Injection Locking Analysis and Simulation of Weakly Coupled Oscillator Networks

Coupled oscillator networks occur in various domains such as biology, astrophysics and electronics. In this paper, we present a comprehensive procedure for rapid and accurate simulation of large coupled oscillator networks using widely accepted, fully-nonlinear perturbation projection vector (PPV) phase macromodels. We validate our method against full simulation of 20times20 coupled network of Brusselator biochemical oscillator and obtain computational speedups of 170x over full simulation. Furthermore, we apply the method to study self-organization phenomenon of Brusselator under asymmetric coupling and time period variations.

Injection-locked oscillators (ILO) have widespread use in wireline and wireless systems as fast-locking clock generators, phase noise filters, and frequency multipliers/ dividers, but their modeling and simulation typically resort to SPICE due to their nonlinear, time-varying characteristics. This paper presents a way to simulate ILOs in an event-driven logic simulator such as SystemVerilog, based on a perturbation projection vector (PPV) model. By expressing the continuous-time signals in analytical forms of which coefficients are updated as events, the nonlinear phase response of the ILO can be simulated in an algebraic fashion without numerical integration. The simulation methodology is demonstrated with the modeling examples of a phase-locked loop and a burst-mode clock recovery loop, each employing a injected-locked frequency divider and oscillator, respectively.

This paper presents an event-driven simulation methodology for injection-locked oscillators. The proposed method adopts a phase-domain macromodel based on a perturbation projection vector (PPV), which expresses the oscillator's phase response using a nonlinear, time-varying differential equation. By expressing the PPV using a piecewise polynomial function, the PPV model can be simulated in an event-driven fashion on a digital logic simulator like SystemVerilog. While this piecewise polynomial approximation introduces a trade-off between the simulation accuracy and speed, our results show that the simulation accuracy quickly reaches its optimal value as the piecewise interval number increases even with first-order polynomials. The experimental results with an LC oscillator, ring oscillator, and burst-mode CDR models operating with both periodic and aperiodic inputs demonstrate that the proposed models achieve speeds up to 2 ~ 30× that of transistor-level Spectre simulations with the same accuracy.

Coupled electronic oscillator networks, under second harmonic injection, have recently been shown to behave as Ising machines capable of solving computationally hard combinatorial optimization problems. In this work, we experimentally investigate the dynamical properties of a reconfigurable network of up to 30 oscillators ( $\equiv $ spins) configured as an Ising machine. Specifically, we analyze the characteristics of the solutions to the Ising model produced by the oscillators and show that as the system evolves towards the ground state through the high-dimensional phase space, it gets trapped in local minima resulting in sub-optimal solutions. Moreover, the exact local minima where the system gets trapped also changes implying that the trajectory of evolution of the system also changes with each trial. Finally, we illustrate experimentally how an appropriately designed annealing scheme can help the coupled oscillators escape a local minimum and attain a lower energy state.

This study presents three EEG/MEG applications in which the modeling of oscillatory signal components offers complementary analysis and an improved explanation of the underlying generator structures. Coupled oscillator networks were used for modeling. Parameters of the corresponding ordinary coupled differential equation (ODE) system are identified using EEG/MEG data and the resulting solution yields the modeled signals. This model-related analysis strategy provides information about the coupling quantity and quality between signal components (example 1, neonatal EEG during quiet sleep), allows identification of the possible contribution of hidden generator structures (example 2, 600-Hz MEG oscillations in somatosensory evoked magnetic fields), and can explain complex signal characteristics such as amplitude-frequency coupling and frequency entrainment (example 3, EEG burst patterns in sedated patients).

Coupled oscillator networks often display transitions between qualitatively different phase-locked solutions—such as synchrony and rotating wave solutions—following perturbation or parameter variation. In the limit of weak coupling, these transitions can be understood in terms of commonly studied phase approximations. As the coupling strength increases, however, predicting the location and criticality of transition, whether continuous or discontinuous, from the phase dynamics may depend on the order of the phase approximation—or a phase description of the network dynamics that neglects amplitudes may become impossible altogether. Here we analyze synchronization transitions and their criticality systematically for varying coupling strength in theory and experiments with coupled electrochemical oscillators. First, we analyze bifurcations analysis of synchrony and splay states in an abstract phase model and discuss conditions under which synchronization transitions with different criticalities are possible. In particular, we show that such conditions can be understood by considering the relative contributions of higher harmonics to the phase dynamics. Second, we illustrate that transitions with different criticality indeed occur in experimental systems. Third, we highlight that the amplitude dynamics observed in the experiments can be captured in a numerical bifurcation analysis of delay-coupled oscillators. Our results showcase that reduced order phase models may miss important features that one would expect in the dynamics of the full system. Published by the American Physical Society 2024

