Quasi-periodic Orbits Research Articles

Multiple Delayed Feedback Control of Discrete-time Quasi-periodic Orbits

The multiple delayed feedback control (MDFC) is proposed for stabilizing discrete-time quasi-periodic orbits (QPOs). There is an inevitable difference between the current and past states in QPOs. In the MDFC, we estimate the current state by the interpolation method using the multiple past states. We apply the MDFC to the sampled-data control of a continuous-time periodic orbit, because the sampled time series is quasi-periodic if the sampling duration is rationally independent of the period of the periodic orbit.

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.

Classification of Bifurcations of Quasi-Periodic Solutions Using Lyapunov Bundles

In continuous-time dynamical systems, a periodic orbit becomes a fixed point on a certain Poincaré section. The eigenvalues of the Jacobian matrix at this fixed point determine the local stability of the periodic orbit. Analogously, a quasi-periodic orbit (2-torus) becomes an invariant closed curve (ICC) on a Poincaré section. From the Lyapunov exponents of an ICC, we can determine the time average of the exponential divergence rate of the orbit, which corresponds to the eigenvalues of a fixed point. We denote the Lyapunov exponent with the smallest nonzero absolute value as the Dominant Lyapunov Exponent (DLE). A local bifurcation manifests as a crossing or touch of the DLE locus with zero. However, the type of bifurcation cannot be determined from the DLE. To overcome this problem, we define the Dominant Lyapunov Bundle (DLB), which corresponds to the dominant eigenvectors of a fixed point. We prove that the DLB of a 1-torus in a map can be classified into four types: A+(annulus and orientation preserving), A-(annulus and orientation reversing), M (Möbius band), and F (focus). The DLB of a 2-torus in a flow can be classified into three types: A+× A+, A-× M (equivalently M × A-and M × M), and F × F. From the results, we conjecture the possible local bifurcations in both cases. For the 1-torus in a map, we conjecture that type A+and A-DLBs correspond to a saddle-node and period-doubling bifurcations, respectively, whereas a type M DLB denotes a double-covering bifurcation, and type F relates to a Neimark–Sacker bifurcation. Similarly, for the 2-torus in a flow, we conjecture that type A+× A+DLBs correspond to saddle-node bifurcations, type A-× M DLBs to double-covering bifurcations, and type F × F DLBs to the Neimark–Sacker bifurcations. After introducing the mathematical concepts, we provide a DLB-calculating algorithm and illustrate all of the above bifurcations by examples.

Versatile low-Reynolds-number swimmer with three-dimensional maneuverability.

We design and simulate the motion of a swimmer, the Quadroar, with three-dimensional translation and reorientation capabilities in low-Reynolds-number conditions. The Quadroar is composed of an I-shaped frame whose body link is a simple linear actuator and four disks that can rotate about the axes of flange links. The time symmetry is broken by a combination of disk rotations and the one-dimensional expansion or contraction of the body link. The Quadroar propels on forward and transverse straight lines and performs full three-dimensional reorientation maneuvers, which enable it to swim along arbitrary trajectories. We find continuous operation modes that propel the swimmer on planar and three-dimensional periodic and quasiperiodic orbits. Precessing quasiperiodic orbits consist of slow lingering phases with cardioid or multiloop turns followed by directional propulsive phases. Quasiperiodic orbits allow the swimmer to access large parts of its neighboring space without using complex control strategies. We also discuss the feasibility of fabricating a nanoscale Quadroar by photoactive molecular rotors.

The phase space of boxy–peanut and X-shaped bulges in galaxies – I. Properties of non-periodic orbits

The investigation of the phase-space properties of structures encountered in a dynamical system is essential for understanding their formation and enhancement. In the present paper we explore the phase space in energy intervals where we have orbits that act as building blocks for boxy-peanut (b/p) and "{\sf X}-shaped" structures in rotating potentials of galactic type. We underline the significance of the rotational tori around the 3D families x1v1 and x1v1$^{\prime}$ that have been bifurcated from the planar x1 family. These tori play a multiple role: (i) They belong to quasi-periodic orbits that reinforce the local density. (ii) They act as obstacles for the diffusion of chaotic orbits and (iii) they attract a large number of chaotic orbits that become sticky to them. There are also bifurcations of unstable families (x1v2, x1v2$^{\prime}$). Their unstable asymptotic curves wind around the x1v1 and x1v1$^{\prime}$ tori generating orbits with hybrid morphologies between that of x1v1 and x1v2. In addition, a new family of multiplicity 2, called x1mul2, is found to be important for the peanut construction. Our work shows also that there are peanut-supporting orbits before the vertical ILR. Non-periodic orbits associated with the x1 family secure this contribution as well as the support of b/p structures at several other energy intervals. Non-linear phenomena associated with complex instability of single and double multiplicity families of periodic orbits show that these structures are not interrupted in regions where such orbits prevail. Depending on the main mechanism behind their formation, boxy bulges exhibit different morphological features. Finally our analysis indicates that "X" features shaped by orbits in the neighbourhood of x1v1 and x1v1$^{\prime}$ periodic orbits are pronounced only in side-on or nearly end-on views of the bar.

