Reverse Period-doubling Bifurcation Research Articles

Energy function and complex dynamics from a jerk system

Jerk, as a mathematical concept, is used in mechanics to describe the rate of change of acceleration and plays a crucial role in the design of control systems for machines and vehicles. Therefore, it is important to master the various states and the energy released during the change of acceleration. This is why a new simple jerk function introduced afterward, energy released, is derived from a Hamilton function using the Helmholtz theorem. The condition of having a stable or unstable rate of change of acceleration is established using Hopf bifurcation theory. Some two-parameter stability charts are then computed for a suitable selection region of the study. Using some nonlinear analysis metrics, in the unstable region of the study, the occurrence of phenomena is found, such as reverse period doubling bifurcation, antimonotonicity, and hysteresis involving the coexistence of the states in the considered jerk system. An electronic circuit is built and used to implement the mathematical expression of the jerk equation and validate the result of the theoretical investigation.

Bifurcation cascade, self-similarity, and duality in the three-rotor problem.

The three-rotor system concerns equally massive point particles moving on a circle subject to attractive cosine potentials of strength g. The quantum theory models chains of coupled Josephson junctions. Classically, it displays order-chaos-order behavior with increasing energy E along with a seemingly globally chaotic phase for 5.33g≲E≲5.6g. It is also known to admit pendulum and isosceles breather families of periodic orbits at all energies. While pendula display a doubly infinite sequence of stability transitions accumulating at their libration to rotation threshold at E=4g, breathers undergo only one stability transition. Here, we show that these stability transitions are associated with forward and reverse fork-like isochronous and period-doubling bifurcations. The new family of periodic orbits born at each of these bifurcations is found using an efficient search algorithm starting from a transverse perturbation to the parent orbit. The graphs of stability indices of various classes of orbits born at pendulum bifurcations meet at E=4g forming "fans." The transitions in the librational and rotational phases are related by an asymptotic duality between bifurcation energies and shapes of newly born periodic orbits. The latter are captured by solutions to a Lamé equation. We also find and numerically validate values of scaling constants for self-similarity in (a) stability indices of librational and rotational pendula and (b) shapes of newly born orbits as E→4g. Finally, we argue that none of the infinitely many families of periodic orbits we have found is stable for 5.33g≲E≲5.6g, providing further evidence for global chaos in this energy band.

Analysis and microcontroller implementation of Josephson junction driven by Wien bridge circuit

This paper recounts the dynamical investigations and microcontroller validation of Josephson junction (JJ) driven by Wien bridge circuit (WBC). Relying on the stability analysis of two equilibrium points found in JJ driven by WBC (JJDWBC), it is revealed that one equilibrium point is unconditionally stable while the other equilibrium point is unstable. The JJDWBC exhibits reverse period-doubling bifurcation, periodic attractors, period-doubling bifurcation, bistable chaotic attractors, and different presentations of monostable chaotic attractors. Partial and total amplitude controls are achieved by adding two controller parameters into the rate equations of JJDWBC. Finally, the microcontroller implementation is used to validate the dynamical behaviors found in JJDWBC.

Dynamic Effects of a MCNN-CS Under Electromagnetic Induction and Its Application

Cellular neural network (CNN) is one of the best artificial neural network models due to its hardware implementability and many applications for image processing. The memristor-based cellular neural network (MCNN) combines the distinct benefits of memristors and neural networks, and it excels in mimicking chaotic systems under the effect of electromagnetic induction. In this work, a novel CNN chaotic system based on a flux-controlled memristor (MCNN-CS) with electromagnetic induction effects is constructed. Dynamical behaviors are examined by controlling the magnetic flux leakage strength and the intensity of the electromagnetic induction related to the memristor. Based on stability analyses and numerical simulation, forward and reverse period-doubling bifurcations, various coexisting attractors, multiperiods, chaos, several periodic windows, and chaotic crisis are observed. Besides, a number of complicated phenomena, including transient chaos, intermittent chaos, sustained chaos and bistability are also observed, demonstrating that this chaotic system has a wide range of dynamic properties. Finally, we generate pseudo-random number sequences based on this new system and apply them to image encryption. The obtained results show that this chaotic system offers superior randomness and is hence appropriate for image encryption applications.

