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Dynamical behavior of the optical traveling pulses for the resonant nonlinear Schrödinger (RNS) equation with external periodic force is studied. Using a complex transformation we obtain an unperturbed dynamical system for the RNS equation. Existence of periodic optical pulses, solitary optical pulses of dark and bright types, breaking optical pulses is dispensed using phase plane analysis of the unperturbed dynamical system. Introducing an external perturbation to the unperturbed dynamical system, quasiperiodicity and chaotic features of the nonlinear optical pulses for the perturbed dynamical system are studied by varying the resonance parameter (c) with special values of other system parameters through different computational tools, like time series plot, phase plot, sensitivity plot, Lyapunov exponent, and Poincare section. The resonance parameter (c) acts as a control parameter on qualitative transition of the nonlinear optical pulses for the perturbed dynamical system from quasiperiodic motion to chaotic motion.

Symmetry is not only used to simplify the analysis of three-phase electrical systems, but it is also used to define its voltages, currents and loads. When the loads are symmetrical, the currents of a three-phase AC system are expected to be symmetrical as well. Given the proper conditions, in converters such as the matrix converter (MC), the output voltages and currents are expected to be sinusoidal with periodic symmetry. However, in some cases this symmetry is broken so that, there appears nonlinear behaviors such as quasiperiodicity and so on. Based on simulations and experiments, this paper focuses on an analysis of a quasiperiodic behavior and the presence of a DC component in the output currents of a Venturini modulated MC. The presence of such behaviors in the output currents indicates that the symmetry in a period is broken. The broken symmetries appear when the input and output frequencies are mismatched. In addition, this paper shows the possibility to recover the symmetry of the output currents of the MC. The method for symmetry recovery is based on a time-delayed feedback control. The simulation and experimental results indicate the possibility of attenuating the quasiperiodic behavior and DC component.

We consider a simple reentrant model of a manufacturing process, consisting of one machine at which two different types of items have to be processed. The model is completely deterministic: all delivery and processing times are fixed, and are generally incommensurate. Dependent on the arrival and processing times, a queue of waiting items grows, remains constant or disappears. We demonstrate that the dynamics of the system crucially depends on the queue type. Complexity is most observed for the case of growing queue. We characterize this dynamics between order and chaos with the T-entropy and the autocorrelation function.

Piecewise smooth maps have been a focus of study for scientists in a wide range of research fields. These maps show qualitatively different types of bifurcations than those exhibited by generic smooth maps. We present a theoretical framework for analyzing three-dimensional piecewise smooth maps by deriving a suitable normal form and then finding the stability criteria for periodic orbits. We also show by numerical simulation different types of border collision bifurcations that can occur in such a map. We have also been able to observe a border collision bifurcation from a period-2 to a quasiperiodic orbit.

This paper considers the processes of torus formation and reconstruction through smooth and nonsmooth bifurcations in a pulse-width modulated DC/DC converter with multilevel control. When operating in a regime of high corrector gain, converters of this type can generate structures of stable tori embedded one into the other and with their basins of attraction delineated by intervening repelling tori. The paper illustrates the coexistence of three stable tori with different resonance behaviors and shows how reconstruction of these tori takes place across the borders of different dynamical regimes. The paper also demonstrates how pairs of attracting and repelling tori emerge through border-collision torus-birth and border-collision torus-fold bifurcations.

We consider the local bifurcations that can occur in a quasiperiodic orbit in a three-dimensional map: (a) a torus doubling resulting in two disjoint loops, (b) a torus doubling resulting in a single closed curve with two loops, (c) the appearance of a third frequency, and (d) the birth of a stable torus and an unstable torus. We analyze these bifurcations in terms of the stability of the point at which the closed invariant curve intersects a "second Poincaré section". We show that these bifurcations can be classified depending on where the eigenvalues of this fixed point cross the unit circle.

The dynamics of the map

The dynamics of a low-dimensional ensemble consisting of a network of five discrete phase oscillators is considered. A two-parameter synchronization picture, which appears instead of the Arnol'd tongues with an increase of the system dimension, is discussed. An appearance of the Arnol'd resonance web is detected on the "frequency–coupling" parameter plane. The cases of attractive and repulsive interactions are discussed.

We consider a class of simple two-dimensional discrete models representative of a system incorporating both mutation and predation. A selection of analytic and numerical results are presented, classifying the dynamic behavior of the system by means of Lyapunov exponents over a biologically-reasonable region of parameter space, and illustrating the occurrence of hyperchaos and a Neimark–Sacker bifurcation producing regions of quasiperiodicity.

This paper reports on an investigation of the two-dimensional parameter-space of a generalized Nosé–Hoover oscillator. It is a mathematical model of a thermostated harmonic oscillator, which consists of a set of three autonomous first-order nonlinear ordinary differential equations. By using Lyapunov exponents to numerically characterize the dynamics of the model at each point of this parameter-space, it is shown that dissipative quasiperiodic structures are present, embedded in a chaotic region. The same parameter-space is also used to confirm the multistability phenomenon in the investigated mathematical model.

We report parameter planes displaying dynamical behaviors of a radio-physical oscillator system, which is modeled by a set of four-parameter three autonomous first-order nonlinear ordinary differential equations. Each parameter plane is numerically computed, and the dynamical behavior of each point is characterized by using the respective Lyapunov exponents spectrum. Chaotic, periodic, and quasiperiodic regions are therefore bounded in parameter planes of a radio-physical oscillator system.