Complex dynamical behaviors in a discrete eco-epidemiological model with disease in prey

In this study, the different dynamical behaviors caused by different parameters of a discrete-time eco-epidemiological model with disease in prey are discussed in ecological perspective. The results indicate that when we choose the same parameters and initial value and only vary the key parameters there appears a series of dynamical behaviors. For example, only varying the death rate of the infected prey (the carrying capacity of the environment for the prey population or the transmission coefficient), there appear chaos, Hopf (flip) bifurcation, local stability, flip (Hopf) bifurcation, and chaos; when only varying the predation coefficient there appear chaos, Hopf bifurcation, local stability, Hopf bifurcation, and chaos. These results are far richer than the corresponding continuous-time model and are rarely seen in previous works. Numerical simulations not only illustrate our results but also exhibit complex dynamical behaviors, such as period-doubling bifurcation in period-2,4,8, quasi-periodic orbits, 3,5,11,16-period orbits and chaotic sets. Moreover, the numerical simulations imply that when the death rate of the infected prey reaches a fixed value the disease dies out. Also, when the predation coefficient parameter reaches some value the disease dies out. These findings indicate that it is practicable to control the disease transmitting in prey by changing the death rate of the infected prey and the predation coefficient parameter.

Delayed feedback control and phase reduction of unstable quasi-periodic orbits.

The delayed feedback control (DFC) is applied to stabilize unstable quasi-periodic orbits (QPOs) in discrete-time systems. The feedback input is given by the difference between the current state and a time-delayed state in the DFC. However, there is an inevitable time-delay mismatch in QPOs. To evaluate the influence of the time-delay mismatch on the DFC, we propose a phase reduction method for QPOs and construct a phase response curve (PRC) from unstable QPOs directly. Using the PRC, we estimate the rotation number of QPO stabilized by the DFC. We show that the orbit of the DFC is consistent with the unstable QPO perturbed by a small state difference resulting from the time-delay mismatch, implying that the DFC can certainly stabilize the unstable QPO.

Hyperchaos and bifurcations in a driven Van der Pol–Duffing oscillator circuit

We investigate the dynamics of a driven Van der Pol–Duffing oscillator circuit and show the existence of higher-dimensional chaotic orbits (or hyperchaos), transient chaos, strange-nonchaotic attractors, as well as quasiperiodic orbits born from Hopf bifurcating orbits. By computing all the Lyapunov exponent spectra, scanning a wide range of the driving frequency and driving amplitude parameter space, we explore in two-parameter space the regimes of different dynamical behaviours.

Complex dynamic behavior of a discrete-time predator-prey system of Holling-III type

In this paper, we investigate the dynamics of a discrete-time predator-prey system of Holling-III type in the closed first quadrant . Firstly, the existence and stability of fixed points of the system is discussed. Secondly, it is shown that the system undergoes a flip bifurcation and a Neimark-Sacker bifurcation in the interior of by using bifurcation theory. Finally, numerical simulations including bifurcation diagrams, phase portraits, and maximum Lyapunov exponents are presented not only to explain our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-6, -7, -9, -15, -16, -22, -23, -32, -35 orbits, a cascade of period-doubling bifurcations in period-2, -4, -8, -16 orbits, quasi-periodic orbits, and chaotic sets. MSC: 37G05, 37G35, 39A28, 39A33.

Discrete prey-predator model with Beddington-DeAngelis functional response: simple vs. complex dynamics

In this paper the dynamics of a discrete-time prey-predator system is investigated in the closed first quadrant . The existence and stability of fixed points are analyzed algebraically .The conditions of existence for flip bifurcation is derived by busing center manifold theorem and bifurcation theory. Numerical simulations not only illustrate our results but also exhibit complex dynamical behaviors of the model, such as the periodic-doubling bifurcation in periods 2,4 and 8 and quasi-periodic orbits and chaotic sets. Keywords : Discrete Model, Beddington-Deangelis Functional Response, Stability, Flip Bifurcation, Center Manifold Theorem, Numerical Simulation.

