Quasi-periodic Solutions Research Articles

QUASI-PERIODIC SOLUTIONS FOR 1D NONLINEAR WAVE EQUATION

In this paper, one-dimensional (1D) nonlinear wave equation $$u_{tt}-u_{xx}+mu+u^{7}=0$$on the finite $x$-interval $[0,\pi]$ with Dirichlet boundary conditions is considered. It is proved that there are many $3$-dimensional elliptic invariant tori, and thus quasi-periodic solutions for the above equation. This is an extension of the previous work [11] by the same author, where many $2$-dimensional elliptic invariant tori for the above equation are obtained. The proof is based on infinite-dimensional KAM theory and partial Birkhoff normal form.

KAM tori for the two-dimensional completely resonant Schrödinger equation with the general nonlinearity

In this paper, two-dimensional completely resonant Schrödinger equation with the general nonlinearityiut−Δu+|u|2pu=0,x∈T2:=R2/(2πZ)2,t∈R,p∈Z+ under periodic boundary conditions is considered. For any given positive integers p and b, it is obtained that a Whitney smooth family of small-amplitude b-quasi-periodic solutions for the equation by developing an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional Hamiltonian systems. The overall strategy in the proof of the KAM theorem is a normal form techniques sparsing angle-dependent terms, which can be achieved by choosing the appropriate tangential sites. The essence of the normal form is the integrable part of the Hamiltonian system after introducing the action-angle variable, which contains both the linear integrable part of the Schrödinger equation and the integrable part from the nonlinearity. Determining such normal form, in general, is not easy task as it requires some novel ideas and large number of complex calculations. This work includes some new ideas and overcomes some technical difficulties, which solves more completely the existence problem of quasi-periodic solutions of the completely resonant Schrödinger equation on the two dimensional torus with the general nonlinearity.

A comparative study on multi- and variable-coefficient harmonic balance methods for quasi-periodic solutions

Quasi-periodic solutions widely exist in nonlinear dynamical systems, which are the one of the most important dynamical behaviors but much more difficult to obtain full- or semi-analytically than equilibria and periodic solutions. Recently, multi-harmonic balance method (MHBM) and variable-coefficient harmonic balance method (VCHBM) have been proposed to determine the quasi-periodic solutions semi-analytically. In this paper, after a brief review on the two methods, that is, MHBM uses a multi-dimensional Fourier series and VCHBM a variable-coefficient Fourier series, the relationship between them is unveiled particularly for the case of quasi-periodic solutions consisting of two frequency components. The transform matrices between the algebraic equations of the Fourier coefficients in the two methods are derived. Furthermore, a new formulation for alternating Frequency-Time method (AFT) and a phase condition for arc-length continuation method (ALC) in VCHBM are proposed to make the method more efficient and robust. Through the application of two examples, it is found that when the two methods choose equal harmonic order and thus have the same number of unknowns, MHBM demonstrates higher computational efficiency but VCHBM shows more robust and accurate, especially for the quasi-periodic solutions in the nonlinear dynamical systems with a single-frequency excitation. This paper provides insight views into the features of MHBM and VCHBM..

Twice Harmonic Balance Method for Stability and Bifurcation Analysis of Quasi-Periodic Responses

Abstract A twice harmonic balance (THB) method is proposed to compute and analyze quasi-periodic (QP) responses of nonlinear dynamical systems, with emphasis on the stability and bifurcation of QP responses. In the first harmonic balancing, the original system is transformed into a truncated system via harmonic balance method with variable-coefficients. The truncated system is further solved via the second harmonic balancing, more specifically the incremental harmonic balance (IHB) method. The equivalence is addressed between the periodic solutions of the truncated system and the QP responses of the original system. According to the relationship, the presented method is in essence to convert the problem of solving the original system for QP responses into a truncated system for periodic solutions. Numerical examples show that the semi-analytical QP solutions obtained by the THB method are in well consistence with the solutions obtained by the Runge–Kutta (RK) method and the IHB method with two time scales, respectively. More importantly, the stability of the attained QP solutions can be analyzed by just applying the Floquet theory to the periodic response of the truncated system. The continuation of the QP responses is generated by the presented method, on which the possible bifurcations resulted from the stability reversal are analyzed in detail. In addition, the evolution of QP responses can also be tracked from periodic solutions, such as that due to the onset of a Neimark–Sacker bifurcation.

