Chronotaxic Systems: A New Class of Self-Sustained Nonautonomous Oscillators

Abstract

Nonautonomous oscillatory systems with stable amplitudes and time-varying frequencies have often been treated as stochastic, inappropriately. We therefore formulate them as a new class and discuss how they generate complex behavior. We show how to extract the underlying dynamics, and we demonstrate that it is simple and deterministic, thus paving the way for a diversity of new systems to be recognized as deterministic. They include complex and nonautonomous oscillatory systems in nature, both individually and in ensembles and networks. DOI: 10.1103/PhysRevLett.111.024101 PACS numbers: 05.45.Xt, 05.65.+b, 05.90.+m, 89.75.Fb Dynamical systems are generally seen as being either deterministic or stochastic. The advent of dynamical chaos several decades ago attracted much attention and illustrated that even complex dynamics can be deterministic. Dynamical systems can also be classified as autonomous (self-contained) or nonautonomous (subject to external influences). In reality, nonautonomous systems are the commoner, but they are far harder to treat. Until now they were mostly treated as stochastic or, alternatively, attempts were made to reformulate them as autonomous. Neither approach captures the characteristic properties of these systems.In this Letter we propose a new class of nonautonomous systems and name them chronotaxic to characterize oscillatory systems with time-varying, but stable, amplitudes and frequencies. Nonautonomous oscillatory systems appear in various fields of research including neuroscience [1,2], cardiovascular dynamics [3], climate [4,5] and evolutionary science [6], as well as in complex systems and networks [7][8][9]. Although we are witnessing a rapid development of the theory of nonautonomous [10,11] and random dynamical systems [12,13], nonautonomous oscillatory systems with stable but time-varying characteristic frequencies have to date not been addressed. When treated in an inverse approach such systems are usually considered as stochastic. In an attempt to cope with the problem, several methods for the inverse approach were introduced, including wavelet-based decomposition [14] Common to all these systems is that they are oscillatory, have stable amplitudes, and frequencies that are resistant to external perturbations. The variety of systems with these characteristics suggests that their dynamics are generated from a universal basis. To date, the description of stable oscillatory dynamics has been based on the model of autonomous self-sustained limit cycle oscillators [23]. While this model provides stable amplitude dynamics, frequencies of oscillations within this model can be easily changed even by weakest external perturbations.The new class of nonautonomous oscillatory dynamical systems that we now propose account for such dynamics. The novelty of these systems is that not only are the amplitude dynamics stable but also the frequencies of the oscillations are time dependent and stable-i.e., their timedependent values cannot be easily altered by external perturbations. Their c...

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