Effect of time delay in a bistable synthetic gene network.

The essence of logical stochastic resonance is the dynamic manipulation of potential wells. The effect of time delay on the depth of potential wells and the width of a bistable region can be inferred by logic operations in the bistable system with time delay. In a time-delayed synthetic gene network, time delay in the synthesis process can increase the depth of the potential wells, while that in the degradation process, it can reduce the depth of the potential wells, which will result in a decrease in the width of the bistable region (the reason for time delay to induce logic operations without external driving force) and the instability of the system (oscillation). These two opposite effects imply stretching and folding, leading to complex dynamical behaviors of the system, including period, chaos, bubble, chaotic bubble, forward and reverse period doubling bifurcation, intermittency, and coexisting attractors.

A novel antimonotic hyperjerk system: Analysis, synchronization and circuit design

We propose a novel chaotic oscillator derived from the generic four-dimensional autonomous jerk systems. By using the Routh–Hurwitz criterion, we analyze the equilibrium point stability. Furthermore, we use the bifurcation diagram and Lyapunov exponents to explore the chaotic regions of the oscillator. One of the novel hyperjerk systems’ main features is the presence of antimonotonicity in a region of the parameter’s space. This phenomenon occurs when periodic orbits are generated and then annihilated via reverse period-doubling bifurcation scenarios as a control parameter is slowly scanned. Next, we propose an electronic design of the hyperjerk system which is compared with the numerical simulations, from which we observe a qualitative agreement among them. Latter, we analyze the synchronization behavior of the chaotic oscillator via feedback control. By using the Lyapunov function methodology, we establish the conditions to achieve synchronization based on the positivity of a matrix written in terms of the system parameters. The numerical simulations support our theoretical results.

Large oscillation of electrostatically actuated curved beams

This paper investigates the responses of initially curved micro-beams subjected to an electrostatic excitation. A Euler–Bernoulli beam theory is utilized to derive the governing equation of motion. A reduced-order model was developed by discretizing the equation of motion using straight beam mode shapes as basis functions in a Galerkin expansion. The results show evidence of the superharmonic resonances of order-three and two in addition to primary resonance. The co-existence of multiple stable orbits observed around only one stable equilibrium and under excitation waveforms with RMS voltage less than the snap-back voltage. These branches are a branch of small orbits within a narrow potential well and two branches of medium-sized and large orbits within a wider potential well. The transition between them results in the characteristic of the double peaks appearing in the frequency-response curve. We also report a bubble structure along the medium-sized branch consists of a cascade of period-doubling bifurcations and a cascade of reverse period-doubling bifurcations. A chaotic attractor develops within that structure at larger excitation levels. It demonstrates evidence of chaos with a wide-based spectrum and an elevated noise-floor. Odd-periodic windows appear also within the attractor including period-three (P-3), period-five (P-5) and period-six (P-6) windows. The chaotic attractor terminates in a cascade of reverse period-doubling bifurcations of period-four (P-4) orbits and period-two (P-2) orbits.

A Unique Jerk System with Abundant Dynamics: Symmetric or Asymmetric Bistability, Tristability, and Coexisting Bubbles

Finding and revealing new features and behaviors of simple chaotic systems have always been an important and attractive research topic. This paper aims to introduce a new 3D autonomous jerk chaotic system and explore its rich dynamical behaviors including period-doubling bifurcation and reverse period-doubling bifurcation routes to chaos, crisis and internal crisis, multiple symmetric coexisting attractors, and antimonotonicity. Especially, the phenomena of asymmetric bistability (e.g., coexistence of a point attractor and chaotic attractor or coexistence of a point attractor and period-5 limit cycle), tristability (e.g., coexistence of a point attractor and a pair of symmetric chaotic or periodic attractors), and coexisting bubbles are found, which have been rarely reported before. By using standard nonlinear analysis methods such as bifurcation diagrams, the largest Lyapunov exponent, phase portraits, Poincare sections, 0–1 test chart and the basin of attraction for an attractor, and the complex dynamical behaviors of the system are investigated in detail. Furthermore, a corresponding hardware electronic circuit is designed to verify the numerical simulations.

