Dynamical analysis of a periodically forced chaotic chemical oscillator

We present a comprehensive dynamical analysis of a chaotic chemical model referred to as the autocatalator, when subject to a periodic administration of one substrate. Our investigation encompasses the dynamical characterization of both unforced and forced systems utilizing isospikes and largest Lyapunov exponents-based parameter planes, bifurcation diagrams, and analysis of complex oscillations. Additionally, we present a phase diagram showing the effect of the period and amplitude of the forcing signal on the system's behavior. Furthermore, we show how the landscapes of parameter planes are altered in response to forcing application. This analysis contributes to a deeper understanding of the intricate dynamics induced by the periodic forcing of a chaotic system.

Synaptic dependence of dynamic regimes when coupling neural populations

In this article we focus on the study of the collective dynamics of neural networks. The analysis of two recent models of coupled "next-generation" neural mass models allows us to observe different global mean dynamics of large neural populations. These models describe the mean dynamics of all-to-all coupled networks of quadratic integrate-and-fire spiking neurons. In addition, one of these models considers the influence of the synaptic adaptation mechanism on the macroscopic dynamics. We show how both models are related through a parameter and we study the evolution of the dynamics when switching from one model to the other by varying that parameter. Interestingly, we have detected three main dynamical regimes in the coupled models: Rössler-type (funnel type), bursting-type, and spiking-like (oscillator-type) dynamics. This result opens the question of which regime is the most suitable for realistic simulations of large neural networks and shows the possibility of the emergence of chaotic collective dynamics when synaptic adaptation is very weak.

Correlation sum scalings from mixed-mode oscillations in weakly coupled molecular lasers

A model for two symmetrically coupled lasers is investigated, in which mixed-mode oscillations arise in the absence of coupling. For small enough coupling, we show that in the time series, certain dynamical transitions from different resonances in the chaotic regime may be explained by the overlap of suitable resonances. These are families of N : N + 1 resonances, which result in isolas as well as isolas consisting of intermediate-phase resonances N : N. It appears that the overlap of resonances can explain the onset of two different scaling regions in the dimension correlation sum, which display an explicit dependence on the optical coupling strength. For very small coupling ranges, there are larger scaling regions that look analogous to that for the uncoupled laser system. For larger coupling, but still well below the synchronization threshold, steeper and larger scaling regions arise, in particular, in the smaller partitions.

Complexity in subnetworks of a peroxidase-oxidase reaction model

The peroxidase-oxidase (PO) reaction is a paradigmatic (bio)chemical system well suited to study the organization and stability of self-sustained oscillatory phases typically present in nonlinear systems. The PO reaction can be simulated by the state-of-the-art Bronnikova-Fedkina-Schaffer-Olsen model involving ten coupled ordinary differential equations. The complex and dynamically rich distribution of self-sustained oscillatory stability phases of this model was recently investigated in detail. However, would it be possible to understand aspects of such a complex model using much simpler models? Here, we investigate stability phases predicted by three simple four-variable subnetworks derived from the complete model. While stability diagrams for such subnetworks are found to be distorted compared to those of the complete model, we find them to surprisingly preserve significant features of the original model as well as from the experimental system, e.g., period-doubling and period-adding scenarios. In addition, return maps obtained from the subnetworks look very similar to maps obtained in the experimental system under different conditions. Finally, two of the three subnetwork models are found to exhibit quint points, i.e., recently reported singular points where five distinct stability phases coalesce. We also provide experimental evidence that such quint points are present in the PO reaction.

Non-quantum chirality in a driven Brusselator

We report the discovery of non-quantum chirality in the a periodically driven Brusselator. In contrast to standard chirality from quantum contexts, this novel type of chirality is governed by rate equations, namely by purely classical equations of motion. The Brusselator chirality was found by computing high-resolution phase diagrams depicting the number of spikes, local maxima, observed in stable periodic oscillations of the Brusselator as a function of the frequency and amplitude of the external drive. We also discuss how to experimentally observe non-quantum chirality in generic oscillators governed by nonlinear sets of rate equations.

