Quasi-periodic Behavior Research Articles

Stationkeeping Analysis for Solar Sailing the Region of Binary Asteroid Systems

Asteroid exploration may largely benefit from solar-sail propulsion that is coupled with satellite miniaturization. There remain, however, several open questions on sail dynamics in the vicinity of small bodies. One such question is the feasibility of solar-sail missions within binary asteroid systems from the perspective of orbit maintenance. Although sail transfer trajectories to small bodies are often discussed, more work is necessary to understand sail stationkeeping dynamics in such environments. This work focuses on solar-sail stationkeeping in the region of binary asteroids within a modified elliptic restricted three-body problem. Quasi-periodic orbits and Kolmogorov–Arnold–Moser theory are employed to construct reference solutions that are naturally bounded and therefore facilitate orbit maintenance. Reference quasi-periodic behaviors may reflect the characteristics of motion within higher-fidelity models more accurately than purely periodic or equilibrium baseline solutions. Then, a turn-and-hold stationkeeping strategy is implemented to offset orbit determination errors, and stationkeeping parameters are investigated. Using the Didymos binary asteroid system as a representative case study, dynamical reference structures for sail trajectories are explored. Additionally, conditions for their controllability are also discussed. Insight from this analysis is applicable to other binary asteroid systems.

Permutation entropy and Lyapunov exponent: Detecting and monitoring the chaotic edge of a closed planar under-actuated system

Abstract Generally, under-actuated mechanism possesses a chaotic and quasi-periodic behavior. The edge of chaos is the transitory region between deterministic and chaotic motion, and the main concern of the design engineers is to acquire the region by changing the parameters of the dynamical system at the edge of chaos. To be different with many existed researches focused on the dynamic analysis of clearance joints, the closed under-actuated systems is analyzed in this paper while very limited study has been done on this subject before. A typical under-actuated sieve is used as a demonstration case in which the driving speed and the length of the link can be changed for the purpose of keeping chaotic motion, as it is good for the work efficiency of sieve machine. The Lagrange equation of the first kind is used to build the numerical model for the sieve system, permutation entropy is employed to detect dynamic change in this complex system, and Lyapunov exponent is applied to judge deterministic and chaotic motion. The equations of the closed under-actuated planar sieve are obtained in this work and the experimental results about the sieve motion have then confirmed that the permutation entropy and Lyapunov exponent provide an effective measure for monitoring and detecting the edge of chaotic motion, which can improve the work efficiency of a planar sieve.

Quasiperiodic behavior in the electrodeposition of Cu/Sn multilayers: extraction of activation energies and wavelet analysis

Living systems are one of the many examples in which self-organizing systems yield more intricate structures than those that can be achieved using a "step-by-step" approach. This phenomenon can be observed in electrochemical organic synthesis, oscillating metal deposition and in chemical clocks. There is a plenitude of temporal instabilities during self-organization, from ordinary period-one oscillations, going through quasiperiodicity, to the onset of chaos. Here, we describe the emergence of quasiperiodic behavior during the oscillatory electro-deposition of Cu/Sn. The time-series were characterized using a continuous wavelet transform in order to extract the oscillation frequency of the process with changes in temperature, and to calculate the apparent activation energies. Two different energy ranges are presented, and these are attributed to an activation barrier (∼50 kJ mol-1) which is closely related to a fast time-scale of the feedback loops responsible for the current oscillations and a diffusional process (∼20 kJ mol-1), connected to the slow modulations of the oscillation amplitude, giving rise to quasiperiodic dynamics. This kinetic information might provide information on the self-organized synthesis of Cu/Sn metallic multilayers which are kept far from the thermodynamic equilibrium. As self-organization in chemical systems is rapidly developing into a powerful strategy for designing new functional materials, the availability of kinetic parameters is of major interest in rational design.