Dynamics of a non-linear Jeffcott rotor in supercritical regime

This work is concerned with the prediction and simulation of limit cycles of a nonlinear Jeffcott rotor. Geometrical nonlinearity (related to large flexural displacements) in both stiffness and internal damping is considered. Classical instabilities due to linear internal damping occurring in the supercritical range are modified by nonlinear effects and lead to oscillating limit cycles at a radius whose value is found analytically as a function of the rotating speed. If the rotor is additionally excited by unbalance, the response spectrum also contains the excitation frequency, and the bifurcation leads to quasi-periodic limit cycles. The dynamics of this system is studied numerically via the harmonic balance method, path continuation and bifurcation tracking. In particular, an original procedure for switching to a branch of quasi-periodic solutions is proposed. To preserve the integrity of the machine post bifurcation, one then considers the possibility of constraining the transverse displacement of the shaft. Such possible rotor–stator interaction generates three-frequency quasi-periodic oscillations. Frequency response curves show that the quasi-periodic solutions created by the bouncing rotor are mainly driven by the friction coefficient.

Algebro-Geometric Quasi-Periodic Solutions to the Bogoyavlensky Lattice 2(3) Equations

Algebro-Geometric Quasi-Periodic Solutions to the Bogoyavlensky Lattice 2(3) Equations

Invariant Tori for a Two-Dimensional Completely Resonant Beam Equation with a Quintic Nonlinear Term

This paper focuses on a two-dimensional completely resonant beam equation with a quintic nonlinear term. This means studying u t t + Δ 2 u + ε f u = 0 , x ∈ T 2 , t ∈ ℝ , under periodic boundary conditions, where ε is a small positive parameter and f u is a real analytic odd function of the form f u = f 5 u 5 + ∑ i ^ ≥ 3 f 2 i ^ + 1 u 2 i ∧ + 1 , f 5 ≠ 0 . It is proved that the equation admits small-amplitude, Whitney smooth, linearly stable quasiperiodic solutions on the phase-flow invariant subspace ℤ † 2 = r = r 1 , r 2 , r 1 ∈ 4 ℤ − 1 , r 2 ∈ 4 ℤ . Firstly, the corresponding Hamiltonian system of the equation is transformed into an angle-dependent block-diagonal normal form by using symplectic transformation, which can be achieved by selecting the appropriate tangential position. Finally, the existence of a class of invariant tori is proved, which implies the existence of quasiperiodic solutions for most values of frequency vector by an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional Hamiltonian systems.

Complex Dynamics and Sliding Bifurcations of the Filippov Lorenz–Chen System

In this paper, we propose a Filippov switching model which is composed of the Lorenz and Chen systems. By employing the qualitative analysis techniques of nonsmooth dynamical systems, we show that the new Filippov system not only inherits the properties of the Lorenz and Chen systems but also presents new dynamics including new chaotic attractors such as four-wing butterfly attractor, Lorenz attractor with sliding segments, etc. In particular, we find that different new attractors can coexist such as the coexistence of two-point attractors and chaotic attractor, the coexistence of two-point attractors and quasi-periodic solution, the coexistence of transient transition chaos and quasi-periodic solution. Furthermore, nonsmooth bifurcations and numerical analyses reveal that the proposed Filippov system has a series of new sliding bifurcations including a symmetric pair of sliding mode bifurcations, a symmetric pair of sliding Hopf bifurcations, and a symmetric pair of Hopf-like boundary equilibrium bifurcations.

Energy Harvesting in a System with a Two-Stage Flexible Cantilever Beam.