Chirality observed in a driven ruthenium-catalyzed Belousov-Zhabotinsky reaction model

Chirality is commonly associated with the spatial geometry of the atoms composing molecules, the biochemistry of living organisms, and spin properties. In sharp contrast, here we report chirality found in numerically computed stability diagrams of a chemical reaction governed by purely classical (that is, not quantum) equations, namely in a photochemically periodically perturbed ruthenium-catalyzed Belousov-Zhabotinsky reaction model. This novel chirality offers opportunities to explore hitherto unsuspected properties of purely classical chemical oscillators.

Ubiquity of ring structures in the control space of complex oscillators

We report the discovery of two types of stability rings in the control parameter space of a vertical-cavity surface-emitting semiconductor laser. Stability rings are closed parameter paths in the laser control space. Inside such rings, laser stability thrives even in the presence of small parameter fluctuations. Stability rings were also found in rather distinct contexts, namely, in the way that cancerous, normal, and effector cells interact under ionizing radiation and in oscillations of an electronic circuit with a junction-gate field-effect transistor (JFET) diode. We argue that stability-enhancing rings are generic structures present in the control parameter space of many complex systems.

Quint points lattice in a driven Belousov-Zhabotinsky reaction model

We report the discovery of a regular lattice of exceptional quint points in a periodically driven oscillator, namely, in the frequency-amplitude control parameter space of a photochemically periodically perturbed ruthenium-catalyzed Belousov-Zhabotinsky reaction model. Quint points are singular boundary points where five distinct stable oscillatory phases coalesce. While spikes of the activator show a smooth and continuous variation, the spikes of the inhibitor show an intricate but regular branching into a myriad of stable phases that have fivefold contact points. Such boundary points form a wide parameter lattice as a function of the frequency and amplitude of light absorption. These findings revise current knowledge about the topology of the control parameter space of a celebrated prototypical example of an oscillating chemical reaction.

Complexity of a peroxidase-oxidase reaction model

The peroxidase-oxidase oscillating reaction was the first (bio)chemical reaction to show chaotic behaviour. The reaction is rich in bifurcation scenarios, from period-doubling to peak-adding mixed mode oscillations. Here, we study a state-of-the-art model of the peroxidase-oxidase reaction. Using the model, we report systematic numerical experiments exploring the impact of changing the enzyme concentration on the dynamics of the reaction. Specifically, we report high-resolution phase diagrams predicting and describing how the reaction unfolds over a quite extended range of enzyme concentrations. Surprisingly, such diagrams reveal that the enzyme concentration has a huge impact on the reaction evolution. The highly intricate dynamical behaviours predicted here are difficult to establish theoretically due to the total absence of an adequate framework to solve nonlinearly coupled differential equations. But such behaviours may be validated experimentally.

Classification of fold/hom and fold/Hopf spike-adding phenomena

The Hindmarsh-Rose neural model is widely accepted as an important prototype for fold/hom and fold/Hopf burstings. In this paper, we are interested in the mechanisms for the production of extra spikes in a burst, and we show the whole parametric panorama in an unified way. In the fold/hom case, two types are distinguished: a continuous one, where the bursting periodic orbit goes through bifurcations but persists along the whole process and a discontinuous one, where the transition is abrupt and happens after a sequence of chaotic events. In the former case, we speak about canard-induced spike-adding and in the second one, about chaos-induced spike-adding. For fold/Hopf bursting, a single (and continuous) mechanism is distinguished. Separately, all these mechanisms are presented, to some extent, in the literature. However, our full perspective allows us to construct a spike-adding map and, more significantly, to understand the dynamics exhibited when borders are crossed, that is, transitions between types of processes, a crucial point not previously studied.

Complex mode-locking oscillations and Stern-Brocot derivation tree in a CSTR reaction with impulsive perturbations

A complex mode-locking (entrainment) topology underlying the continuous stirred tank reactor reaction model subjected to impulsive perturbations is identified. Employing high-resolution stability diagrams, we exhibit the global structure of mode-locking oscillations and describe how they are interconnected and how their complexity unfolds with control parameters varying. The scenarios shown in the bi-parametric planes revealed that the skeleton of Arnold's tongues is organized according to the symmetric Stern-Brocot sum tree. Moreover, the mode-locking organization is controlled by an invariant torus (a pair of frequencies) initiated from Hopf bifurcations. Interestingly, the mode-locking order is unfolded in an elusive way, that is, in perfect agreement with the reciprocal of the Stern-Brocot sum tree. The findings reported here contribute to providing a description and classification of mode-locking oscillations for the impulsive system.