Design of a New Seven-Dimensional Hyperchaotic Circuit and Its Application in Secure Communication

In the fields of physics, mathematics, communication, control, biology, etc., chaos is widely used, and it plays an important role in modern science and technology. In this paper, a new seven-dimensional hyperchaotic system is proposed and some related features of the hyperchaotic system are analyzed, including equilibrium stability, dissipation, bifurcation and Lyapunov exponent. The results show that the proposed hyperchaotic system exhibits hyperchaotic, chaotic, quasi-periodic and periodic dynamic behaviors. The seven-dimensional hyperchaotic system is implemented through experiments, and the phase diagrams of the circuit simulation are consistent with those of the numerical simulation, which verifies the physical realization of the hyperchaotic system. In view of chaos in some important fields, such as communication security and complex systems, the seven-dimensional hyperchaotic system is applied to signal encryption and decryption circuit design. Firstly, the proposed seven-dimensional hyperchaotic system is scaled and transformed into a hyperchaotic system that is easy to implement. Then an improved chaotic circuit module design is adopted. Finally, the seven-dimensional hyperchaotic circuit is designed to form a hyperchaotic secure communication circuit. The electronic circuit realizes the confidential communication and its experimental results prove the effectiveness of the designed scheme.

A hierarchy of long wave-short wave type equations: quasi-periodic behavior of solutions and their representation

Based on the Lenard recursion relation and the zero-curvature equation, we derive a hierarchy of long wave-short wave type equations associated with the 3 × 3 matrix spectral problem with three potentials. Resorting to the characteristic polynomial of the Lax matrix, a trigonal curve is defined, on which the Baker-Akhiezer function and two meromorphic functions are introduced. Analyzing some properties of the meromorphic functions, including asymptotic expansions at infinite points, we obtain the essential singularities and divisor of the Baker-Akhiezer function. Utilizing the theory of algebraic curves, quasi-periodic solutions for the entire hierarchy are finally derived in terms of the Riemann theta function.

An inclined cable excited by a non-ideal massive moving deck: an asymptotic formulation

A moving deck is an important (kinematic) excitation source for the inclined cable in cable-stayed bridges. In ideal cases, the deck motion is assumed to be harmonic oscillation and cable’s dynamic effects on the deck are neglected. As a refined version, an inclined cable excited by a massive non-ideal moving deck, i.e., the deck’s oscillation, is slowly modulated by the cable and thus not exactly harmonic is investigated in an asymptotically coupled formulation for understanding cable–deck dynamic interactions. More explicitly, by ordering the deck/cable mass ratio as a large parameter, the coupled system is reduced using asymptotic approximations and multi-scale expansions. After neglecting the reduced model’s nonlinear terms, firstly, cable–deck linear coupled modes are obtained, leading to two different kinds of linear modal dynamics, i.e., the cable-dominated one and the deck-dominated one, whose asymptotic characteristics are also revealed. Then cable’s forced nonlinear vibrations, excited by the deck’s modulated oscillation (i.e., non-ideal moving deck), are fully investigated. Nonlinear frequency responses of the cable–deck coupled system are found, and the dynamic effects on the cable’s periodic and quasi-periodic behaviors, due to cable–deck coupling (characterized by the deck/cable mass ratio), cable’s inclinations, and boundary damping, are also presented.

Initial conditions-related dynamical behaviors in PI-type memristor emulator-based canonical Chua’s circuit

PurposeThe purpose of this paper is to construct a proportion-integral-type (PI-type) memristor, which is different from that of the previous memristor emulator, but the constructing memristive chaotic circuit possesses line equilibrium, leading to the emergence of the initial conditions-related dynamical behaviors.Design/methodology/approachThis paper presents a PI-type memristor emulator-based canonical Chua’s chaotic circuit. With the established mathematical model, the stability region for the line equilibrium is derived, which mainly consists of stable and unstable regions, leading to the emergence of bi-stability because of the appearance of a memristor. Initial conditions-related dynamical behaviors are investigated by some numerically simulated methods, such as phase plane orbit, bifurcation diagram, Lyapunov exponent spectrum, basin of the attraction and 0-1 test. Additionally, PSIM circuit simulations are executed and the seized results validate complex dynamical behaviors in the proposed memristive circuit.FindingsThe system exhibits the bi-stability phenomenon and demonstrates complex initial conditions-related bifurcation behaviors with the variation of system parameters, which leads to the occurrence of the hyperchaos, chaos, quasi-periodic and period behaviors in the proposed circuit.Originality/valueThese memristor emulators are simple and easy to physically fabricate, which have been increasingly used for experimentally demonstrating some interesting and striking dynamical behaviors in the memristor-based circuits and systems.

