5 Normal Hyperbolicity for Non-autonomous Oscillators and Oscillator Networks
77
circle cross time, being a perturbation of the product of a m : n synchronised limit
cycle for the autonomous system and time. In this situation the non-autonomous pair
of oscillators can be replaced by a single one.
So again, synchronisation of two oscillators is a dimension-reduction.
5.4 What is Coupling?
In the previous section I used the standard dynamical systems notion for coupling
as a perturbation of the product of two vector fields. One might want, however, to
allow more general forms of coupling, for example incorporating time-delays or
coupling via an intermediate dynamical system. Furthermore, suppose one achieved
a dimension-reduction as in Sect. 5.2 or 5.3 and then wants to consider how the new
effective oscillator is coupled to others that originally were coupled to one or both
of the pair of oscillators. This is no longer describable as a standard perturbation of
the product of vector fields.
So I generalise the notion of coupling of two non-autonomous oscillators. As
already defined, a non-autonomous oscillator is a non-autonomous system with NH
cylinder on which the dynamics can be described by one phase θ with ˙
θ = f (θ, t). A
coupling of two non-autonomous oscillators is a non-autonomous system with a NH
torus-cylinder on which the dynamics can be described by two phases θ = (θ 1 , θ 2 )
with ˙
θ i = ˜
f i (θ, t) and ˜
f i (θ, t) close to f i (θ i , t) for some f i .
Then the dynamics on the NH torus-cylinder may contain a NH attracting circlecylinder, as in the more restricted case of the previous section. If the trajectory is in
its basin of attraction, I say the two oscillators synchronise.
5.5 Synchronisation of N Oscillators
Not too strong coupling of N non-autonomous oscillators produces a NH N -toruscylinder. The dynamics on it might contain an attracting NH d-torus-cylinder for
some d < N . If d = 1 the whole group is synchronised and can be replaced by a
single effective non-autonomous oscillator. If d = 0 the whole group is phase-locked
to its inputs and can be eliminated.
Once again, synchronisation, whether partial or complete, means dimensionreduction.
5.6 Hierarchical Aggregation
In a network of oscillators, the above dimension-reductions can in principle be iterated. First one identifies groups of oscillators which synchronise or phase-lock to
77
circle cross time, being a perturbation of the product of a m : n synchronised limit
cycle for the autonomous system and time. In this situation the non-autonomous pair
of oscillators can be replaced by a single one.
So again, synchronisation of two oscillators is a dimension-reduction.
5.4 What is Coupling?
In the previous section I used the standard dynamical systems notion for coupling
as a perturbation of the product of two vector fields. One might want, however, to
allow more general forms of coupling, for example incorporating time-delays or
coupling via an intermediate dynamical system. Furthermore, suppose one achieved
a dimension-reduction as in Sect. 5.2 or 5.3 and then wants to consider how the new
effective oscillator is coupled to others that originally were coupled to one or both
of the pair of oscillators. This is no longer describable as a standard perturbation of
the product of vector fields.
So I generalise the notion of coupling of two non-autonomous oscillators. As
already defined, a non-autonomous oscillator is a non-autonomous system with NH
cylinder on which the dynamics can be described by one phase θ with ˙
θ = f (θ, t). A
coupling of two non-autonomous oscillators is a non-autonomous system with a NH
torus-cylinder on which the dynamics can be described by two phases θ = (θ 1 , θ 2 )
with ˙
θ i = ˜
f i (θ, t) and ˜
f i (θ, t) close to f i (θ i , t) for some f i .
Then the dynamics on the NH torus-cylinder may contain a NH attracting circlecylinder, as in the more restricted case of the previous section. If the trajectory is in
its basin of attraction, I say the two oscillators synchronise.
5.5 Synchronisation of N Oscillators
Not too strong coupling of N non-autonomous oscillators produces a NH N -toruscylinder. The dynamics on it might contain an attracting NH d-torus-cylinder for
some d < N . If d = 1 the whole group is synchronised and can be replaced by a
single effective non-autonomous oscillator. If d = 0 the whole group is phase-locked
to its inputs and can be eliminated.
Once again, synchronisation, whether partial or complete, means dimensionreduction.
5.6 Hierarchical Aggregation
In a network of oscillators, the above dimension-reductions can in principle be iterated. First one identifies groups of oscillators which synchronise or phase-lock to