Amplitude Death: The emergence of stationarity in coupled nonlinear systems

When nonlinear dynamical systems are coupled, depending on the intrinsic dynamics and the manner in which the coupling is organized, a host of novel phenomena can arise. In this context, an important emergent phenomenon is the complete suppression of oscillations, formally termed amplitude death (AD). Oscillations of the entire system cease as a consequence of the interaction, leading to stationary behavior. The fixed points that the coupling stabilizes can be the otherwise unstable fixed points of the uncoupled system or can correspond to novel stationary points. Such behaviour is of relevance in areas ranging from laser physics to the dynamics of biological systems. In this review we discuss the characteristics of the different coupling strategies and scenarios that lead to AD in a variety of different situations, and draw attention to several open issues and challenging problems for further study.Comment: Physics Reports (2012

Conjugate coupling induced symmetry breaking and quenched oscillations

Spontaneous symmetry breaking (SSB) is essential and plays a vital role many natural phenomena, including the formation of Turing pattern in organisms and complex patterns in brain dynamics. In this work, we investigate whether a set of coupled Stuart-Landau oscillators can exhibit spontaneous symmetry breaking when the oscillators are interacting through dissimilar variables or conjugate coupling. We find the emergence of SSB state with coexisting distinct dynamical states in the parametric space and show how the system transits from symmetry breaking state to out-of-phase synchronized (OPS) state while admitting multistabilities among the dynamical states. Further, we also investigate the effect of feedback factor on SSB as well as oscillation quenching states and we point out that the decreasing feedback factor completely suppresses SSB and oscillation death states. Interestingly, we also find the feedback factor completely diminishes only symmetry breaking oscillation and oscillation death (OD) states but it does not affect the nontrivial amplitude death (NAD) state. Finally, we have deduced the analytical stability conditions for in-phase and out-of-phase oscillations, as well as amplitude and oscillation death states.Comment: Accepted for publication in Europhysics Letter

Impact of environmental inputs on reverse-engineering approach to network structures

Background: Uncovering complex network structures from a biological system is one of the main topic in system biology. The network structures can be inferred by the dynamical Bayesian network or Granger causality, but neither techniques have seriously taken into account the impact of environmental inputs. Results: With considerations of natural rhythmic dynamics of biological data, we propose a system biology approach to reveal the impact of environmental inputs on network structures. We first represent the environmental inputs by a harmonic oscillator and combine them with Granger causality to identify environmental inputs and then uncover the causal network structures. We also generalize it to multiple harmonic oscillators to represent various exogenous influences. This system approach is extensively tested with toy models and successfully applied to a real biological network of microarray data of the flowering genes of the model plant Arabidopsis Thaliana. The aim is to identify those genes that are directly affected by the presence of the sunlight and uncover the interactive network structures associating with flowering metabolism. Conclusion: We demonstrate that environmental inputs are crucial for correctly inferring network structures. Harmonic causal method is proved to be a powerful technique to detect environment inputs and uncover network structures, especially when the biological data exhibit periodic oscillations

The uncoupling limit of identical Hopf bifurcations with an application to perceptual bistability

We study the dynamics arising when two identical oscillators are coupled near a Hopf bifurcation where we assume a parameter $\epsilon$ uncouples the system at $\epsilon=0$. Using a normal form for $N=2$ identical systems undergoing Hopf bifurcation, we explore the dynamical properties. Matching the normal form coefficients to a coupled Wilson-Cowan oscillator network gives an understanding of different types of behaviour that arise in a model of perceptual bistability. Notably, we find bistability between in-phase and anti-phase solutions that demonstrates the feasibility for synchronisation to act as the mechanism by which periodic inputs can be segregated (rather than via strong inhibitory coupling, as in existing models). Using numerical continuation we confirm our theoretical analysis for small coupling strength and explore the bifurcation diagrams for large coupling strength, where the normal form approximation breaks down

Chaos at the border of criticality

The present paper points out to a novel scenario for formation of chaotic attractors in a class of models of excitable cell membranes near an Andronov-Hopf bifurcation (AHB). The mechanism underlying chaotic dynamics admits a simple and visual description in terms of the families of one-dimensional first-return maps, which are constructed using the combination of asymptotic and numerical techniques. The bifurcation structure of the continuous system (specifically, the proximity to a degenerate AHB) endows the Poincare map with distinct qualitative features such as unimodality and the presence of the boundary layer, where the map is strongly expanding. This structure of the map in turn explains the bifurcation scenarios in the continuous system including chaotic mixed-mode oscillations near the border between the regions of sub- and supercritical AHB. The proposed mechanism yields the statistical properties of the mixed-mode oscillations in this regime. The statistics predicted by the analysis of the Poincare map and those observed in the numerical experiments of the continuous system show a very good agreement.Comment: Chaos: An Interdisciplinary Journal of Nonlinear Science (tentatively, Sept 2008