Antimonotonicity, Chaos and Multidirectional Scroll Attractor in Autonomous ODEs Chaotic System

Three-dimensional autonomous ordinary differential equations (ODEs) are the simplest and most important chaotic systems in nonlinear dynamics. In fact, they have been applied in many fields. In this paper, a systematic methodology for analyzing complex behavior of the ODEs chaotic system, as one of the ODEs chaotic systems, the improved TCS which satisfies the condition $a_{12}a_{21} = 0$ , is proposed. It is dissipative, chaos, symmetric, antimonotonicity and can generate multiple directional ( $M\times N\times L$ ) scroll attractors. Then, bifurcation diagrams, Lyapunov exponents, time series, Poincare sections, and Hausdroff dimensions are analyzed by setting the parameters and initial value. More interestingly, antimonotonicity (named reverse period-doubling bifurcation) and coexisting bifurcations are also reported. Finally, the results of theoretical analyses may be verified by electric experimental.

Bifurcations and chaos in two-coupled periodically driven four-well Duffing-van der Pol oscillators

Bifurcations and chaos in two-coupled periodically driven four-well Duffing-van der Pol (DVP) oscillators are numerically investigated as a function of the strength (δ) of nonlinear coupling and the amplitude (f) of the periodic driving force. The influence of weak coupling (δ ≪ 1) and strong coupling (δ ≫ 1) is analyzed with and without the external periodic force. For strong coupling (δ ≫ 1), the system shows completely chaotic behaviour but for weak coupling the system exhibits period-doubling and period-tripling routes to chaos, quasiperiodic orbit, reverse period-doubling bifurcation, periodic windows, chaotic orbit and intermittent behaviours for specific set of values of the parameters. The effect of phase shift (ϕ) of the system is also analyzed. Numerical results are demonstrated by constructing the bifurcation diagram, phase portrait and Poincare´ map.

Dynamics of a cracked rotor system with oil-film force in parameter space

An organization structure of global oscillation with respect to a cracked rotor system with oil-film force is investigated in this paper. We profit from GPU cluster parallel computing to present a number of high-quality phase diagrams, and exhibit global dynamic characteristics of the system. An interesting scenario, “eye” of chaos, is discovered in this cracked rotor system, emerging as the accumulation limit of forward and reverse period-doubling bifurcation cascades. In this system, it is a common phenomenon that the vibration response of the rotor presents three typical characteristics in parameter space with the rotation speed increasing. Moreover, these phase diagrams assist us to identify multi-attractor coexisting that makes the dynamics behavior of this system become more enrich and complex. These results we represent get us better to understand the nonlinear response of the cracked rotor system and are beneficial to control and diagnose the crack.

Effects of system parameter and fractional order on dynamic behavior evolution in fractional-order Genesio-Tesi system

The object of this paper is to reveal the relationship between dynamics of the fractional-order Genesio-Tesi system and its parameter, especially the order of fractional derivatives. At first, the transcritical bifurcation is carried out based on the stability analysis of equilibrium points. Then, effects of system parameter and fractional order on the dynamics have been investigated deeply and systematically. For commensurate fractional-order system, period-doubling bifurcation, reverse period-doubling bifurcation, period window, transient chaos and steady chaos are found with parameter and fractional order varying simultaneously. For incommensurate fractional-order system, there also exists the route to chaos by period-doubling bifurcation. Furthermore, the effect of different fractional orders on dynamical behavior evolution has been compared. These new findings contribute to successful selecting the most efficient control function, which has been validated by some numerical tools.

Analysis on Chaotic Vibrations of the Magneto-Rheological Suspension System

The magneto-rheological(MR) suspension system has been established, by employing the modified Bouc-wen force-velocity (F-v) model of magneto-rheological damper(MRD). The possibility of chaotic motions in MR suspension is discovered by employing nonlinear systems stability theory. With the bifurcation diagram and corresponding Lyapunov exponent spectrum diagrams detected through numerical calculation, we can observe the complex dynamical behaviors and oscillating mechanism of alternating periodic oscillations, quasi-periodic oscillations and chaotic oscillations with different profiles of road excitation, as well as the dynamical evolution to chaos by period-doubling bifurcations, saddle-node bifurcations and reverse period-doubling bifurcations.

