PPV modelling of memristor-based oscillator

In this letter, we propose for the first time a method of abstracting the PPV (Perturbation Projection Vector) characteristic of the up-to-date memristor-based oscillators. Inspired from biological oscillators and its characteristic named PRC (Phase Response Curve), we build a bridge between PRC and PPV. This relationship is verified rigorously using the transistor level simulation of Colpitts and ring oscillators, i.e., comparing the PPV converted from PRC and the PPV obtained from accurate PSS+PXF simulation. Then we apply this method to the PPV calculation of the memristor-based oscillator. By keeping the phase dynamics of the oscillator and dropping the details of voltage/current amplitude, the PPV modelling is highly efficient to describe the phase dynamics due to the oscillator coupling, and will be very suitable for the fast simulation of large scale oscillatory neural networks

Effects of parametric uncertainties in cascaded open quantum harmonic oscillators and robust generation of Gaussian invariant states

This paper is concerned with the generation of Gaussian invariant states in cascades of open quantum harmonic oscillators governed by linear quantum stochastic differential equations. We carry out infinitesimal perturbation analysis of the covariance matrix for the invariant Gaussian state of such a system and the related purity functional subject to inaccuracies in the energy and coupling matrices of the subsystems. This leads to the problem of balancing the state-space realizations of the component oscillators through symplectic similarity transformations in order to minimize the mean square sensitivity of the purity functional to small random perturbations of the parameters. This results in a quadratic optimization problem with an effective solution in the case of cascaded one-mode oscillators, which is demonstrated by a numerical example. We also discuss a connection of the sensitivity index with classical statistical distances and outline infinitesimal perturbation analysis for translation invariant cascades of identical oscillators. The findings of the paper are applicable to robust state generation in quantum stochastic networks.Comment: 41 pages, 3 figures, 1 tabl

Smale-Williams Solenoids in a System of Coupled Bonhoeffer-van der Pol Oscillators

The principle of constructing a new class of systems with hyperbolic chaotic attractors is proposed. It is based on using oscillators, the transfer of excitation between which is provided resonantly due to the difference in the frequencies of small and large oscillations by an integer number of times being accompanied by phase transformation according to Bernoulli nap. We consider a system with Smale-Williams attractor, which is based on two coupled Bonhoeffer-van der Pol oscillators. The oscillators manifest activity and suppression turn by turn. With appropriate selection of the modulation, relaxation oscillations occur at the end of each activity stage, the frequency of which is by an integer factor $M = 2,3,4,\ldots$ smaller than that of small oscillations. When the partner oscillator enters the activity stage, the oscillations start being stimulated by the M-th harmonic of the relaxation oscillations, so that the transformation of phase during the modulation period corresponds to the M-fold Bernoulli map. In the state space of the Poincar\'e map this corresponds to Smale-Williams attractor, constructed with M-fold increase in the number of turns of the winding at each step of the mapping. The results of numerical studies confirming the occurrence of the hyperbolic attractors in certain parameter domains are presented, including the portraits of attractors, diagrams illustrating the phase transformation according to the Bernoulli map, Lyapunov exponents, and charts of regimes in parameter planes. The hyperbolic nature of the attractors is verified by numerical tests that confirm absence of tangencies of stable and unstable manifolds for trajectories on the attractor (criterion of angles). An electronic circuit is proposed that implements this principle of obtaining the hyperbolic chaos and its functioning is demonstrated using the software package Multisim

Parametric Generator of Robust Chaos: Circuit Implementation and Simulation Using the Program Product MULTISIM

A scheme is suggested of the parametric generator of chaotic oscillations with attractor represented by a kind of Smale-Williams solenoid that operates under a periodic sequence of pump pulses at two different frequencies. Simulation of chaotic dynamics using the software product Multisim is provided.Comment: 14 pages, 10 figure

Networked oscillator based modeling and control of unsteady wakes

A networked oscillator based analysis is performed for periodic bluff body flows to examine and control the transfer of kinetic energy. Spatial modes extracted from the flow field with corresponding amplitudes form a set of oscillators describing unsteady fluctuations. These oscillators are connected through a network that captures the energy exchanges amongst them. To extract the network of interactions among oscillators, amplitude and phase perturbations are impulsively introduced to the oscillators and the ensuing dynamics are analyzed. Using linear regression techniques, a networked oscillator model is constructed that reveals energy transfers and phase interactions among the modes. The model captures the nonlinear interactions amongst the modal oscillators through a linear approximation. A large collection of system responses are aggregated into a network model that captures interactions for general perturbations. The networked oscillator model describes the modal perturbation dynamics better than the empirical Galerkin reduced-order models. A model-based feedback controller is then designed to suppress modal amplitudes and the resulting wake unsteadiness leading to drag reduction. The strength of the proposed approach is demonstrated for a canonical example of two- dimensional unsteady flow over a circular cylinder. The present formulation enables the characterization of modal interactions to control fundamental energy transfers in unsteady vortical flows.Comment: 33 pages, 16 figure