Characterization of a 3DOF aeroelastic system with freeplay and aerodynamic nonlinearities – Part I: Higher-order spectra

Abstract The identification of nonlinear systems in aeroelasticity poses a significant challenge for practitioners, often hampered by the complex nature of aeroelastic response data which may contain multiple forms of nonlinearity. Characterizing and quantifying nonlinearities is further hampered when the response is obtained at a location which is away from the nonlinear source and/or the response is contaminated by noise. In the present paper, a three-degree-of-freedom airfoil with a freeplay nonlinearity located in the control surface and exposed to transonic flow is investigated. In this Part I paper the main form of analysis is via higher-order spectra techniques to unveil features of the nonlinear mechanism which result from i) structural nonlinearities (freeplay) in isolation and ii) freeplay with Euler derived nonlinear inviscid aerodynamic phenomena (transition between Tijdeman Type-A and Type-B shock motion). It is shown that the control surface structural freeplay nonlinearity is characterized by strong cubic phase-coupling between linear and nonlinear modes. On the other hand, nonlinear inviscid flow phenomena are shown to be characterized by quadratic phase-coupling between linear and nonlinear modular modes, the strength of which is related to the strength of the aerodynamic nonlinearity (amplitude of the shock motion). The nonlinear inviscid flow phenomena do not appear to affect the identification of the freeplay nonlinearity. Conjectures are made which address the transition between aperiodic, quasi-periodic and periodic behavior (pre-flutter), further physical support towards these conjectures is provided in Part II [1] . The limitations of the higher-order spectra approach are assessed, in particular, the analysis demonstrates the difficulty in extracting natural frequencies with this approach.

Hopf Bifurcation, Hopf-Hopf Bifurcation, and Period-Doubling Bifurcation in a Four-Species Food Web

Complex dynamics of a four-species food web with two preys, one middle predator, and one top predator are investigated. Via the method of Jacobian matrix, the stability of coexisting equilibrium for all populations is determined. Based on this equilibrium, three bifurcations, i.e., Hopf bifurcation, Hopf-Hopf bifurcation, and period-doubling bifurcation, are analyzed by center manifold theorem, bifurcation theorem, and numerical simulations. We reveal that, influenced by the three bifurcations, the food web can exhibit very complex dynamical behaviors, including limit cycles, quasiperiodic behaviors, chaotic attractors, route to chaos, period-doubling cascade in orbits of period 2, 4, and 8 and period 3, 6, and 12, periodic windows, intermittent period, and chaos crisis. However, the complex dynamics may disappear with the extinction of one of the four populations, which may also lead to collapse of the food web. It suggests that the dynamical complexity and food web stability are determined by the food web structure and existing populations.

Nonlinear dynamics of a flexible rotor on tilting pad journal bearings experiencing rub–impact

Rub–impact phenomenon occurring in hydrodynamic journal bearings is one of the main malfunctions in rotating machines and causes undesirable dynamic behavior. In order to investigate such a phenomenon, nonlinear dynamics due to rub–impact within tilting pad journal bearings supporting a flexible rotor is studied. In simulating rub–impact between the journal and the associated pads, this paper employs mixed lubrication theory along with elasto-plastic asperity contact model between pads and journal. Periodic, quasiperiodic and chaotic vibrational behaviors of system are studied by varying the unbalanced load magnitude and the rotational speed of the rotor as control parameters. Phase plane orbits, waterfall frequency response spectra and bifurcation diagrams are used to show various dynamic responses of rotor when the control parameters are changed. The Poincare maps are used to determine the onset of irregular motions. Presented results provide better understanding of strongly nonlinear vibrations occurring due to rub–impact in TPJBs supporting industrial rotating machines.

Study of a periodically forced magnetohydrodynamic system using Floquet analysis and nonlinear Galerkin modelling

We present a detailed study of Rayleigh–Benard magnetoconvection with periodic gravity modulation and uniform vertical magnetic field. Linear stability analysis is carried out using Floquet theory to construct the stability boundaries in order to estimate the magnitude of forcing amplitude $$\epsilon $$ required for having convection in the system for a fixed Rayleigh number Ra, wave number k and modulating frequency $$\Omega $$ . The effects of varying Prandtl number Pr and Chandrasekhar number Q on the threshold of convection are also investigated. A higher Pr value reduces the value of the threshold, whereas a higher Q value increases it. Bicritical states are also observed at which the minimum forcing amplitude needed for convection to begin occurs at two different k values in harmonic and sub-harmonic regions, respectively. We also construct a nonlinear Galerkin model and compare the results with those obtained from linear stability analysis. Two-dimensional (2D) oscillatory convection is observed at the onset, while quasiperiodic and chaotic behaviours are found at higher Ra values. 2D as well as nonlinear convective flow patterns are observed for primary and higher-order instabilities, respectively. Bifurcation diagrams with respect to different parameters such as $$\epsilon $$ , Ra and Q are provided for thorough understanding of the forced nonlinear system.

