A modified Oregonator model exhibiting complicated limit cycle behavior in a flow system

Slow-fast oscillatory dynamics and phantom attractors in stochastic modeling of biochemical reactions

A problem of the probabilistic analysis of stochastic phenomena in slow-fast dynamical systems modeling biochemical reactions is considered. We study how multiplicative noise induces systematic shifts of probabilistic distributions and forms "phantom" attractors in nonlinear enzymatic models. The mathematical analysis of the underlying probabilistic mechanism of such stochastic transformations is performed by the "freeze-and-average" method. Our theoretical results are supported by direct numerical simulation.

Period-doubling route to mixed-mode chaos

Mixed-mode oscillations (MMOs) are a complex dynamical behavior in which each cycle of oscillation consists of one or more large amplitude spikes followed by one or more small amplitude peaks. MMOs typically undergo period-adding bifurcations under parameter variation. We demonstrate here, in a set of three identical, linearly coupled van der Pol oscillators, a scenario in which MMOs exhibit a period-doubling sequence to chaos that preserves the MMO structure, as well as period-adding bifurcations. We characterize the chaotic nature of the MMOs and attribute their existence to a master-slave-like forcing of the inner oscillator by the outer two with a sufficient phase difference between them. Simulations of a single nonautonomous oscillator forced by two sine functions support this interpretation and suggest that the MMO period-doubling scenario may be more general.

Emergence of Mixed Mode Oscillations in Random Networks of Diverse Excitable Neurons: The Role of Neighbors and Electrical Coupling

In this paper, we focus on the emergence of diverse neuronal oscillations arising in a mixed population of neurons with different excitability properties. These properties produce mixed mode oscillations (MMOs) characterized by the combination of large amplitudes and alternate subthreshold or small amplitude oscillations. Considering the biophysically plausible, Izhikevich neuron model, we demonstrate that various MMOs, including MMBOs (mixed mode bursting oscillations) and synchronized tonic spiking appear in a randomly connected network of neurons, where a fraction of them is in a quiescent (silent) state and the rest in self-oscillatory (firing) states. We show that MMOs and other patterns of neural activity depend on the number of oscillatory neighbors of quiescent nodes and on electrical coupling strengths. Our results are verified by constructing a reduced-order network model and supported by systematic bifurcation diagrams as well as for a small-world network. Our results suggest that, for weak couplings, MMOs appear due to the de-synchronization of a large number of quiescent neurons in the networks. The quiescent neurons together with the firing neurons produce high frequency oscillations and bursting activity. The overarching goal is to uncover a favorable network architecture and suitable parameter spaces where Izhikevich model neurons generate diverse responses ranging from MMOs to tonic spiking.

Mechanism and Phenomenology of an Oscillating Chemical Reaction

Chemical reactions, which are far from equilibrium, are capable of displaying oscillations in species concentrations and hence in colour, electrode potential, pH and/or temperature. The oscillations arise from the interplay between positive and negative kinetic feedback. Mechanisms for such reactions are presented, along with the rich phenomenology that these systems exhibit, from complex oscillations and chemical waves, to stationary concentration patterns. This review will focus on the Belousov-Zhabotinksy reaction but reference to other reactions will be made where appropriate.

The Tris(2,2'-Bipyridyl)Ruthenium-Catalysed Belousov–Zhabotinsky Reaction

The Belousov – Zhabotinsky (BZ) reaction is the prototypical oscillating chemical reaction. The tris(2,2'-bipyridine)ruthenium-catalysed BZ reaction (often simply referred to as the ruthenium-catalysed BZ reaction) displays photosensitivity and has been widely exploited for examination of the effects of illumination on nonlinear reaction kinetics. In this review, we investigate the behaviour of the ruthenium-catalysed BZ reaction. The mechanism of the reaction is analysed and we examine how light sensitivity is incorporated into kinetic models of the reaction. The temporal dynamics of the photosensitive reaction is presented and, finally, we discuss the extraordinary wealth of behaviour that has been observed in the spatially-distributed system when perturbed by visible light.