Periodicity characterization of the nonlinear magnetization dynamics

In this work, we study numerically the periodicity of regular regions embedded in chaotic states for the case of an anisotropic magnetic particle. The particle is in the monodomain regime and subject to an applied magnetic field that depends on time. The dissipative Landau-Lifshitz-Gilbert equation models the particle. To perform the characterization, we compute several two-dimensional phase diagrams in the parameter space for the Lyapunov exponents and the isospikes. We observe multiple transitions among periodic states, revealing complex topological structures in the parameter space typical of dynamic systems. To show the finer details of the regular structures, iterative zooms are performed. In particular, we find islands of synchronization for the magnetization and the driven field and several shrimp structures with different periods.

Feedforward attractor targeting for non-linear oscillators using a dual-frequency driving technique

A feedforward control technique is presented to steer a harmonically driven, non-linear system between attractors in the frequency-amplitude parameter plane of the excitation. The basis of the technique is the temporary addition of a second harmonic component to the driving. To illustrate this approach, it is applied to the Keller-Miksis equation describing the radial dynamics of a single spherical gas bubble placed in an infinite domain of liquid. This model is a second-order, non-linear ordinary differential equation, a non-linear oscillator. With a proper selection of the frequency ratio of the temporary dual-frequency driving and with the appropriate tuning of the excitation amplitudes, the trajectory of the system can be smoothly transformed between specific attractors; for instance, between period-3 and period-5 orbits. The transformation possibilities are discussed and summarized for attractors originating from the subharmonic resonances and the equilibrium state (absence of external driving) of the system.

Chaotic convection in an Oldroyd viscoelastic fluid in saturated porous medium with feedback control

The control effects on the convection dynamics in a viscoelastic fluid-saturated porous medium heated from below and cooled from above are studied. A truncated Galerkin expansion was applied to balance equations to obtain a four-dimensional generalized Lorenz system. The dynamical behavior is mainly characterized by the Lyapunov exponents, bifurcation, and isospike diagrams. The results show that within a range of moderate and high Rayleigh numbers, proportional controller gain is found to enhance the stabilization and destabilization effects on the thermal convection. Furthermore, due to the effect of viscoelasticity, the system exhibits remarkable topological structures of regular regions embedded in chaotic domains.

A numerical investigation of the effect of external resistance and applied potential on the distribution of periodicity and chaos in the anodic dissolution of nickel

Low density, elongation, and suppression of the shrimp-like structures in the resistance-potential phase diagrams have been observed in the oscillatory dissolution of nickel.

Phase diagrams and dynamical evolution of the triple-pathway electro-oxidation of formic acid on platinum

A detailed numerical study including stability phase diagrams for the dynamical evolution of the electro-oxidation of formic acid on platinum was reported. The study evidences the existence of intertwined stability phases and the absence of chaos.

Optimal Phase-Control Strategy for Damped-Driven Duffing Oscillators

Phase-control techniques of chaos aim to extract periodic behaviors from chaotic systems by applying weak harmonic perturbations with a suitably chosen phase. However, little is known about the best strategy for selecting adequate perturbations to reach desired states. Here we use experimental measures and numerical simulations to assess the benefits of controlling individually the three terms of a Duffing oscillator. Using a real-time analog indicator able to discriminate on-the-fly periodic behaviors from chaos, we reconstruct experimentally the phase versus perturbation strength stability areas when periodic perturbations are applied to different terms governing the oscillator. We verify the system to be more sensitive to perturbations applied to the quadratic term of the double-well Duffing oscillator and to the quartic term of the single-well Duffing oscillator.

