Generalized chronotaxic systems: time-dependent oscillatory dynamics stable under continuous perturbation

Stabilization of dynamics of oscillatory systems by nonautonomous perturbation

Synchronization and stability under periodic oscillatory driving are well understood, but little is known about the effects of aperiodic driving, despite its abundance in nature. Here, we consider oscillators subject to driving with slowly varying frequency, and investigate both short-term and long-term stability properties. For a phase oscillator, we find that, counterintuitively, such variation is guaranteed to enlarge the Arnold tongue in parameter space. Using analytical and numerical methods that provide information on time-variable dynamical properties, we find that the growth of the Arnold tongue is specifically due to the growth of a region of intermittent synchronization where trajectories alternate between short-term stability and short-term neutral stability, giving rise to stability on average. We also present examples of higher-dimensional nonlinear oscillators where a similar stabilization phenomenon is numerically observed. Our findings help support the case that in general, deterministic nonautonomous perturbation is a very good candidate for stabilizing complex dynamics.

Modelling chronotaxicity of cellular energy metabolism to facilitate the identification of altered metabolic states

Altered cellular energy metabolism is a hallmark of many diseases, one notable example being cancer. Here, we focus on the identification of the transition from healthy to abnormal metabolic states. To do this, we study the dynamics of energy production in a cell. Due to the thermodynamic openness of a living cell, the inability to instantaneously match fluctuating supply and demand in energy metabolism results in nonautonomous time-varying oscillatory dynamics. However, such oscillatory dynamics is often neglected and treated as stochastic. Based on experimental evidence of metabolic oscillations, we show that changes in metabolic state can be described robustly by alterations in the chronotaxicity of the corresponding metabolic oscillations, i.e. the ability of an oscillator to resist external perturbations. We also present a method for the identification of chronotaxicity, applicable to general oscillatory signals and, importantly, apply this to real experimental data. Evidence of chronotaxicity was found in glycolytic oscillations in real yeast cells, verifying that chronotaxicity could be used to study transitions between metabolic states.

Maximum amplitude of limit cycles in Liénard systems

We establish sufficient criteria for the existence of a limit cycle in the Liénard system x[over ̇]=y-ɛF(x),y[over ̇]=-x, where F(x) is odd. In their simplest form the criteria lead to the result that, for all finite nonzero ɛ, the amplitude of the limit cycle is less than ρ and 0≤a≤ρ≤u, where F(a)=0 and ∫(0)(u)F(x)dx=0. We take the van der Pol oscillator as a specific example and establish that for all finite, nonzero ɛ, the amplitude of its limit cycle is less than 2.0672, a value whose precision is limited by the capacity of our symbolic computation software package. We show how the criterion for the upper bound can be extended to establish a bound on the amplitude of a limit cycle in systems where F(x) contains both odd and even components. We also show how the criteria can be used to establish bounds for bifurcation sets.

Homogeneous delays in the Kuramoto model with time-variable parameters

The Kuramoto model with time-varying parameters has been extended to consider the effect of delay in couplings. A collective dynamics arises from the interplay between the time scales of the original system, the external forcing, and the delays. This complex low-dimensional dynamics is described, uncovering an echo effect near the synchronization threshold. Hence, the delayed couplings substantially alter the dynamics of what is an open system and should be taken into consideration, depending on the ensemble's evolution time scale. We also introduce a first-harmonic approximation for the evolution of the mean field under harmonic forcing, valid for any delays and forcing of a coherent population.