Period-doubling Sequence Research Articles

On bifurcations of chaotic attractors in a pulse width modulated control system

This paper discusses bifurcational phenomena in a control system with pulse-width modulation of the first kind. We show that the transition from a regular dynamics to chaos occurs in a sequence of classical supercritical period doubling and border collision bifurcations. As a parameter is varied, one can observe a cascade of doubling of the cyclic chaotic intervals, which are associated with homoclinic bifurcations of unstable periodic orbits. Such transition are also refereed as merging bifurcation (known also as merging crisis). At the bifurcation point, the unstable periodic orbit collides with some of the boundaries of a chaotic attractor and as a result, the periodic orbit becomes a homoclinic. This condition we use for obtain equations for bifurcation boundaries in the form of an explicit dependence on the parameters. This allow us to determine the regions of stability for periodic orbits and domains of the existence of four-, two- and one-band chaotic attractors in the parameter plane.

Infinite products involving the period-doubling sequence

Infinite products involving the period-doubling sequence

Dynamical Analysis of a Memristive Chua’s Oscillator Circuit

In this work, a novel memristive Chua’s oscillator circuit is presented. In the proposed circuit, a linear negative resistor, which is parallel coupled with a first-order memristive diode bridge, is used instead of the well-known Chua’s diode. Following this, an extensive theoretical and dynamical analysis of the circuit is conducted. This involves numerical computations of the system’s phase portraits, bifurcation diagrams, Lyapunov exponents, and continuation diagrams. A comprehensive comparison is made between the numerical simulations and the circuit’s simulations performed in Multisim. The analysis reveals a range of intriguing phenomena, including the route to chaos through a period-doubling sequence, antimonotonicity, and coexisting attractors, all of which are corroborated by the circuit’s simulation in Multisim.

Effect of a modulated acoustic field on the dynamics of a vibrating charged bubble

We investigate the effect of amplitude-modulated acoustic irradiation on the dynamics of a charged bubble vibrating in a liquid. We show that the potential V(x) of the bubble, and the number and stability of its equilibria, depend on the magnitude of the charge it carries. Under high-frequency amplitude-modulation, a modulation threshold, Gth, was found for the onset of increased bubble amplitude oscillations. For some pressure field values, charge can facilitate the control of chaotic dynamics via reversed period-doubling bifurcation sequences. There is evidence for peak-shouldering and shock waves. The Mach number increases rapidly with the drive amplitude G. In the supersonic regime, for G>1.90Pa, the high-frequency modulation raises both Blake’s and the transient cavitation thresholds. We found a decrease in the bubble’s maximum charge threshold, and threshold modulation amplitude for the occurrence Vibrational resonance (VR). VR occurs due to the modulated oscillatory pressure field, and the influence on VR of the electrostatic charge, and other parameters of the system are investigated. In contrast to the cases of VR reported earlier, where the amplitude G of the high-frequency driving is typically much higher than the amplitude of the low-frequency driving (Ps), the VR resonance peaks occur here at relatively low G values (0<G<10Pa) compared to the acoustic driving pressure Ps∼105 Pa. The optimal parameter values for enhanced response could be useful in acoustic cavitation applications.

Algebraic Automatic Continued Fractions in Characteristic 2

Abstract We present two families of automatic sequences that define algebraic continued fractions in characteristic $2$. The period-doubling sequence belongs to the first family $\mathcal {P}$, and its sum modulo $2$, the Thue–Morse sequence, belongs to the second family $\mathcal {G}$. The family $\mathcal {G}$ contains all the iterated sums of sequences from the $\mathcal {P}$ and more.

Period-doubling bifurcations and islets of stability in two-degree-of-freedom Hamiltonian systems.

In this paper, we show that the destruction of the main Kolmogorov-Arnold-Moser (KAM) islands in two-degree-of-freedom Hamiltonian systems occurs through a cascade of period-doubling bifurcations. We calculate the corresponding Feigenbaum constant and the accumulation point of the period-doubling sequence. By means of a systematic grid search on exit basin diagrams, we find the existence of numerous very small KAM islands ("islets") for values below and above the aforementioned accumulation point. We study the bifurcations involving the formation of islets and we classify them in three different types. Finally, we show that the same types of islets appear in generic two-degree-of-freedom Hamiltonian systems and in area-preserving maps.

