − 1

L)y).

Setting ρ = 1, L = 43 , DF(0, 0, 1, 4

3 ) has null space N spanned by (sin t, 0), (0, sin t2). Moreover, these

functions span the subspace R⊥ that is the orthogonal complement to the range R of DF(0, 0, 1, 43). Let

P2 : X2 × Y2 → N be the orthogonal projection, and define V2 = (I − P2)(X2 × Y2). Similarly, let

P : X × Y → R⊥, so that I − P : X × Y → R. Then

(I − P )DF(0, 0, 1,4

3)

is a bijection from V2 to R. Now we consider perturbing x and y near zero, ρ near ρ = 1, and L near L = 43 .

In the Lyapunov-Schmidt process, we solve

(I − P )F(a sin t + vx, b sint

2+ vy, ρ, L) = 0 (3.16)

for v = (vx, vy)(a, b, ρ, L) ∈ V2 using the Implicit Function Theorem. Since F is C∞, this yields a smooth

function v : Bǫ(0, 0, 1,43 ) → V2 (where Bǫ(0, 0, 1,

43) ⊂ R

4 is the open ball of radius ǫ, with center

(0, 0, 1, 43)).The function v has the additional properties:

• There is a neighborhood B of zero in V2 such that for each (a, b, ρ, L) ∈ Bǫ(0, 0, 1,43), v = v(a, b, ρ, L)

is the unique solution of (3.16) in B.

• v(0, 0, ρ, L) ≡ 0, ∂∂a

v(0, 0, 1, 43) = 0, ∂

∂bv(0, 0, 1, 4

3) = 0.

• Moreover, v = (vx, vy) inherits the symmetries of F :

7

Lemma 3.1 There are C∞ functions wx, wy from Bǫ(0, 0, 1,43) to V2 that are even in a and in b, such that

vx(a, b, ρ, L) = awx(a, b, ρ, L), vy(a, b, ρ, L) = bwy(a, b, ρ, L).

Proof: The symmetry properties follow from uniqueness of v. Thus, vx is odd in a, and even in b, while vy

is odd in b and even in a. The definition of wx, wy is then a standard construction:

wx =

{

1avx(a, b, ρ, L) if a 6= 0∂∂a

vx(0, b, ρ, L) if a = 0.

Then vx = awx, and wx is C∞. The function wy is defined similarly.

The problem of solving the equation

F(a sin t + vx, b sint

2+ vy, ρ, L) = 0

is now reduced to the two–dimensional problem

PF

(

a[

sin t + wx(a, b, ρ, L)]

, b[

sint

2+ wy(a, b, ρ, L)

]

, ρ, L

)

= 0,

for (a, b, ρ, L) ∈ Bǫ(0, 0, 1,43). Using Taylor series expansions for f, g about (0, 0), this equation can be

expressed as two simultaneous equations for a, b, with parameters ρ, L :

−ρa = −a +ab2

2L3+ af(a2, b2, ρ, L)

−ρb

4= −(1 − 1

L)b +

a2b

2L3− 3b3

16L3+ bg(a2, b2, ρ, L),

(3.17)

where f , g contain terms of higher order. Dropping the higher order terms, we can solve the cubic equations

explicitly. Let

σ = 1 − 1

L− ρ

4.

Then we have solutions

a = 0, b = 0; a = 0, b2 = 16L3

3 σ; b = 0, ρ = 1; (i)

8a2 − 3b2 = σ, b2 = 2(1 − ρ)L3. (ii)(3.18)

In the (a, b) plane, equations (3.18)(ii) represent the intersection of a horizontal line and a hyperbola. The

cases σ > 0 and σ < 0 are shown in Fig. 3.2. Together with the restriction ρ ≤ 1, the intersections shown

in the figure lead to the solvability conditions

ρ ≤ 11

5− 5

5L, and ρ < 1.

These are illustrated in Fig. 3.3(left).