Nonlinear dynamics of a viscoelastic sandwich beam with parametric excitations and internal resonance

Nonlinear dynamical behaviors of an axially accelerating viscoelastic sandwich beam subjected to three-to-one internal resonance and parametric excitations resulting from simultaneous velocity and tension fluctuations are investigated. The direct method of multiple scales is adopted to obtain a set of first-order ordinary differential equations and associated boundary conditions. The frequency and amplitude response curves along with their stability and bifurcation are numerically studied. A great number of dynamic behaviors are presented in the form of phase portraits, time traces, Poincare sections, and FFT power spectra. Due to modal interaction, various periodic, quasiperiodic, and chaotic behaviors are displayed, depending on the initial conditions. The largest Lyapunov exponent is carried out to determine the midly chaotic response by the convergent form of exponents. Numerical results show various oscillatory behaviors indicating the influence of internal resonance and coupled effects of fluctuating axial velocity and tension.

Five new 4-D autonomous conservative chaotic systems with various type of non-hyperbolic and lines of equilibria

Very little research is available in the field of 4-D autonomous conservative chaotic systems. This paper presents five new 4-D autonomous conservative chaotic systems having non-hyperbolic equilibria with various characteristics. The proposed systems have different numbers of non-hyperbolic equilibrium points. One of the new systems has four non-hyperbolic equilibria points along with lines of equilibria. Hence, this system may belong to the category of hidden attractors chaotic system. The first, second, fourth and fifth type of the systems exhibit coexistence of chaotic flow, whereas the third type of the system exhibits coexistence of chaotic flows with quasi-periodic behaviour. The chaotic behaviours of the proposed systems are verified by using phase portrait plot, Poincaré map, local Lyapunov spectrum, bifurcation diagram and frequency spectrum plots. The conservative nature of the proposed systems is proved by finding the sum of finite-time local Lyapunov exponents, finite-time local Lyapunov dimensions and divergence of the vector field. The sum of the finite-time local Lyapunov exponents and divergence of the vector field are equal to zero, and local Lyapunov dimension is equal to the order of the system confirm the conservative nature of the new chaotic systems.

On the ionospheric propagation of VLF waves generated by currents in the lower ionosphere

The consistent description of ELF/VLF wave propagation in the ionosphere is essential for understanding the properties of the wave phenomena observed both at the ground and satellites. Due to strong inhomogeneity of the lower ionosphere coupled with collisional damping of a wave, the geometrical optics approach is not valid, while the full-wave approach suffers from the problem of numerical instability. In this paper, we implement a rather general and powerful method avoiding the numerical instability to study propagation of the wave generated in the lower ionosphere. Our calculations demonstrate the quasi-periodic behaviour of the field in the upper ionosphere, which is associated with the Earth-ionosphere waveguide. The field distribution varies significantly with frequency, from being nearly sign-definite to oscillating with comparable positive and negative values, producing the ring structure in wave-vector space. The field distribution at the ground is studied and shown to have resonant structure defined by two parameters: number of the resonance, which depends on wave frequency, and spatial scale of the Earth-ionosphere waveguide.

Variations of the Energy of Free Particles in the pp-Wave Spacetimes

We consider the action of exact plane gravitational waves, or pp-waves, on free particles. The analysis is carried out by investigating the variations of the geodesic trajectories of the particles, before and after the passage of the wave. The initial velocities of the particles are non-vanishing. We evaluate numerically the kinetic energy per unit mass of the free particles and obtain interesting, quasi-periodic behavior of the variations of the kinetic energy with respect to the width λ of the Gaussian that represents the wave. The variation of the energy of the free particle is expected to be exactly minus the variation of the energy of the gravitational field, and therefore provides an estimation of the local variation of the gravitational energy. The investigation is carried out in the context of short bursts of gravitational waves, and of waves described by normalized Gaussians, that yield impulsive waves in a certain limit.