On a family of 2-automatic sequences generating algebraic continued fractions in characteristic 2

We present family of automatic sequences that define algebraic continued fractions in characteristic 2. This family is constructed from ultimately period words and contains the period-doubling sequence.

Behaviour and onset of low-dimensional chaos with a periodically varying loss in single-mode homogeneously broadened laser

Abstract In this work, we numerically study the behaviour of the single-mode homogeneously broadened laser versus a large perturbation of the steady state. Periodic behaviour develops for suitable values of the ratio of the population decay rates ℘ \wp , which has been analytically predicted by the analytical approach developed in the previous works, and for a pumping parameter 2 C 2C situated below and above the instability threshold 2 C 2 th 2{C}_{2{\rm{th}}} . We here show that the adiabatic elimination of the polarisation with sinusoidal time-dependent perturbation of the cavity rate leads the single-mode homogenously broadened laser to exhibit chaotic emission, if the frequency ω \omega is equal to the natural frequency Δ P {\Delta }_{P} . Increasing the pumping parameter 2 C 2C , the laser undergoes a period-doubling sequence. This cascade of period doubling drives the laser towards chaos. We also propose a reformulation of our analytical procedure, which describes the self-pulsing regime of the single-mode homogeneously broadened laser. We here report that asymmetric aspect that appears in the time evolution of the electric field is due to phase effects between the electric field components.

An Insight into the Dynamical Behaviour of the Swing Equation

Motivated by the nonlinear dynamics of mathematical models encountered in power systems, an investigation into the dynamical behaviour of the swing equation is carried out. This paper examines analytically and numerically the development of oscillatory periodic solutions, whereby increases of the control parameter, lead to a cascade of period doubling bifurcations, before eventually loss in stability is exhibited and effective forerunners to chaos revealed. Gaining an understanding on the dynamical behaviour of the system can help to produce a deeper insight of the bifurcations entailed, with the appearance of the triggered sequence of the first period doubling’s acting as precursors of imminent danger and difficult operations of a practical system.

Bistability, bifurcations and chaos in the Mackey-Glass equation

&lt;p style='text-indent:20px;'&gt;Numerical bifurcation analysis, and in particular two-parameter continuation, is used in consort with numerical simulation to reveal complicated dynamics in the Mackey-Glass equation for moderate values of the delay close to the onset of chaos. In particular a cusp bifurcation of periodic orbits and resulting branches of folds of periodic orbits effectively partition the parameter space into regions where different behaviours are seen. The cusp bifurcation leads directly to bistability between periodic orbits, and subsequently to bistability between a periodic orbit and a chaotic attractor. This leads to two different mechanisms by which the chaotic attractor is destroyed in a global bifurcation with a periodic orbit in either an interior crisis or a boundary crisis. In another part of parameter space a sequence of subcritical period-doublings is found to give rise to bistability between a periodic orbit and a chaotic attractor. Torus bifurcations, and a codimension-two fold-flip bifurcation are also identified, and Lyapunov exponent computations are used to determine chaotic regions and attractor dimension.&lt;/p&gt;

On the modeling of some triodes-based nonlinear oscillators with complex dynamics: case of the Van der Pol oscillator

In this work, the dynamic of the triode-based Van der Pol oscillator coupled to a linear circuit is investigated (Triode-based VDPCL oscillator). Towards this end, we present a mathematical model of the triode, chosen from among the many different ones present in literature. The dynamical behavior of the system is investigated using classical tools such as two-parameter Lyapunov exponent, one-parameter bifurcation diagram associated with the graph of largest Lyapunov exponent, phase portraits, and time series. Numerical simulations reveal rather rich and complex phenomena including chaos, transient chaos, the coexistence of solutions, crisis, period-doubling followed by reverse period-doubling sequences (bubbles), and bursting oscillation. The coexistence of attractors is illustrated by the phase portraits and the cross-section of the basin of attraction. Such triode-based nonlinear oscillators can find their applications in many areas where ultra-high frequencies and high powers are demanded (radio, radar detection, satellites communication, etc) since triode can work with these performances and are often used in the aforementioned areas. In contrast to some recent work on triode-based oscillators, LTSPICE simulations, based on real physical consideration of the triode, are carried out in order to validate the theoretical results obtained in this paper as well as the mathematical model adopted for the triode.