From b = 0 we have ρ = 1, so that

a2 = 2L3

(

3

4− 1

L

)

, (3.19)

8

0 a

b

b = 2(1 - )L2

22

3ρ

b = 2(1 - )L2 3ρ

8a - 3b = σ

0 a

b

b = 2(1 - )L2

22

3ρ

b = 2(1 - )L2 3ρ

8a - 3b = σ

Figure 3.2: Solutions of equations (3.18(ii)). Left: 1 − 1L− ρ

4 > 0; Right: 1 − 1L− ρ

4 < 0.

1.0 2.52.01.50.5

3/4

1/L

0

1/L = 1 - ρ

ρ

/4

1/L = 11/8 - 5ρ /8

1.0 2.52.01.50.5

3/4

1/L

0 ρ

1/L = 11/8 - 5ρ /8

b = 0

a = 0

Figure 3.3: (Left) Parameter values for which equations (3.18(ii)) have solutions. (Right) Parameter values

for which equations (3.17) have solutions.

9

which is also the total energy. Each point of the curve, shown in comparison with other approximations in

Fig. 4.2, represents a period doubling bifurcation point in a constant energy surface H = const.

To include the higher order terms, we consider only solutions in which a and b are positive, and divide the

two equations by a, b respectively. Letting A = a2, B = b2, we thereby reduce equations (3.17) to finding

zeroes of the functions

F1(A,B, ρ, L) = ρ − 1 +B

2L3+ f(A,B, ρ, L)

F2(A,B, ρ, L) =1

4ρ − (1 − 1

L) +

A

2L3− 3B

16L3+ g(A,B, ρ, L).

(3.20)

Then F1 and F2 are zero at ρ = 1, L = 4/3, A = B = 0, and the Jacobian(∂F1, ∂F2)

(∂ρ, ∂L)is clearly nonsin-

gular. Consequently, there are uniquely defined functions ρ = ρ(A,B), L = L(A,B) defined and smooth

near A = B = 0, such that:

Fk(A,B, ρ(A,B), L(A,B)) = 0, k = 1, 2, (3.21)

and ρ(0, 0) = 1, L(0, 0) = 4/3. Since we need A ≥ 0, B ≥ 0, it is significant to know values of ρ, L where

A or B is zero. These are the curves Γ1 = {ρ(A, 0), L(A, 0)} (where B = 0), and Γ2 = {ρ(0, B), L(0, B)}(where A = 0). Differentiating (3.21) with respect to A,B at A = B = 0, we find

∂ρ

∂A(0, 0) = 0,

∂ρ

∂B(0, 0) = − 27

128,

∂L

∂A(0, 0) = −3

8,

∂L

∂B(0, 0) = − 9

64.

These gradients of course relate to the lines in Fig 3.3(left). More precisely, the region in which A and Bare non-negative is bounded by curves that are tangent at ρ = 1, 1/L = 3/4 to the lines shown in Fig 3.3.

A schematic showing the region is in Fig 3.3(right).

4 Perturbations of simple harmonic motion

In this section, we explore the period doubling bifurcation in which the simple harmonic motion on the

x-axis does not necessarily have small amplitude, and coexists with a nearby periodic solution with twice

the period. The solutions we seek are represented in Fig. 4.1. By symmetry, x(t) has half the period of y(t).In numerical experiments, we chose a value of L and then varied the angle θ in the initial conditions

(x(0), y(0)) = (0, 0), (x′(0), y′(0)) = (cos θ, sin θ) (4.1)

until we found a periodic solution of the general figure eight shape shown in Fig. 4.1. From this procedure,

we were able to find a value of θ if L was in the range 0.9 ≤ L ≤ 1.50695. In Fig. 4.1(right), we record

the values of L and θ, showing a clear trend θ → 0 as L approaches a limit of approximately 1.50695. As θapproaches zero, the amplitude of the y(t) oscillation decreases to zero. Thus, it appears that L ≈ 1.50695is a bifurcation point for period doubling.