Getting It Right Matters: Climate Spectra and Their Estimation

In many recent publications, climate spectra estimated with different methods from observed, GCM-simulated, and reconstructed time series contain many peaks at time scales from a few years to many decades and even centuries. However, respective spectral estimates obtained with the autoregressive (AR) and multitapering (MTM) methods showed that spectra of climate time series are smooth and contain no evidence of periodic or quasi-periodic behavior. Four order selection criteria for the autoregressive models were studied and proven sufficiently reliable for 25 time series of climate observations at individual locations or spatially averaged at local-to-global scales. As time series of climate observations are short, an alternative reliable nonparametric approach is Thomson’s MTM. These results agree with both the earlier climate spectral analyses and the Markovian stochastic model of climate.

Megastability in a quasi-periodically forced system exhibiting multistability, quasi-periodic behaviour, and its analogue circuit simulation

In this paper, we present a new quasi-periodically forced system which exhibits coexistence of an infinite number of nested attractors (megastability) along with quasi-periodic and multistable behaviours. The dynamical behaviour of the system is analysed using the Lyapunov exponents, Lyapunov spectrum, bifurcation diagrams, and phase plots. An analogue circuit of the proposed system is designed and simulated using the NI Multisim simulation environment to access the unusual dynamical behaviour of the proposed system.

Zitterbewegung in time-reversal Weyl semimetals

We perform a systematic study of the Zitterbewegung effect of fermions, which are described by a Gaussian wave with broken spatial-inversion symmetry in a three-dimensional low-energy Weyl semimetal. Our results show that the motion of fermions near the Weyl points is characterized by rectilinear motion and Zitterbewegung oscillation. The ZB oscillation is affected by the width of the Gaussian wave packet, the position of the Weyl node, and the chirality and anisotropy of the fermions. By introducing a one-dimensional cosine potential, the new generated massless fermions have lower Fermi velocities, which results in a robust relativistic oscillation. Modulating the height and periodicity of periodic potential demonstrates that the ZB effect of fermions in the different Brillouin zones exhibits quasi-periodic behavior. These results may provide an appropriate system for probing the Zitterbewegung effect experimentally.

Three-Dimensional Flow Behavior Inside the Submerged Entry Nozzle

According to various authors, the surface quality of steel depends on the dynamic conditions that occur within the continuous casting mold’s upper region. The meniscus, found in that upper region, is where the solidification process begins. The liquid steel is distributed into the mold through a submerged entry nozzle (SEN). In this paper, the dynamic behavior inside the SEN is analyzed by means of physical experiments and numerical simulations. The particle imaging velocimetry technique was used to obtain the vector field in different planes and three-dimensional flow patterns inside the SEN volume. Moreover, large eddy simulation was performed, and the turbulence model results were used to understand the nonlinear flow pattern inside the SEN. Using scaled physical and numerical models, quasi-periodic behavior was observed due to the interaction of two three-dimensional vortices that move inside the SEN lower region located between the exit ports of the nozzle.

Chaos in a second-order non-autonomous Wien-bridge oscillator without extra nonlinearity

PurposeThe purpose of this paper is to develop a simple chaotic circuit. The circuit can be fabricated by less discrete electronic components, within which complex dynamical behaviors can be generated.Design/methodology/approachA second-order non-autonomous inductor-free chaotic circuit is presented, which is obtained by introducing a sinusoidal voltage stimulus into the classical Wien-bridge oscillator. The proposed circuit only has two dynamic elements, and its nonlinearity is realized by the saturation characteristic of the operational amplifier in the classical Wien-bridge oscillator. After that, its dynamical behaviors are revealed by means of bifurcation diagram, Lyapunov exponent and phase portrait and further confirmed using the 0-1 test method. Moreover, an analog circuit using less discrete electronic components is implemented, and its experimental results are measured to verify the numerical simulations.FindingsThe equilibrium point located in a line segment varies with time evolution, which leads to the occurrence of periodic, quasi-periodic and chaotic behaviors in the proposed circuit.Originality/valueUnlike the previously published works, the significant values of the proposed circuit with simple topology are inductor-free realization and without extra nonlinearity, which make the circuit can be used as a paradigm for academic teaching and experimental illustraction for chaos.