Global computation of phase-amplitude reduction for limit-cycle dynamics

Abstract

Recent years have witnessed increasing interest in phase-amplitude reduction of limit-cycle dynamics. Adding an amplitude coordinate to the phase coordinate allows us to take into account the dynamics transversal to the limit cycle and thereby overcome the main limitations of classic phase reduction (strong convergence to the limit cycle and weak inputs). While previous studies, mostly focus on local quantities such as infinitesimal responses, a major and limiting challenge of phase-amplitude reduction is to compute amplitude coordinates globally, in the basin of attraction of the limit cycle. In this paper, we propose a method to compute the full set of phase-amplitude coordinates in the large. Our method is based on the so-called Koopman (composition) operator and aims at computing the eigenfunctions of the operator through Laplace averages (in combination with the harmonic balance method). This yields a forward integration method that is not limited to two-dimensional systems. We illustrate the method by computing the so-called isostables of limit cycles in two-, three-, and four-dimensional state spaces, as well as their responses to strong external inputs.

Isostable reduction of oscillators with piecewise smooth dynamics and complex Floquet multipliers

“…Each complex conjugate pair of Floquet multipliers λ k and λ * k is associated with two isostable coordinates ψ M k and ψ P k defined in Eqs. (56) and (57), respectively. Near the periodic orbit, on a given isochron, level sets of ψ M are well approximated by ellipses, with ψ M k corresponding to the location on a given elipse.…”

“…(5) and (70) remain valid. For this reason, others have investigated the possibility of calculating phase and amplitude coordinates in R n [56], [57], [58], however, this is difficult to accomplish numerically for models of more than n = 4 dimensions and ultimately does not reduce the number of state variables which must be considered.…”

“…(63) it will be shown that the coordinates obtained from Eqs. (56) and (57) are related to the coordinates obtained from the definition from Eq. (8) by the relationships ψ M k = 2|ψ k | and ψ P k = arg(ψ k ).…”

“…where 1 can be calculated using the definitions from Eqs. (56) and (57), respectively, and are shown for different cross sections of X 3 in panels D and E. Some initial conditions (shown in white) do not converge to the limit cycle, but rather to other attractors (c.f., [52]). In this example, because the pendulum receives an external time dependent forcing, the phase is only dependent on the variable X 3 and the relationship θ(t) = 2πX3(t) T can be used to relate the X 3 coordinate to the isochrons.…”

Colored Noise in Oscillators. Phase-Amplitude Analysis and a Method to Avoid the itô-Stratonovich Dilemma

“…We shall derive an equivalent description of system (25) (or (26)) in terms of phase and amplitude deviation variables, analogous to the one derived in [18], [26], [27]. The phase function used in our description coincides locally, in the neighborhood of the limit cycle, with the asymptotic phase defined in [6], [13], [28]. As a second step, we shall derive a phase reduced model, that describes the oscillator dynamics in terms of the phase variable alone.…”