René Orlando Medrano-Torricos

We develop an operator-description for the linear stability in finite networks of Kuramoto oscillators on circulant graphs. This mathematical approach offers analytical predictions for the linear stability of q-states, which include phase synchronization (q=0) and phase-locked states with different spatial frequencies (|q|>0). This approach seamlessly incorporates the presence of time delays (represented by phase lags in the coupling). With this, we are able to determine the specific combination of connectivity and time delays (phase lags) that leads to any given q-state to be linearly stable. We apply our framework to a variety of networks, including k-ring graphs, distance-dependent graphs, and random circulant graphs. This approach offers a geometric perspective of linear stability in finite networks in terms of the connectivity and delays (phase lag), and it opens a path to designing and controlling the spatiotemporal dynamics of individual and finite oscillator networks.

In this study, we identify an intricate class of periodic behavior, coined as "fishbone," within the chaotic dynamics induced by external periodic disturbances in a neuronal system. Alongside this structure, we observe well-known patterns such as shrimps and cockroaches, revealing the system's sensitivity to external perturbations. Notably, our work highlights the significant impact of altering biophysical parameters, showing that changes in the ionic conductance of neurons can lead to the cohesion or detachment of complex periodic structures from the primary fishbone-shaped structure. This finding underscores the role of ionic channels in shaping neuronal responses, which is relevant given their involvement in various neuronal disorders. Furthermore, our study provides insights into how external perturbations influence neuron signal processing.

We study a network of identical Kuramoto oscillators with higher-order interactions that also break the rotational symmetry of the system. To gain analytical insights into this model, we use the Watanabe–Strogatz Ansatz, which allows us to reduce the dimensionality of the original system of equations. The study of stability and bifurcations of the reduced system reveals a codimension two Bogdanov–Takens bifurcation and several other associated bifurcations. Such analysis is corroborated by numerical simulations of the associated Kuramoto system, which, in turn, unveils a variety of collective behaviors such as synchronized motion, oscillation death, chimeras, incoherent states, and traveling waves. Importantly, this system displays a case where alternating chimeras emerge in an indistinguishable single population of oscillators, which may offer insights into the unihemispheric slow-wave sleep phenomenon observed in mammals and birds.

This paper presents a recursive method for identifying extreme and superstable curves in the parameter space of dissipative one-dimensional maps. The method begins by constructing an Archimedean spiral with a constant arc length. Subsequently, it identifies extreme and superstable curves by calculating an observable ψ. The spiral is used to locate a region where ψ changes sign. When this occurs, a bisection method is applied to determine the first point on the desired superstable or extreme curve. Once the initial direction is established, the recursive method identifies subsequent points using an additional bisection method, iterating the process until the stopping conditions are met. The logistic-Gauss map demonstrates each step of the method, as it exhibits a wide variety of periodicity structures in the parameter space, including cyclic extreme and superstable curves, which contribute to the formation of period-adding structures. Examples of extreme and superstable curves obtained by the recursive method are presented. It is important to note that the proposed method is generalizable and can be adapted to any one-dimensional map.

We consider the standard nontwist map with strong dissipation that leads the system to a 1D circular map with a quadratic sinusoidal oscillation and two control parameters. The 2D Lyapunov and isoperiodic diagrams reveal a complex interplay between domains of periodicity embedded in regions dominated by quasiperiodic and chaotic behaviors. Arnold tongues and shrimp-like, among other sets of periodicities, compose this rich dynamical scenario in the parameter space. Cobwebs and bifurcation diagrams help reveal the behavior of attractors, including multistability, period-doubling, pitchfork bifurcations, as well as boundary, merging, and interior crises that influence the structures of periodicity. Furthermore, we bring to light the global organization of shrimp-like structures by carrying out a new concept of orbits, the extreme orbits, and announce that the fractal dimension, believed to be universal in the parameter space for decades, has its symmetry breaking in the vicinity of shrimp-like cascades.

This study focuses on the analysis of a unique composition between two well-established models, known as the Logistic–Gauss map. The investigation cohesively transitions to an exploration of parameter space, essential for unraveling the complexity of dissipative mappings and understanding the intricate relationships between periodic structures and chaotic regions. By manipulating control parameters, our approach reveals intriguing patterns, with findings enriched by extreme orbits, trajectories that connect local maximum and minimum values of one-dimensional maps. This theory enhances our perception of structural organization and offers valuable perceptions of the system behaviors, contributing to an expanded understanding of chaos and periodicity in dynamic systems. The analysis reveals Complex Sets of Periodicity (CSP) in the parameter space, characterized by superstable curves that traverse their main bodies. The exploration of different combinations of parameters shows cascades of CSP structures with added periods and are organized based on extreme curves. This investigation offers valuable discoveries of the dynamics of dissipative mappings, opening avenues for future explorations in chaotic systems.

We report remarkable pattern formation of quasiperiodic domains in the two-dimensional parameter space of an intrinsically coupled system, comprising a rotor and a Duffing oscillator. In our analysis, we characterize the system using Lyapunov exponents, identifying self-similar islands composed of intricate regions of chaotic, quasiperiodic, and periodic behaviors. These islands form structures with an accumulation arrangement, denominated here as metamorphic tongues. Inside the islands, we observe Arnold tongues corresponding to periodic solutions. In addition, we surprisingly identify quasiperiodic shrimp-shaped domains that have been typically observed for periodic solutions. Similar features to the periodic case, such as period-doubling and secondary-near shrimp with three times the period, are observed in quasiperiodic shrimp as torus-doubling and torus-tripling.

We report the peculiar organization of oscillations in the forced Brusselator system, found in the parameter space as a nested structure of regular and chaotic phases. To this end, we apply the winding number concept, conceived for nonlinear driven oscillators, to expose all oscillatory phases in the nested structure. First, we use the period and torsion of orbits to describe every periodic oscillation in the parameter spaces, describing the nested structure in high-resolution phase diagrams. Next, we propose a basic structure organizing the periodicity, a “skeletal set” whose properties elucidate the genealogy and composition of oscillations in the nested structure. Finally, we discuss the application of the skeletal structure in a diversity of Brusselator’s oscillatory regimes.