Self-organization of pulsing and bursting in a CO2 laser with opto-electronic feedback

We report a detailed investigation of the stability of a CO2 laser with feedback as described by a six-dimensional rate-equations model which provides satisfactory agreement between numerical and experimental results. We focus on experimentally accessible parameters, like bias voltage, feedback gain, and the bandwidth of the feedback loop. The impact of decay rates and parameters controlling cavity losses are also investigated as well as control planes which imply changes of the laser physical medium. For several parameter combinations, we report stability diagrams detailing how laser spiking and bursting is organized over extended intervals. Laser pulsations are shown to emerge organized in several hitherto unseen regular and irregular phases and to exhibit a much richer and complex range of behaviors than described thus far. A significant observation is that qualitatively similar organization of laser spiking and bursting can be obtained by tuning rather distinct control parameters, suggesting the existence of unexpected symmetries in the laser control space.

Nested arithmetic progressions of oscillatory phases in Olsen's enzyme reaction model

We report some regular organizations of stability phases discovered among self-sustained oscillations of a biochemical oscillator. The signature of such organizations is a nested arithmetic progression in the number of spikes of consecutive windows of periodic oscillations. In one of them, there is a main progression of windows whose consecutive number of spikes differs by one unit. Such windows are separated by a secondary progression of smaller windows whose number of spikes differs by two units. Another more complex progression involves a fan-like nested alternation of stability phases whose number of spikes seems to grow indefinitely and to accumulate methodically in cycles. Arithmetic progressions exist abundantly in several control parameter planes and can be observed by tuning just one among several possible rate constants governing the enzyme reaction.

Complex dynamics of a dc glow discharge tube: Experimental modeling and stability diagrams

We report a detailed experimental study of the complex behavior of a dc low-pressure plasma discharge tube of the type commonly used in commercial illuminated signs, in a microfluidic chip recently proposed for visible analog computing, and other practical devices. Our experiments reveal a clear quasiperiodicity route to chaos, the two competing frequencies being the relaxation frequency and the plasma eigenfrequency. Based on an experimental volt-ampere characterization of the discharge, we propose a macroscopic model of the current flowing in the plasma. The model, governed by four autonomous ordinary differential equations, is used to compute stability diagrams for periodic oscillations of arbitrary period in the control parameter space of the discharge. Such diagrams show self-pulsations to emerge remarkably organized into intricate mosaics of stability phases with extended regions of multistability (overlap). Specific mosaics are predicted for the four dynamical variables of the discharge. Their experimental observation is an open challenge.

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.

Zig-zag networks of self-excited periodic oscillations in a tunnel diode and a fiber-ring laser

We report numerical evidence showing that periodic oscillations can produce unexpected and wide-ranging zig-zag parameter networks embedded in chaos in the control space of nonlinear systems. Such networks interconnect shrimplike windows of stable oscillations and are illustrated here for a tunnel diode, for an erbium-doped fiber-ring laser, and for the Hénon map, a proxy of certain CO(2) lasers. Networks in maps can be studied without the need for solving differential equations. Tuning parameters along zig-zag networks allows one to continuously modify wave patterns without changing their chaotic or periodic nature. In addition, we report convenient parameter ranges where such networks can be detected experimentally.

Chaotic dynamics of a magnetic nanoparticle

We study the deterministic spin dynamics of an anisotropic magnetic particle in the presence of a magnetic field with a constant longitudinal and a time-dependent transverse component using the Landau-Lifshitz-Gilbert equation. We characterize the dynamical behavior of the system through calculation of the Lyapunov exponents, Poincaré sections, bifurcation diagrams, and Fourier power spectra. In particular we explore the positivity of the largest Lyapunov exponent as a function of the magnitude and frequency of the applied magnetic field and its direction with respect to the main anisotropy axis of the magnetic particle. We find that the system presents multiple transitions between regular and chaotic behaviors. We show that the dynamical phases display a very complicated structure of intricately intermingled chaotic and regular phases.

Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows

Infinite cascades of periodicity hubs were predicted and very recently observed experimentally to organize stable oscillations of some dissipative flows. Here we describe the global mechanism underlying the genesis and organization of networks of periodicity hubs in control parameter space of a simple prototypical flow, namely a Rössler's oscillator. We show that spirals associated with periodicity hubs emerge and accumulate at the folding of certain fractal-like sheaves of Shilnikov homoclinic bifurcations of a common saddle-focus equilibrium. The specific organization of hub networks is found to depend strongly on the interaction between the homoclinic orbits and the global structure of the underlying attractor.