To understand how the bifurcation point is selected, we seek solutions in a perturbation series of the form

x(t) = x0(t) + ǫx1(t) + O(ǫ2), y(t) = ǫy1(t) + O(ǫ2), (4.2)

10

−1 0 1−1

−0.5

0

0.5

1φ = 0 r = 0

Position−1 0 1

−0.5

0

0.5 θ = 0.46485 ρ = 1

Velocity0.8 1 1.2 1.4 1.60

0.1

0.2

0.3

0.4

0.5

L

θ

Figure 4.1: Left: A periodic solution of system (2.1) with L = 1.1. Right: Values of L, θ or which there is

a similar solution.

in which ǫ represents the amplitude of the y(t) oscillation. Substituting this ansatz into the ODE system

(2.3), we obtain at leading order the pair of equations

x′′0 = −2x0, y′′1 = y1G(x2

0, 0). (4.3)

Consequently, x0(t) = a sin√

2t, with amplitude a, and period√

2π. The equation for y = y1 becomes

y′′(t) + f(t)y(t) = 0, f(t) = 2

(

1 − L

L2 − a2 12(1 − cos 2

√2t)

)

. (4.4)

We seek solutions of this equation that are periodic with period 2√

2π, i.e., twice the period of x0(t). These

will exist only for certain values of the parameters L and a. Our goal is to establish a curve of values of

(L, a) for which there is a nontrivial 2√

2π periodic solution. We use two techniques to find this curve. The

first technique is a numerical shooting method, and the second uses Fourier series.

First, observe that we wish to find a solution y(t) with boundary conditions

y(0) = 0, y(√

2π) = 0. (4.5)

By symmetry, the solution will be even about the midpoint t = π/√

2, so that the odd periodic extension

also satisfies the equation, and has the correct period. In the shooting method, we set initial conditions

y(0) = 0, y′(0) = 1, (4.6)

solve the initial value problem and evaluate the solution at t =√

2π. This defines a function g = g(a,L) of

the two parameters. Zeros of g give the desired curve, shown as the thick solid curve in Fig. 4.2(right).

In the second approach, we use a Fourier series and Galerkin method. By symmetry, the 2√

π periodic

solution of (4.4) has the form

y(t) =∞∑

k=1

bk sin(2k − 1)t√2. (4.7)

11

We rewrite equation (4.4) as

(L2 − 12a2 + 1

2a2 cos 2√

2t)y′′ + 2(L2 − 12a2 − L + a2 cos 2

√2t)y = 0. (4.8)

Substituting the Fourier series (4.7) into the equation and using the trig identity

cos a sin b = 12(sin(a + b) − sin(a − b)),

we equate coefficients in the resulting Fourier sine series. This gives an infinite set of equations for the

coefficients bk, k = 1, 2, .... Let

αk = −1

4

[

(2L2 − a2)(2k − 3)(2k + 1) + 8L]

.

Then the Fourier series takes the form, after relabeling indices,

∞∑

k=1

αkbk sin(2k−1)t√2−a2

8

∞∑

k=3

(4k2−20k+17)bk−2 sin(2k−1)t√2−a2

8

∞∑

k=−1

(4k2+12k+1)bk+2 sin(2k−1)t√2.

Equating coefficients to zero, we obtain a system of equations

α1b1 +5a2

8b2 −

21a2

8b3 = 0

α2b2 −3a2

8b1 −

45a2

8b4 = 0

αkbk − a2

8(4k2 − 20k + 17)bk−2 −

a2

8(4k2 + 12k + 1)bk+2 = 0, k ≥ 3.

(4.9)

The Galerkin procedure is to take the first N terms in the series (4.7), assume the remaining coefficients are

negligibly small, and the remaining equations in (4.9) are negligible. The resulting system of N equations

is linear in b1, ..., bN . It therefore has a solution if and only if the determinant is zero, giving an equation

gN (a,L) = 0,

whose solution approximates the curve g(a,L) = 0 obtained by the shooting method.

