5 Normal Hyperbolicity for Non-autonomous Oscillators and Oscillator Networks
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omorphic to the product of a Euclidean space with a general topological space. It
decomposes into leaves, which are locally submanifolds but in general only injectively immersed, so a leaf may accumulate onto itself. The theory of NH laminations
requires a C
1 -structure in addition, but is basically the same as for NH submanifolds.
In particular, a NH lamination persists under C
1 -small perturbation.
This means one can treat some chaotic attractors in greater detail. In particular,
imagine we start with a non-trivial uniformly hyperbolic attractor of an autonomous
system, for example a suspension of a Plykin attractor [23]. This is perhaps less
familiar than Rössler’s attractor but deserves to be better known, as the simplest uniformly hyperbolic attractor after equilibria and periodic orbits. The Plykin attractor
was constructed for a discrete-time system, but the map is isotopic to the identity so
one can realise it as the first return map of an associated continuous-time system.
My Ph.D. student Tim Hunt showed an explicit way to realise it in a system of three
ODEs, extended by another Ph.D. student Linling Ru, and less cumbersome ways
have been proposed (though not yet with rigorous justification) [16]. It is a NH lamination, whose leaves are its unstable manifolds (of dimension two: one expanding
dimension and one time-dimension) and they form a Cantor set transversally. Under
time-dependent forcing, it persists to a Cantor set of 3D leaves whose tangent space
is spanned by one expanding dimension and two near-neutral dimensions. The persistence is highly robust, requiring only that any tangential contraction be slower
than any transverse contraction. Then one can ask what happens on the leaves. The
dynamics might collapse onto a 2D subleaf with the same expanding dimension-one
neutral dimension. I would say the attractor has synchronised to the forcing.
Similarly, one could couple a suspended Plykin attractor to a limit-cycle oscillator.
It produces an attractor with a Cantor set of 3D leaves (the product of the 2D leaves of
the chaotic attractor with the limit cycle). The dynamics of each leaf might collapse
onto 2D subleaves. I would say the Plykin attractor and limit cycle synchronise
together.
More generally, one could couple a continuous-time autonomous uniformly hyperbolic attractor with M unstable dimensions to N limit cycle oscillators and obtain an
attractor with a mixture of chaos and nearly quasiperiodic behaviour. It would have
M unstable dimensions, N nearly quasiperiodic dimensions, and the flow dimension,
with the remaining dimensions contracting onto the leaves. By the theory of NH laminations, such attractors persist for small smooth perturbations, though the dynamics in
the quasiperiodic dimensions cannot be expected to remain quasiperiodic. Nonetheless, it will have small Lyapunov exponents for those dimensions and perhaps there
is a non-autonomous KAM theory that would even give truly quasiperiodic motion
for a set of nearly full measure of parameters. I propose this as an explanation of the
scenario reported recently by [30].
As a final note, one might ask about physical realisation of attractors like Rössler’s.
I designed an electronic oscillator back in 1981, principally to demonstrate perioddoubling sequences [18], but moving the parameter further it exhibited a Rössler
type of attractor. Model equations for the voltages at three points have the form
81
omorphic to the product of a Euclidean space with a general topological space. It
decomposes into leaves, which are locally submanifolds but in general only injectively immersed, so a leaf may accumulate onto itself. The theory of NH laminations
requires a C
1 -structure in addition, but is basically the same as for NH submanifolds.
In particular, a NH lamination persists under C
1 -small perturbation.
This means one can treat some chaotic attractors in greater detail. In particular,
imagine we start with a non-trivial uniformly hyperbolic attractor of an autonomous
system, for example a suspension of a Plykin attractor [23]. This is perhaps less
familiar than Rössler’s attractor but deserves to be better known, as the simplest uniformly hyperbolic attractor after equilibria and periodic orbits. The Plykin attractor
was constructed for a discrete-time system, but the map is isotopic to the identity so
one can realise it as the first return map of an associated continuous-time system.
My Ph.D. student Tim Hunt showed an explicit way to realise it in a system of three
ODEs, extended by another Ph.D. student Linling Ru, and less cumbersome ways
have been proposed (though not yet with rigorous justification) [16]. It is a NH lamination, whose leaves are its unstable manifolds (of dimension two: one expanding
dimension and one time-dimension) and they form a Cantor set transversally. Under
time-dependent forcing, it persists to a Cantor set of 3D leaves whose tangent space
is spanned by one expanding dimension and two near-neutral dimensions. The persistence is highly robust, requiring only that any tangential contraction be slower
than any transverse contraction. Then one can ask what happens on the leaves. The
dynamics might collapse onto a 2D subleaf with the same expanding dimension-one
neutral dimension. I would say the attractor has synchronised to the forcing.
Similarly, one could couple a suspended Plykin attractor to a limit-cycle oscillator.
It produces an attractor with a Cantor set of 3D leaves (the product of the 2D leaves of
the chaotic attractor with the limit cycle). The dynamics of each leaf might collapse
onto 2D subleaves. I would say the Plykin attractor and limit cycle synchronise
together.
More generally, one could couple a continuous-time autonomous uniformly hyperbolic attractor with M unstable dimensions to N limit cycle oscillators and obtain an
attractor with a mixture of chaos and nearly quasiperiodic behaviour. It would have
M unstable dimensions, N nearly quasiperiodic dimensions, and the flow dimension,
with the remaining dimensions contracting onto the leaves. By the theory of NH laminations, such attractors persist for small smooth perturbations, though the dynamics in
the quasiperiodic dimensions cannot be expected to remain quasiperiodic. Nonetheless, it will have small Lyapunov exponents for those dimensions and perhaps there
is a non-autonomous KAM theory that would even give truly quasiperiodic motion
for a set of nearly full measure of parameters. I propose this as an explanation of the
scenario reported recently by [30].
As a final note, one might ask about physical realisation of attractors like Rössler’s.
I designed an electronic oscillator back in 1981, principally to demonstrate perioddoubling sequences [18], but moving the parameter further it exhibited a Rössler
type of attractor. Model equations for the voltages at three points have the form