Natural extension of fast-slow decomposition for dynamical systems

Modeling and parameter estimation to capture the dynamics of physical systems are often challenging because many parameters can range over orders of magnitude and are difficult to measure experimentally. Moreover, selecting a suitable model complexity requires a sufficient understanding of the model's potential use, such as highlighting essential mechanisms underlying qualitative behavior or precisely quantifying realistic dynamics. We present an approach that can guide model development and tuning to achieve desired qualitative and quantitative solution properties. It relies on the presence of disparate time scales and employs techniques of separating the dynamics of fast and slow variables, which are well known in the analysis of qualitative solution features. We build on these methods to show how it is also possible to obtain quantitative solution features by imposing designed dynamics for the slow variables in the form of specified two-dimensional paths in a bifurcation-parameter landscape.

Nonchaos-Mediated Mixed-Mode Oscillations in an Enzyme Reaction System

We report numerical evidence of a new type of wide-ranging organization of mixed-mode oscillations (MMOs) in a model of the peroxidase-oxidase reaction, in the control parameter plane defined by the supply of the reactant NADH and the pH of the medium. In classic MMOs, the intervals of distinct periodic oscillations are always separated from each other by windows of chaos. In contrast, in the new unfolding, such windows of chaos do not exist. Chaos-mediated and nonchaos-mediated MMO phases are separated by a continuous transition boundary in the control parameter plane. In addition, for low pH values, we find an exceptionally wide and intricate mosaic of MMO phases that is described by a detailed phase diagram.

Complex dynamical behavior in the highly photosensitive cerium-bromate-1,4-benzoquinone reaction

Experimental and numerical investigations are carried out on the nonlinear dynamics of the cerium-bromate-1,4-benzoquinone reaction in which a unique kinetic feature is that the reduction of Ce(IV) is through bromide ions rather than by organic substrates. Nonlinear phenomena including both simple and sequential oscillations have been observed, and the system could oscillate for longer than a week. Significantly, fluorescent ceiling light with an intensity of less than 20 μW/cm(2) exhibited strong influence on the frequency, lifetime, and complexity of the spontaneous oscillations. The transient oscillations lasted for a longer period of time at a low light intensity, were quenched by a moderate illumination, and then became long-lived again at a higher light intensity. Characterizations with (1)H NMR and GC/MS spectroscopy and ion selective electrode suggest that 2-bromo-1,4-benzoquinone is an important unstable intermediate product that undergoes photoaccelerated decomposition to produce hydroxy-1,4-benzoquinone and bromide ions. Simulations successfully reproduced the occurrence of oscillatory behavior in the studied system.

Mixed-mode oscillations via canard explosions in light-emitting diodes with optoelectronic feedback

Chaotically spiking attractors in semiconductor lasers with optoelectronic feedback have been recently observed to be the result of canard phenomena in three-dimensional phase space (incomplete homoclinic scenarios). Since light-emitting diodes display the same dynamics and are much more easily controllable, we use one of these systems to complete the attractor analysis demonstrating experimentally and theoretically the occurrence of complex sequences of periodic mixed-mode oscillations. In particular, we investigate the transition between periodic and chaotic mixed-mode states and analyze the effects of the unavoidable experimental noise on these transitions.

Introduction to focus issue: mixed mode oscillations: experiment, computation, and analysis

Mixed mode oscillations (MMOs) occur when a dynamical system switches between fast and slow motion and small and large amplitude. MMOs appear in a variety of systems in nature, and may be simple or complex. This focus issue presents a series of articles on theoretical, numerical, and experimental aspects of MMOs. The applications cover physical, chemical, and biological systems.

