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order reduction - Coggle Diagram
order reduction
epsilon=order
epsilon^1: pairwise
epsilon^0: uncoupled
epsilon^2: three body
paper 5: exhibits even when oscillators are not connected (through intermediate nodes)
tends to prolong hysteresis (p9) by decreasing backward transition point
epsilon^n>2: higher body
stuart landau
complex form
polar form gives amplitude and phase
amplitude given by ODE converges to 1
phase sensitivity function (Z) (paper 6)
in ROC of phase, multiplied to the external perturbation, usually the coupling effects; can be negative, 0, positive
PSF * velocity = 1
change of Z = Jacobien * Z
the fokker-planck equation describes the density in a PDE (p12)
density is a function of phase, natural freq. and time
ROC density on time = (diffusion): D x ROC^2 density on phase - ROC of velocity (kuramoto)xdensity on phase
used to measure synchronization too.
reduction methods
amplitude slaving
self-consistent method (p8)
decomposition of coupling input to oscillator is the sum of fast small wave and a slow big wave
phase reduction: ROC of phase is instant. freq. + sensitivity to slow changes x ROC of slow wave + sens. to fast chagnes x fast cahnge + O(1/amplitude relaxation)
amplitude relaxation measures how quickly a variable "converges" to the natural steady state;
they are comparatively much faster than the coupling changes
the family of limit cycles has its period, frequency, orbit, and amplitude relaxation rate
parameterized by the coupling input
ROC slow wave depends on ROC of phase
standard: input is a function involving all current state vectors X(I)
with a low-pass filter: the slow wave for input depends on the past trajectories and past phases (integration)
the bold assumption: assumes that the
amplitude varies strictly and instantaneously based on the limit cycle
because it is a fast variable compared to the phases.
The slow wave input are just functions of the instantaneous phases?
iteration: assumes the model has been running with q1^(i) forever, constructs the past trajectories based on q1^(i) and gets q1^(i+1) and so on recursively
rotational symmetry
z, |z|^2z, |z|^4z follows
f(z x e^(itheta))=f(z) x e^(itheta)
asymptotic phase (phi) is phase (theta) - c * ln (amplitue)
the roc is constantly, the natural freq
weaknesses of 1st order
fails to capture partial coherence (p7)
non-isochronous <=> coupled
focus:
supercritical hopf bifurcation
amplitude ROC is often r^(0)+er^(1)+e^2r^(2)+...
isochrons: sets of equal phases
amplitude is often 1+er^(1)+e^2r^(2)+...
r^(n) nth order is linked (proportional?) to Zn=1/N sum (e^(i x
n
x theta))
mean field: some model directly computes the average instead of individual sin(phase deltas)
irrelevent: inertia (roc^2phase)
MATH JARGONS