Kurt Wiesenfeld

Huygens's clocks

The 336–year–old synchronization observations of Christiaan Huygens are re–examined in modern experiments. A simple model of synchronization is proposed.

Neural correlates of the complexity of rhythmic finger tapping.

Using functional magnetic resonance imaging (fMRI), we studied the neural correlates of the complexity of rhythmic finger tapping. Our experiments measured the brain activity of 13 subjects performing rhythmic tapping on a response box with multistable rhythms of 1 to 5 different interresponse intervals. From the button press response times, we constructed phase portraits where we identified the number of clusters of periodic points in a rhythm that corresponded to the number of different beats of the rhythm performed. We then constructed a statistical model for correlation analysis involving the following behavioral parameters: rate of tapping and number of beats in a rhythm. The tapping rate correlated with the brain activity in the ipsilateral pre/postcentral gyrus, and the number of beats (complexity) was correlated with activations in the primary motor cortex, supplementary motor area, basal ganglia, thalamus, and cerebellum. A region of interest (ROI) average analysis showed that the complexity of a rhythm had a differential correlation with the activity in these regions. The cerebellum and the thalamus showed increasing activity, and the basal ganglia showed decreasing activity with complexity of a rhythm. These results identify the areas involved in a rhythm generation and the modulation of brain activity with the complexity.

Dynamics of a globally coupled oscillator array

Abstract We study a set of N globally coupled ordinary differential equations of the form encountered in circuit analysis of superconducting Josephson junction arrays. Particular attention is paid to two kinds of simple time-periodic behavior, known as in-phase and splay phase states. Some results valid for general N , as well as further results for N = 2 and N → ∞, are presented; a recurring theme is the appearance of very weak dynamics near the periodic states. The implications for Josephson junction arrays are discussed.

Phase‐locked oscillator optimization for arrays of Josephson junctions

An overview of phase locking in two‐dimensional (2D) arrays of identical Josephson junctions is presented. General design criteria are discussed for optimization of power and linewidth. A harmonic balance technique is used to derive an analytic expression for the fundamental power as a function of bias voltage for a single shunted tunnel junction with an external shunt resistor having parasitic inductance. A linear stability analysis is performed on the in‐phase state of 2D arrays in the absence of any external load. Most excitation modes in the 2D array are damped, leading to stable phase locking between parallel junctions within each row; however, within the theoretical model, no mechanisms intrinsic to the array were found to induce phase locking between rows of junctions. The results of these calculations and their impact on and relevance to the design of phase‐locked Josephson oscillators are discussed.

Elimination of chaos in an intracavity-doubled Nd:YAG laser

We predict theoretically a stable configuration for the operation of a multimode, intracavity-doubled, diode-pumped Nd:YAG laser. Experimental results are presented that demonstrate the elimination of chaotic amplitude fluctuations by rotatory alignment of the KTP crystal.

Averaging of globally coupled oscillators

Abstract We study a specific system of symmetrically coupled oscillators using the method of averaging. The equations describe a series array of Josephson junctions. We concentrate on the dynamics near the splay-phase state (also known as the antiphase state, ponies on a merry-go-round, or rotating wave). We calculate the Floquet exponents of the splay-phase periodic orbit in the weak-coupling limit, and find that all of the Floquet exponents are purely imaginary; in fact, all the Floquet exponents are zero except for a single complex conjugate pair. Thus, nested two-tori of doubly periodic solutions surround the splay-phase state in the linearized averaged equations. We numerically integrate the original system, and find startling agreement with the averaging results on two counts: The observed ratio of frequencies is very close to the prediction, and the solutions of the full equations appear to be either periodic or doubly periodic, as they are in the averaged equations. Such behavior is quite surprising from the point of view of generic dynamical systems theory-one expects higher-dimensional tori and chaotic solutions. We show that the functional form of the equations, and not just their symmetry, is responsible for this nongeneric behavior.

Stability results for in-phase and splay-phase states of solid-state laser arrays

We present a theoretical study of synchronization in N-element solid-state laser arrays. We carry out the linear stability analysis for three types of solution: the nonlasing state, the in-phase periodic state, and the splayphase state. Both nearest-neighbor (on a ring) coupling and global (all-to-all) coupling are treated; the system symmetries enable us to solve the linear stability problem for arbitrary N. We consider the general case in which the coupling coefficient iκ is complex and find that stability depends crucially on the sign of the imaginary part of κ. In the case of global coupling, we discover a surprising result: the existence of an N − 2 parameter family of frequency-locked neutrally stable states. These states should display substantial phase diffusion in the presence of noise.

Phase locking of Josephson junction arrays

We report the results of a stability analysis of coherent oscillations in series arrays of Josephson junctions with a matched resistive load. We find that arbitrarily large, dc biased arrays of Josephson junctions will phase lock most strongly when the capacitance parameter βc ≊1, and the bias current is about twice the critical current of the individual junctions.

Disorder-enhanced synchronization

Abstract We find that an increase in the disorder of an array of Josephson junctions can lead to significant improvement in the synchronization of the array. Both this effect and the opposite, more expected behavior are seen over a broad parameter range.

A physicist's sandbox

We discuss some recent results suggesting that certain spatially extended dynamical systems naturally evolve toward a state characterized by domains of all length scales. The analogy with second-order phase transitions has prompted the name “self-organized criticality” specific results are available for cellular automaton models, which can be thought of as caricatures of a sandpile undergoing avalances. The potential generality of the results stems from the very simple (nonlinear) diffusion dynamics governing the system.