OSCILLATOR TODA

'Oscillator Toda' is special kind of nonlinear oscillator; it is vulgarization of the Toda field theory, which rerers to a continuous limit of Toda's chain, of chain of particles,
with exponential potential of interaction between neibours
^[1].
The 'oscillator Toda' is used as simple model to understand the phenomenon of self-pulsation, which is quasi-periodic pulsation of
the output intensity of a solid-state lasers in the
transient regime.

Contents

Definition

Physical meaning

Energy

Period of pulsation

Decay of pulsation

References

Definition

'Oscillator Toda' is a dynamical system of any origin,
which can be described with dependent coordinate $~x~$,
and independent coordinate $~z~$,
characterized in that the evolution along independent coordinate $~z~$ can be approximated with equaiton
$$
rac{{
m d^{2}}x}{{
m d}z^{2}}+
D(x)rac{{
m d}x}{{
m d}z}+
Phi'(x) =0
,
where
$~D(x)=u\; e^\{x\}+v~$,
$~Phi(x)=e^x-x-1~$
and prime denotes the derivative.

Physical meaning

The independent coordinate $~z~$ has sense of time.
Indeed, it may be proportional to time $~t~$ with some
relation like
$~z=t/t\_0~$, where $~t\_0~$ is constant.
The derivative
m d}x}{{
m d}z}
may have sense of velocity of particle with coordinate
$~x~$;
then
m d}^2x}{{
m d}z^2}~
can be interpreted as acceleration; and the mass of such a particle is equal to unity.
The dissipative funciton $~D~$ may have sense of coefficient
of the speed-proportional friction.
Usually, both parameters $~u~$ and $~v~$ are
supposed to be positive; then this speed-proportional friction coefficient
grows exponentially at large positive values of coordinate $~x~$.
The potential $~Phi(x)=e^x-x-1~$ is fixed funciton, which
also shows exponential grow at large positive values of coordinate $~x~$.
In the application in laser physics, $~x~$ may have
sense of logarithm of number of photons in the laser cavity,
related to its steady-state value. Then, the output power of such
laser is proportional to $~exp(x)~$ and may show pulsation
at oscillation of $~x~$.
Both analogies, with a unity mass particle and logarithm of number of photons are useful in the analysis of behavior of 'Oscillator Toda'.

Energy

Rigorously, the oscillation is periodic only at $~u=v=0~$.
Indeed in realization of 'oscillator Toda' as self-pulsing laser,
these parameters may have values of order of $~10^\{-4\}~$;
during several pulses, the amplutude of pulsation does not change much.
In this case, we can speak about period of pulsation, function
$~x=x(t)~$ is almost periodic.
In the case $~u=v=0~$, the energy of oscillator

m d}x}{{
m d}z}
ight)^{2}+Phi(x)~
does not depend on $~z~$, and can be treated as constant of motion.
Then, during one period of pulsation, the relation between
$~x~$ and
$~z~$ can be expressed analytically
^[2],
^[3]:
$$
z=pmint_{x}^{x_{max}}!!rac{{
m d}a}
{sqrt{2}sqrt{E-Phi(a)}}

wehere
$~x\_\{min\}~$ and
$~x\_\{max\}~$ are minimal and maximal values of $~x~$; this solution is written for the case when $dot\; x(0)=0$.
however, other soluitons may be obtained using the translational invariance.

The ratio $~x\_\{max\}/x\_\{max\}=2gamma~$ is convenient parameter to characterize the amplitude of pulsation,
then, the median value
$$
delta=rac{x_max -x_min}{1}

can be expressed as
$$
delta=
lnrac{sin(gamma)}{gamma}
;
and the energy
$$
E=E(gamma)=rac{gamma}{ anh(gamma)}+lnrac{sinh gamma}{gamma}-1

also is elementary funciton of $~gamma~$.
For the case $~gamma=5~$, an example of pulsation of the oscillator toda is shown in Fig.1.
In application, the quantity $E$ have no need to be physical energy of the system;
in these cases, this dimension-less quantity may be called quasienergy.

Period of pulsation

The period of pulsation is increasing function of the amplitude $~gamma~$.
At $~gamma\; ll\; 1~$,
the period
$~T(gamma)=2pi$
left(
1 + rac{gamma^2} {24} + O(gamma^4)
ight)
~
At $~gamma\; gg\; 1~$, the period
$~T(gamma)=$
4gamma^{1/2}
left(1+O(1/gamma)
ight) ~
In the whole range
$~gamma\; >\; 0~$, the period $~T=T(gamma)~$
and the frequency can be approximated with
$$
k_{
m fit}(gamma)=
rac{2pi}
{T_{
m fit}(gamma)}=

$$
left(
rac
{
10630
+ 674gamma
+ 695.2419gamma^2
+ 191.4489gamma^3
+ 16.86221gamma^4
+ 4.082607gamma^5 + gamma^6
}
{10630 + 674gamma + 2467gamma^2 + 303.2428 gamma^3+164.6842gamma^4 + 36.6434gamma^5 + 3.9596gamma^6 +
0.8983gamma^7 +rac{16}{pi^4} gamma^8}
ight)^{1/4}

with at least 8 significant figures;
The relative error of this approximation does not exceed $22\; imes\; 10^\{-9\}$.

Decay of pulsation

At small (but still positive) values of
$~u~$ and $~v~$, the pulsation decays slowly,
and this decay can be described analytically.
In the first approximation parameters $~u~$ and $~v~$
hive additive contribution to the decay;
the decay rate, as well as the amplitude and phase of the nonlinear oscillation
can be approximated with elementary functions in the similar manner, as the period above.
This allows to approximate the solution of the initial equation; and the error of such approximation
is small compared to the difference between behavior of the idealized oscillator Toda and
nehavior of the experimental realization of oscillator Toda as self-pulsing laser at the
optical bench, alghough, qualitatively, a self-pulsing laser shows very similar behavior.
^[3]

References

1. Dynamics of chain with exponential interaction between neibours, M.Toda, , , Physics Reports, 1975
2. Toda potential in laser equations, G.L.Oppo, , , Zeitschrift fur Physik B, 1985
3. Self-pulsing laser as oscillator Toda: Approximation through elementary functions, D.Kouznetsov, , , Journal of Physics A, 2007
4. Self-pulsing laser as oscillator Toda: Approximation through elementary functions, D.Kouznetsov, , , Journal of Physics A, 2007