We present a method for optimizing mutual coupling functions to achieve fast and global synchronization between a pair of weakly coupled limit-cycle oscillators. Our method is based on phase ...reduction that provides a concise low-dimensional representation of the synchronization dynamics of mutually coupled oscillators, including the case where the coupling depends on past time series of the oscillators. We first describe a method for a pair of identical oscillators and then generalize it to the case of slightly nonidentical oscillators. The coupling function is designed in two optimization steps for the functional form and amplitude, where the amplitude is numerically optimized to minimize the average convergence time under a constraint on the total power. We perform numerical simulations of the synchronization dynamics with the optimized coupling functions using the FitzHugh–Nagumo and Rössler oscillators as examples. We show that the coupling function optimized by the present method can achieve global synchronization more efficiently than those obtained by the previous methods.
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Synchronization of two dissipatively coupled Van der Pol oscillators in the quantum regime is studied. Due to quantum noise strict frequency locking is absent and is replaced by a crossover from weak ...to strong frequency entrainment. The differences to the behavior of one quantum Van der Pol oscillator subject to an external drive are discussed. Moreover, a possible experimental realization of two coupled quantum Van der Pol oscillators in an optomechanical setting is described.
Synchronization of two dissipatively coupled Van der Pol oscillators in the quantum regime is studied. Due to quantum noise strict frequency locking is absent and is replaced by a crossover from weak to strong frequency entrainment. The differences to the behavior of one quantum Van der Pol oscillator subject to an external drive are discussed. Moreover, a possible experimental realization of two coupled quantum Van der Pol oscillators in an optomechanical setting is described.
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An overview is given on two representative methods of dynamical reduction known as
and
. These theories are presented in a somewhat more unified fashion than the theories in the past. The target ...systems of reduction are coupled limit-cycle oscillators. Particular emphasis is placed on the remarkable structural similarity existing between these theories. While the two basic principles, i.e. (i) reduction of dynamical degrees of freedom and (ii) transformation of reduced evolution equation to a canonical form, are shared commonly by reduction methods in general, it is shown how these principles are incorporated into the above two reduction theories in a coherent manner. Regarding the phase reduction, a new formulation of perturbative expansion is presented for discrete populations of oscillators. The style of description is intended to be so informal that one may digest, without being bothered with technicalities, what has been done after all under the word
. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.
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We propose a method for designing 2-D limit-cycle oscillators with prescribed periodic trajectories and phase response properties based on the phase reduction theory, which gives a concise ...description of weakly perturbed limit-cycle oscillators and is widely used in the analysis of synchronization dynamics. We develop an algorithm for designing the vector field with a stable limit cycle that possesses a given shape and also a given phase sensitivity function. The vector field of the limit-cycle oscillator is approximated by polynomials whose coefficients are estimated by convex optimization. The linear stability of the limit cycle is ensured by introducing an upper bound to the Floquet exponent. The validity of the proposed method is verified numerically by designing several types of 2-D existing and artificial oscillators. As applications, we first design a limit-cycle oscillator with an artificial star-shaped periodic trajectory and demonstrate global entrainment. We then design a limit-cycle oscillator with an artificial high-harmonic phase sensitivity function and demonstrate multistable entrainment caused by a high-frequency periodic input.
The conditions for explosive death transitions in complex networks of oscillators having generalized network topology, coupled via mean-field diffusion, are derived analytically. The network ...behaviour is characterized using three order parameters that define the average amplitude of oscillations, the mean state and the fraction of dead oscillators. As the mean field coupling is changed adiabatically, the amplitude order parameter undergoes explosive death transitions and the nodes in the network collectively cease to oscillate. The transition points in the parameter space and the boundaries of amplitude death regime are derived analytically. Sub-critical Hopf bifurcations are shown to be responsible for amplitude death transition. Explosive death transitions are characterized by hysteresis on adiabatic change of the bifurcation parameter and surprisingly, the backward transition point is shown to be independent of network topology. The theoretical developments have been validated through numerical examples, involving networks of limit cycle systems — such as the Van der Pol oscillator and chaotic systems, such as the Rossler attractor for both random and small world networks.
•Analytical conditions for explosive death transitions are derived.•Sub-critical Hopf bifurcations responsible for explosive death.•Explosive death transitions exhibit hysteresis on adiabatic change in parameters.•Forward death transition dependent on network topology.•Backward death transition independent of network topology.
