Optimal synchronization of oscillatory chemical reactions with complex pulse, square, and smooth waveforms signals maximizes Tsallis entropy

Abstract

We show that the mathematical structure of Tsallis entropy underlies an important and ubiquitous problem in nonlinear science related to an efficient synchronization of weakly forced nonlinear oscillators. The maximization of the locking range of oscillators with the use of phase models is analyzed with general constraints that encompass forcing waveform power, magnitude, or area. The optimization problem is then recasted as a general form of Tsallis entropy maximization. The solution of these optimization problems is shown to be a direct consequence from Hölder's inequality. The resulting new maximization principle is confirmed in numerical simulations and experiments with chemical oscillations with nickel electrodissolution. While weakly nonlinear oscillators have generic optimal waveforms (sinusoidal, 50% duty cycle square wave, and equally paced bipolar pulses for power-, area-, and magnitude-constraints, respectively), strongly nonlinear oscillators require more complex waveforms such as smooth, square, and pulse ones.

Phase reduction and synchronization of a network of coupled dynamical elements exhibiting collective oscillations

“…Though we have treated only a simple example of two identical networks of neural oscillators, the theory is general and can be applied to analyzing, controlling, and designing networks of dynamical elements exhibiting collective oscillations. Several interesting directions would be optimization of injection locking of the collective oscillation of a network [27][28][29][30][31][32], optimization of mutual coupling between the networks for synchronization [33][34][35], and design of network structures that lead to desirable phase response properties. Because the theory does not require homogeneity of the dynamical elements nor smallness of the coupling of the network, the theory can be tested by real experimental systems, such as the system of coupled electrochemical oscillators developed by John L. Hudson.…”

A phase model approach for thermostatically controlled load demand response

“…The phase model representation of nonlinear oscillators becomes highly valuable in this case because the phase sensitivity function Z(φ) can be used to provide the theoretical linearized limits of the entrainment region commonly referred to as Arnold tongue (see Fig. 11) [48][49][50]. For more details see Appendix C.…”

“…ing in a maximum entrainment range [50] or fast entrainment [51]. While maximizing the width of the Arnold tongue is good for entrainment, for control of TCL ensembles it is to be avoided.…”

Optimizing stability of mutual synchronization between a pair of limit-cycle oscillators with weak cross coupling

“…The phase reduction theory, originally developed for finite-dimensional smooth limit-cycle oscillators, has recently been generalized to non-conventional limitcycling systems such as collectively oscillating populations of coupled oscillators [26], systems with time delay [27][28][29], reaction-diffusion systems [30], oscillatory fluid convection [31], and hybrid dynamical systems [32]. Recently, methods for optimizing periodic external driving signals for efficient injection locking and controlling of a single nonlinear oscillator (or a population of uncoupled oscillators) have also been proposed on the basis of the phase reduction theory [33][34][35][36][37][38][39][40][41][42][43][44]. In this study, we consider a pair of coupled limit-cycle oscillators and try to optimize the linear stability of the synchronized state * nakao@mei.titech.ac.jp (corresponding author) using the phase reduction theory.…”