Emergent oscillations in unidirectionally coupled overdamped bistable systems

Abstract

It is well known that overdamped unforced dynamical systems do not oscillate. However, well-designed coupling schemes, together with the appropriate choice of initial conditions, can induce oscillations when a control parameter exceeds a threshold value. In a recent publication [Phys. Rev. E 68, 045102 (2003)]], we demonstrated this behavior in a specific prototype system, a soft-potential mean-field description of the dynamics in a hysteretic "single-domain" ferromagnetic sample. The previous analysis of this work showed that N (odd) unidirectionally coupled elements with cyclic boundary conditions would, in fact, oscillate when a control parameter-in this case the coupling strength-exceeded a critical value. These oscillations are now finding utility in the detection of very weak "target" signals, via their effect on the oscillation characteristics, e.g., the frequency and asymmetry of the oscillation wave forms. In this paper we explore the underlying dynamics of this system. Scaling laws that govern the oscillation frequency in the vicinity of the critical point, as well as the zero-crossing intervals in the presence of a symmetry-breaking target dc signal, are derived; these quantities are germane to signal detection and analysis.

Transient dynamics around unstable periodic orbits in the generalized repressilator model

“…A recent example is the study of UPOs in relation to the emergence of chaotic behavior in rings of Duffing oscillators. 7 Closest to the dynamics of the gene system studied here are the rings of unidirectionally coupled magnetic cores 9 that have been not only studied numerically but also implemented experimentally by Bulsara et al In the antiferromagnetic case, the system of coupled magnetic cores is described by the following ODEs (Ref . 9):…”

“…Loops or ring structures of nonlinear elements [5][6][7] constitute one of the simplest topologies of practical interest in biology and engineering. [8][9][10][11] For instance, in a recent series of papers, 3,9,12,13 Bulsara et al reported the implementation through electronics of rings of unidirectionally coupled magnetic flux gates and their successful experimental use as a highly sensitive magnetic field sensors. The mathematical analysis of the steady-state dynamics of a variety of such ring systems has also revealed the richness of their dynamical attractors, including multistability, unstable periodic solutions, quasiperiodicity and even chaos and hyperchaos.…”

“…The mathematical analysis of the steady-state dynamics of a variety of such ring systems has also revealed the richness of their dynamical attractors, including multistability, unstable periodic solutions, quasiperiodicity and even chaos and hyperchaos. 7,9,13 These classic design concepts have found a new applicability with the recent emergence of Synthetic Biology, a field which encompasses ideas from electrical engineering, computer science, and physics to engineer circuits made up of biological parts, in many cases through the design of gene regulatory networks. 14,15 Such synthetic circuits of interconnected genes have been implemented in living microbes to perform low-level functions (e.g., logical gates, 16 switches, 17 and genetic oscillators 10 ) by exploiting the steady-state behavior of the gene network.…”

Symmetry-Breaking as a Paradigm to Design Highly-Sensitive Sensor Systems

“…This network symmetry is critical in the creation of a heteroclinic cycle and, more importantly, in the accompanying infinite-period bifurcations of limit cycle oscillations. Indeed, a bifurcation analysis [3,6] reveals that the system (4) displays oscillatory behavior with the following features:…”

“…The oscillations emerge through a global branch of heteroclinic connections between steady states, so that near the onset of the cycle, the accompanying oscillations exhibit long periods, which in turn renders their waveform highly sensitive to symmetry-breaking effects caused by very small external signals. This critical observation has led to the fabrication of highly-sensitive sensors, magnetic-and electric-field ones, whose operation relies mainly on the coupling signal that travels from one bistable system to the next one [3][4][5][6]. That is, in the absence of coupling, each bistable unit cannot oscillate.…”