Derived Sequences and the Factor Spectrum of the Period-Doubling Sequence

Derived Sequences and the Factor Spectrum of the Period-Doubling Sequence

Dynamical invariants and inverse period-doubling cascades in multi-delay systems.

We investigate transitions to simple dynamics in first-order nonlinear differential equations with multiple delays. With a proper choice of parameters, a single delay can destabilize a fixed point. In contrast, multiple delays can both destabilize fixed points and promote high-dimensional chaos but also induce stabilization onto simpler dynamics. We show that the dynamics of these systems depend on the precise distribution of the delays. Narrow spacing between individual delays induces chaotic behavior, while a lower density of delays enables stable periodic or fixed point behavior. As the dynamics become simpler, the number of unstable roots of the characteristic equation around the fixed point decreases. In fact, the behavior of these roots exhibits an astonishing parallel with that of the Lyapunov exponents and the Kolmogorov-Sinai entropy for these multi-delay systems. A theoretical analysis shows how these roots move back toward stability as the number of delays increases. Our results are based on numerical determination of the Lyapunov spectrum for these multi-delay systems as well as on permutation entropy computations. Finally, we report how complexity reduction upon adding more delays can occur through an inverse period-doubling sequence.

Period-doubling route to mixed-mode chaos.

Mixed-mode oscillations (MMOs) are a complex dynamical behavior in which each cycle of oscillation consists of one or more large amplitude spikes followed by one or more small amplitude peaks. MMOs typically undergo period-adding bifurcations under parameter variation. We demonstrate here, in a set of three identical, linearly coupled van der Pol oscillators, a scenario in which MMOs exhibit a period-doubling sequence to chaos that preserves the MMO structure, as well as period-adding bifurcations. We characterize the chaotic nature of the MMOs and attribute their existence to a master-slave-like forcing of the inner oscillator by the outer two with a sufficient phase difference between them. Simulations of a single nonautonomous oscillator forced by two sine functions support this interpretation and suggest that the MMO period-doubling scenario may be more general.

Criteria for apwenian sequences

In 1998, Allouche, Peyrière, Wen and Wen showed that the Hankel determinant Hn of the Thue-Morse sequence over {−1,1} satisfies Hn/2n−1≡1(mod2) for all n≥1. Inspired by this result, Fu and Han introduced apwenian sequences over {−1,1}, namely, ±1 sequences whose Hankel determinants satisfy Hn/2n−1≡1(mod2) for all n≥1, and proved with computer assistance that a few sequences are apwenian. In this paper, we obtain an easy to check criterion for apwenian sequences, which allows us to determine all apwenian sequences that are fixed points of substitutions of constant length. Let f(z) be the generating functions of such apwenian sequences. We show that for all integer b≥2 with f(1/b)≠0, the real number f(1/b) is transcendental and its irrationality exponent is equal to 2.Besides, we also derive a criterion for 0-1 apwenian sequences whose Hankel determinants satisfy Hn≡1(mod2) for all n≥1. We find that the only 0-1 apwenian sequence, among all fixed points of substitutions of constant length, is the period-doubling sequence. Various examples of apwenian sequences given by substitutions with projection are also provided. Furthermore, we prove that all Sturmian sequences over {−1,1} or {0,1} are not apwenian. And we conjecture that fixed points of substitution of non-constant length over {−1,1} or {0,1} can not be apwenian.

Geometric analysis of mixed-mode oscillations in a model of electrical activity in human beta-cells

Human pancreatic beta-cells may exhibit complex mixed-mode oscillatory electrical activity, which underlies insulin secretion. A recent biophysical model of human beta-cell electrophysiology can simulate such bursting behavior, but a mathematical understanding of the model’s dynamics is still lacking. Here we exploit time-scale separation to simplify the original model to a simpler three-dimensional model that retains the behavior of the original model and allows us to apply geometric singular perturbation theory to investigate the origin of mixed-mode oscillations. Changing a parameter modeling the maximal conductance of a potassium current, we find that the reduced model possesses a singular Hopf bifurcation that results in small-amplitude oscillations, which go through a period-doubling sequence and chaos until the birth of a large-scale return mechanism and bursting dynamics. The theory of folded node singularities provide insight into the bursting dynamics further away from the singular Hopf bifurcation and the eventual transition to simple spiking activity. Numerical simulations confirm that the insight obtained from the analysis of the reduced model can be lifted back to the original model.