The subject of the research contained in this paper is a new design solution for an energy harvesting system resulting from the combination of a quasi-zero-stiffness energy harvester and a two-stage flexible cantilever beam. Numerical tests were divided into two main parts-analysis of the dynamics of the system due to periodic, quasiperiodic, and chaotic solutions and the efficiency of energy generation. The results of numerical simulations were limited to zero initial conditions as they are the natural position of the static equilibrium. The article compares the energy efficiency for the selected range of the dimensionless excitation frequency. For this purpose, three cases of piezoelectric mounting were analyzed-only on the first stage of the beam, on the second and both stages. The analysis has been carried out with the use of diagrams showing difference of the effective values of the voltage induced on the piezoelectric electrodes. The results indicate that for effective energy harvesting, it is advisable to attach piezoelectric energy transducers to each step of the beam despite possible asynchronous vibrations.

The Spin–Spin Problem in Celestial Mechanics

We study the dynamics of two homogeneous rigid ellipsoids subject to their mutual gravitational influence. We assume that the spin axis of each ellipsoid coincides with its shortest physical axis and is perpendicular to the orbital plane. Due to such assumptions, the problem is planar and depends on particular parameters of the ellipsoids, most notably, the equatorial oblateness and the flattening with respect to the shortest physical axes. We consider two models for such configuration: while in the full model, there is a coupling between the orbital and rotational motions, in the Keplerian model, the centers of mass of the bodies are constrained to move on coplanar Keplerian ellipses. The Keplerian case, in the approximation that includes the coupling between the spins of the two ellipsoids, is what we call spin–spin problem, that is a generalization of the classical spin–orbit problem. In this paper we continue the investigations of Misquero (Nonlinearity 34:2191–2219, 2021) on the spin–spin problem by comparing it with the spin–orbit problem and also with the full model. Beside detailing the models associated to the spin–orbit and spin–spin problems, we introduce the notions of standard and balanced resonances, which lead us to investigate the existence of periodic and quasi-periodic solutions. We also give a qualitative description of the phase space and provide results on the linear stability of solutions for the spin–orbit and spin–spin problems. We conclude by providing a comparison between the full and the Keplerian models with particular reference to the interaction between the rotational and orbital motions.

Quasi-Periodic Solutions of Derivative Beam Equation on Flat Tori

Quasi-Periodic Solutions of Derivative Beam Equation on Flat Tori

Steady state responses of an infinite beam resting on a tensionless visco-elastic foundation under a harmonic moving load

In this paper we study the steady state responses of an infinite beam resting on a tensionless visco-elastic foundation under a harmonic point force moving with a constant speed. The equation of motion is formulated with respect to a coordinate system moving with the point load. The beam and the foundation are defined as in contact when both the deflection and the “contact force” are negative. We replace the infinite beam by a finite beam with sufficient length and expand the beam deflection into a harmonic series. Newmark method is used to integrate the discretized equation. For a tensionless foundation, the contact condition between the beam and the foundation changes with time. Therefore, the matrices involved in the discretized equation of motion must be updated at every time step. In the bifurcation diagram we change the excitation frequency and record the beam deflection at the loading point via Poincaré sampling. When the dimensionless amplitude of the harmonic force is high, say 15, one can observe periodic, quasi-periodic, and chaotic solutions. We have also found that reducing the amplitude of the harmonic component or adding a time-invariant component can simplify the bifurcation diagram to include only periodic solutions.

Dynamics, multistability, and crisis analysis of a sine-circle nontwist map.

We propose a one-dimensional dynamical system, the sine-circle nontwist map, that can be considered a local approximation of the standard nontwist map and an extension of the paradigmatic sine-circle map. The map depends on three parameters, exhibiting a simple mathematical form but with a rich dynamical behavior. We identify periodic, quasiperiodic, and chaotic solutions for different parameter sets with the Lyapunov exponent and Slater's theorem. From the bifurcation analysis, we determine two bifurcation lines, those that depend on just two of the control parameters, for which the bifurcation that occurs is of the saddle-node type. In order to investigate multistability, we analyze the bifurcation diagrams in the two directions of parameter variation and we observe some regions of hysteresis, representing the coexistence of different attractors. We also analyze different multistable scenarios, as single attractor, coexistence of periodic attractors, coexistence of chaotic and periodic attractors, chaotic behavior, and coexistence of different chaotic bands, by the Lyapunov exponent and the analysis of the domain occupied by the solutions. From the parameter spaces constructed, we observe the prevalence of single attractor and only chaotic behavior scenarios. The multistable scenario is, mostly, formed by different periodic attractors. Lastly, we analyze the crisis in chaotic attractors and we identify the interior and the boundary crisis. From our results, the boundary crisis plays a key role for the extinction of multistability.

