Developing a Symplectic Integrator to explore the

Fermi-Pasta-Ulam problem and related models

Joachim Worthington

April 20, 2012

1 Introduction

The Fermi-Pasta-Ulam, or FPU model, is a family of Hamiltonians describing systems ofvibrating particles with nearest-neighbour interactions that introduce nonlinear forces.The results of the analysis of these systems, some of the earliest in computer simulationnumerics, were very surprising at the time. Later work elucidated the significance ofthese results in terms of general nonlinear systems. In this paper, I will first discussthe background and history of the FPU problem, and the related Toda lattice, beforedeveloping a symplectic integrator and running a variety of experiments to recreate theresults for both these systems.

2 Historical Discussion

2.1 The FPU problem

In the early 1950s, computers were becoming more accessible to scientists and mathe-maticians. The MANIAC 1 (Mathematical Analyzer, Numerical Integrator and Com-puter, model 1) was a one-of-a-kind computer built and run at the Los Alamos ScientificLaboratory from 1952, until 1957 when it was succeeded by MANIAC II. At Los Alamos,in New Mexico, four scientists, Enrico Fermi, John Pasta, Stanislaw Ulam, and MaryTsingou began researching the idea of nonlinear physics. Fermi, Pasta and Ulam intro-duced a small nonlinear perturbation to a physical system. They developed a series ofextremely surprising and counter-intuitive numerical results. These results led to newparadigms in the analysis of nonlinear systems and demonstrated the importance ofnumerical computer simulations for such systems.

The genesis of Fermi, Pasta and Ulam’s research was Fermi’s belief that future theo-retical physics would involve nonlinear operators and equations, and his desire to beginanalysing such systems numerically. To this end, they developed a simple model demon-strating such nonlinearity; a one-dimensional dynamical system of particles such thateach particle is affected only by forces from the neighboring particles, and no external

1

forces. Without the nonlinear perturbation, these equations of motion are simply

xi = xi+1 + xi−1 − 2xi, i = 1, 2, 3...N (1)

which corresponds to the usual form of Hooke’s Law between N particles. The nonlinearterms were introduced either as a quadratic term,

xi = xi+1 + xi−1 − 2xi + α[(xi+1 − xi)2 − (xi − xi−1)2], i = 1, 2, 3...N (2)

or a cubic term,

xi = xi+1 + xi−1 − 2xi + β[(xi+1 − xi)3 − (xi − xi−1)3], i = 1, 2, 3...N (3)

with small values of α and β. The original FPU paper[1] considered systems of N = 64particles, a variety of initial conditions and values of α and β, and fixed end points asboundary conditions. The system was always taken to be at rest at t = 0. The equationswere discretised and coded into MANIAC by Mary Tsingou. The results were analysedin Fourier space; the energy in each vibrational mode was calculated by way of Fouriertransform, then plotted.

The results of these experiments were immediately surprising. It was expected thatthe system would undergo thermalisation; that is, the energy would gradually and con-tinuously decrease in the lower modes and increase in the higher modes until the energyis more or less equally distributed amongst all modes (essentially amounting to noise).However, an entirely and profoundly different behaviour was observed.

An archetypal example was the result for a small quadratic force, with an initialdisplacement of a pure sine wave. Initially, as expected, it was observed that the energyin the fundamental mode increased, followed by the energy in the next mode, and soon. However, instead of continuing in this manner, the second mode began to increasein energy until it had more energy than all other modes combined. The third modethen began to increase in energy, becoming the dominant mode. Instead of continuingto higher and higher modes, peaks were observed for only the lowest few modes, withhigher modes remaining at negligible energy levels. Most surprisingly though is thateventually the first mode returned to within one percent of its original energy. In theoriginal 1955 paper[1], this behaviour of the system was described as “almost periodic”.