With N = 1, the equation is α1 = 0, so that g1(a,L) = 8L − 6L2 + 3a2. Consequently, the first approxi-

mation is

L =1

3

(

2 +

√

4 +9

2a2

)

. (4.10)

As might be expected, this curve emanates from L = 4/3, a = 0, the value of L predictable from the

linear equations (3.9), with frequencies√

2 for x(t), and

√

2 − 2L

for y(t). Thus, in the limit of vanishing

amplitude in x and y, we find that y(t), has twice the period of x(t) precisely when L = 4/3. This is the 1:2

resonance studied for general hamiltonian systems with symmetry by Montaldi, Roberts and Stewart [4].

In Fig. 4.2 we show the approximate bifurcation curves with N = 1, 2, 3, and the comparisons with the

shooting method and the approximation (3.19). Th bifurcation point in Fig. 4.1(right), where θ = 0, L ≈1.50695 corresponds to a = 1/

√2, since x′(0) = 1. This agrees with the result from the shooting method

(the heavy curve in Fig. 4.2). Estimate (3.19) yields L = 1.48, while the Galerkin approximations with

N = 1, 2, 3 yield L = 1.500, L = 1.503, L = 1.512, respectively.

12

1.3 1.4 1.5 1.60

0.2

0.4

0.6

0.8

1 1

2

L

a

3

Figure 4.2: Value of L near 4/3 vs. a near zero, for which there is a 1 : 2 resonance bifurcation. Comparison

of Galerkin approximations with n = 1, 2, 3 (numbered light solid curves) with the exact solution from the

shooting method of §4 (thick solid curve). The dashed curve comes from eqn. (3.19).

5 Periodic Solutions off the Coordinate Axes, L > 1.

We locate periodic solutions by varying parameters in the initial conditions

[

x(0)y(0)

]

= r

[

cos φsin φ

]

,

[

x′(0)y′(0)

]

= ρ

[

cos θsin θ

]

. (5.1)

The following lemma suggests a procedure for detecting periodic solutions that pass through the origin.

Lemma 5.1 Consider a solution (x(t), y(t)) of (2.3) with initial conditions (5.1) having r = 0, ρ > 0.

(i) If there is T > 0 such that (x(T ), y(T )) = (0, 0), then the solution is periodic, with period T or 2T.

(ii) If there is τ > 0 such that x′(τ) = 0 and y(τ) = 0, then the solution is periodic, with period 4τ.

Proof: (i) For T ≤ t ≤ 2T, define

x(t) = −x(2T − t), y(t) = −y(2T − t).

Then

x(T ) = y(T ) = 0, x′(T ) = x′(T ), y′(T ) = y′(T ), (5.2)

13

and

x(2T ) = y(2T ) = 0, x′(2T ) = x′(0), y′(2T ) = y′(0). (5.3)

Thus, from (5.3), it remains to show that x(t) = x(t), y(t) = y(t), T ≤ t ≤ 2T. At t = T, the functions

are zero, and their derivatives are equal, by (5.2). Moreover, (x(t), y(t)) satisfies the ODE system (2.3) for

T ≤ t ≤ 2T :

[

xy

]

(t) =

[

−x′′

−y′′

]

(2T − t) =

[

−xF (x2, y2)−yG(x2, y2)

]

(2T − t) =

[

xF (x2, y2)yG(x2, y2)

]

(t).

Therefore, since the initial conditions at t = T are the same, the two solutions of (2.3) coincide.

In case (ii), the solution is continued from the interval [0, τ ] to [0, 4τ ] in a similar way as in case (i). Let

(x(t), y(t)) =

(x(t), y(t)), 0 ≤ t ≤ τ

(x(2τ − t),−y(2τ − t)), τ ≤ t ≤ 2τ

(−x(t − 2τ), y(t − 2τ)) 2τ ≤ t ≤ 3τ

(−x(4τ − t),−y(4τ − t)), 3τ ≤ t ≤ 4τ.

