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6.3.2 Finite-Time
In this section, we present methods that are time-resolved counterparts of the ones
described above. Here, the output of the methods is the evolution of measured properties over time, allowing one to observe the time-variable dynamics.
6.3.2.1 Finite-Time Lyapunov Exponents
Stability can be considered over finite time. In particular, given a time-window
[t, t + T ], for a trajectory x(·) and a unit vector v t corresponding to a direction
of perturbation at time t, we define the corresponding FTLE as follows: letting x δ (·)
denote the solution coinciding with x(t) + δv t at time t for each δ ∈ R, the FTLE
λ T (t) is given by
λ T (t) = lim
δ→0
1
T
ln
x δ (t) − x(t)
|δ|
.
(6.6)
The length T of the time-window is of crucial importance, as different lengths
will reveal features of the dynamics over different timescales. The choice of windowlength thus depends on the system at hand and the timescale of interest; a longer timewindow will average out faster-timescale variations of time-localised stability. For
T → ∞, one recovers the ALE; for T → 0, one obtains an instantaneous Lyapunov
exponent (ILE). In numerical computation of an ALE via direct simulation, one is
really just computing a FTLE over a very long time-window.
Contrary to ALEs, FTLEs are not coordinate-invariant. The FTLE has the advantage that it does not rely on the aforementioned assumptions (i) and (ii) relative to
the definition of the model and the existence of the limit t → ∞. The FTLE can be
applied to any model, autonomous or non-autonomous, defined over finite or infinite
time.
Note that an ALE for a direction of initial perturbation v 0 is an infinite-time average
of the ILE λ(t) in the direction v t := lim δ→0+
x δ (t)−x(t)
x δ (t)−x(t)
where x δ (0) = x(0) + δv 0 .
(In one dimension, this simply means that the single ALE is the infinite-time average
of the ILE.) Therefore, the maximal ALE only measures the average stability of a
trajectory, but does not give any indication about how it might change over time.
Often, in real-world systems, and especially in living systems, finite-time properties
are of crucial importance to their maintaining of vital functions and of life itself. For
the heart to keep beating, the mutual synchrony of its cells must be stable at all times,
not just on average.
M. Lucas et al.
6.3.2 Finite-Time
In this section, we present methods that are time-resolved counterparts of the ones
described above. Here, the output of the methods is the evolution of measured properties over time, allowing one to observe the time-variable dynamics.
6.3.2.1 Finite-Time Lyapunov Exponents
Stability can be considered over finite time. In particular, given a time-window
[t, t + T ], for a trajectory x(·) and a unit vector v t corresponding to a direction
of perturbation at time t, we define the corresponding FTLE as follows: letting x δ (·)
denote the solution coinciding with x(t) + δv t at time t for each δ ∈ R, the FTLE
λ T (t) is given by
λ T (t) = lim
δ→0
1
T
ln
x δ (t) − x(t)
|δ|
.
(6.6)
The length T of the time-window is of crucial importance, as different lengths
will reveal features of the dynamics over different timescales. The choice of windowlength thus depends on the system at hand and the timescale of interest; a longer timewindow will average out faster-timescale variations of time-localised stability. For
T → ∞, one recovers the ALE; for T → 0, one obtains an instantaneous Lyapunov
exponent (ILE). In numerical computation of an ALE via direct simulation, one is
really just computing a FTLE over a very long time-window.
Contrary to ALEs, FTLEs are not coordinate-invariant. The FTLE has the advantage that it does not rely on the aforementioned assumptions (i) and (ii) relative to
the definition of the model and the existence of the limit t → ∞. The FTLE can be
applied to any model, autonomous or non-autonomous, defined over finite or infinite
time.
Note that an ALE for a direction of initial perturbation v 0 is an infinite-time average
of the ILE λ(t) in the direction v t := lim δ→0+
x δ (t)−x(t)
x δ (t)−x(t)
where x δ (0) = x(0) + δv 0 .
(In one dimension, this simply means that the single ALE is the infinite-time average
of the ILE.) Therefore, the maximal ALE only measures the average stability of a
trajectory, but does not give any indication about how it might change over time.
Often, in real-world systems, and especially in living systems, finite-time properties
are of crucial importance to their maintaining of vital functions and of life itself. For
the heart to keep beating, the mutual synchrony of its cells must be stable at all times,
not just on average.