Later research, on faster, more modern computers, gave further insight into thebehaviour of the system. J Tuck and M Menzel (the married name of the aforementionedMary Tsingou) ran the same simulations, but to higher accuracies and for longer times,and were able to more closely study the “energy return” of the system, the observedreturn of energy to the first node. As had been earlier observed, most, but not all, ofthe energy returned to the initial vibrational mode and the amount of energy in thefirst mode after each subsequent cycle decreased initially. However, counter-intuitively,eventually the energy in the first mode increased again, returning to within one percentof the initial energy. Such a “supercycle” added another level of intrigue to the problem.

Subsequent work revealed more and more about the true nature of the problem. Inthe 1955 paper, Fermi, Pasta and Ulam had difficulties describing their results analyt-ically. Questions remained as to whether the phenomenons observed were in part, or

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in whole, influenced by the numerical computations, which were highly experimental atthe time. In 1961, N J Zabusky (see [2]) wrote a paper entitled Exact Solutions forthe Vibrations of a nonlinear continuous string, which gave an analytical basis to theearlier numerical observations. Zabusky solved exactly a perturbed differential equationcorresponding to the equations of motions given by (2) and (3), but with periodic, ratherthan fixed, boundary conditions. A later paper by Zabusky and M D Kruskal (see [3])related the behaviour to that of solitons, directly deriving the Korteweg-de Vries equa-tion wτ + 124wξξξ + αwwξ = 0, which is known to have soliton solutions. The results ofthe FPU experiments were recast in position space rather than energy space. By relatingthe numerical problem with this family of related soliton solutions, many of the resultsof the 1955 experiments were now much better understood.

2.2 The Toda Lattice and Conservation Laws

After the work on the FPU problem, and the growth in tandem of soliton mathemat-ics, mathematicians searched for a system which was completely integrable. In general,models of the form

H(q,p) =

N∑i=1

(1

2p2i + V (qn+1 − qn)) (4)

were considered, and a choice of potential V (r) such that the system is completelyintegrable was sought after.

Japanese physicist Marikazu Toda came up with exactly such a potential,

V (r) = e−r + r − 1 (5)

which leads to a system known as the Toda lattice. Explicitly, the Hamiltonian for thissystem is given by

H(q,p) =N∑i=1

(1

2p2i + eqi+1−qi) (6)

As noted above, this system is completely integrable. Thus, a set of canonical co-ordinatescould be calculated using the Hamilton-Jacobi method. In such a set of co-ordinates, theconjugate momenta are all conserved, which gives N independent conserved quantities(see [12] for more details).

It was demonstrated by Flaschka in 1974 (see [8]) that the Toda lattice lattice is afinite-dimensional analog of the previously mentioned Korteweg-de Vries (KdV) equa-tion. And as with the KdV equation, the Toda lattice admits soliton solutions; wave-likesolutions that interact like particles. Several stable wave solutions exist, and interactwithout losing their shape or energy. The Toda lattice is remarkable in this regard, andexhibits no chaos.

A variant of the Toda lattice is the mass diatomic Toda lattice, given by

H(q,p) =

N∑i=1

(1

2mip2i + eqi+1−qi) (7)

3

where the masses of odd and even particles are slightly different, m2i = 1 + ε andm2i = 1 − ε for some small ε. Experiments performed by Casati and Ford (see [10])showed that this system demonstrates chaotic behaviour.

3 Symplectic Integrator

To analyse the various problems mentioned above, we begin by developing a symplecticintegrator. In this context, a symplectic integrator is one that preserves the Hamiltonianstructure of the system. We wish to implement a general integrator for any separableHamiltonian; that is, Hamiltonians of the form

H(q,p) = T (p) + V (q) (8)

where T is the kinetic energy and V is the potential energy. Given a Hamiltonian ofthis form (though, it should be noted, not necessarily of this form; this technique can beapplied to many other Hamiltonians), we can apply a split-step method, as implementedbelow in MATLAB:

SymplecticIntegrator.m

1 dt=0.125; %Timestep Size (parameter)2 TS=10000; %Number of timesteps to take (parameter)3 N=32; %Number of coordinates (parameter)4

5 X=zeros(2*N,TS);6 X(1:N,1)=sin((1:N).*(2*pi)./(N)); %Initial Condition For Coordinates7 X(N+1:2*N,1)=0; %Initial Condition for Momenta8

9 J=[zeros(N) +eye(N); −eye(N) zeros(N)]; %The symplectic matrix10

11 for ct=2:TS12

13 X(:,ct)=X(:,ct−1);14 X(:,ct)=X(:,ct)+J*delT(X(:,ct)).*dt./2; %Half timestep on the ...

positions15 X(:,ct)=X(:,ct)+J*delV(X(:,ct)).*dt; %Full timestep on the momenta16 X(:,ct)=X(:,ct)+J*delT(X(:,ct)).*dt/2; %Half timestep on the positions17

18 end;

This provides a simple, three-step integrator for separable Hamiltonians. Most cru-cial is the definition of the functions delT and delV; they take a vector x and returnthe vectors

∇T (x) =

(∂T

∂q1(x)...

∂T

∂qN(x),

∂T

∂p1(x)...

∂T

∂pN(x)

)∇V (x) =

(∂V

∂q1(x)...

∂V

∂qN(x),

∂V

∂p1(x)...

∂V

∂pN(x)

)

4

Note that, for our separable Hamiltonians, ∂Tqi

= ∂Vpi

= 0 i = 1...N . After delT.m anddelV.m been defined in a separate function files, the parameters in lines 1 - 3 and theinitial conditions in lines 6 & 7 can be modified to suit the problem. We will be furthermodifying this base program to suit our purposes later in this paper.

The first section of note is line 9, defining the symplectic matrix. The symplecticmatrix is defined as

J =

0 0 · · · 0 +1 0 · · · 00 0 · · · 0 0 +1 · · · 0...

.... . .

......

.... . .

...0 0 · · · 0 0 0 · · · +1−1 0 · · · 0 0 0 · · · 00 −1 · · · 0 0 0 · · · 0...

.... . .

......

.... . .

...0 0 · · · −1 0 0 · · · 0

=

(0 +I−I 0

)

Then Hamilton’s equations can be succinctly as

d

dtx = J∇H(x) (9)

J has the properties that JT = −J , JTJ = I.The actual integration occurs on lines 14 - 16. On line 14, a dt/2 integration step is

taken, but only for the qi values. Then a full dt step is taken, this time only for the pivalues. Finally, another dt/2 step is taken for the qi values.

4 Reproducing the FPU experiments

We now turn our attention back to the FPU Hamiltonian,

H =

N∑i=1

[1

2p2i +

1

2(qi+1 − qi)2 +

α

3(qi+1 − qi)3 +

β

4(qi+1 − qi)4

](10)

(Note that in this paper we will only consider α 6= 0, or β 6= 0, but not both.)Our first step in reproducing the 1955 results is to build the delT and delV, which

are included in appendix A.

4.1 Verifying the linear results

We begin by ensuring that our program calculates the expected results for the linearproblem, that is, with α = β = 0. We set the initial condition to a full period sine waveconsisting of 64 particles, with all momentum zero, and run the code. Results are shownin Figure 1.

As expected, we see the normal vibration of a string, and we can verify that thiscontinues, implying our integrator is stable, at least for this simplified problem. (Note

5

0 10 20 30 40 50 60−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure 1: Displacement of the particles at t=0, t=15, t=30 and t=55

that, rather than individual particles, the diagrams show a continuous string; this choiceis purely aesthetic, for easier visualisation.)

