Abstract
Chronotaxic systems represent deterministic nonautonomous oscillatory systems which are capable of resisting continuous external perturbations while having a complex time-dependent dynamics. Until their recent introduction in Phys. Rev. Lett. 111, 024101 (2013) chronotaxic systems had often been treated as stochastic, inappropriately, and the deterministic component had been ignored. While the previous work addressed the case of the decoupled amplitude and phase dynamics, in this paper we develop a generalized theory of chronotaxic systems where such decoupling is not required. The theory presented is based on the concept of a time-dependent point attractor or a driven steady state and on the contraction theory of dynamical systems. This simplifies the analysis of chronotaxic systems and makes possible the identification of chronotaxic systems with time-varying parameters. All types of chronotaxic dynamics are classified and their properties are discussed using the nonautonomous Poincaré oscillator as an example. We demonstrate that these types differ in their transient dynamics towards a driven steady state and according to their response to external perturbations. Various possible realizations of chronotaxic systems are discussed, including systems with temporal chronotaxicity and interacting chronotaxic systems.
Highlights
Complex dynamics, observed in real physical systems, is often modeled as stochastic or chaotic, or high-dimensional autonomous
In this paper we extend the theory of chronotaxic systems: we develop a mathematical definition of high-dimensional chronotaxic systems where the amplitude and phase dynamics do not need to be separated, in contrast to previous work [26,27]
In this work we have developed a generalized model of chronotaxic systems which are high dimensional and cannot be split into the decoupled one-dimensional dynamical systems, in contrast to previous works [26,27]
Summary
Complex dynamics, observed in real physical systems, is often modeled as stochastic or chaotic, or high-dimensional autonomous Such a description is inappropriate when these systems are open (which is almost always the case), i.e., when they depend on time explicitly or when they are exposed to continuous perturbation originating from the external environment which cannot be considered as a part of a system. [26,27], together with corresponding inverse approach methods [31], make it possible to identify the underlying deterministic dynamics within the complex stochastic-like dynamics It will be useful in various research fields, especially in living systems, where the identification of systems with complex stochastic-like dynamics as chronotaxic can help us in understanding their structure and function and their interactions with the external environment.
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