Periodicity characterization of the nonlinear magnetization dynamics

Performance analysis of indicators of chaos for nonlinear dynamical systems

The efficient detection of chaotic behavior in orbits of a complex dynamical system is an active domain of research. Several indicators have been proposed, and new ones have recently been developed in view of improving the performance of chaos detection by means of numerical simulations. The challenge is to predict chaotic behavior based on the analysis of orbits of limited length. In this paper the performance analysis of past and recent indicators of chaos, in terms of predictive power, is carried out in detail using the dynamical system characterized by a symplectic Hénon-like cubic polynomial map.

Chaotic dynamics from coupled magnetic monodomain and Josephson current

The ordinary (superconductor-insulator-superconductor) Josephson junction cannot exhibit chaos in the absence of an external ac drive, whereas in the superconductor-ferromagnet-superconductor Josephson junction, known as the φ_{0} junction, the magnetic layer effectively provides two extra degrees of freedom that can facilitate chaotic dynamics in the resulting four-dimensional autonomous system. In this work, we use the Landau-Lifshitz-Gilbert model for the magnetic moment of the ferromagnetic weak link, while the Josephson junction is described by the resistively capacitively shunted-junction model. We study the chaotic dynamics of the system for parameters surrounding the ferromagnetic resonance region, i.e., for which the Josephson frequency is reasonably close to the ferromagnetic frequency. We show that, due to the conservation of magnetic moment magnitude, two of the numerically computed full spectrum Lyapunov characteristic exponents are trivially zero. One-parameter bifurcation diagrams are used to investigate various transitions that occur between quasiperiodic, chaotic, and regular regions as the dc-bias current through the junction, I, is varied. We also compute two-dimensional bifurcation diagrams, which are similar to traditional isospike diagrams, to display the different periodicities and synchronization properties in the I-G parameter space, where G is the ratio between the Josephson energy and the magnetic anisotropy energy. We find that as I is reduced the onset of chaos occurs shortly before the transition to the superconducting state. This onset of chaos is signaled by a rapid rise in supercurrent (I_{S}⟶I) which corresponds, dynamically, to increasing anharmonicity in phase rotations of the junction.

Bifurcation structure and chaos in dynamics of nanomagnet coupled to Josephson junction

Irregular easy axis reorientation features are observed in numerical simulations of the nanomagnet coupled to the Josephson junction. We study magnetization bifurcations and chaos that appear in this system due to the interplay of superconductivity and magnetism. The bifurcation structure of magnetization under the variation of Josephson to magnetic energy ratio as a control parameter demonstrates several precessional motions that are related to chaotic behavior and orbits with different periodicities in the ferromagnetic resonance region. The effect of an external periodic signal on the bifurcation structure is also investigated. The results demonstrate high-frequency modes of a periodic motion and a chaotic response near resonance. Far from the ferromagnetic resonance, we observe a quasiperiodic behavior. The obtained results explain the irregular reorientation of the easy axis and the transitions between different types of motion.

Influence of higher-order modes on ferroconvection

Using Fourier representations, an elaborate study of regular cellular-convective and chaotic motions in a ferrofluid is made. Investigation is made on the adequacy or otherwise of the minimal mode in studying such motions. Higher-order modes are also considered by adding modes (vertical/horizontal/combined extension). For higher modes, the extensions yield a dynamical system of order greater than three. The characteristic features of extended ferromagnetic-Lorenz models are analyzed using the largest Lyapunov exponent(LE), second largest LE, bifurcation diagram, and phase-space plots. The effect of additional modes on critical modal-Rayleigh (infinitesimal and finite-amplitude ones) numbers and the Rayleigh number at which transition to chaos occurs are examined to report features of ferroconvection hitherto unseen in previous studies. As both horizontal and vertical modes are increased, our findings infer that the dynamical system displays advanced onset of regular convection and delayed chaotic motion. Vigorous-chaotic motion is seen on adding vertical modes, whereas on adding horizontal modes, intense chaos appears with decreased intensity for large values of the scaled Rayleigh number. Most important finding from the study is that as modes are increased (vertical/horizontal), the transition from regular to chaotic motion is greatly modified and leads the system to a hyper-chaotic state. Conventionally, the chaotic or hyper-chaotic state is intermittent with a periodic/quasi-periodic state but it can be retained in the chaotic or hyper-chaotic state by considering moderate values of the Prandtl number and/or by bringing in the ferromagnetic effect.

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.

Introduction to Focus Issue: Instabilities and nonequilibrium structures

This Focus Issue on instabilities and nonequilibrium structures includes invited contributions from leading researchers across many different fields. The issue was inspired in part by the "VII Instabilities and Nonequilibrium Structures 2019" conference that took place at the Pontifica Universidad Católica de Valparaiso, Chile in December 2019. The conference, which is devoted to nonlinear science, is one of the oldest conferences in South America (since December 1985). This session has an exceptional character since it coincides with the 80th anniversary of Professor Enrique Tirapegui. We take this opportunity to highlight Tirapegui's groundbreaking contributions in the field of random perturbations experienced by macroscopic systems and in the formation of spatiotemporal structures in such systems operating far from thermodynamic equilibrium. This issue addresses a cross-disciplinary area of research as can be witnessed by the diversity of systems considered from inert matter such as photonics, chemistry, and fluid dynamics, to biology.