Mixed-mode oscillations in a homogeneous pH-oscillatory chemical reaction system

We examine experimentally a chemical system in a flow-through stirred reactor, which is known to provide large-amplitude oscillations of the pH value. By systematic variation of the flow rate, we find that the system displays hysteresis between a steady state and oscillations, and more interestingly, a transition to chaos involving mixed-mode oscillations. The basic pattern of the measured pH in the mixed-mode regime includes a large-scale peak followed by a series of oscillations on a much smaller scale, which are usually highly irregular and of variable duration. The bifurcation diagram shows that chaos sets in via a period-doubling route observed on the large-amplitude scale, but simultaneously small-amplitude oscillations are involved. Beyond the apparent accumulation of period doubling bifurcations, a mixed-mode regime with irregular oscillations on both scales is observed, occasionally interrupted by windows of periodicity. As the flow rate is further increased, chaos turns into quasiperiodicity and later to a simple small-amplitude periodic regime. Dynamics of selected typical regimes were examined with the tools of nonlinear time-series analysis, which include phase space reconstruction of an attractor and calculation of the maximal Lyapunov exponent. The analysis points to deterministic chaos, which appears via a period doubling route from below and via a route involving quasiperiodicity from above, when the flow rate is varied.

Line-defects-mediated complex-oscillatory spiral waves in a chemical system

In this paper, we summarize our experimental observations on complex-oscillatory spiral waves that arise in a Belousov-Zhabotinsky (BZ) reaction-diffusion system. The observed wave structures generically bear line defects across which the phase of local oscillation changes by a multiple of 2 pi. The local oscillation at every spatial point along a line defect of period-2 (P-2) oscillatory media is period-1 (P-1) oscillatory. For the homogeneous BZ reaction can be excitable, simply periodic, complex periodic, or chaotic as the control parameters are tuned, a number of different complex wave states are revealed. A two-dimensional phase diagram, which includes domains of P-2 oscillatory spirals, intermittently breathing spirals, period-3 (P-3) oscillatory spirals, two different types of mixed-mode periodic spirals, and line-defect-mediated turbulence, is constructed. Several different transitions among different dynamic states are described systematically. In all cases, line defects are found to play an important role.

Origin of bursting oscillations in an enzyme model reaction system

The transition from simple periodic to bursting behavior in a three-dimensional model system of the hemin-hydrogen-peroxide-sulfite pH oscillator is investigated. A two-parameter continuation in the flow rate and the hemin decay rate is performed to identify the region of complex dynamics. The bursting oscillations emerge subsequent to a cascade of period-doubling bifurcations and the formation of a chaotic attractor in parameter space where they are found to be organized in periodic-chaotic progressions. This suggests that the bursting oscillations are not associated with phase-locked states on a two-torus. The bursting behavior is classified by a bifurcation analysis using the intrinsic slow-fast structure of the dynamics. In particular, we find a slowly varying quasispecies (i.e., a linear combination of two species) which acts as an "internal" or quasistatic bifurcation parameter for the remaining two-dimensional subsystem. A systematic two-parameter continuation in the internal parameter and one of the external bifurcation parameters reveals a transition in the bursting mechanism from sub-Hopf/fold-cycle to fold/sub-Hopf type. In addition, the slow-fast analysis provides an explanation for the origin of quasiperiodic behavior in the hemin system, even though the underlying mechanism might be of more general importance.

Deterministic chaos in the Belousov-Zhabotinsky reaction: Experiments and simulations

An account of the experimental discovery of complex dynamical behavior in the continuous-flow, stirred tank reactor (CSTR) Belousov-Zhabotinsky (BZ) reaction, as well as numerical simulations based on the BZ chemistry are given. The most recent four- and three-variable models that are deduced from the well-accepted, updated chemical mechanism of the BZ reaction and which exhibit robust chaotic states are summarized. Chaos has been observed in experiments and simulations embedded in the regions of complexities at both low and high flow rates. The deterministic nature of the observed aperiodicities at low flow rates is unequivocally established. However, controversy still remains in the interpretation of certain aperiodicities observed at high flow rates.