Quint points lattice and multistability in a damped-driven curved carbon nanotube oscillator model.

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.

Transient dynamics and multistability in two electrically interacting FitzHugh-Nagumo neurons.

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.

Parameter space of experimental chaotic circuits with high-precision control parameters.

We report high-resolution measurements that experimentally confirm a spiral cascade structure and a scaling relationship of shrimps in the Chua's circuit. Circuits constructed using this component allow for a comprehensive characterization of the circuit behaviors through high resolution parameter spaces. To illustrate the power of our technological development for the creation and the study of chaotic circuits, we constructed a Chua circuit and study its high resolution parameter space. The reliability and stability of the designed component allowed us to obtain data for long periods of time (∼21 weeks), a data set from which an accurate estimation of Lyapunov exponents for the circuit characterization was possible. Moreover, this data, rigorously characterized by the Lyapunov exponents, allows us to reassure experimentally that the shrimps, stable islands embedded in a domain of chaos in the parameter spaces, can be observed in the laboratory. Finally, we confirm that their sizes decay exponentially with the period of the attractor, a result expected to be found in maps of the quadratic family.

Lyapunov exponent diagrams of a 4-dimensional Chua system.

We report numerical results on the existence of periodic structures embedded in chaotic and hyperchaotic regions on the Lyapunov exponent diagrams of a 4-dimensional Chua system. The model was obtained from the 3-dimensional Chua system by the introduction of a feedback controller. Both the largest and the second largest Lyapunov exponents were considered in our colorful Lyapunov exponent diagrams, and allowed us to characterize periodic structures and regions of chaos and hyperchaos. The shrimp-shaped periodic structures appear to be malformed on some of Lyapunov exponent diagrams, and they present two different bifurcation scenarios to chaos when passing the boundaries of itself, namely via period-doubling and crisis. Hyperchaos-chaos transition can also be observed on the Lyapunov exponent diagrams for the second largest exponent.

Some two-dimensional parameter spaces of a Chua system with cubic nonlinearity.

In this paper we investigate three two-dimensional parameter spaces of a three-parameter set of autonomous differential equations used to model the Chua oscillator, where the piecewise-linear function usually taken to describe the nonlinearity of the Chua diode has been replaced by a cubic polynomial. It is made by using three independent two-dimensional cross sections of the three-dimensional parameter space generated by the model, which contains three parameters. We show that, independent of the parameter set considered in plots, all the diagrams present periodic structures embedded in a large chaotic region, and we also show that these structures organize themselves in period-adding cascades. We argue that these selected two-dimensional cross sections can be representative of the three-dimensional parameter space as a whole, in the range of parameters here investigated.

High-resolution parameter space of an experimental chaotic circuit.

A high-resolution codimension-two parameter space showing the abundance of complex periodic structures of an experimental chaotic circuit is reported. Such resolution was propitiated by the use of a 0.5 mV step dc voltage source as one of the control parameters. Those complex periodic structures organize themselves in a period-adding bifurcation cascade that accumulates in a chaotic region. Numerical investigations on the dynamical model were also carried out to corroborate several new features observed in the experimental high-resolution parameter space.