In modern times, satellite-based global positioning and navigation systems, such as the GPS, include precise time-keeping devices, e.g. atomic clocks, which are crucial for navigation and for a wide range of economic and industrial applications. However, precise timing might not be available when the environment renders satellite equipment inoperable. In response to this critical need, we have been carrying out, over the past three years, theory and preliminary experiments [Buono et al., 2018a; Buono et al., 2018b; Palacios et al., 2020], towards developing a novel and inexpensive precision timing device that can function independently of GPS availability. The fundamental idea is to exploit collective behavior generated by networks of coupled nonlinear oscillators. Common sense may suggest that synchronized oscillations may lead to higher accuracy. Previous works show, however, that it is not synchronization but rather, traveling wave patterns, in which consecutive oscillators are out of phase by a constant amount, that can better reduce the negative effects of noise and material imperfections which cause phase drift. In this work we advance the state-of-art in the network-based concept by studying, mainly computationally, collective behavior in networks of Colpitts oscillators. These type of oscillators are chosen because they offer a wide range of advantages (such as the ability to tune up the oscillations over a broad frequency range). The results highlight the regions of parameter space, including coupling strength, where traveling wave patterns have the largest basins of attraction and the ability to reduce phase drift by a [Formula: see text] scaling law, where [Formula: see text] is the number of oscillators in the network. The results should also provide guidelines for follow-up design and fabrication tasks of a network-based technology for precision timing.

A recurrent hippocampal network model was constructed using mapped clock oscillators (MCOs) and shown to reliably generate spontaneous epileptiform activity under simulated hyperexcitable conditions. Such a network may prove useful as a platform for launching investigations into seizure control, and for complementing in vitro and animal models of epilepsy. More importantly, the therapeutic potential of MCO networks in the treatment of neurological disorders like epilepsy is only beginning to be realized

Phase Pattern Switching by Pulse Stimulation in Coupled Oscillators Networks

This paper focuses on the collective dynamics of multisynchronization among heterogeneous genetic oscillators under a partial impulsive control strategy. The coupled nonidentical genetic oscillators are modeled by differential equations with uncertainties. The definition of multisynchronization is proposed to describe some more general synchronization behaviors in the real. Considering that each genetic oscillator consists of a large number of biochemical molecules, we design a more manageable impulsive strategy for dynamic networks to achieve multisynchronization. Not all the molecules but only a small fraction of them in each genetic oscillator are controlled at each impulsive instant. Theoretical analysis of multisynchronization is carried out by the control theory approach, and a sufficient condition of partial impulsive controller for multisynchronization with given error bounds is established. At last, numerical simulations are exploited to demonstrate the effectiveness of our results.

Supply and ground noise sensitivity of a wide variety of ring and LC oscillators has been analyzed based on the perturbation projection vector (PPV) technique. The resulting PPV provides an understanding of how specific frequency content of supply/ground noise is converted to oscillator phase noise. Based on this analysis oscillators that are tolerant of supply/ground noise can be identified and used for low noise oscillator design.

Improving the efficiency and utilization of battery systems can increase the viability and cost-effectiveness of existing technologies for electric vehicles (EVs). Developing smarter battery management systems and advanced sensing technologies can circumvent problems arising due to capacity fade and safety concerns. This paper describes how efficient simulation techniques and improved algorithms can alleviate some of these problems to help electrify the transportation industry by improving the range of variables that are predictable and controllable in a battery in real-time within an electric vehicle. The use of battery models in a battery management system (BMS) is reviewed. The effect of different simulation techniques on computational cost and accuracy are also compared, and the validity of implementation in a microcontroller environment for model predictive control (MPC) is addressed. Using mathematical techniques to add more physics without losing efficiency is also discussed.

A novel sub-harmonic injection-locked balanced oscillator is proposed. The circuit provides two outputs with a 180° phase difference by employing a transmission line section for impedance transformation to meet the oscillation conditions. A coupling network is connected at the mid-point of the transmission line to inject the sub-harmonic frequency. This eliminates the need for a circulator or balun. The circuit is small and consumes low DC power. Under the locking state, the circuit provides double the injection frequency and also the phase noise of the two outputs is substantially improved.

A set of N identical oscillators (clocks) are connected to form a mutually synchronous network. General matrix expressions are derived for the phase error spectrum at each oscillator when the network operates in the linear regime. For certain topologies of practical interest the phase error variance and the correlation between the errors at different oscillators is computed and compared in their behavior for large N . The results presented assume no thermal noise and no delay between the oscillators. However, the theory can be easily extended to include these two effects.

In this chapter we describe phenomenon of ragged synchronizability. First we give a theoretical introduction about detecting of the complete synchronization in the network of coupled systems. As an example we choose a network of non-diagonally coupled forced van der Pol oscillators.We present construction of the single electrical van der Pol oscillator and array of such systems. We show numerical and experimental results of the existence of the complete synchronization in the array and we discuss it in the very detailed way. We also consider the synchronization in clusters, experimentally in the small array and numerically in the larger network. Our numerical simulations show good agreement with the experimental observations.