72
R. S. MacKay
some variables for a dynamical system might contract relatively rapidly onto some
invariant submanifold in the state space, and then it suffices to study the dynamics
on the submanifold. Two key results of normal hyperbolicity theory are: (i) conditions guaranteeing existence of such a submanifold, called a normally hyperbolic
(NH) submanifold, and (ii) their smooth persistence of normally hyperbolic (NH)
submanifolds as parameters are varied smoothly. It was developed before Haken’s
slaving principle and deserves to be better known in the physics community. It is a
generalisation of centre manifold theory, which is the main mathematical tool Haken
used, but has much wider scope. An obstacle is that it demands considerable technical expertise in mathematical analysis. Yet the obstacles are genuine: it turns out that
NH submanifolds are differentiable some number r times, depending on the ratio
between normal and tangential contraction rates, but typically not more than r times.
This is important to recognise, as there is a tendency in physics to consider such
functions as pathologies (though physicists do understand that there can be fractal
functions).
It is a project on which I have been working for many years, notably with Ph.D.
student Stephen Gin (2006–13). It was prompted initially by Mohammad Ghaffari
Saadat in 2003, who had formulated a limit-cycle model for a bipedal robot walking
down a slope [28] and asked me how much non-uniformity of slope it could cope
with. I proposed to tackle this problem by fitting it into the framework of the nonautonomous version of the theory of NH submanifolds, where the result of a not too
large forcing function on an oscillator is a circle of possible trajectories. Gin and
I attempted to develop good versions of the proofs of normal hyperbolicity results
to produce realistic conditions guaranteeing the outcome [12]. Our approach is still
incomplete, but I present here the key ideas.
In the world of conservative dynamics, an oscillator is considered to be a Hamiltonian system with an elliptic equilibrium point; this view has fundamental importance
but is not the appropriate one for present purposes.
Outside the world of conservative dynamics, an oscillator is usually considered to
be an autonomous dynamical system with an attracting periodic orbit. The concept
has been extended to cater for chaotic oscillators, but I will postpone treating that
extension until near the end of this chapter.
This concept of oscillator as a system with an attracting limit-cycle, however, fails
to include the many situations where it is subject to time-dependent forcing. Also, in a
network of oscillators, each is subject to input from others, in general time-dependent,
so even if the network is autonomous it is useful to consider time-dependent forcing
on each of its oscillators.
So I propose a view of an oscillator as a mapping from input functions f of time to
a circle’s worth of solutions for its state x as a function of time. Each input function
f (possibly with more than one component) causes a response x θ (a function of time)
with a phase θ ∈ S
1 labelling the different possible responses. This view is justified
by the theory of normal hyperbolicity, at least for not too strong forcing. It is also
my interpretation of chronotaxic systems.
The idea is to consider a non-autonomous system ˙
x = v(x, t) on a state space X
as an autonomous system in the extended state space X × R, with the real line R
R. S. MacKay
some variables for a dynamical system might contract relatively rapidly onto some
invariant submanifold in the state space, and then it suffices to study the dynamics
on the submanifold. Two key results of normal hyperbolicity theory are: (i) conditions guaranteeing existence of such a submanifold, called a normally hyperbolic
(NH) submanifold, and (ii) their smooth persistence of normally hyperbolic (NH)
submanifolds as parameters are varied smoothly. It was developed before Haken’s
slaving principle and deserves to be better known in the physics community. It is a
generalisation of centre manifold theory, which is the main mathematical tool Haken
used, but has much wider scope. An obstacle is that it demands considerable technical expertise in mathematical analysis. Yet the obstacles are genuine: it turns out that
NH submanifolds are differentiable some number r times, depending on the ratio
between normal and tangential contraction rates, but typically not more than r times.
This is important to recognise, as there is a tendency in physics to consider such
functions as pathologies (though physicists do understand that there can be fractal
functions).
It is a project on which I have been working for many years, notably with Ph.D.
student Stephen Gin (2006–13). It was prompted initially by Mohammad Ghaffari
Saadat in 2003, who had formulated a limit-cycle model for a bipedal robot walking
down a slope [28] and asked me how much non-uniformity of slope it could cope
with. I proposed to tackle this problem by fitting it into the framework of the nonautonomous version of the theory of NH submanifolds, where the result of a not too
large forcing function on an oscillator is a circle of possible trajectories. Gin and
I attempted to develop good versions of the proofs of normal hyperbolicity results
to produce realistic conditions guaranteeing the outcome [12]. Our approach is still
incomplete, but I present here the key ideas.
In the world of conservative dynamics, an oscillator is considered to be a Hamiltonian system with an elliptic equilibrium point; this view has fundamental importance
but is not the appropriate one for present purposes.
Outside the world of conservative dynamics, an oscillator is usually considered to
be an autonomous dynamical system with an attracting periodic orbit. The concept
has been extended to cater for chaotic oscillators, but I will postpone treating that
extension until near the end of this chapter.
This concept of oscillator as a system with an attracting limit-cycle, however, fails
to include the many situations where it is subject to time-dependent forcing. Also, in a
network of oscillators, each is subject to input from others, in general time-dependent,
so even if the network is autonomous it is useful to consider time-dependent forcing
on each of its oscillators.
So I propose a view of an oscillator as a mapping from input functions f of time to
a circle’s worth of solutions for its state x as a function of time. Each input function
f (possibly with more than one component) causes a response x θ (a function of time)
with a phase θ ∈ S
1 labelling the different possible responses. This view is justified
by the theory of normal hyperbolicity, at least for not too strong forcing. It is also
my interpretation of chronotaxic systems.
The idea is to consider a non-autonomous system ˙
x = v(x, t) on a state space X
as an autonomous system in the extended state space X × R, with the real line R