A model of quasiperiodic-chaotic neural networks is proposed on the basis of chaotic neural networks. A quasiperiodic-chaotic neuron exhibits quasiperiodic dynamics that an original chaotic neuron does not have. Quasiperiodic and chaotic solutions are exclusively isolated in the parameter space. The chaotic domain can be identified by the presence of a folding structure of an invariant closed curve. Using the property that the influence of perturbation is conserved in the quasiperiodic solution, we demonstrate short-term visual memory in which real numbers are acceptable for representing colors. The quasiperiodic solution is sensitive to dynamical noise when images are restored. However, the quasiperiodic synchronization among neurons can reduce the influence of noise. Short-term analog memory using quasiperiodicity is important in that it can directly store analog quantities. The quasiperiodic-chaotic neural networks are shown to work as large-scale analog storage arrays. This type of analog memory has potential applications to analog computation such as deep learning.

The paper describes some aspects of sudden transformations of closed invariant curves in a 2D piecewise smooth map. In particular, using detailed numerically calculated phase portraits, we discuss transitions from smooth to piecewise smooth closed invariant curves. We show that such transitions may occur not only when a closed invariant curve collides with a border but also via a homoclinic bifurcation. Furthermore, we describe an unusual transformation from a closed invariant curve to a large amplitude chaotic attractor and demonstrate that this transition occurs in two steps, involving a small amplitude closed-invariant-curve-like chaotic attractor.

An innovative box-counting method is developed in this research for diagnozing the nonlinear characteristics of dynamical systems. With the method developed, an approach that depicts the evolutionary process on Poincaré maps is established such that the nonlinear dynamical characteristics of the transient and stable process of the system can be graphically and quantitatively identified. A Duffing–van der Pol system is adopted in the research to demonstrate an application of the method. A diagram graphically describing the periodic, quasiperiodic, chaotic, and transient chaotic regions of the system’s responses is constructed based on the method. Furthermore, the nature of different box-point curves is explained based on the topology of chaos and quasiperiodicity. The method developed shows innovation and efficiency in diagnozing nonlinear dynamical systems based on the topological properties of general nonlinear systems.

The nearest-neighbor Aubry–André quasiperiodic localization model is generalized to include power-law translation-invariant hoppings $T_l\propto t/l^a$ or power-law Fourier coefficients $W_m\propto w/m^b$ in the quasiperiodic potential. The Aubry–André duality between $T_l$ and $W_m$ manifests when the Hamiltonian is written in the real-space basis and in the Fourier basis on a finite ring. The perturbative analysis in the amplitude $t$ of the hoppings yields that the eigenstates remain power-law localized in real space for $a>1$ and are critical for $a_c=1$ where they follow the strong multifractality linear spectrum, as in the equivalent model with random disorder. The perturbative analysis in the amplitude $w$ of the quasiperiodic potential yields that the eigenstates remain delocalized in real space (power-law localized in Fourier space) for $b>1$ and are critical for $b_c=1$ where they follow the weak multifractality Gaussian spectrum in real space (or strong multifractality linear spectrum in the Fourier basis). This critical case $b_c=1$ for the Fourier coefficients $W_m$ corresponds to a periodic function with discontinuities, instead of the cosinus function of the standard self-dual Aubry–André model.

We review our recent rigorous results on renormalization in a variety of quasiperiodically forced systems. Our results include a description of (i) self-similar fluctuations of localized states in the Harper equation, including the renormalization strange set (known as the orchid) in the generalized Harper equation; and (ii) self-similarities in the correlations of strange nonchaotic attractors, barrier billiards, and quantum two-level systems.

Dielectric Elastomers (DEs) belong to a class of electrostatically driven ElectroActive Polymers (EAPs) employed effectively in electromechanical transduction technology. Inflated Dielectric Elastomer Actuators (DEAs) represent a special class of electromechanical transducers that exploit electro-elastic instability phenomena to generate out-of-plane axial symmetric deformations of circular membranes made up of non-conductive rubbery materials. Practically, DE membrane-based Inflated DEAs execute the transient motion which is significantly affected by the viscoelastic behavior of the DE membrane. In this paper, we present a nonlinear theoretical electromechanical framework to interrogate the effect of membrane viscoelasticity and strain stiffening effect on the dynamic behavior of the Inflated DEAs. The developed dynamic model investigates the DC and AC response, effects of membrane viscoelasticity on the stability transition, and natural frequencies of the Inflated DEAs. Further, a qualitative corroboration with experimental observations is articulated, highlighting the practical feasibility of the developed model. The results revealed in this study can find its potential use in the efficient design and development of inflated DEAs subjected to a time-dependent motion.

The layer of water at temperatures close to the point of the density maximum is considered. The density–temperature dependence for water can be approximated by quadratic function in the vicinity of the point density maximum. If the point of density maximum is inside the layer then the stable and unstable regions are formed. We studied transition of convective regimes from static to stochastic in two-dimensional horizontal plane layer in case of equal heights of stable and unstable layers in the conductive state. The stability of two-dimensional steady and periodic patterns was investigated to choose a horizontal scale for the study of transitional processes. Regions of hysteresis were obtained. The most realizable scenario of transition to chaos is through bifurcation of the steady solution to the periodic one, then there happens the secondary subcritical Hopf bifurcation to quasiperiodic torus, but initially the motion is doubly periodic. With the increase of supercriticality, intermittency arises with strong turbulent bursts on the background of base quasiperiodic motion. Then, there is a window of quasiperiodic solutions inside intermittent solutions which are different from initial quasiperiodic motion.