(5.4)

Stepping through the four parts of the solution, we see that (x(t), y(t)) and (x(t), y(t)) coincide over

the entire interval [0, 4τ ]. Moreover, by construction, (x(4τ), y(4τ)) = −(x(0), y(0)) = (0, 0), and

(x′(4τ), y′(4τ)) = (x′(0), y′(0)), so the solution is periodic with period 4τ. This completes the proof.

To locate periodic solutions of the form suggested by the lemma, consider initial conditions (5.1) with r = 0.For fixed ρ > 0, we capture all solutions with the same energy (determined by ρ) and passing through the

origin, by allowing θ to vary in the interval [0, π/2]. This corresponds to varying the initial velocity on a

quarter circle of radius ρ in the first quadrant of the velocity plane. For each θ ∈ (0, π/2), let tj(θ) denote

the jth time that the trajectory crosses the x axis, i.e. y(tj(θ)) = 0, and tj < tj+1, j = 1, 2, ... It turns out

that part (ii) of the lemma gives visually sharper results than part (i), so we explain how it is used. We record

the values x′j(θ) = x′(tj(θ)), j = 1, 2, ... We are interested in values θ = θ for which x′

J(θ) = 0 for some

(smallest) J. Then the lemma shows that the solution with initial condition specified by θ = θ is periodic

with period 4tJ (θ). Moreover, since x′(2n−1)J (θ) = 0, n = 1, 2, ... the graphs of x′

(2n−1)J (θ) will all cross

x′ = 0 at θ = θ.

The procedure is easily visualized using numerical simulations. The result for L = 1.9, ρ = 1 is shown in

Fig. 5.1, for the first few values of j. To generate the figure, trajectories were calculated to high accuracy

in MATLAB, recording crossings of the x axis, as described above. In the figure, many crossings of the

graphs of x′j(θ) may be observed. Note that many of the x′

j(θ) cross zero numerically. In this case, it is

the first crossing (j = 1) that gives the first positive time the periodic trajectory passes through the origin.

Since x′1(θ) depends continuously on θ, and changes sign near θ = 0.25, there must be a zero of x′

1(θ) near

θ = 0.25. This can be found systematically using a root finding algorithm for x′1(θ).

In Fig. 5.2 we show the corresponding periodic solution. In this figure, the trajectory (x(t), y(t)) is displayed

in two panels, one with position (x, y), the second with velocity (x′, y′). The solution, from time zero to

14

0 0.5 1 1.5

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

θ

x

-

-

-

-

-

'

Figure 5.1: Return Map with L=1.9.

0 0.4 0.844

3

-0.

-0.

-0.- 0. 5- 0.

2

-0.1

- 0. 2

- 0. 1

0

0.1

0. 2

0. 3

0

0.1

0. 2

0. 3

x

y

-1 0 0. 5 1- 0. 4

- 0. 3

u

v

position (x,y) velocity (u,v)

Figure 5.2: A periodic solution exhibiting symmetry; L = 1.9, ρ = 1.0, θ = 0.25018.

15

time τ , when it first crosses the x axis, is shown as a dashed line in the figure. Then we construct the entire

solution from this dashed curve, using symmetry, as in (5.4).

In Fig 5.3, we show a more systematic bifurcation diagram that captures branches of periodic solutions. To

generate this figure, initial data are chosen with r = 0, ρ = 1, so that the trajectories all start from the origin,

and have the same kinetic energy initially. More precisely, values for L and θ are taken from a 1000× 1000grid. For each choice of L and θ leads to a different initial velocity (cos θ, sin θ)); for each corresponding

solution, the minimum distance to the origin d(L, θ) = min0.1<t<1

√

x(t)2 + y(t)2 is recorded.