On the automaticity of sequences defined by the Thue–Morse and period-doubling Stieltjes continued fractions

Continued fraction expansions of automatic numbers have been extensively studied during the last few decades. The research interests are, on one hand, in the degree or automaticity of the partial quotients following the seminal paper of Baum and Sweet in 1976, and on the other hand, in calculating the Hankel determinants and irrationality exponents, as one can find in the works of Allouche–Peyrière–Wen–Wen, Bugeaud, and the first author. This paper is motivated by the converse problem: to study Stieltjes continued fractions whose coefficients form an automatic sequence. We consider two such continued fractions defined by the Thue–Morse and period-doubling sequences, respectively, and prove that they are congruent to algebraic series in [Formula: see text] modulo [Formula: see text]. Consequently, the sequences of the coefficients of the power series expansions of the two continued fractions modulo [Formula: see text] are [Formula: see text]-automatic.

Unexpected Behaviors in a Single Mesh Josephson Junction Based Self-Reproducing Autonomous System

In the literature, existing Josephson junction based oscillators are mostly driven by external sources. Knowing the different limits of the external driven systems, we propose in this work a new autonomous one that exhibits the unusual and striking multiple phenomena among which coexist the multiple hidden attractors in self-reproducing process under the effect of initial conditions. The eight-term autonomous chaotic system has a single nonlinearity of sinusoidal type acting on only one of the state variables. A priori, the simplicity of the system does not predict the richness of its dynamics. We also find that a limit cycle attractor widens to a parameter controlling coexisting multiple-scroll attractors through the splitting and the inverse splitting of periods. Multiple types of bifurcations are found including period-doubling and period-splitting (antimonotonicity) sequences to chaos, crisis and Hopf type bifurcation. To the best of our knowledge, some of these interesting phenomena have not yet been reported in similar class of autonomous Josephson junction based circuits. Moreover, analytical investigations based on the Hopf theory analysis lead to the expressions that determine the direction of appearance of the Hopf bifurcation, confirming the existence and determining the stability of bifurcating periodic solutions. To observe this latter bifurcation and to illustrate the theoretical analysis, numerical simulations are performed. Chaos can be easily controlled by the frequency of the linear oscillator, the superconducting junction current, as well as the gain of the amplifier or circuit component values. The circuit and Field Programmable Gate Arrays (FPGA)-based implementation of the system are presented as well.

Sum-free sets generated by the period-[formula omitted]-folding sequences and some Sturmian sequences

First, we show that the sum-free set generated by the period-doubling sequence is not κ-regular for any κ≥2. Next, we introduce a generalization of the period-doubling sequence, which we call the period-k-folding sequences. We show that the sum-free sets generated by the period-k-folding sequences also fail to be κ-regular for all κ≥2. Finally, we study the sum-free sets generated by Sturmian sequences that begin with ‘11’, and their difference sequences.

The Thomas Attractor with and without Delay: Complex Dynamics to Amplitude Death

Bifurcations in the Thomas cyclic system leading from simple dynamics into chaotic regimes are considered. In particular, the existence of only one non-trivial fixed point of the system in parameter space implies that this point attractor may only be destabilized via a Hopf bifurcation as the single system parameter is varied. Saddle-node, transcritical and pitchfork bifurcations are precluded. The periodic orbit immediately following the Hopf bifurcation is constructed analytically by the method of multiple scales, and its stability is determined from the resulting normal form and verified by numerical simulations. The dynamically rich range of parameters past the Hopf bifurcation is next systematically explored. In particular, the period-doubling sequences there are found to be more complex than noted previously, and include period-three-like windows for instance. As the system parameter is decreased below these period-incrementing bifurcations, various additional features of the subsequent crises are also carefully tracked. Finally, we consider the effect of delay on the system, leading to the suppression of both the Hopf bifurcation as well as all of the subsequent complex dynamics. In modern terminology, this is an example of Amplitude Death, rather than Oscillation Death, as the complex system dynamics is quenched, with all of the variables settling to a fixed point of the original system.