90
M. Lucas et al.
harder to treat analytically in general. Before closing this section, we mention that
we consider, in this chapter, cases where the nonautonomicity comes from either
time-varying frequencies or coupling strength.
6.3 Methods
We now briefly present existing methods for the analysis of non-autonomous dynamical systems. We discuss the assumptions made in each case, and describe their
strengths and limitations when dealing with time-varying systems.
6.3.1 Asymptotic
First, we present traditional time-asymptotic methods that were historically developed for autonomous dynamical systems. For these methods, it is assumed that the
model considered is defined over infinite time. We restrict ourselves to methods relevant for the rest of the present work. Other methods can be found e.g. in [54] for a
theoretical approach, and in [10] for a signal processing approach.
6.3.1.1 Lyapunov Exponents
The stability of a trajectory x(t) of a dynamical system (which could be autonomous,
noise-diven, or nonautonomous) is often assessed in terms of the asymptotic Lyapunov exponents (ALEs) associated to it [48]. For a trajectory x(t) and a unit vector
v 0 corresponding to a direction of initial perturbation at time 0, we define the corresponding ALE as follows: letting x δ (t) denote the solution starting at x(0) + δv 0 for
each δ ∈ R, the ALE λ
(∞) is given by
λ
(∞)
= lim
t→∞
lim
δ→0
1
t
ln
x δ (t) − x(t)
|δ|
(6.4)
if this limit exists. This measures the exponential separation rate between x(t) and a
nearby trajectory: a positive value corresponds to exponentially growing separation,
while a negative value corresponds to exponential attraction between the trajectories.
This heuristic notion of “infinitely close trajectories” can be made mathematically
rigorous in terms of suitable calculus notions. If the dynamical system has a ddimensional state space, we define a spectrum of d ALEs associated to a trajectory
x(t), typically ordered as λ max = λ 1 ≥ λ 2 ≥ · · · ≥ λ d , each one corresponding to a
different direction of the initial infinitesimal separation δx(0). Assuming the system
is sufficiently well behaved, the maximal ALE λ max is obtained in Eq. (6.4) for a
generic v 0 . The maximal ALE indicates the stability of the trajectory x(t): if λ max > 0
M. Lucas et al.
harder to treat analytically in general. Before closing this section, we mention that
we consider, in this chapter, cases where the nonautonomicity comes from either
time-varying frequencies or coupling strength.
6.3 Methods
We now briefly present existing methods for the analysis of non-autonomous dynamical systems. We discuss the assumptions made in each case, and describe their
strengths and limitations when dealing with time-varying systems.
6.3.1 Asymptotic
First, we present traditional time-asymptotic methods that were historically developed for autonomous dynamical systems. For these methods, it is assumed that the
model considered is defined over infinite time. We restrict ourselves to methods relevant for the rest of the present work. Other methods can be found e.g. in [54] for a
theoretical approach, and in [10] for a signal processing approach.
6.3.1.1 Lyapunov Exponents
The stability of a trajectory x(t) of a dynamical system (which could be autonomous,
noise-diven, or nonautonomous) is often assessed in terms of the asymptotic Lyapunov exponents (ALEs) associated to it [48]. For a trajectory x(t) and a unit vector
v 0 corresponding to a direction of initial perturbation at time 0, we define the corresponding ALE as follows: letting x δ (t) denote the solution starting at x(0) + δv 0 for
each δ ∈ R, the ALE λ
(∞) is given by
λ
(∞)
= lim
t→∞
lim
δ→0
1
t
ln
x δ (t) − x(t)
|δ|
(6.4)
if this limit exists. This measures the exponential separation rate between x(t) and a
nearby trajectory: a positive value corresponds to exponentially growing separation,
while a negative value corresponds to exponential attraction between the trajectories.
This heuristic notion of “infinitely close trajectories” can be made mathematically
rigorous in terms of suitable calculus notions. If the dynamical system has a ddimensional state space, we define a spectrum of d ALEs associated to a trajectory
x(t), typically ordered as λ max = λ 1 ≥ λ 2 ≥ · · · ≥ λ d , each one corresponding to a
different direction of the initial infinitesimal separation δx(0). Assuming the system
is sufficiently well behaved, the maximal ALE λ max is obtained in Eq. (6.4) for a
generic v 0 . The maximal ALE indicates the stability of the trajectory x(t): if λ max > 0