No abstract

Publication informationThe European Physical Journal B, 85 (3):Publisher Abstract. We study the emergence of oscillatory self-sustained behavior in a nonequilibrium Nambu system that features an exchange between different kinetical and potential energy forms. To this end, we study the Yamaleev oscillator in a canonical-dissipative framework. The bifurcation diagram of the nonequilibrium Yamaleev oscillator is derived and different bifurcation routes that are leading to limit cycle dynamics and involve pitchfork and Hopf bifurcations are discussed. Finally, an analytical expression for the probability density of the stochastic nonequilibrium oscillator is derived and it is shown that the shape of the density function is consistent with the oscillator properties in the deterministic case.

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.

Two widely used concepts in physics and the life sciences are combined: mean field theory and time-discrete time series modeling. They are merged within the framework of strongly nonlinear stochastic processes, which are processes whose stochastic evolution equations depend self-consistently on process expectation values. Explicitly, a generalized autoregressive (AR) model is presented for an AR process that depends on its process mean value. Criteria for stationarity are derived. The transient dynamics in terms of the relaxation of the first moment and the stationary response to fluctuations in terms of the autocorrelation function are discussed. It is shown that due to the stochastic feedback via the process mean, transient and stationary responses may exhibit qualitatively different temporal patterns. That is, the model offers a time-discrete description of many-body systems that in certain parameter domains feature qualitatively different transient and stationary response dynamics.

No abstract