Bifurcations of periodic solutions and chaos in Josephson system with parametric excitation

Josephson system with parametric excitation is investigated. Using second-order averaging method and Melnikov function, we analyze the existence and bifurcations for harmonic, (2, 3, n-order) subharmonics and (2, 3-order) superharmonics and the heterocilinic and homoclinic bifurcations for chaos under periodic perturbation. Using numerical simulation, we check our theoretical analysis and further study the effect of the parameters on dynamics. We find the complex dynamics, including the jumping behaviors, symmetry-breaking, chaos converting to periodic orbits, interior crisis, non-attracting chaotic set, interlocking (reverse) period-doubling bifurcations from periodic orbits, the processes from interlocking period-doubling bifurcations of periodic orbits to chaos after strange non-chaotic motions when the parameter β increases, etc.

Effect of Different Shape of Periodic Forces on Chaotic Oscillation in Brusselator Chemical System

Bifurcations of periodic orbits and chaos in the Brusselator chemical reaction with different shape of external periodic forces are studied numerically. The external periodic forces considered are sine wave, square-wave, rectified sine wave, symmetric saw-tooth wave, asymmetric saw-tooth wave and modulus of sine wave. Period doubling bifurcations, chaos, reverse period doubling bifurcations, quasiperiodic bifurcations are found to occur due to the applied forces. Parametric regimes where suppression of chaos occurs are predicted. Numerical investigations including bifurcation diagram, maximal Lyapunov exponent, Poincare map, Phase portrait and time series plot are used to detect chaos, quasiperiodic and periodic orbits.

Nonlinear Dynamics Analysis of the Semiactive Suspension System with Magneto-Rheological Damper

This paper examines dynamical behavior of a nonlinear oscillator which models a quarter-car forced by the road profile. The magneto-rheological (MR) suspension system has been established, by employing the modified Bouc-Wen force-velocity (F-v) model of magneto-rheological damper (MRD). The possibility of chaotic motions in MR suspension is discovered by employing the method of nonlinear stability analysis. With the bifurcation diagrams and corresponding Lyapunov exponent (LE) spectrum diagrams detected through numerical calculation, we can observe the complex dynamical behaviors and oscillating mechanism of alternating periodic oscillations, quasiperiodic oscillations, and chaotic oscillations with different profiles of road excitation, as well as the dynamical evolutions to chaos through period-doubling bifurcations, saddle-node bifurcations, and reverse period-doubling bifurcations.

Bifurcations and Chaos of Duffing–van der Pol Equation with Nonsymmetric Nonlinear Restoring and Two External Forcing Terms

Bifurcations and chaos of Duffing–van der Pol equation with nonsymmetric nonlinear restoring and two external forcing terms are investigated. The threshold values of the existence of chaotic motion are obtained under periodic perturbation. By the second-order averaging method, we prove the criteria of the existence of chaos in an averaged system under quasi-periodic perturbation for ω2 = nω1 + εσ, n = 1, 2, 3, 5, and cannot prove the criterion of existence of chaos in an averaged system under quasi-periodic perturbation for ω2 = nω1 + εσ, n = 4, 6, 7, …, where σ is not rational to ω1, but can show the occurrence of chaos in the original system by numerical simulation. Numerical simulation including homoclinic or heteroclinic bifurcation surfaces, bifurcation diagrams, maximal Lyapunov exponents, phase portraits and Poincaré maps, not only show the consistence with the theoretical analysis but also exhibit more new complex dynamical behaviors. We show that cascades of interlocking period-doubling and reverse period-doubling bifurcations lead to interleaving occurrence of chaotic behaviors and quasi-periodic orbits, symmetry-breaking of periodic orbits in chaotic regions, onset of chaos occurring more than once, chaos suddenly disappearing to periodic orbits, strange nonchaotic attractor, nonattracting chaotic set and nice chaotic attractors.

