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

Direct transition to high-dimensional chaos through a global bifurcation

In the present work we report on a genuine route by which a high-dimensional (with d>4) chaotic attractor is created directly, i.e., without a low-dimensional chaotic attractor as an intermediate step. The high-dimensional chaotic set is created in a heteroclinic global bifurcation that yields an infinite number of unstable tori.The mechanism is illustrated using a system constructed by coupling three Lorenz oscillators. So, the route presented here can be considered a prototype for high-dimensional chaotic behavior just as the Lorenz model is for low-dimensional chaos.Comment: 7 page

Flexible dynamics of two quorum-sensing coupled repressilators

Genetic oscillators play important roles in cell life regulation. The regulatory efficiency usually depends strongly on the emergence of stable collective dynamic modes, which requires designing the interactions between genetic networks. We investigate the dynamics of two identical synthetic genetic repressilators coupled by an additional plasmid which implements quorum sensing (QS) in each network thereby supporting global coupling. In a basic genetic ring oscillator network in which three genes inhibit each other in unidirectional manner, QS stimulates the transcriptional activity of chosen genes providing for competition between inhibitory and stimulatory activities localized in those genes. The “promoter strength”, the Hill cooperativity coefficient of transcription repression, and the coupling strength, i.e., parameters controlling the basic rates of genetic reactions, were chosen for extensive bifurcation analysis. The results are presented as a map of dynamic regimes. We found that the remarkable multistability of the antiphase limit cycle and stable homogeneous and inhomogeneous steady states exists over broad ranges of control parameters. We studied the antiphase limit cycle stability and the evolution of irregular oscillatory regimes in the parameter areas where the antiphase cycle loses stability. In these regions we observed developing complex oscillations, collective chaos, and multistability between regular limit cycles and complex oscillations over uncommonly large intervals of coupling strength. QS coupling stimulates the appearance of intrachaotic periodic windows with spatially symmetric and asymmetric partial limit cycles which, in turn, change the type of chaos from a simple antiphase character into chaos composed of pieces of the trajectories having alternating polarity. The very rich dynamics discovered in the system of two identical simple ring oscillators may serve as a possible background for biological phenotypic diversification, as well as a stimulator to search for similar coupling in oscillator arrays in other areas of nature, e.g., in neurobiology, ecology, climatology, etc

Stochastic resonance in electrical circuits—II: Nonconventional stochastic resonance.

Stochastic resonance (SR), in which a periodic signal in a nonlinear system can be amplified by added noise, is discussed. The application of circuit modeling techniques to the conventional form of SR, which occurs in static bistable potentials, was considered in a companion paper. Here, the investigation of nonconventional forms of SR in part using similar electronic techniques is described. In the small-signal limit, the results are well described in terms of linear response theory. Some other phenomena of topical interest, closely related to SR, are also treate

Multistability in Bursting Patterns in a Model of a Multifunctional Central Pattern Generator.

A multifunctional central pattern generator (CPG) can produce bursting polyrhythms that determine locomotive activity in an animal: for example, swimming and crawling in a leech. Each rhythm corresponds to a specific attractor of the CPG. We employ a Hodgkin-Huxley type model of a bursting leech heart interneuron, and connect three such neurons by fast inhibitory synapses to form a ring. This network motif exhibits multistable co-existing bursting rhythms. The problem of determining rhythmic outcomes is reduced to an analysis of fixed points of Poincare mappings and their attractor basins, in a phase plane defined by the interneurons\u27 phase differences along bursting orbits. Using computer assisted analysis, we examine stability, bifurcations of attractors, and transformations of their basins in the phase plane. These structures determine the global bursting rhythms emitted by the CPG. By varying the coupling synaptic strength, we examine the dynamics and patterns produced by inhibitory networks

Complex Dynamics in Digital Nonlinear Oscillators: Experimental Analysis and Verification

A specific topology of Digital Nonlinear Oscillators (DNOs) has been implemented by using commercial off-the-shelf digital components to experimentally verify and demonstrate the capability of these circuits to support complex dynamics, independently from their implementation technology. In detail, a direct experimental evidence of the DNO dynamical behavior is presented at the analog level with a bifurcation diagram analysis, investigation of periodic and chaotic attractors, and dynamical stability. The autonomous circuit has been investigated as a source of entropy, adopting different figures of merit, including the Lempel–Ziv Complexity, to evaluate the dynamics measured under different operating conditions

Symmetry in Chaotic Systems and Circuits

Symmetry can play an important role in the field of nonlinear systems and especially in the design of nonlinear circuits that produce chaos. Therefore, this Special Issue, titled “Symmetry in Chaotic Systems and Circuits”, presents the latest scientific advances in nonlinear chaotic systems and circuits that introduce various kinds of symmetries. Applications of chaotic systems and circuits with symmetries, or with a deliberate lack of symmetry, are also presented in this Special Issue. The volume contains 14 published papers from authors around the world. This reflects the high impact of this Special Issue