We now consider the energy in each vibrational mode. We begin by defining xi =qi+1 − qi, the distance between neighbouring particles, and Xk and Pk, the Fouriertransforms of xi and pi, respectively. Then we can calculate the energy in vibrationalmode k by

Ek =1

2|Pk|2 +

1

2|Xk|2

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1

2

3

4

5

6

Time

Ene

rgy

0 2000 4000 6000 8000 100001

2

3

4

5

6

7

8

9

10

11

Time

Ene

rgy

k=2k=3k=4k=5

Figure 2: Energy in the first few vibrational modes for a single period sine wave (left) and amore complicated initial displacement (right)

By plotting this energy, we can verify that energy is conserved across vibrationalmodes for the linear problem. In Figure 2, we see that the energy levels are constantover a long period of time, confirming our expectations.

4.2 Recreating the nonlinear results

We now move onto the nonlinear problem, and the main FPU results. We begin byintroducing the variables alpha and beta into SymplecticIntegrator.m, and setalpha=0.4, a small quadratic force. We leave the initial displacement as a single-periodsine wave,with zero momentum. Figure 3 shows the energy of the first few vibrationalmodes, over a long period of time. The striking FPU result is immediately obvious; the

6

0 0.5 1 1.5 2

x 104

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time

Ene

rgy

k=2k=3k=4k=5

Figure 3: Energy in the first few modes, with a single-period sine wave for initial condition, forα = 0.4

energy moves into the other modes, but remains primarily in the first few. Rather thanundergo thermalisation, after a period of time, almost all the energy returns to the initialmode. We repeat the experiment for a more complicated initial condition in Figure 4,which gives similar results, except with energy present initially in modes other than thefundamental.

For a higher value of α, a larger quadratic force, we see the same general behaviour,but a much higher degree of thermalisation, and much less energy returning to the initialmode. It is unclear at this point as to whether this is a valid result, or a consequenceof compounded numerical errors. If we try to integrate over a longer period of time, thecalculations “explode”, giving infinite energy in each mode. By observing the energyjust before this happens, it appears that the system starts to approach a state of equalenergy across all modes, but without being able to integrate further, nothing can beconcluded. Altering dt and TS allow us to integrate over somewhat longer intervals, butmemory costs become prohibitive quickly. Plots of the energy as far as can be feasiblycalculated are shown in Figure 5.

Our next step is to explore a cubic force. With the same parameters and initialconditions, we now set beta=8 (and set alpha=0 again), a cubic force. In Figure 6,we observe very similar behaviour to that for the quadratic force , though with the

7

0 500 1000 1500 2000 2500 3000 3500 40000

1

2

3

4

5

6

7

8

9

10

Time

Ene

rgy

k=2k=3k=4

Figure 4: Energy in the first few modes, with a superposition of two sine waves for initialcondition, for α = 0.2

0 2000 4000 6000 8000 100000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time

Ene

rgy

k=2k=3k=4k=5

0 0.5 1 1.5 2 2.5

x 104

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time

Ene

rgy

Figure 5: Energy in the first few modes, with a single-period sine wave for initial condition, forα = 2. At left, the first few vibrational modes are shown; at right, the first mode is shown untilthe calculations break down.

difference that energy only builds in every second vibrational mode. The cubic forceacts symmetrically on the particles, so the string keeps its symmetry, and the other

8

0 0.5 1 1.5 2

x 104

0

1

2

3

4

5

6

Time

Ene

rgy

k=2k=4k=6

Figure 6: Energy in every second mode, with a single-period sine wave for initial condition, forβ = 8 (a cubic force)

modes will have zero energy at all times.

4.3 Exploring the supercycle

As noted in earlier, the energy dynamics actually undergo a supercycle; the level ofenergy to which the initial mode returns decreases, and then increases, in a fairly regularmanner. By plotting just the first energy mode over a very long time, we can observethis supercycle (see Figure 7; appendix B shows the code for generating this curve.). Themaximum energy level peaks fairly clearly approximately every 3000 time units, movingbetween about 2.8 and 4.9. There appears to be some fairly significant dynamics evenwithin the supercycle, but the enormous volume of data involved in this analysis makesit impractical for the scope of this paper.

