Elsevier

Physics Letters A

Volume 374, Issue 27, 14 June 2010, Pages 2709-2722
Physics Letters A

Basins of attraction in a ring of overdamped bistable systems with delayed coupling

https://doi.org/10.1016/j.physleta.2010.04.060Get rights and content

Abstract

Theoretical and experimental works reveal that coupling similar overdamped bistable systems can lead, under certain conditions that depend on the topology of connections and the number of units, to self-induced large-amplitude oscillations that emerge through a global bifurcation of heteroclinic connections between saddle-node equilibria. This critical observation has led to new mechanisms for weak (compared to the energy barrier height) signal detection and amplification. While the mathematical models and related devices governed by bistable potential functions may assume instantaneous coupling, in practice we must account for the fact that even high-speed, high-precision, circuit components can introduce a delay in the coupling signal. Thus, in this manuscript we investigate the behavior of a ring of overdamped bistable systems with delayed nearest-neighbor connections. Related work, without delay, shows that large-amplitude oscillations and nontrivial synchronous equilibria can coexist near the onset of the oscillations. Our study shows that a delay-induced Hopf bifurcation occurs from the synchronous equilibria but, generically, the small amplitude oscillations that appear are unstable. Thus, delay has the effect of decreasing the size of the basin of attraction of nontrivial synchronous equilibria, which in turn, makes the basin of attraction of the stable large-amplitude oscillations larger. Collectively, this is a positive effect because the sensor device depends mainly on large amplitude oscillations, so a small delay can make it easier to induce the device to oscillate on its own. As a “test bed”, we use the model equations of a CCFM device with N fluxgates. The results are, however, generic and applicable to all rings of overdamped bistable units unidirectionally coupled.

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 x˙=U(x), where x(t) is the state variable of the device at time t, e.g., magnetization state, and U(x) is the bistable potential function. In the absence of an external force, the state point x(t) 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 x(t) 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 110pT/Hz [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 x(t):x˙(t)=xU(x) in terms of the potential energy function U(x,t)=x2(t)/2c1lncoshc[x(t)+Asinωt+ε(t)], 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:x˙i(t)=xi(t)+tanh(c(xi(t)+λxi+1(tτi)+ε)), where i=1,2,,N mod N and τ1,τ2,,τN, denote the corresponding delays in the connectivity scheme. Recall that we are primarily interested in the case where λ<0, 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.

References (26)

  • P. Ripka

    J. Magn. Magn. Mat.

    (2000)
  • Waterloo Maple Software

    Maple 13

    (2001)
  • S.A. Campbell

    Calculating centre manifolds for delay differential equations using Maple

  • H.T. Banks et al.

    SIAM J. Control Optim.

    (1981)
  • F. Primdahl

    Fluxgate magnetometers

  • P. Ripka

    Sensors Actuators A

    (1996)
  • V. In et al.

    Phys. Rev. E

    (2003)
    A. Bulsara et al.

    Phys. Rev. E

    (2004)
  • A. Bulsara et al.

    Measurement Sci. Technol.

    (2008)
  • V. In et al.

    Phys. Rev. E

    (2006)
  • A. Palacios et al.

    Phys. Rev. E

    (2006)
  • A. Bulsara et al.

    Measurement Sci. Technol.

    (2008)
  • W. Bornhofft et al.P. Ripka

    Magnetic Sensors and Magnetometers

    (2001)
  • D. Gordon et al.

    IEEE Trans. Magn.

    (1972)
    C. Russell et al.

    Science

    (1979)
    J. Lenz

    IEEE Proc.

    (1990)
    R. Snare et al.

    IEEE Trans. Magn.

    (1997)
  • View full text