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R. S. MacKay
their inputs. One reduces to a new network of effective oscillators. Then one repeats,
if possible. The end result is a decomposition into synchronised clusters.
Although I did not find out about his work until after I’d proposed this, it is a
direct example of Willems’ “tearing, zooming, linking” approach [29].
One should note that the end result is not necessarily complete synchronisation.
Indeed, it could well be a chimera [1], meaning a system in which some of the
oscillators are synchronised but others behave chaotically. The chaotic ones force
the synchronised ones and the synchronised ones force the chaotic ones, but our
approach of non-autonomous oscillators caters for both of these. There is now a
huge literature on chimera. To me the phenomenon was not a surprise because it
fits in my framework, but without the framework it can admittedly be considered
surprising.
5.7 Normal Hyperbolicity Estimates
To achieve the above dimension-reductions requires good normal hyperbolicity estimates, i.e. results guaranteeing existence of NH submanifolds.
The easiest case, namely, 1D submanifolds, which are just uniformly hyperbolic
trajectories of non-autonomous systems, was already treated in [4] (incidentally, it
was formulated with attracting trajectories in mind, but another application would
be to the unstable trajectories of geophysical flows that form boundaries between
trajectories of different classes, e.g. [11]). So that takes care of the case of phaselocking.
Higher-dimensional NH submanifolds, however, require more theory. The classic
references are [10, 15]. They are not particularly well adapted to producing practical
estimates. Thus I set Stephen Gin onto developing a better way. His Ph.D. thesis [12]
gives the outcome, but it is not a complete treatment. So here, I sketch an approach
to NH estimates that I believe will be useful. It is in the classic dynamical systems
setting of a vector field on the product of state space and time, but hopefully could
be extended to take care of the more general forms of coupling that I have described
here.
I restrict attention to submanifolds that are torus-cylinders, but of arbitrary dimension m + 1. So suppose
˙
θ = (θ, r, t)
(5.5)
˙
r = R(θ, r, t),
for θ ∈ T
m , r ∈ U , a neighbourhood of 0 ∈ R
p . I suppose that the product |R θ || r |
is small (where subscript denotes derivatives), the r -dynamics is hyperbolic, and the
Green function for linearised normal dynamics decays faster than any contraction
that may occur in θ-dynamics.
R. S. MacKay
their inputs. One reduces to a new network of effective oscillators. Then one repeats,
if possible. The end result is a decomposition into synchronised clusters.
Although I did not find out about his work until after I’d proposed this, it is a
direct example of Willems’ “tearing, zooming, linking” approach [29].
One should note that the end result is not necessarily complete synchronisation.
Indeed, it could well be a chimera [1], meaning a system in which some of the
oscillators are synchronised but others behave chaotically. The chaotic ones force
the synchronised ones and the synchronised ones force the chaotic ones, but our
approach of non-autonomous oscillators caters for both of these. There is now a
huge literature on chimera. To me the phenomenon was not a surprise because it
fits in my framework, but without the framework it can admittedly be considered
surprising.
5.7 Normal Hyperbolicity Estimates
To achieve the above dimension-reductions requires good normal hyperbolicity estimates, i.e. results guaranteeing existence of NH submanifolds.
The easiest case, namely, 1D submanifolds, which are just uniformly hyperbolic
trajectories of non-autonomous systems, was already treated in [4] (incidentally, it
was formulated with attracting trajectories in mind, but another application would
be to the unstable trajectories of geophysical flows that form boundaries between
trajectories of different classes, e.g. [11]). So that takes care of the case of phaselocking.
Higher-dimensional NH submanifolds, however, require more theory. The classic
references are [10, 15]. They are not particularly well adapted to producing practical
estimates. Thus I set Stephen Gin onto developing a better way. His Ph.D. thesis [12]
gives the outcome, but it is not a complete treatment. So here, I sketch an approach
to NH estimates that I believe will be useful. It is in the classic dynamical systems
setting of a vector field on the product of state space and time, but hopefully could
be extended to take care of the more general forms of coupling that I have described
here.
I restrict attention to submanifolds that are torus-cylinders, but of arbitrary dimension m + 1. So suppose
˙
θ = (θ, r, t)
(5.5)
˙
r = R(θ, r, t),
for θ ∈ T
m , r ∈ U , a neighbourhood of 0 ∈ R
p . I suppose that the product |R θ || r |
is small (where subscript denotes derivatives), the r -dynamics is hyperbolic, and the
Green function for linearised normal dynamics decays faster than any contraction
that may occur in θ-dynamics.