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The dynamics of a proposed microelectromechanical system (MEMS) consisting of an array of limit cycle oscillators (LCOs) are analyzed. The LCOs have dissimilar limit cycle frequencies and are coupled ...in a nearest-neighbor configuration via electrostatic fringing fields. The emergence of synchrony in the array is outlined for two cases: self-synchronization of the array to a single frequency, and entrainment of the array to an external inertial drive. Numerical analysis is used to study the dependence of synchrony on system parameters such as the coupling strength, detuning in the array, inertial drive strength, and frequency of the inertial drive. It is shown that the route to synchrony is complex due to the formation of frequency clusters. The limit cycle frequency of a single equivalent oscillator, with parameters averaged over the array is used as an estimate for the frequency of locking for the array. This equivalent oscillator is used to approximate the entire array and perturbation methods are applied to it. The perturbation method qualitatively captures the entrainment characteristics of the externally-driven array. This analysis is also used to track the complex sequence of bifurcations that occur as the drive strength changes, and to estimate the threshold drive strength for entrainment.
•The path to self-synchronization for coupled oscillators is non-monotonic.•The stability of frequency clusters depends sensitively on initial conditions.•A self-synchronized oscillator array behaves like a single equivalent oscillator.•Drive threshold for a self-synchronized system is independent of the coupling.
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•Theory of interplay of common noise and global coupling for limit-cycle oscillators.•Strong enough common noise synchronizes ensembles with repulsive coupling.•Frequency repulsion accompanies the ...synchronization for a negative coupling.•Phase deviation distribution always possesses power-law heavy tails.
We construct an analytical theory of interplay between synchronizing effects by common noise and by global coupling for a general class of smooth limit-cycle oscillators. Both the cases of attractive and repulsive coupling are considered. The derivation is performed within the framework of the phase reduction, which fully accounts for the amplitude degrees of freedom. Firstly, we consider the case of identical oscillators subject to intrinsic noise, obtain the synchronization condition, and find that the distribution of phase deviations always possesses lower-law heavy tails. Secondly, we consider the case of nonidentical oscillators. For the average oscillator frequency as a function of the natural frequency mismatch, limiting scaling laws are derived; these laws exhibit the nontrivial phenomenon of frequency repulsion accompanying synchronization under negative coupling. The analytical theory is illustrated with examples of Van der Pol and Van der Pol–Duffing oscillators and the neuron-like FitzHugh–Nagumo system; the results are also underpinned by the direct numerical simulation for ensembles of these oscillators.
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This paper presents a modified astrocyte model that allows a convenient digital implementation. This model is aimed at reproducing relevant biological astrocyte behaviors, which provide appropriate ...feedback control in regulating neuronal activities in the central nervous system. Accordingly, we investigate the feasibility of a digital implementation for a single astrocyte and a biological neuronal network model constructed by connecting two limit-cycle Hopf oscillators to an implementation of the proposed astrocyte model using oscillator-astrocyte interactions with weak coupling. Hardware synthesis, physical implementation on field-programmable gate array, and theoretical analysis confirm that the proposed astrocyte model, with considerably low hardware overhead, can mimic biological astrocyte model behaviors, resulting in desynchronization of the two coupled limit-cycle oscillators.
Intracellular oscillators entrain to periodic signals by adjusting their phase and frequency. However, the low copy numbers of key molecular players make the dynamics of these oscillators ...intrinsically noisy, disrupting their oscillatory activity and entrainment response. Here, we use a combination of computational methods and experimental observations to reveal a functional distinction between the entrainment of individual oscillators (e.g., inside cells) and the entrainment of populations of oscillators (e.g., across tissues). We demonstrate that, in the presence of intracellular noise, weak periodic cues robustly entrain the population averaged response, even while individual oscillators remain un-entrained. We mathematically elucidate this phenomenon, which we call stochastic population entrainment, and show that it naturally arises due to interactions between intrinsic noise and nonlinear oscillatory dynamics. Our findings suggest that robust tissue-level oscillations can be achieved by a simple mechanism that utilizes intrinsic biochemical noise, even in the absence of biochemical couplings between cells.
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•Intracellular oscillators often need to entrain to periodic stimuli•However, biochemical noise can disrupt the entrainment of individual oscillators•We show that population averages of noisy uncoupled oscillators entrain robustly•This may explain how noisy peripheral clocks entrain nicely at the tissue level
Gupta et al. show that intrinsic biochemical noise can interact with dynamic nonlinearities to cause entrainment of the population mean of uncoupled intracellular oscillators, even though these oscillators may not be individually entrained. They call this effect stochastic population entrainment (SPE), and they demonstrate it both theoretically and computationally.
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We investigate the double Hopf bifurcation at zero equilibrium point. Firstly, we give the critical values of Hopf and double Hopf bifurcations. Secondly, we implement the normal form method and the ...center manifold theory for delay-coupled limit cycle oscillators, and derive the universal unfolding and a complete bifurcation diagram of the system. Thirdly, many interesting phenomena, such as attractive periodic motion and three-dimensional invariant torus, are observed using numerical simulation. Finally, the normal forms of several strong resonant cases are listed.
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