9

0 0.5 1 1.5 2

x 105

2.5

3

3.5

4

4.5

5

Time

Max

imum

Ene

rgy

Figure 7: The “Maximum Energy” in the first mode over a very long time. Shows the levelto which the the energy in the first mode returns, demonstrating a supercycle. α = 0.5, initialcondition a single-period sine wave.

5 The Toda Lattice

We now turn our attention to the Toda lattice. The Toda lattice has the followingHamiltonian

H(q,p) =N∑i=1

(1

2p2i + eqi+1−qi) (11)

and, as noted in the preamble, is completely integrable. We thus have N conservedquantities, some of which we will verify here.

5.1 Conservation Laws of the Toda Lattice

First, we can check that the behaviour is as we expect. Our delV.m program is modified(see appendix C for delVToda.m) and the program is run for a Gaussian as an initialcondition (X(1:N,1)=exp(−(((1:N)−N/4)./4).ˆ2);). Unfortunately, the results us-ing our Symplectic Integrator were not found to be accurate enough, even with verysmall timesteps, so the code was rewritten to use MATLAB’s built in ode45 function, aRunge-Kutta based integrator (this code is also in Appendix C). The results are shown

10

10 20 30 40 50 60

0

0.2

0.4

0.6

0.8

1

1.2

10 20 30 40 50 60

0

0.2

0.4

0.6

0.8

1

1.2

10 20 30 40 50 60

0

0.2

0.4

0.6

0.8

1

1.2

10 20 30 40 50 60

0

0.2

0.4

0.6

0.8

1

1.2

10 20 30 40 50 60

0

0.2

0.4

0.6

0.8

1

1.2

Figure 8: The displacement of the particles for the Toda Hamiltonian, initial condition a Gaus-sian, at (from top, left to right) t = 0, t = 7, t = 28, t = 31, t = 39. Soliton behaviour isobserved. From the initial condition, a Gaussian, two solitons are created, travelling in oppositedirections. At t = 60, they begin to interact, like waves, but after interacting, they continue asthey were traveling previously unscathed, like particles.

in Figure 8.Next, we analyse the conserved quantities. These quantities are discussed in detail in

the paper by Henon (see [12]). The first of these is our energy ; as the system is closed,the total energy is conserved. The energy is given by the same expression as for theHamiltonian itself. The code below calculates the energy for a typical system, over along period of time:

1 EDU>> max(Energy)2 ans =3 128.15474 EDU>> min(Energy)5 ans =6 128.1547

Seeing the above, the energy is conserved, to a fairly high degree of accuracy. Ournext step is to check the total momentum, which should also be conserved for our closedsystem. The total momentum is given by

∑Ni pi, and we check this for a typical execution

of the lattice

1 EDU>> max(TotalMomentum)2 ans =3 7.08984 EDU>> min(TotalMomentum)5 ans =6 7.0898

11

The total momentum is also conserved, to a similar level of accuracy. A third con-served quantity we can check is given in Henon’s paper; unlike the other two, though, itdoes not have a clear physical interpretation. The quantity is given

I =∑i,j,i6=j

(pipj)−N∑i=1

(e−(qi+1−qi)) (12)

where the first summation is over all possible pairs of momenta. As above, we calculatethis quantity

1 EDU>> max(Conserved3)2 ans =3 −41.52854 EDU>> min(Conserved3)5 ans =6 −41.6199

This quantity does not seem to be as closely conserved as the other two; this canprobably be associated with the (initially small) errors which we observe with the resultsfor our program.

