5 Normal Hyperbolicity for Non-autonomous Oscillators and Oscillator Networks
79
Given a Lipschitz graph r = ρ(θ, t), a candidate for an invariant submanifold,
construct a new one, T ρ, by the following steps:
(1) For all (θ 0 , t 0 ), let θ() be the trajectory of ˙
θ(t) = (θ, ρ(θ, t), t) from θ(t 0 ) = θ 0 .
(2) Solve ˙
r (t) = R(θ(t), r (t), t) for the unique function r () such that r (t) is near
ρ(θ(t), t) for all t.
(3) Set (T ρ)(θ 0 , t 0 ) = r (t 0 ).
To achieve the second step, I assume that L : C
1
(R, R
p
) → C
0
(R, R
p
) defined by
L[x](t) = ˙
x(t) − R r (θ(t), r (t), t)x(t)
on infinitesimal displacements x in r has bounded inverse. This is equivalent to
the first part of the NH condition, namely a splitting of the normal bundle into
exponentially contracting backwards and forwards subspaces.
Having thus constructed the “graph transform” T , I want to prove that it is a
contraction on a suitable space of graphs and hence has a unique fixed point there,
which will be an invariant graph. In the direction of achieving this, define a slope to be
a linear map σ from displacements in θ to displacements in r . For an approximation
˜
σ to the expected derivative ρ θ , define M ˜
σ : W
1,∞
(R, R
mp
) → W
0,∞
(R, R
mp
) by
M ˜
σ [σ] = ˙
σ − R r σ + σ(( θ + r ˜
σ)
on slope functions σ of t, where W
s,∞ are the spaces of functions with essentially
bounded s
th derivative. Suppose that M ˜
σ has bounded inverse. This is the second part
of the NH condition, namely faster normal contraction than tangential contraction.
Then T should be a contraction in the space of C
0 functions with an a priori
Lipschitz constant. So it would have a unique fixed point ρ. Any fixed point is
invariant and actually C
1 with slope ρ θ being the fixed point of the contraction map
σ → M
−1
σ [R θ ].
To complete this programme requires detailed estimates. Formulated in terms
of contraction maps as here, it should be possible to obtain excellent estimates,
along the lines of the uniformly hyperbolic case in [4]. We might do best to follow
the approach of [14] (cf. [8]), but replacing their exponential hypotheses by our
hypotheses of invertibility of L and M and modifying their exponentially weighted
norm to use the linearised tangential flow. I would like to finish this one day.
5.8 Extension to Class 1 Neurons
So far, I have considered the simplest type of oscillator, namely limit cycles, but
the treatment can be extended to class I neurons (or excitable oscillators). These are
dynamical systems with an attracting invariant cylinder in the autonomous case and
dynamics on it in simplest form given by
79
Given a Lipschitz graph r = ρ(θ, t), a candidate for an invariant submanifold,
construct a new one, T ρ, by the following steps:
(1) For all (θ 0 , t 0 ), let θ() be the trajectory of ˙
θ(t) = (θ, ρ(θ, t), t) from θ(t 0 ) = θ 0 .
(2) Solve ˙
r (t) = R(θ(t), r (t), t) for the unique function r () such that r (t) is near
ρ(θ(t), t) for all t.
(3) Set (T ρ)(θ 0 , t 0 ) = r (t 0 ).
To achieve the second step, I assume that L : C
1
(R, R
p
) → C
0
(R, R
p
) defined by
L[x](t) = ˙
x(t) − R r (θ(t), r (t), t)x(t)
on infinitesimal displacements x in r has bounded inverse. This is equivalent to
the first part of the NH condition, namely a splitting of the normal bundle into
exponentially contracting backwards and forwards subspaces.
Having thus constructed the “graph transform” T , I want to prove that it is a
contraction on a suitable space of graphs and hence has a unique fixed point there,
which will be an invariant graph. In the direction of achieving this, define a slope to be
a linear map σ from displacements in θ to displacements in r . For an approximation
˜
σ to the expected derivative ρ θ , define M ˜
σ : W
1,∞
(R, R
mp
) → W
0,∞
(R, R
mp
) by
M ˜
σ [σ] = ˙
σ − R r σ + σ(( θ + r ˜
σ)
on slope functions σ of t, where W
s,∞ are the spaces of functions with essentially
bounded s
th derivative. Suppose that M ˜
σ has bounded inverse. This is the second part
of the NH condition, namely faster normal contraction than tangential contraction.
Then T should be a contraction in the space of C
0 functions with an a priori
Lipschitz constant. So it would have a unique fixed point ρ. Any fixed point is
invariant and actually C
1 with slope ρ θ being the fixed point of the contraction map
σ → M
−1
σ [R θ ].
To complete this programme requires detailed estimates. Formulated in terms
of contraction maps as here, it should be possible to obtain excellent estimates,
along the lines of the uniformly hyperbolic case in [4]. We might do best to follow
the approach of [14] (cf. [8]), but replacing their exponential hypotheses by our
hypotheses of invertibility of L and M and modifying their exponentially weighted
norm to use the linearised tangential flow. I would like to finish this one day.
5.8 Extension to Class 1 Neurons
So far, I have considered the simplest type of oscillator, namely limit cycles, but
the treatment can be extended to class I neurons (or excitable oscillators). These are
dynamical systems with an attracting invariant cylinder in the autonomous case and
dynamics on it in simplest form given by