5 Normal Hyperbolicity for Non-autonomous Oscillators and Oscillator Networks
75
Perhaps this direction is not what Aneta had in mind, but I believe it provides
a self-consistent theory for non-autonomous oscillators and I hope that it will be
useful.
5.2 Phase-Locking
It is well-known that an oscillator may phase-lock to some features of its inputs.
Indeed, this is the principle of phase-locked loops in electronic engineering [5] and
of synchronous generators and motors in AC electrical networks.
My definition of phase-locking of an oscillator to forcing is that the NH cylinder
(assumed attracting) has an attracting trajectory on it and the initial condition is in
its basin of attraction.
Any discussion of attractors for non-autonomous systems requires care because
the dynamics is unbounded in the time-direction of extended state-space, so there
are inequivalent choices of neighbourhoods of a trajectory. For example, for the
2D system ˙
x = x, ˙
s = 1, any trajectory has a neighbourhood of attraction, despite
looking unstable, e.g. for the solution x = 0 just take neighbourhood of the form
|x| < εe
2s . So I make precise here that by “attracting trajectory” I mean the case
with zero unstable space of a uniformly hyperbolic trajectory in the non-autonomous
sense. To explain what this means would take some space, so I refer the reader to
[4] (with my Ph.D. student Zahir Bishnani), but the important feature is to choose a
notion of distance in extended state-space that is uniform in time (so that one does
not allow neighbourhoods like that in the above example). There might be a bundle
of trajectories which all converge together in forward time, but in general there
is only one trajectory in the bundle that has a uniform-in-time neighbourhood of
attraction. It is a pullback attractor (for this concept, see the contribution by Kloeden
in this volume). My concept of attracting trajectory is distinct, however, from that of
pullback attractor, because it can also occur that a pullback attractor is not uniformly
hyperbolic (it may be repelling after some time).
An alternative way to describe phase-locking is that the oscillator is synchronised
to its inputs. I use “synchronise” in a weak sense: that to a given input function of
time there is a locally unique forwards asymptotic solution (the strong sense applies
to systems of identical oscillators with a symmetry of the coupling that maps any
oscillator to any other, and consists in all oscillators doing the same; for an example,
see [31]). Note that a forced oscillator may have more than one such attracting
trajectory; this would allow different synchronisations to the same input.
This is in contrast to non-synchronisation, where there is a circle’s worth of solutions that do not converge asymptotically to a discrete subset. The strongest version
of non-synchronisation is when there is a time-dependent choice of C
1 coordinate φ
around the cylinder, replacing an initial coordinate θ, such that ˙
φ = ω(t), a positive
function of t only, and
∂φ
∂θ
and its inverse are bounded. Then with a new time τ
defined by dτ /dt = ω(t), we obtain dφ/dτ = 1. It would be interesting to investigate the probability of this case with respect to a distribution of oscillator frequencies
75
Perhaps this direction is not what Aneta had in mind, but I believe it provides
a self-consistent theory for non-autonomous oscillators and I hope that it will be
useful.
5.2 Phase-Locking
It is well-known that an oscillator may phase-lock to some features of its inputs.
Indeed, this is the principle of phase-locked loops in electronic engineering [5] and
of synchronous generators and motors in AC electrical networks.
My definition of phase-locking of an oscillator to forcing is that the NH cylinder
(assumed attracting) has an attracting trajectory on it and the initial condition is in
its basin of attraction.
Any discussion of attractors for non-autonomous systems requires care because
the dynamics is unbounded in the time-direction of extended state-space, so there
are inequivalent choices of neighbourhoods of a trajectory. For example, for the
2D system ˙
x = x, ˙
s = 1, any trajectory has a neighbourhood of attraction, despite
looking unstable, e.g. for the solution x = 0 just take neighbourhood of the form
|x| < εe
2s . So I make precise here that by “attracting trajectory” I mean the case
with zero unstable space of a uniformly hyperbolic trajectory in the non-autonomous
sense. To explain what this means would take some space, so I refer the reader to
[4] (with my Ph.D. student Zahir Bishnani), but the important feature is to choose a
notion of distance in extended state-space that is uniform in time (so that one does
not allow neighbourhoods like that in the above example). There might be a bundle
of trajectories which all converge together in forward time, but in general there
is only one trajectory in the bundle that has a uniform-in-time neighbourhood of
attraction. It is a pullback attractor (for this concept, see the contribution by Kloeden
in this volume). My concept of attracting trajectory is distinct, however, from that of
pullback attractor, because it can also occur that a pullback attractor is not uniformly
hyperbolic (it may be repelling after some time).
An alternative way to describe phase-locking is that the oscillator is synchronised
to its inputs. I use “synchronise” in a weak sense: that to a given input function of
time there is a locally unique forwards asymptotic solution (the strong sense applies
to systems of identical oscillators with a symmetry of the coupling that maps any
oscillator to any other, and consists in all oscillators doing the same; for an example,
see [31]). Note that a forced oscillator may have more than one such attracting
trajectory; this would allow different synchronisations to the same input.
This is in contrast to non-synchronisation, where there is a circle’s worth of solutions that do not converge asymptotically to a discrete subset. The strongest version
of non-synchronisation is when there is a time-dependent choice of C
1 coordinate φ
around the cylinder, replacing an initial coordinate θ, such that ˙
φ = ω(t), a positive
function of t only, and
∂φ
∂θ
and its inverse are bounded. Then with a new time τ
defined by dτ /dt = ω(t), we obtain dφ/dτ = 1. It would be interesting to investigate the probability of this case with respect to a distribution of oscillator frequencies