Kwok Yeung Tsang

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.

Attractor crowding in Josephson junction arrays

Large arrays of coupled nonlinear oscillators can suffer a noise sensitivity due to competition between huge numbers of coexisting states. We have found direct numerical evidence that a Josephson junction series array can exhibit attractor crowding. Thus, for the parameter values considered, no matter how small the noise level, there is a limit to the size of the array beyond which noise corrupts the in‐phase dynamical state.

Analytical calculation of invariant distributions on strange attractors

Abstract We obtain analytically by an iterative procedure the equilibrium invariant distribution for a class of strange attractors found in two-dimensional perturbed twist maps. The zeroth order (phase-averaged) distribution is found by solving the appropriate Fokker-Planck equation. Repeated iterations over the map yield the higher order distributions that reveal successively finer structures. We develop a quantitative method to compare the maps thus obtained with the maps computed numerically. Using this method, the analytical theory has been verified for the dissipative Fermi, zaslavskii and quadratic Chirikov maps, to the order that the numerical results can be obtained.

On the comparison between Josephson‐junction array variations

Recent work has shown that the expected performance of a given Josephson‐junction series array depends greatly on whether the coupling load is a resistor or an inductor‐capacitor combination. This raises the question of how arrays using different loads can be compared in a quantitatively meaningful way, for example, with respect to the effects of noise. We show that it is possible to choose the circuit parameters so that the in‐phase output for different loads is, though not identical, nearly so. The key lies in the development of a nonlinear generalization of the load impedance.

Invariant distribution on strange attractors in highly dissipative systems

Abstract We obtain analytically by an iterative procedure the equilibrium invariant distribution for a class of strange attractors in highly dissipative systems. The zeroth order distribution is found by solving the phase-averaged Markov equation. Repeated iterations over the map yield the higher order distributions that reveal successively finer structures. The analytical results have been compared quantitatively to numerical results for the Zaslavskii map.

RECURRING ANTI-PHASE SIGNALS IN COUPLED NONLINEAR OSCILLATORS: CHAOTIC OR RANDOM TIME SERIES?

We discovered the existence of a new type of high-dimensional attractor in coupled nonlinear oscillator systems. Due to the presence of neutrally stable directions on the attractor, there can be noise-driven phase space diffusion. Recurring anti-phase states are observed as coherent portions of the time series. The observed time series looks coherent for a while, then incoherent, and then coherent again. Although the time series “looks” chaotic, the Lyapunov exponents are not positive.