Holokx A. Albuquerque

We investigate the nonlinear dynamics of a linearly coupled van der Pol–Duffing system using numerical continuation, time-domain simulations, stochastic analysis, and analog circuit experiments. The model exhibits a rich variety of dynamical regimes, including periodic oscillations, period-doubling cascades, and chaotic attractors arising from the interplay between self-excitation and nonlinear stiffness. Numerical continuation is employed to reconstruct the bifurcation structure, enabling the identification of equilibrium branches, periodic solutions, and their stability in parameter space. The time-domain numerical results reveal the mechanisms governing transitions between regular and chaotic dynamics. To assess robustness under realistic conditions, intrinsic stochastic perturbations are introduced, showing that increasing noise intensity progressively erodes fine periodic structures, while larger dynamical domains remain comparatively robust. Experimental results obtained from an analog circuit implementation confirm the main dynamical regimes predicted numerically. Overall, the combined computational and experimental approach provides a systematic characterization of the system’s bifurcation structure and its robustness to noise. The results support the concept of chaos-based sensing and are consistent with previous findings in chaotic bioimpedance detection, indicating that maximum sensitivity occurs near regions of high bifurcation complexity, where small parameter variations induce significant qualitative changes in the system dynamics.

In this work, we employ simulations to investigate deterministic and stochastic dynamics of a system governed by four coupled nonlinear ordinary differential equations (ODEs) derived from the canonical three-dimensional Chua model, replacing the piecewise nonlinearity with a cubic function. We employed two simulation approaches: (i) using software — where we performed numerical simulations (digital computation) in the ODEs model, and (ii) using hardware — where we performed experimental studies (analog computation) in the ODEs model. In the initial software-based approach for simulations, we performed stability and bifurcation analyses of the trivial equilibrium point, which were analytically conducted, along with the numerical continuation method. We solved the model using digital computation and explored its intricate dynamical behavior through nonlinear dynamics techniques applied to the parameter planes. In the second approach, employing hardware for simulations through analog computation, we designed an electronic circuit using analog electronics to solve the model in a continuous-time integration. The experimental setup mirrored the numerical simulations, yielding comparable results in terms of dynamical behaviors. This study underscores significant agreement between numerical and experimental findings, highlighting the robustness of the model across digital and analog contexts and offering practical insights into the various nonlinear dynamics presented in this system.

The three-dimensional Muthuswamy–Chua–Ginoux (MCG) circuit model is a generalization of the paradigmatic canonical Muthuswamy–Chua circuit, where a physical memristor assumes the role of a thermistor, and it is connected in series with a linear passive capacitor, a linear passive inductor, and a nonlinear resistor. The physical memristor presents an electrical resistance which is a function of temperature. Nowadays, the MCG circuit model has gained considerable attention due to the lack of extensive numerical explorations and their distinct dynamical properties, exemplified by phenomena such as the transition from torus breakdown to chaos, giving rise to a double spiral attractor associated to independent period-doubling cascades. In this contribution, the complex dynamics of the MCG circuit model is studied in terms of the Lyapunov exponents spectra, Kaplan–Yorke (KY) dimension, and the number of local maxima (LM) computed in one period of oscillation, as two parameters are simultaneously varied. Using the Lyapunov spectra to distinguish different dynamical regimes, KY dimension to estimate the attractors’ dimension, and the number of LM to characterize different periodic attractors, we construct high-resolution two-dimensional stability diagrams considering specific ranges of the parameter pairs [ ]. These parameters are associated with the inverse of the capacitance in the passive capacitor, and the heat capacitance and dissipation constant of the thermistor, respectively. Unexpectedly, we identify sequences of infinite self-organized generic stable periodic structures (SPSs) and Arnold tongues-like structures (ATSs) merged into chaotic dynamics domains, and the coexistence of different attracting sets (attractors) for the same parameter combinations and different initial conditions (multistability). We explore the multistable dynamics using the stability analysis and computation of Lyapunov coefficients, the inspection of the coexisting attractors, bifurcations diagrams, and basins of attraction. The periods of the ATSs and a particular sequence of shrimp-shaped SPSs obey specific generating and recurrence rules responsible for the bifurcation cascades. As the MCG circuit model has the crucial properties presented by the usual Muthuswamy–Chua circuit model, specific properties explored in our study should be helpful in real problems involving circuits with the presence of physical memristor playing the role of thermistors.

Single-walled carbon nanotubes (SWCNTs) can undergo arbitrarily large nonlinear deformations without permanent damage to the atomic structure and mechanical properties. The dynamic response observed in curved SWCNTs under externally driven forces has fundamental implications in science and technology. Therefore, it is interesting to study the nonlinear dynamics of a damped-driven curved SWCNT oscillator model if two control parameters are varied simultaneously, e.g., the external driven strength and damping parameters. For this purpose, we construct high-resolution two-dimensional stability diagrams and, unexpectedly, we identify (i) the existence of a quint points lattice merged in a domain of periodic dynamics, (ii) the coexistence of different stable states for the same parameter combinations and different initial conditions (multistability), and (iii) the existence of infinite self-organized generic stable periodic structures (SPSs) merged into chaotic dynamics domains. The quint points lattice found here is composed of five distinct stability domains that coalesce and are associated with five different periodic attractors. The multistability is characterized by the coexistence of three different multi-attractors combinations for three exemplary parameter sets: two periodic attractors, two chaotic attractors, or one periodic and one chaotic attractor. This study demonstrates how complex the dynamics of a damped-driven curved SWCNT oscillator model can be when parameters and initial conditions are varied. For this reason, it may have a relevant impact on new theoretical and experimental applications of damped-driven curved SWCNTs.

We analyze the existence of chaotic and regular dynamics, transient chaos phenomenon, and multistability in the parameter space of two electrically interacting FitzHugh–Nagumo (FHN) neurons. By using extensive numerical experiments to investigate the particular organization between periodic and chaotic domains in the parameter space, we obtained three important findings: (i) there are self-organized generic stable periodic structures along specific directions immersed in a chaotic portion of the parameter space; (ii) the existence of transient chaos phenomenon is responsible for long chaotic temporal evolution preceding the asymptotic (periodic) dynamics for particular parametric combinations in the parameter space; and (iii) the existence of various multistable domains in the parameter space with an arbitrary number of attractors. Additionally, we also prove through numerical simulations that chaos, transient chaos, and multistability prevail even for different coupling strengths between identical FHN neurons. It is possible to find multistable attractors in the phase and parameter spaces and to steer them apart by increasing the asymmetry in the coupling force between neurons. Such a strategy can be essential to experimental matters, as setting the right parameter ranges. As the FHN model shares the crucial properties presented by the more realistic Hodgkin–Huxley-like neurons, our results can be extended to high-dimensional coupled neuron models.