For periodic solutions of period not exceeding 1, we have d(L, θ) = 0. Here, a threshold value of 10−3 is

used to differentiate between periodic (and “near periodic”) solutions and non periodic ones. In other words,

the grey areas in Fig 5.3 correspond to d(L, θ) < 10−3. Larger values of the threshold mask the structure

of the bifurcation diagram while smaller values are ill-adapted to the already fine 1000 × 1000 grid. The

figure exhibits various bifurcations from the one-dimensional solutions, the oscillations on the coordinate

axes. The capital letters in the Figure correspond to parameter values for which the periodic solutions are

shown in the lower figure. In particular, the transition from A to A1, A2 is a symmetry-breaking secondary

bifurcation. The periodic solution and bifurcation curve of Fig.4.1 are included as case A. Interestingly,

many of the large amplitude solutions of the full nonlinear system resemble Lissajous figures.

In Figs. 5.4 we show a variety of additional shapes of solution for various choices of parameter. These

solutions were obtained by choosing values for L, ρ, r, φ and varying θ in the initial conditions (5.1) until a

periodic solution is found. On the left are solutions with L > 1, including two trajectories that do not pass

through the origin in position or velocity. On the right are some extreme shapes for L = 0.1, including a final

trajectory that oscillates about the equilibrium an oscillation about the equilibrium x = 0, y = ±√

1 − L2

(at which the springs have zero tension), with L < 1. This figure can be considered a perturbation of the

nonlinear oscillations on the y-axis, with a superimposed oscillation in x, resulting in a periodic solution

with period four times the period of the underlying oscillation in y.

16

L

θ

A

BCD

EF G

H

IJ

K

N A1

A2

L

M

0.5 1 1.5 2 2.50

0.5

1

1.5

A A1

A2 B

C D E F

G H I J

K L M N

Figure 5.3: Bifurcation diagram for periodic solutions with unit kinetic energy.

17

−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5L = 2.1 φ = 1 r = 1

Position−2 0 2

−1.5

−1

−0.5

0

0.5

1

1.5 θ = 0.5338 ρ = 1

Velocity−2 0 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2L = 0.1 φ = 1.0306 r = 2

Position−2 0 2

−1.5

−1

−0.5

0

0.5

1

1.5 θ = 0 ρ = 0

Velocity

−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5L = 3 φ = 1 r = 1

Position−1 0 1

−1.5

−1

−0.5

0

0.5

1

1.5 θ = 1.182 ρ = 1

Velocity−5 0 5

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8L = 0.1 φ = 0 r = 0

Position−2 0 2

−0.4

−0.2

0

0.2

0.4

0.6 θ = 0.262 ρ = 1

Velocity

−1 0 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2L = 1.9 φ = 0 r = 0

Position−2 0 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2 θ = 0.9983 ρ = 2

Velocity−1 0 10

0.5

1

1.5

2L = 0.1 φ = 1.03 r = 1.5

Position−0.5 0 0.5−1

−0.5

0

0.5

1 θ = 0 ρ = 0

Velocity

Figure 5.4: Periodic solutions of system (2.1). Left: L > 1. Right: L = 0.1.

18

References

[1] S. S. Antman. The equations for large vibrations of strings. Amer. Math. Monthly, 87:359–370, 1980.

[2] N. Bowditch. On the motion of a pendulum suspended from two points. Memoirs of the American

Academy of Arts and Sciences, 1815.

[3] A. Deprit. The Lissajous transformation. I. Basics. Celestial Mech. Dynam. Astronom., 51:201–225,

1991.

[4] J. Montaldi, M. Roberts, and I. Stewart. Existence of nonlinear normal modes of symmetric Hamiltonian

systems. Nonlinearity, 3:695–730, 1990.

[5] M. Shearer. Elementary wave solutions of the equations describing the motion of an elastic string. SIAM

J. Math. Anal., 16:447–459, 1985.

[6] A. F. Vakakis and R. H. Rand. Normal modes and global dynamics of a two-degree-of-freedom nonlin-

ear system. I. Low energies. Internat. J. Non-Linear Mech., 27:861–874, 1992.

[7] R. Young. Wave interactions in nonlinear elastic strings. Arch. Ration. Mech. Anal., 161:65–92, 2002.

19