Basins of attraction in a ring of overdamped bistable systems with delayed coupling
Introduction
Bistability, the attribute which indicates that two possible states of a system are possible, can be found in many physical systems, including: optical devices, Schmitt triggers, neural integrators, and ferroelectric and ferromagnetic sensors. The behavior of these, and many other systems, is governed, generically, by a gradient system of the form , where is the state variable of the device at time t, e.g., magnetization state, and is the bistable potential function. In the absence of an external force, the state point will rapidly relax to one of the two stable states, each associated with a local minimum of the potential function. Which state is actually observed depends on the actual initial conditions. In the presence of an external force, however, the system can exhibit a hysteresis loop in which the state point can be induced to oscillate between the two stable states. The hysteresis behavior is normally employed in the standard, spectral-based [1], [2], [3], detection mechanism of small target signals (dc or low frequency signals) wherein a known periodic bias signal is applied to induce a sensor device to oscillate between two stable attractors. In the absence of a target signal, the power spectral density contains only the odd harmonics of the bias frequency. In the presence of a weak target signal, however, the potential energy function is skewed, resulting in the appearance of even harmonics. The response at the second harmonic is then used to detect and quantify the target signal.
In recent works [4], [5], we have demonstrated, theoretically and experimentally, that coupling similar bistable systems can lead, under certain conditions, to self-induced oscillations that mimic the hysteresis loop of a single element. The conditions depend, mainly, on the topology of connections and the number of units. But, more importantly, since the oscillations appear without the need of an external signal, they provide an alternative signal detection scheme as well as, potentially, enhanced sensitivity. These results have been incorporated into a “coupled core fluxgate magnetometer (CCFM)” [4] and in a “coupled electric field sensor (CEFS)” [6]. The analysis of the effects of different coupling schemes has also led to the prediction of novel cooperative behavior in a ring of coupled SQUID (Superconducting Quantum Intereference Devices) devices [7].
In this Letter we concentrate on the CCFM system as the “test-bed” to study the effects of time delay in generic formulations of coupled bistable systems. While the laboratory realization of the CCFM employs high-speed, high-precision, operational amplifiers, a comprehensive analysis of the effects of time delay is still needed. In particular, to help us understand the potential effects of delay in the signal processing module that reads out and processes the output of each individual fluxgate before it can be used to drive the dynamics of another fluxgate.
This Letter is organized as follows. In Section 2, we start with an overview of fluxgate magnetometers, which should be valuable to readers not familiar with the technology. In Section 3 we provide a description of the dynamics of the coupled bistable model equation for a CCFM without delay, comprised of an odd number N of wound ferromagnetic cores. We note that N (unless taken to be quite large) must be odd [4], [8]. Then in Section 4 we present results of an analytic and computational study of the effects of time delay. Before we advance to the next section we wish to emphasize that the results of this Letter, while relevant to the CCFM as a “test-bed”, are also applicable to almost all overdamped bistable dynamic systems.
Section snippets
Overview of fluxgate magnetometers
Fluxgate magnetometers [9] are considered to be the most cost-efficient magnetic field sensors for applications that require measuring relatively small magnetic fields in the 0.01 mT regime. Originally developed around 1928, today's highly specialized devices can measure magnetic fields in the range of [11] for a variety of magnetic remote sensing applications [10]. In its most basic form, the fluxgate magnetometer consists of two detection coils wound around two ferromagnetic cores
Cooperative behavior without delay
A conventional (i.e., single core) fluxgate magnetometer can be treated as a nonlinear dynamical system by assuming the core to be approximately single-domain [13], and writing down an equation for the evolution of the (suitably normalized) macroscopic magnetization parameter in terms of the potential energy function , where c is a nonlinear temperature-dependent parameter, which controls the topology of the potential function: the
Model equations
We now investigate the behavior of an N-dimensional (N odd for negative feedback) unidirectionally coupled ring of overdamped bistable systems with delayed nearest-neighboor connections. As a test case, we use the model equations (1) of a CCFM device with N fluxgates: where mod N and , denote the corresponding delays in the connectivity scheme. Recall that we are primarily interested in the case where , which is a negative
Discussion
In this manuscript, we have expanded our earlier work on coupled overdamped bistable systems to investigate the effects of delayed coupling on synchronous equilibrium points and on collective oscillations that emerge from a global branch of heteroclinic connections between different saddle-node equilibria. These oscillations have shown to be excellent detectors of small external dc and ac signals because the long period, which occurs near the onset of the heteroclinic connection, renders their
Acknowledgements
We gratefully acknowledge support from the Office of Naval Research (Code 331) and SPAWAR internal S&T program. A.P. and D.L. were supported in part by National Science Foundation grants CMS-0625427 and CMMI-0923803, and by DoD – SPAWAR Command grant N66001-08-D-0154.
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