BIFURCATIONS OF PERIODIC SOLUTIONS AND CHAOS IN DUFFING-VAN DER POL EQUATION WITH ONE EXTERNAL FORCING

The Duffing-Van der Pol equation withfifth nonlinear-restoring force and one external forcing term isinvestigated in detail: the existence and bifurcations of harmonicand second-order subharmonic, and third-order subharmonic,third-order superharmonic and $m$-order subharmonic under smallperturbations are obtained by using second-order averaging methodand subharmonic Melnikov function; the threshold values of existenceof chaotic motion are obtained by using Melnikov method. Thenumerical simulation results including the influences of periodicand quasi-periodic and all parameters exhibit more new complexdynamical behaviors. We show that the reverse period-doublingbifurcation to chaos, period-doubling bifurcation to chaos,quasi-periodic orbits route to chaos, onset of chaos, and chaossuddenly disappearing, and chaos suddenly converting to periodorbits, different chaotic regions with a great abundance of periodicwindows (periods:1,2,3,4,5,7,9,10,13,15,17,19,21,25,29,31,37,41, andso on), and more wide period-one window, and varied chaoticattractors including small size and maximum Lyapunov exponentapproximate to zero but positive, and the symmetry-breaking ofperiodic orbits. In particular, the system can leave chaotic regionto periodic motion by adjusting the parameters $p, \beta, \gamma, f$and $\omega$, which can be considered as a control strategy.

COMPLEX DYNAMICS IN A PENDULUM EQUATION WITH A PHASE SHIFT

Pendulum equation with a phase shift, parametric and external excitations is investigated in detail. By applying Melnikov's method, we prove the criteria of existence of chaos under periodic perturbation. Numerical simulations, including bifurcation diagrams of fixed points, bifurcation diagrams of the system in three- and two-dimensional spaces, homoclinic and heteroclinic bifurcation surfaces, Maximum Lyapunov exponents (ML), Fractal Dimension (FD), phase portraits, Poincaré maps are plotted to illustrate the theoretical analysis, and to expose the complex dynamical behaviors including the onset of chaos, sudden conversion of chaos to period orbits, interior crisis, periodic orbits, the symmetry-breaking of periodic orbits, jumping behaviors of periodic orbits, new chaotic attractors including two-three-four-five-six-eight-band chaotic attractors, nonchaotic attractors, period-doubling bifurcations from period-1, 2, 3 and 5 to chaos, reverse period-doubling bifurcations from period-3 and 5 to chaos, and so on.By applying the second-order averaging method and Melnikov's method, we obtain the criteria of existence of chaos in an averaged system under quasi-periodic perturbation for Ω = nω + ϵν, n = 1, 2, 4, but cannot prove the criteria of existence of chaos in the averaged system under quasi-periodic perturbation for Ω = nω + ϵν, n = 3, 5 – 15, by Melnikov's method, where ν is not rational to ω. By using numerical simulation, we have verified our theoretical analysis and studied the effect of parameters of the original system on the dynamical behaviors generated under quasi-periodic perturbations, such as the onset of chaos, jumping behaviors of quasi-periodic orbits, interleaving occurrence of chaotic behaviors and nonchaotic behaviors, interior crisis, quasi-periodic orbits to chaotic attractors, sudden conversion of chaos to quasi-periodic behaviors, nonchaotic attractors, and so on. However, we did not find period-doubling and reverse period-doubling bifurcations. We found that the dynamical behaviors under quasi-periodic perturbations are different from that under periodic perturbations, and the dynamics with a phase shift are different from the dynamics without phase shift.

New Period-Doubling and Equiperiod Bifurcations of the Reversible Area-Preserving Map

Two types of period-doubling and equiperiod bifurcations of the reversible areapreserving map are studied. Ordinary period-doubling bifurcation means that the eigenvalue of the mother elliptic periodic orbit (u) is −1, u becomes a saddle periodic orbit with reflection, and an elliptic daughter periodic orbit (v) appears, where the period of v is twice that of u. The other period-doubling bifurcation named the reverse period-doubling bifurcation means that the eigenvalue of the mother saddle periodic orbit with reflection (u′) is −1, u′ becomes an elliptic orbit, and a daughter periodic orbit (v′) appears, where the period of v′ is twice that of u′. The daughter periodic orbit is a saddle with reflection. We prove that both the daughters v and v′ exist in the reversible Smale horseshoe. The forcing relation of the ordinary and reverse period-bifurcations is obtained. Similarly, the ordinary equiperiod and reverse equiperiod bifurcations are also discussed.