76
R. S. MacKay
for given weak forcing, perhaps obtaining a sort of non-autonomous KAM result,
2
extending the theory of reducibility of cocycles (see [7] for an early example).
The main conclusion of this section is that synchronisation of an oscillator to its
inputs is dimension-reduction. In particular, if there is no immediate feedback from
the oscillator to any of its inputs, then one could delete that oscillator, replacing its
outputs by some modifications of the outputs from its inputs.
5.3 Synchronisation of Two Oscillators
Let us start with two autonomous oscillators x i = v i (x i ), i = 1, 2, meaning each has
a limit cycle γ i , and couple them in the standard sense of a modification to the vector
field of the product system, depending on the state of each but not too strongly, so
˙
x i = v i (x i ) + g i (x 1 , x 2 ),
(5.4)
with g i C
1 -small. Then the product system has a NH 2-torus, being a small perturbation of γ 1 × γ 2 .
If the difference of the frequencies of the uncoupled limit cycles is smaller in a
suitable dimensionless sense than the coupling then the NH torus has an attracting
limit cycle on it, which makes one turn in the γ 2 direction for each turn in the
γ 1 direction. I say the two oscillators have gone into 1 : 1 synchronisation. Recall
Huygens’ clocks. The torus may have more than one attracting limit cycle on it, in
which case several synchronised solutions are possible. It may also have unstable
limit cycles on it.
Similarly, if the frequencies are close to being in (coprime) integer ratio m : n
then coupling might produce an attracting m : n limit cycle on the NH torus, which
makes m revolutions in the γ 1 direction and n in the γ 2 direction per period. On the
other hand, for weak coupling and smooth enough dynamics, the non-synchronised
situation occurs with high probability. More precisely, if one adds a free parameter
varying the unperturbed frequency ratio, then KAM theory gives a set of parameter
values of nearly full measure for which the dynamics is conjugate to a constant vector
field on a 2-torus with irrational frequency ratio (e.g. [17] for a version by my Ph.D.
student João Lopes Dias). Thus synchronisation does not always result.
Now consider the non-autonomous situation, where one or both of the oscillators
is subject to external forcing. If the forcing is not too strong then the resulting system
has a NH submanifold in extended state space, diffeomorphic to γ 1 × γ 2 × R, which
I call a torus-cylinder. More generally, for any manifold M I define an M-cylinder
to be a manifold diffeomorphic to M × R. Thus an ordinary cylinder can be called
a circle-cylinder. If the unperturbed frequencies are close to integer ratio m : n then
the NH submanifold might contain a NH attracting submanifold diffeomorphic to a
2 The original KAM theory gives a set of invariant tori for near-integrable Hamiltonian systems, the
measure of whose complement goes to zero as the perturbation from integrability goes to zero.
R. S. MacKay
for given weak forcing, perhaps obtaining a sort of non-autonomous KAM result,
2
extending the theory of reducibility of cocycles (see [7] for an early example).
The main conclusion of this section is that synchronisation of an oscillator to its
inputs is dimension-reduction. In particular, if there is no immediate feedback from
the oscillator to any of its inputs, then one could delete that oscillator, replacing its
outputs by some modifications of the outputs from its inputs.
5.3 Synchronisation of Two Oscillators
Let us start with two autonomous oscillators x i = v i (x i ), i = 1, 2, meaning each has
a limit cycle γ i , and couple them in the standard sense of a modification to the vector
field of the product system, depending on the state of each but not too strongly, so
˙
x i = v i (x i ) + g i (x 1 , x 2 ),
(5.4)
with g i C
1 -small. Then the product system has a NH 2-torus, being a small perturbation of γ 1 × γ 2 .
If the difference of the frequencies of the uncoupled limit cycles is smaller in a
suitable dimensionless sense than the coupling then the NH torus has an attracting
limit cycle on it, which makes one turn in the γ 2 direction for each turn in the
γ 1 direction. I say the two oscillators have gone into 1 : 1 synchronisation. Recall
Huygens’ clocks. The torus may have more than one attracting limit cycle on it, in
which case several synchronised solutions are possible. It may also have unstable
limit cycles on it.
Similarly, if the frequencies are close to being in (coprime) integer ratio m : n
then coupling might produce an attracting m : n limit cycle on the NH torus, which
makes m revolutions in the γ 1 direction and n in the γ 2 direction per period. On the
other hand, for weak coupling and smooth enough dynamics, the non-synchronised
situation occurs with high probability. More precisely, if one adds a free parameter
varying the unperturbed frequency ratio, then KAM theory gives a set of parameter
values of nearly full measure for which the dynamics is conjugate to a constant vector
field on a 2-torus with irrational frequency ratio (e.g. [17] for a version by my Ph.D.
student João Lopes Dias). Thus synchronisation does not always result.
Now consider the non-autonomous situation, where one or both of the oscillators
is subject to external forcing. If the forcing is not too strong then the resulting system
has a NH submanifold in extended state space, diffeomorphic to γ 1 × γ 2 × R, which
I call a torus-cylinder. More generally, for any manifold M I define an M-cylinder
to be a manifold diffeomorphic to M × R. Thus an ordinary cylinder can be called
a circle-cylinder. If the unperturbed frequencies are close to integer ratio m : n then
the NH submanifold might contain a NH attracting submanifold diffeomorphic to a
2 The original KAM theory gives a set of invariant tori for near-integrable Hamiltonian systems, the
measure of whose complement goes to zero as the perturbation from integrability goes to zero.