Complex dynamics in the improved Koren–Feingold cloud–rain system

In this paper, we study the improved aerosol–cloud–precipitation system. Since in our previous article, we guessed that the delayed term y(t−τ) in the denominator tends to cause singularity. So we remove the denominator by time transformation, the original cloud–rain model is transformed into a simpler polynomial system under certain conditions, namely, the improved Koren–Feingold cloud–rain system. Fortunately, the new model meets our expectations. The main work of the present paper is to study the mechanism of cloud and rain interaction by double Hopf bifurcation analysis via the methods of multiple scales. We find some complicated dynamical behaviors of the system via analytical method, such as stable quasi-periodic solutions and chaos. The numerical simulations agree well with those of theoretical analysis, which shows that we are improving the model in the right direction, meanwhile perhaps it suggests that the MMS employed in this paper provides us an effective and accurate method of analyzing double Hopf bifurcation. The above complex dynamical phenomena are of great significance for the researchers to understand the mechanism of aerosol–cloud–precipitation system. And it is also significant to ecosystem and climate control.

Boundedness of solutions for some impulsive pendulum-type equations

In the present paper, we study the following impulsive pendulum-type equation where , . Under suitable impulses, by Moser's twist theorem, we prove that any solution of the impulsive pendulum-type equation is bounded, and there are infinitely many quasi-periodic solutions for the impulsive pendulum-type equation.

Resonant quasi-periodic near-rectilinear Halo orbits in the Elliptic-Circular Earth-Moon-Sun Problem

Motivated by the near-future re-exploration of the cislunar space, this paper investigates dynamical substitutes of the Earth-Moon’s resonant Near-Rectilinear Halo Orbits (NRHOs) under the Elliptic-Circular Restricted Four-Body Problem formulation of the Earth-Moon-Sun system. This model considers that the Earth and Moon move in elliptical orbits about each other and that a third body, the Sun, moves in a circular orbit about the Earth-Moon barycenter. By making use of this higher-fidelity dynamical model, we are able to incorporate the Sun’s influence and the Moon’s eccentricity, two of the most significant perturbations of the cislunar environment. As a result of these perturbations, resonant periodic NRHOs of the Earth-Moon Circular Restricted Three-Body Problem (CR3BP) are hereby replaced by two-dimensional quasi-periodic tori that better represent the dynamical evolution of satellites near the vicinity of the Moon. We present the steps and algorithms needed to compute these dynamical structures in the Elliptic-Circular model and subsequently assess their utility for spacecraft missions. We focus on the planned orbit for the NASA-led Lunar Gateway mission, a 9:2 synodic resonant L2 southern NRHO, as well as on the 4:1 synodic and 4:1 sidereal resonances, due to the proximity to the nominal orbit and their advantageous dynamical properties. We verify that the dynamical equivalents of these orbits preserve key dynamical attributes such as eclipse avoidance and near-linear stability. Furthermore, we find that the higher dimensionality of quasi-periodic solutions offers interesting alternatives to mission designers in terms of phasing maneuvers and low-altitude scientific observations.

Multi-Mode Motion Dynamics of a Top Tensioned Riser Excited by Vortices and Varying Tension