On hyperchaos in a small memristive neural network

This paper studies a small Hopfield neural network with a memristive synaptic weight. We show that the previous stable network after one weight replaced by a memristor can exhibit rich complex dynamics, such as quasi-periodic orbits, chaos, and hyperchaos, which suggests that the memristor is crucial to the behaviors of neural networks and may play a significant role. We also prove the existence of a saddle periodic orbit, and then present computer-assisted verification of hyperchaos through a homoclinic intersection of the stable and unstable manifolds, which gives a positive answer to an interesting question that whether a 4D memristive system with a line of equilibria can demonstrate hyperchaos.

Detecting Rotating Stall Precursors in Axial Compressors via Perturbations Part 1: Theory

The nonlinear dynamics of fluid instabilities such as rotating stall and surge in axial compressors are typically modeled as subcritical Hopf bifurcations with hysteresis. In this paper, nonlinear dynamical systems with subcritical Hopf bifurcations under periodic forcing are analyzed, and the theoretic results are applied to prediction of rotating stall precursors. Depending on the forcing amplitude and frequency, there are possibly two types of features for the periodic solutions as the bifurcation parameter approaches the critical values. For the first type, the sensitivity of the amplitude of the periodic orbits to the bifurcation parameter becomes infinitely large while the amplitude itself remains finite, which corresponds to the double-cycle bifurcation of the periodic orbits, whereas, in the second case, though the sensitivity is not as steep, there are secondary resonant peaks in the power spectrum in addition to those of the multiples of periodic orbits. These secondary resonant spikes correspond to the Neimark–Sacker bifurcation of periodic orbits, which typically leads to quasi-periodic orbits. These two types of prominent features can serve as precursors to subcritical Hopf bifurcations. The theoretic results are applied to the Moore–Greizter model for rotating stall in axial compressors. In the second part of the paper, experiments are carried out on a single-stage low-pressure axial compressor with inlet circumferential distortions, and it is demonstrated that the theory can yield identification of rotating stall precursors much earlier than the case without distortions.

Quasi-periodic solutions of the spatial lunar three-body problem

In this paper, we consider the spatial lunar three-body problem in which one body is far away from the other two. By applying a well-adapted version of KAM theorem to Lidov–Ziglin’s global study of the quadrupolar approximation of the spatial lunar three-body problem, we establish the existence of several families of quasi-periodic orbits in the spatial lunar three-body problem.

Long-Term Station Keeping of Space Station in Lunar Halo Orbits

Optimization methods are presented for long-term station keeping of a low-thrust space station in lunar halo orbits for unstable collinear libration points. The methods use discretization of each half-revolution into segments and a set of pseudoimpulses for each segment. A matrix inequality of the sum of the characteristic velocities for the pseudoimpulses is used to transform the problem into a large-scale linear programming form. Terminal constraints are presented as a linear matrix equation using numerical partial derivatives along a reference orbit. Possible station-keeping strategies for quasi-periodic orbits in the vicinity of the Earth–moon collinear libration point are considered. An iterative shooting algorithm, based on the full ephemeris model and differential correction, is used. Preliminary simulation and comparative results of long-term station-keeping strategies for one year are presented. As application examples, two quasi-periodic orbits are considered. The first is similar to a planar Lyapunov orbit. The second is a halo orbit, with continuous visibility of the space station from the Earth. An algorithm for very low-thrust station keeping with long-duration burns is also presented. The algorithm can be used to estimate the required thrust-to-weight ratio of the space station.

A dynamical system with a strange attractor and invariant tori

Abstract This paper describes a simple three-dimensional time-reversible system of ODEs with quadratic nonlinearities and the unusual property that it is exhibits conservative behavior for some initial conditions and dissipative behavior for others. The conservative regime has quasi-periodic orbits whose amplitude depend on the initial conditions, while the dissipative regime is chaotic. Thus a strange attractor coexists with an infinite set of nested invariant tori in the state space.

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.

