Evaluación de las rutas marítimas por cada puerto evaluado

Direct methods differ conceptually from adjoint methods in that they do not im- mediately solve for the phase sensitivity functionZ. Instead, direct methods aim at quantifying the phase response to an (arbitrarily small or large) stimulus p of the limit cycle trajectoryxc(t) at a particular phaseθ. We introduced this type of response as the phase response functionG(θ,p) inSection 2.1 and presented an ex- act description how to determineG. This direct method can also be implemented experimentally, which dates back to the work by Glass, Mackey and co-workers155 in the 1980s. Despite its simplicity, the experimental procedure is not very ac- curate when it comes to infinitesimal perturbations. That is why direct methods [29]_{An alternative proof to establish phase equations for oscillatory neural networks is given in} Theorem 9.178_{by Hoppensteadt and Izhikevich. They focus on the phase dynamics of (2.85)} and use normal form theory as presented inSection 2.2.1 to describe perturbationsP off the invariant manifold of (the product of) hyperbolic limit cycles. Ad-hoc they interpret their choice P≡0 as an ‘infinite attraction’ to the invariant manifold and thus link their result to Ermentrout and Kopell’s work.

have been avoided to compute the phase sensitivity functionZ via G(θ,p) in the limit of infinitesimal perturbations kpk 1. Recently, however, Noviˇcenko and Pyragas proposed an algorithm based on the same idea of the oscillator’s response to short finite pulses at different phases of the limit cycle156_{. Their algorithm} does not require any backward integration nor a numerical interpolation of the Jacobian. Moreover, it is faster than the algorithms implemented in XPPAUT, seeSection 2.3.1 above. This is especially true when the limit cycle is only weakly stable. The idea behind this algorithm builds on the (linearized) dynamics (2.81) of infinitesimal deviations u from the limit cycle C = {xc_{(t) : 0 ≤ t ≤ T} as the} adjoint method,

˙

u(t) = ∇f(x)|_x=xc_(t)u(t) , (2.89) wherexc(t) denotes theT-periodic limit cycle solution of ˙x=f(x) with initial con- ditionx(0) =xc(θ) (initial phaseθ). To obtain thej-th componentZjof the phase sensitivity function Z, we choose the initial condition u(0) = (u1(0), . . . , un(0))| withuk(0) =δkj whereδkj denotes the Kronecker-δ. Then, it can be found156that

Zj(θ) = lim p→∞

f(xc(θ))·u(pT)

f(xc_(θ))·f(xc_(θ)) . (2.90) To improve this algorithm, the authors replaced the vector u by the fundamen- tal matrix Φ and eventually extract the phase sensitivity function Z from Φ. For more details, we refer to their instructive work156_{, which includes numerical} demonstrations of the algorithm and a comparison with the standard algorithm as implemented in XPPAUT.

For our purposes, we tested the standard algorithm, both using XPPAUT as well as our own adjoint solver implemented in Matlab (The Mathworks Inc., Natwick, MA), against the one presented here. We found a very good agreement between all methods, such that we use either of them interchangeably as “the” numerical method unless stated otherwise.

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3

Collective behavior of coupled

oscillators and their reduced phase

models

Having a battery of analytic and numerical phase reduction techniques introduced and explicated in the previous Chapter, we duly apply them to two classic exam- ples. The first is a network of Brusselators, which is one of the most discussed chemical oscillators. The second example comprises a more elaborate interdisci- plinary model of coupled Wilson-Cowan oscillators. Both of them illustrate the benefits and pitfalls of the different phase reduction techniques. A point-by-point application further allows for a thorough comparison between the techniques. The reduction of complex oscillatory systems is crucial for numerical analyses but more so for analytical estimates and model prediction. The most common reduction is towards phase oscillator networks that have proven successful in describing not only the transition between incoherence and global synchronization, but in predict- ing the existence of non-trivial network states. Many of these predictions have been confirmed in experiments. The phase dynamics, however, depends to large extent on the employed phase reduction technique.

Adapted from: Pietras B., Daffertshofer A. (2018). Network dynamics of coupled oscillators and phase reduction techniques, (Sections 5 – 7). Under review.

3.1 Networks of identical Brusselators

The Brusselator is a theoretical model of oscillating chemical reactions. It perfectly serves to illustrate our approaches to phase reduction introduced inChapter2 since it exhibits a supercritical Hopf bifurcation. The system comprises four hypotheti- cal chemical reactions and has been developed by the Brussels school around Ilya Prigogine and Ren´e Lefever157 – hence the name. For a long time, reports on oscillating chemical reactions were facing harsh skepticism. Despite the strong interest in biological and biochemical oscillations in the 1950s and 60s, the dis- covery of oscillatory patterns in a closed chemical system by Belousov158 _{in 1951} had to be meticulously reproduced and investigated for years by Zhabotinsky159 until the nowadays so famous Belousov-Zhabotinsky reaction found its way into the scientific community160_{; for an overview of oscillating chemical reactions see} also38,161,162_{. In a way, the Belousov-Zhabotinsky reaction was conceived as a man-} ageable model of more complex systems, which simultaneously bore a close analogy to biology: Strogatz describes this analogy where “propagating waves of oxidation [. . . ] annihilate upon collision just like waves of excitation in neural or cardiac tis- sue. [. . . ] spiral waves are now an ubiquitous feature of chemical, biological, and physical excitable media”162_{. The original Belousov-Zhabotinsky reaction, which} involves more than twenty elementary reaction steps, could effectively be rewrit- ten in three differential equations. From a similar perspective, one can regard the Brusselator as a simplified chemical oscillator, which can be described in two dif- ferential equations. Despite its ability to exhibit oscillatory dynamics, as found in the Belousov-Zhabotinsky reaction, the Brusselator is a mere hypothetical model and is not based on a particular chemical reaction. Nonetheless, it serves as an exquisite example to apply the arsenal of phase reduction techniques presented in the previous section.[1]

[1]_{There exists also a natural extension of the Brusselator model into a two-component reaction-} diffusion system, which allows for so-called chemical waves and other pattern formation, such as, e.g., traveling fronts or rotating spirals in an extended medium38_{. It is not only possible} to define a phase for rhythmic patterns in extended media, but also to derive the correspond- ing phase dynamics from the underlying spatio-temporal dynamics, as has been successfully demonstrated by Nakao, Kawamura and co-workers87,163,164_{. This strategy can be used to} determine a meaningful phase dynamics of periodic fluid flows165_{. It has been extended to} reduce the phase dynamics of limit cycle solutions to general partial differential equations166_. In the same way, the phase dynamics of collective oscillations of globally coupled noisy ele- ments can be derived, given that these oscillations are solutions to a nonlinear Fokker-Planck equation167,168_.