A coupled model of a top tensioned riser (TTR) under combined excitations from vortices and time-varying tension is proposed. And the van der Pol oscillator is introduced to simulate the time-varying characteristics of vortex shedding. The first three mode approximations as a six degree-of-freedom (DOF) system of vibrations of the fluid-structure interaction system are derived. Two cases of resonances are analyzed by using the numerical method. One case is on the vortex induced vibration (VIV) of the six DOF system, where the 1-to-1 internal resonance (also named “lock-in” phenomenon) between frequencies of the first modes of structure and the wake oscillator are investigated. Bifurcation curves between dynamic responses in terms of the shedding frequency are derived. Comparing to the vibrations of the first order mode, the “lock-in” phenomenon for vibrations of the higher order modes can occur in the wider parameter range with large values. The second case is on the combined resonance between the 1-to-1 internal and 1-to-2 parametric resonance. Response curves versus the shedding frequency under different amplitude ratios are calculated by the Runge–Kutta method. Effects of system parameters including the amplitude ratio of the parametric excitation, shedding frequency on the response curves are investigated. Comparing to the case when only considering the 1-to-1 internal resonance in the VIV, results of the combined resonance show those large-amplitude responses of the structure and wake oscillator can be induced. Responses will increase rapidly as the amplitude of the varying tension increases. Unstable responses can occur within a large frequency band of nondimensional shedding frequency. Furthermore, the dynamic characteristics of the responses of the six DOF system including the bifurcation diagrams, time histories, phase-plane projection and Poincaré sections are computed for different amplitude ratios. Results indicate that complex dynamics such as the quasi-periodic and chaotic solutions can occur for different amplitude ratios. The analysis can be helpful for the parameter design of deep-water risers.

A Robust Lyapunov Criterion for Nonoscillatory Behaviors in Biological Interaction Networks

We introduce the notion of nonoscillation, propose a constructive method for its robust verification, and study its application to biological interaction networks (also known as, chemical reaction networks). We begin by revisiting Muldowney’s result on the nonexistence of periodic solutions based on the study of the variational system of the second additive compound of the Jacobian of a nonlinear system. We show that exponential stability of the latter rules out limit cycles, quasi-periodic solutions, and broad classes of oscillatory behavior. We focus then on nonlinear equations arising in biological interaction networks with general kinetics, and we show that the dynamics of the aforementioned variational system can be embedded in a linear differential inclusion. We then propose algorithms for constructing piecewise linear Lyapunov functions to certify global robust nonoscillatory behavior. Finally, we apply our techniques to study several regulated enzymatic cycles, where available methods are not able to provide any information about their qualitative global behavior.

Complicated Dynamics of a Delayed Photonic Reservoir Computing System

In this paper, we consider the complicated dynamics of a delay-based photonic reservoir computing system. Since conventional computer architectures are approaching their limit, it is imperative to find new, efficient and fast ways of data processing. Photonic reservoir computing (RC) is a promising way which combines the computational capabilities of recurrent neural networks with high processing speed and energy efficiency of photonics. As the RC system is very promising, we analyze its dynamics so that we can make better use of it. In this paper, we mainly focus on its double Hopf bifurcation. We first analyze the existence of double Hopf bifurcation points. Then we use DDE-BIFTOOL to draw the bifurcation diagrams with respect to two bifurcation parameters, i.e. feedback strength [Formula: see text] and delay [Formula: see text], and give a clear picture of the double Hopf bifurcation points of the system. These figures show stability switches and the existence of double Hopf bifurcation points. Finally, we employ the method of multiple scales to obtain their normal forms, use the method of normal form to unfold and classify their local dynamics. The classification and unfolding of these double Hopf bifurcation points are obtained. Three types of double Hopf bifurcations are found. We verify the results by numerical simulations and find its complicated behavioral dynamics. For example, there exist stable equilibrium, stable periodic and quasi-periodic solutions in distinct regions. The discovered rich dynamical phenomena can help us to choose suitable values of parameters to achieve excellent performance of RC.

Normal forms of double Hopf bifurcation for a reaction-diffusion system with delay and nonlocal spatial average and applications

In this paper, we are concerned with a reaction-diffusion model incorporating delay and nonlocal effects. The normal form of double Hopf bifurcation is derived. The diffusive model of pollen tube tip growth is discussed and numerical simulations show that spatially homogeneous and inhomogeneous periodic solutions can be both stable or connected by a heteroclinic orbit under certain conditions. In addition, the diffusive Lotka-Volterra model with delay and nonlocality is considered and spatially inhomogeneous quasi-periodic solution is obtained.