Tangential Interpolatory Projection for Model Reduction of Linear Quantum Stochastic Systems

This paper presents a model reduction method for the class of linear quantum stochastic systems often encountered in quantum optics and their related fields. The approach is proposed on the basis of an interpolatory projection ensuring that specific input-output responses of the original and the reduced-order systems are matched at multiple selected points (or frequencies). Importantly, the physical realizability property of the original quantum system imposed by the law of quantum mechanics is preserved under our tangential interpolatory projection. An error bound is established for the proposed model reduction method and an avenue to select interpolation points is proposed. A passivity preserving model reduction method is also presented. Examples of both active and passive systems are provided to illustrate the merits of our proposed approach.Comment: 28 pages, 8 figures. A preliminary version of Section 4 will appear in Proceedings of the 54th IEEE Conference on Decision and Control (CDC) (Osaka, Japan, December 15-18, 2015

Oscillator-based Ising Machine

Many combinatorial optimization problems can be mapped to finding the ground states of the corresponding Ising Hamiltonians. The physical systems that can solve optimization problems in this way, namely Ising machines, have been attracting more and more attention recently. Our work shows that Ising machines can be realized using almost any nonlinear self-sustaining oscillators with logic values encoded in their phases. Many types of such oscillators are readily available for large-scale integration, with potentials in high-speed and low-power operation. In this paper, we describe the operation and mechanism of oscillator-based Ising machines. The feasibility of our scheme is demonstrated through several examples in simulation and hardware, among which a simulation study reports average solutions exceeding those from state-of-art Ising machines on a benchmark combinatorial optimization problem of size 2000.Comment: Added an example for invertible logic computation; added results on G22 with central frequency variations; fixed some typo

OIM: Oscillator-based Ising Machines for Solving Combinatorial Optimisation Problems

We present a new way to make Ising machines, i.e., using networks of coupled self-sustaining nonlinear oscillators. Our scheme is theoretically rooted in a novel result that establishes that the phase dynamics of coupled oscillator systems, under the influence of sub-harmonic injection locking, are governed by a Lyapunov function that is closely related to the Ising Hamiltonian of the coupling graph. As a result, the dynamics of such oscillator networks evolve naturally to local minima of the Lyapunov function. Two simple additional steps (i.e., adding noise, and turning sub-harmonic locking on and off smoothly) enable the network to find excellent solutions of Ising problems. We demonstrate our method on Ising versions of the MAX-CUT and graph colouring problems, showing that it improves on previously published results on several problems in the G benchmark set. Our scheme, which is amenable to realisation using many kinds of oscillators from different physical domains, is particularly well suited for CMOS IC implementation, offering significant practical advantages over previous techniques for making Ising machines. We present working hardware prototypes using CMOS electronic oscillators

Dynamics, numerical analysis, and some geometry

Geometric aspects play an important role in the construction and analysis of structure-preserving numerical methods for a wide variety of ordinary and partial differential equations. Here we review the development and theory of symplectic integrators for Hamiltonian ordinary and partial differential equations, of dynamical low-rank approximation of time-dependent large matrices and tensors, and its use in numerical integrators for Hamiltonian tensor network approximations in quantum dynamics.Comment: prepared for the Proceedings of ICM 2018 (Christian Lubich's plenary talk

Dissecting the Phase Response of a Model Bursting Neuron

We investigate the phase response properties of the Hindmarsh-Rose model of neuronal bursting using burst phase response curves (BPRCs) computed with an infinitesimal perturbation approximation and by direct simulation of synaptic input. The resulting BPRCs have a significantly more complicated structure than the usual Type I and Type II PRCs of spiking neuronal models, and they exhibit highly timing-sensitive changes in the number of spikes per burst that lead to large magnitude phase responses. We use fast-slow dissection and isochron calculations to analyze the phase response dynamics in both weak and strong perturbation regimes.Comment: 38 page