Dynamical analysis and chaos control of a discrete SIS epidemic model

The dynamical behaviors of a discrete-time SIS epidemic model are investigated in this paper. The result indicates that the model undergoes a flip bifurcation and a Hopf bifurcation, as found by using the center manifold theorem and bifurcation theory. Numerical simulations not only illustrate our results, but they also exhibit the complex dynamical behaviors, such as the period-doubling bifurcation in period-2, -4, -8, quasi-periodic orbits and the chaotic sets. Specifically, when the parameters A, , , r, λ are fixed at some values and the bifurcation parameter h changes with different values, there exist local stability, Hopf bifurcation, 3-periodic orbits, 7-periodic orbits, period-doubling bifurcation and chaotic sets. These results reveal far richer dynamical behaviors of the discrete epidemic model compared with the continuous epidemic models although the discrete epidemic model is simple. Finally, the feedback control method is used to stabilize chaotic orbits at an unstable endemic equilibrium.

The Stark problem in the Weierstrassian formalism

We present a new general, complete closed-form solution of the Stark problem in terms of Weierstrass elliptic and related functions. With respect to previous treatments of the problem, our analysis is exact and valid for all values of the external force field, and it is expressed via unique formul\ae{} valid for all initial conditions and parameters of the system. The simple form of the solution allows us to perform a thorough investigation of the properties of the dynamical system, including the identification of quasi-periodic and periodic orbits, the formulation of a simple analytical criterion to determine the boundness of the trajectory, and the characterisation of the equilibrium points.

High-order solutions around triangular libration points in the elliptic restricted three-body problem and applications to low energy transfers

High-order series expansions around triangular libration points in the elliptic restricted three-body problem (ERTBP) are constructed first, and then with the aid of the series solutions, two-impulse and low-thrust low energy transfers to the triangular point orbits of the Earth–Moon system are designed in this paper. The equations of motion of ERTBP in the pulsating synodic reference frame have the same symmetries as the ones in circular restricted three-body problem (CRTBP), and also have five equilibrium points. Considering the stable dynamics of triangular libration points, the analytical solutions of the motion around them in ERTBP are expressed as formal series of four amplitudes: the orbital eccentricity of the primary, the long, short and vertical periodic amplitudes. The series expansions truncated at arbitrary order are constructed by means of Lindstedt–Poincaré method, and then the quasi-periodic orbits around triangular libration points in ERTBP can all be parameterized. In particular, when the eccentricity of the primary is zero, the series expansions constructed can be reduced to describe the motion around triangular libration points in CRTBP. In order to check the validity of the series expansions constructed, the domain of convergence corresponding to different orders is studied by using numerical integration. After obtaining the analytical solutions of the bounded orbits around triangular points, the target orbits in practical missions can be expressed by several related parameters. Thanks to the series expansions constructed, two missions are planned to transfer a spacecraft from the Earth to the short periodic orbits around triangular libration points of Earth–Moon system. To complete the missions with less fuel cost, low energy transfers (two-impulse and low-thrust) are investigated by means of numerical optimization methods (both global and local optimization techniques). Simulation results indicate that (a) the low-thrust, low energy transfers outperform the corresponding two-impulse, low energy transfers in terms of propellant fraction, and (b) compared with the traditional Hohmann-like transfers, both the two-impulse, low energy and low-thrust, low energy transfers perform very efficiently, at the cost of flight time.

Quasi-Periodic Frequency Analysis Using Averaging-Extrapolation Methods

We present a new approach to the numerical computation of the basic frequencies of a quasi-periodic signal. Although a complete toolkit for frequency analysis is presented, our methodology is better understood as a refinement process for any of the frequencies, provided we have a rough approximation of the frequency that we wish to compute. The cornerstone of this work is a recently developed method for the computation of Diophantine rotation numbers of circle diffeomorphisms, based on suitable averages of the iterates and Richardson extrapolation. This methodology was successfully extended to compute rotation numbers of quasi-periodic invariant curves of planar maps. In this paper, we address the case of a signal with an arbitrary number of frequencies. The most outstanding aspect of our approach is that frequencies can be calculated with high accuracy at a moderate computational cost, without simultaneously computing the Fourier representation of the signal. The method consists in the construction of a new quasi-periodic signal by appropriate averages of phase-shifted iterates of the original signal. This allows us to define a quasi-periodic orbit on the circle in such a way that the target frequency is the rotation frequency of the iterates. This orbit is well suited for the application of the aforementioned averaging-extrapolation methodology for computing rotation numbers. We illustrate the presented methodology with the study of the vicinity of the Lagrangian equilibrium points of the restricted three body problem (RTBP), and we consider the effect of additional planets using a multicircular model.