Frequency and wavelet based analyses of partial and complete measure synchronization in a system of three nonlinearly coupled oscillators.

Measure Synchronization (MS) is the generalization of synchrony to Hamiltonian Systems. Partial measure synchronization (PMS) and complete measure synchronization in a system of three nonlinearly coupled one-dimensional oscillators have been investigated for different initial conditions on the basis of numerical computation. The system is governed by the classical SU(2) Yang-Mills-Higgs (YMH) Hamiltonian with three degrees of freedom. Various transitions in the quasiperiodic (QP) region, namely, QP unsynchronized to PMS, PMS to PMS, and PMS to chaos are identified through the average bare energies and interaction energies route maps as the coupling strength is varied. The transition from quasiperiodicity to chaos is seen to be associated with a gradual transition to complete chaotic measure synchronization (CMS) which is followed by chaotic unsynchronized states, the most stable state in this case. The analyses illustrate the dependence on initial conditions. The explanation of the behavior in the QP regime is sought from the power spectral analysis. The existence of PMS is confirmed using the order parameter M (here M α ß for different combination pairs of oscillators), best suited to identify MS in coupled two-oscillator systems, and this definition is extended to obtain a new order parameter, M 3 , aiding to distinguish complete MS of three oscillators from other forms of motion. The study of wavelet coefficient spectra sheds new light on the relative phase information of the oscillators in the QP PMS regions, also highlighting the intertwined role played by the various frequency components and their amplitudes as they vary temporally. Furthermore, this technique helps to draw a sharp distinction between CMS and chaotic unsynchronized states. Based on the Continuous Wavelet Transform coefficients of the three oscillators, an order parameter M w a v is defined to indicate the extent of synchronization of the various scales (frequencies) for different coupling strengths in the chaotic regime.

Exploring the route to measure synchronization in non-linearly coupled Hamiltonian systems.

Measure Synchronization is a general term used for weak synchronization in Hamiltonian systems. Route to measure synchronization in a system of two non-linearly coupled one-dimensional oscillators, the potential of which is represented by the Pullen-Edmonds Potential is investigated on the basis of numerical computation. Transitions to measure synchronization and unsynchronization, both quasiperiodic and chaotic, are investigated and distinguished on the basis of the variation of average bare energies, average interaction energy, root-mean-square value of oscillations, phase difference, and frequencies with the coupling strength. A suitable order parameter to identify and characterize both quasiperiodic and chaotic measure synchronous states is sought, and drawbacks of the various order parameters, suggested previously, are discussed.

Dynamic multiscaling in stochastically forced Burgers turbulence.

We carry out a detailed study of dynamic multiscaling in the turbulent nonequilibrium, but statistically steady, state of the stochastically forced one-dimensional Burgers equation. We introduce the concept of interval collapse time, which we define as the time taken for a spatial interval, demarcated by a pair of Lagrangian tracers, to collapse at a shock. By calculating the dynamic scaling exponents of the moments of various orders of these interval collapse times, we show that (a) there is not one but an infinity of characteristic time scales and (b) the probability distribution function of the interval collapse times is non-Gaussian and has a power-law tail. Our study is based on (a) a theoretical framework that allows us to obtain dynamic-multiscaling exponents analytically, (b) extensive direct numerical simulations, and (c) a careful comparison of the results of (a) and (b). We discuss possible generalizations of our work to higher dimensions, for the stochastically forced Burgers equation, and to other compressible flows that exhibit turbulence with shocks.

Using wavelet analysis to investigate synchronization.

Wavelet analysis is shown to be a more robust technique than previously used methods in the investigation of synchronization. The highlight of the technique is that it encompasses most of the information obtained by conventional methods into a single picture, while giving a deeper insight into the dynamics of the system. Order parameters derived from continuous wavelet transform coefficients are proposed, which can be used in the quantification of measure synchronization in Hamiltonian systems and identical synchronization in dissipative systems, irrespective of the nature of coupling, the nature of synchronization (complete or partial, quasiperiodic or chaotic), and the number of coupled subsystems.