Amplitude Equations and Bifurcation Diagrams for Multifrequency Synchronization of Canonical-Dissipative Oscillators

Abstract

<jats:p> Coupled systems of two canonical-dissipative limit cycle oscillators are considered in the general case and for the case of monofrequency and multifrequency synchronization. Specifically, the oscillator frequency ratios of 1:1, 1:2, and 1:3 are examined and modeled by a hybrid Rayleigh–van der Pol oscillator and an oscillator model suggested by Holt as well as an oscillator model suggested by Fokas and Lagerstrom. It is shown that all three systems exhibit a unique bifurcation diagram that describes limit cycle attractors of monofrequency and multifrequency synchronization. In particular, the relative phase describing the lag between the two oscillators both in the monofrequency and multifrequency case can be tuned by an appropriately defined bifurcation parameter [Formula: see text]. For [Formula: see text] two limit cycle attractors with different relative phases exist that merge at [Formula: see text] into pitchfork bifurcations and give rise to single limit cycle attractors that continue to exist for [Formula: see text]. Similarities and differences to bifurcation diagrams published in previous work of a similar coupled oscillator system, one based on Smorodinsky–Winternitz potentials and exhibiting 1:1 synchronization, are noted. </jats:p>

Unstable eigenvectors and reduced amplitude spaces specifying limit cycles of coupled oscillators with simultaneously diagonalizable matrices: with applications from electric circuits to gene regulation

“…5.2 and the appendix 7.3 , the basic CD framework developed in Ref. [ 55 ] for the example of a 1:2 polyrhythm has been presented, a more sophisticated and more flexible CD framework has been presented in Ref. [ 56 ].…”

“…CD models exhibit the advantage that they can be treated to a large extent analytically and include a variety of special cases that have been previously discussed in the literature (for details, see Refs. [55,56] again). To illustrate by a numerical example the general ideas leading from Eq.…”

“…A class of relative simple multifrequency models, so-called canonical-dissipative (CD) coupled multifrequency oscillators, has been recently introduced [ 55 , 56 ]. CD models exhibit the advantage that they can be treated to a large extent analytically and include a variety of special cases that have been previously discussed in the literature (for details, see Refs.…”

“…The Holt 1:2 multifrequency oscillator within the CD framework is presented in Ref. [ 55 ]. Using the variables and as in Sect.…”

“…On the other hand, coupled oscillator systems have been comprehensively studied [21,22], in particular, to identify their characteristic a e-mail: till.frank@uconn.edu (corresponding author) attractors. A frequently used technique is the phase dynamics approach in which phase dynamics equations are derived from coupled oscillator state equations and stability features of the oscillatory dynamics are primarily discussed on the level of the phase dynamics [22][23][24][25][26][27]. Alternatively, eigenvalues of coupled oscillator systems are determined to address the nature of the systems' fixed points and/or extensive numerical simulations are conducted to identify attractors beyond fixed points.…”