5.2 The Mass Diatomic Toda Lattice

The mass diatomic Toda Lattice is a closely related model; the only difference being asmall difference in mass between the particles; explicitly, we change the kinetic energyto T =

∑∗Ni=1

12mi

p2i , where m2i = 1 + ε and m2i+1 = 1 − ε, for some small ε. Weexplore this initially for a very small mass perturbation, ε = 0.02. Running this, andchecking the energy, gives us

1 EDU>> max(Energy)2 ans =3 128.15484 EDU>> min(Energy)5 ans =6 128.1546

The energy still seems to be conserved, but to a lower degree of accuracy than before.Increasing the value of ε seems to decrease the degree to which the energy is observed tobe conserved. For ε = 0.5, we get max(Energy)=128.1887 and min(Energy)=128.1532.Integrating over a longer period of time increases these discrepencies as well, for moreobvious reasons. The total momentum seems to fare better; for ε = 0.02, we get

1 EDU>> max(TotalMomentum)2 ans =3 7.08984 EDU>> min(TotalMomentum)5 ans =

12

6 7.0898

and increasing ε seems to have little effect on the total momentum; indeed, the totalmomentum calculation does not seem to be affected by the value of ε very much at all.Checking the third conserved quantity, we observe that the inclusion of a value for εagain has little effect on the results:

1 EDU>> max(Conserved3)2 ans =3 −41.52854 EDU>> min(Conserved3)5 ans =6 −41.6579

However, unlike the above, the conservation is less and less strict for larger values of ε,but is still generally fairly close, until ε becomes very close to 1, where it starts varyingby nearly 20%.

6 Conclusion

Using only a personal computer, we quite quickly reproduced all of the original 1955FPU experiments, and see the same striking results. In fact, the extra computing powerafforded to us by half a century of progress allowed us to quickly and easily calculatefar beyond the original experiments, and to see the existence of a super-cycle in thedynamics of the system. Our experiments on the Toda lattice also provided us withexcellent results, showing clear soliton-like behaviour, though they also demonstratedthe limits of the usefulness of our simple symplectic integrator. The analysis of theconserved quantities of the Toda Lattice and the mass diatomic Toda Lattice also gavevery clear results with very little code required. In future experiments, a more accurateintegrator could be developed, and the FPU results extended for larger quadratic andcubic forces.

A Appendix: delT.m and delV.m for the FPU problem

The code to calculate ∇T and ∇V for equation (10).

delT.m

1 function [ dT ] = delT( X )2 %delT returns the partial derivatives of T for vector X3 N=length(X)/2; %Number of coordinates4 dT=zeros(2*N,1);5 dT(N+1:2*N)=X(N+1:2*N);6 end

13

delV.m

1 function [ dV ] = delV( X, alpha, beta )2 %delH returns the partial derivatives of H for vector X3 N=length(X)/2; %Number of coordinates4 dV=zeros(2*N,1);5 dV(1:N−1)=dV(1:N−1)−(X(2:N)−X(1:N−1));6 dV(1:N−1)=dV(1:N−1)−alpha*(X(2:N)−X(1:N−1)).ˆ2;7 dV(1:N−1)=dV(1:N−1)−beta*(X(2:N)−X(1:N−1)).ˆ3;8 dV(N)=−(X(1)−X(N))−alpha*(X(1)−X(N)).ˆ2−beta*(X(1)−X(N)).ˆ3;9 dV(2:N)=dV(2:N)+(X(2:N)−X(1:N−1));

10 dV(2:N)=dV(2:N)+alpha*(X(2:N)−X(1:N−1)).ˆ2;11 dV(2:N)=dV(2:N)+beta*(X(2:N)−X(1:N−1)).ˆ3;12 dV(1)=dV(1)+(X(1)−X(N))+alpha*(X(1)−X(N)).ˆ2+beta*(X(1)−X(N)).ˆ3;13 end

B Appendix: Supercycle Code

The following code generates the maximum points for each energy cycle, to demonstratethe existence of the supercycle.

