118
J. M. I. Newman et al.
• if A < A ∗ then the trajectories of (7.3) exhibit neutral stability,
• but if A > A ∗ then the trajectories of (7.3) exhibit stability.
In physical terms, sufficiently broadly time-variable forcing induces phase stability
in an oscillatory process evolving according to the model (7.3). The chapter [24]
presents the stabilisation phenomenon from the point of view of the overall region
of stability in parameter space. The value A ∗ represents the critical A-value for the
transition between neutral stability and stability.
7.2.5 Numerics for the Non-autonomous Adler Equation
To aid physical intuition, in the presentation of all our numerics, time t is considered
as real time measured in units of seconds, and all other parameters and variables have
the appropriate units accordingly. Throughout this chapter, solutions of Eq. (7.1) are
simulated by numerical integration using a 4th order Runge-Kutta scheme, with a
time step of 0.01 s. For the reverse-time bifurcation diagrams, to obtain the initial
condition θ(0) for a given final state θ(T ), the value of θ(T ) was used as the initial
condition θ back (0) of a forward-time simulation of the differential equation ˙
θ back (t) =
a sin(θ back (t)) − G(T − t), and then θ(0) was taken as θ back (T ).
7.3 A Toy Model of “inherently Finite-Time” Dynamics
In this section, we use the stabilisation phenomenon described in Sect. 7.2.4 to illustrate that stability and neutral stability can be central among the physical properties
of a finite-time process even if the process’s dynamics cannot be described by the
long-term behaviour of an infinite-time mathematical model of the process. In this
case, such stability properties cannot be formalised and quantified by the traditional
mathematical formalisms such as asymptotic Lyapunov exponents.
Specifically, as a finite-time adaptation of the indefinite-time ergodic Gaussian
process considered in [14], we consider here the non-autonomous Adler equation (7.3) with g(t) being a low-pass-filtered sample realisation of a Brownian bridge,
as illustrated in Fig. 7.1.
7.3.1 Definition and Basic Properties of a Brownian Bridge
A Brownian bridge is a finite-time stochastic process that models the inhomogeneity of a large sample of random times selected mutually independently from the
uniform distribution on a pre-specified finite time-interval. Namely, fixing a finite
time-interval [0, τ ] and a value σ > 0 (analogous to the diffusion parameter of
J. M. I. Newman et al.
• if A < A ∗ then the trajectories of (7.3) exhibit neutral stability,
• but if A > A ∗ then the trajectories of (7.3) exhibit stability.
In physical terms, sufficiently broadly time-variable forcing induces phase stability
in an oscillatory process evolving according to the model (7.3). The chapter [24]
presents the stabilisation phenomenon from the point of view of the overall region
of stability in parameter space. The value A ∗ represents the critical A-value for the
transition between neutral stability and stability.
7.2.5 Numerics for the Non-autonomous Adler Equation
To aid physical intuition, in the presentation of all our numerics, time t is considered
as real time measured in units of seconds, and all other parameters and variables have
the appropriate units accordingly. Throughout this chapter, solutions of Eq. (7.1) are
simulated by numerical integration using a 4th order Runge-Kutta scheme, with a
time step of 0.01 s. For the reverse-time bifurcation diagrams, to obtain the initial
condition θ(0) for a given final state θ(T ), the value of θ(T ) was used as the initial
condition θ back (0) of a forward-time simulation of the differential equation ˙
θ back (t) =
a sin(θ back (t)) − G(T − t), and then θ(0) was taken as θ back (T ).
7.3 A Toy Model of “inherently Finite-Time” Dynamics
In this section, we use the stabilisation phenomenon described in Sect. 7.2.4 to illustrate that stability and neutral stability can be central among the physical properties
of a finite-time process even if the process’s dynamics cannot be described by the
long-term behaviour of an infinite-time mathematical model of the process. In this
case, such stability properties cannot be formalised and quantified by the traditional
mathematical formalisms such as asymptotic Lyapunov exponents.
Specifically, as a finite-time adaptation of the indefinite-time ergodic Gaussian
process considered in [14], we consider here the non-autonomous Adler equation (7.3) with g(t) being a low-pass-filtered sample realisation of a Brownian bridge,
as illustrated in Fig. 7.1.
7.3.1 Definition and Basic Properties of a Brownian Bridge
A Brownian bridge is a finite-time stochastic process that models the inhomogeneity of a large sample of random times selected mutually independently from the
uniform distribution on a pre-specified finite time-interval. Namely, fixing a finite
time-interval [0, τ ] and a value σ > 0 (analogous to the diffusion parameter of