SCycleEnergy.m

1 %Finds the Maximums of the Energy vector2 thresh1=1.5; %Double thresholding to reduce noise3 thresh2=2;4 flag=true;5 currMax=0;6 mUpTo=1;7 for m=1:TS8 if (E(2,m)<thresh1 && flag==true)9 mUpTo=mUpTo+1;

10 currMax=0;11 flag=false;12 end;13 if (E(2,m)>thresh2 && flag==false)14 flag=true;15 end;16 if (flag==true && E(2,m)>currMax)17 currMax=E(2,m);18 maxEnergyPt(mUpTo)=m;19 maxEnergyLvl(mUpTo)=E(2,m);20 end;21 end;22

23 plot(maxEnergyPt*dt,maxEnergyLvl,'k')

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C Appendix: code for the Toda Lattice

The code to calculate ∇V for the Toda Lattice. (Note that the code for ∇T is thesame as for delT.m above for the equal-mass Toda Lattice; only minor modifications arerequired for the mass diatomic lattice)

delVToda.m

1 function [ dV ] = delVToda( X )2 %delH returns the partial derivatives of H for vector X3 N=length(X)/2; %Number of coordinates4 dV=zeros(2*N,1);5 dV(1:N−1)=dV(1:N−1)−exp(X(2:N)−X(1:N−1));6 dV(N)=−exp(X(1)−X(N));7 dV(2:N)=dV(2:N)+exp(X(2:N)−X(1:N−1));8 dV(1)=dV(1)+exp(X(1)−X(N));9 end

Our rewritten integrator, used for the Toda Lattice

TodaMain.m

1 dt=0.001; %Timestep Size2 TS=100000; %Number of timesteps3 N=64; %Number of particles4

5 X0=zeros(2*N,1);6 X0(1:N,1)=exp(−(((1:N)−N/4)/4).ˆ2); %Initial Condition7

8 oldOpts = odeset;9 newOpts = odeset(oldOpts,'RelTol',1e−10); %Option for greater accuracy

10 tspan = linspace(0, TS*dt, TS); %Time to integrate over11 fname='dxdt'; %Our total derivative function12

13 [t,h] = ode45(fname,tspan,X0);

A function required for the ode45 function; returns the total derivative of X as asingle vector

dxdt.m

1 function [ derv ] = dxdt( t, X )2 N=length(X)/2;3 J=[zeros(N) eye(N); −eye(N) zeros(N)];4 derv=J*(delV(X)+delT(X,0));5 end

15

References

[1] E. Fermi, J. Pasta, S. Ulam Studies of Non-Linear Problems Document LA-1940(May 1955)

[2] N. J. Zabusky Exact Solutions for the Vibrations of a nonlinear continuous stringJournal of Mathematical Physics, Vol. 3, p.1028-1039 (October 1961)

[3] N. J. Zabusky, M D Kruskal Interaction of “Solitons” in a Collisionless Plasma andthe Recurrence of Initial States Phys. Rev. Lett, Volume 15 (August 1965)

[4] Thierry Dauxois and Stefano Ruffo Fermi-Pasta-Ulam nonlinear lattice oscillationsScholarpedia, 3(8):5538. (2008)

[5] Thierry Dauxois Fermi, Pasta, Ulam and a mysterious lady Physics Today (January2008)

[6] M. Toda Theory of Nonlinear Lattices, 2nd enl. ed. Springer, Berlin (1989)

[7] Gerald Teschl Almost Everything You Always Wanted to Know About the Toda Equa-tion Jahresber. Deutsch. Math.-Verein. 103, no. 4, 149-162 (2001)

[8] H Flaschka The Toda Lattice. II. Existence of Integrals Phys. Rev B 9 (1974)

[9] Eric W. Weisstein Eric Weisstein’s World of Physics: Toda Latticehttp://scienceworld.wolfram.com/physics/TodaLattice.html (2007)

[10] G Casati and J Ford Stochastic Transition in the Unequal-Mass Toda Lattice Phys.Rev. A 12, 1702-1709 (1975)

[11] Christophe Gole Symplectic Maps Scholarpedia, 5(3):3722 2010

[12] M. Henon Integrals of the Toda Lattice Phys. Rev. B 9 1974

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