Quasi-periodic Solutions Research Articles

On the reducibility of linear quasi-periodic systems with Liouvillean basic frequencies and multiple eigenvalues

In this paper we consider the linear quasi-periodic systemx˙=(A+ϵP(t))x,x∈Rd, where A is a d×d constant matrix of elliptic type and has multiple eigenvalues, P(t) is analytic quasi-periodic with respect to t with basic frequencies ω=(1,α), where α is irrational, and ϵ is a small perturbation parameter. Under suitable non-resonant condition, non-degeneracy condition and 0≤β(α)<r⁎, where β(α)=limsupn→∞ln⁡qn+1qn, qn is the sequence of denominations of the best rational approximations for α∈R∖Q, 0<r⁎<r, r is the initial radius of analytic strip, it is proved that for most sufficiently small ϵ, this system can be reduced to a constant system x˙=A⁎x, where A⁎ is a constant matrix close to A. As some applications, we apply our results to quasi-periodic Hill's equations, three dimensional skew symmetric systems and n coupled Schrödinger equations to study the Lyapunov stability of the equilibrium and the existence of quasi-periodic solutions.

Dynamical behaviors for generalized pendulum type equations with p-Laplacian

We consider a pendulum type equation with p-Laplacian (ϕp(x′))′ + G′x(t, x) = p(t), where ϕp(u) = |u|p−2u, p > 1, G(t, x) and p(t) are 1-periodic about every variable. The solutions of this equation present two interesting behaviors. On the one hand, by applying Moser’s twist theorem, we find infinitely many invariant tori whenever \(\int_0^1 {p(t){\rm{d}}t = 0} \), which yields the boundedness of all solutions and the existence of quasi-periodic solutions starting at t = 0 on the invariant tori. On the other hand, if p(t) = 0 and G′x(t, x) has some specific forms, we find a full symbolic dynamical system made by solutions which oscillate between any two different trivial solutions of the equation. Such chaotic solutions stay close to the trivial solutions in some fixed intervals, according to any prescribed coin-tossing sequence.

Real Lax spectrum implies spectral stability

AbstractWe consider the dynamical stability of periodic and quasiperiodic stationary solutions of integrable equations with 2 2 Lax pairs. We construct the eigenfunctions and hence the Floquet discriminant for such Lax pairs. The boundedness of the eigenfunctions determines the Lax spectrum. We use the squared eigenfunction connection between the Lax spectrum and the stability spectrum to show that the subset of the real line that gives rise to stable eigenvalues is contained in the Lax spectrum. For non‐self‐adjoint members of the AKNS hierarchy admitting a common reduction, the real line is always part of the Lax spectrum and it maps to stable eigenvalues of the stability problem. We demonstrate our methods work for a variety of examples, both in and not in the AKNS hierarchy.

Stability and bifurcations in oscillations of composite laminates with curvilinear fibres under a supersonic airflow

Nonlinear flutter of variable stiffness composite plates—using a reduced-order model benefiting from a large-enough number of modes or degrees of freedom—is the subject of this research. Dynamic of such plates is enriched by different scenarios including period-n (un)stable limit cycle oscillations, quasi-periodic or chaotic solutions, along with Hopf, symmetry breaking and period doubling bifurcations. The fibre path orientation, in any individual ply of these rectangular plates, changes linearly from the left edge to the right one. The plates are subjected to a supersonic airflow. The plate’s displacement field is modelled using a third-order shear deformation theory; a p-version finite element is used to discretize the displacement components. Aerodynamic pressure due to the airflow is approximated using linear piston theory. The self-exciting vibrational full-order model (FOM) is formed by employing the principle of virtual work, and then, the ROM is extracted from the FOM using two reduction techniques, namely static condensation and modal summation method. The inclusion of a large-enough number of modes of vibration in vacuum in the latter technique is critical for the observation of, often overlooked, complex dynamic behaviour of these plates. The equations of motion representing the ROM are solved using the shooting method (to obtain stable and unstable LCOs) and using Newmark method (for stable or non-oscillatory solutions). The rich dynamics of these plates are explored with the help of vibration and flutter mode shapes, phase-plane plots, bifurcation diagrams, fast Fourier transform diagrams and Poincare sections.

Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency

In this paper, one dimensional nonlinear wave equation \begin{document}$ u_{tt}-u_{xx} +mu +\varepsilon f(\omega t,x,u;\xi) = 0 $\end{document} with Dirichlet boundary condition is considered, where \begin{document}$ \varepsilon $\end{document} is small positive parameter, \begin{document}$ \omega = \xi \bar{\omega}, $\end{document} \begin{document}$ \bar{\omega} $\end{document} is weak Liouvillean frequency. It is proved that there are many quasi-periodic solutions with Liouvillean frequency for the above equation. The proof is based on an infinite dimensional KAM Theorem.

Quasi-periodic waves and irregular solitary waves of the AB system

Investigated in this paper is the AB system, which is used to describe the baroclinic instability processes in the geophysical flows. By introducing an auxiliary function , we construct the quasi-periodic wave solutions of the AB system, and find that behaviors of the quasi-periodic waves are similar to those of the periodic waves of the AB system except for certain phase shift in each evolution period. Especially, when , the quasi-periodic waves develop into the periodic waves. Besides, it is found that under certain conditions the quasi-periodic waves degenerate into the irregular solitary waves, which are characterized by shape variations or appearances, splits, combinations and disappearances of certain bulges in the evolution.

Bäcklund transformations and Riemann–Bäcklund method to a (3 + 1)-dimensional generalized breaking soliton equation

In this paper, the bilinear method is employed to investigate the N-soliton solutions of a (3 + 1)-dimensional generalized breaking soliton equation. Three sets of bilinear Backlund transformations are obtained by means of gauge transformation. The Riemann–Backlund method is further extended to the (3 + 1)-dimensional nonlinear integrable systems. The quasiperiodic wave solutions of the (3 + 1)-dimensional generalized breaking soliton equation are systematically analyzed. The asymptotic properties of the quasiperiodic solutions are discussed by using a limiting procedure. The one-periodic and two-periodic waves tend to the 1-soliton and 2-soliton under a small amplitude limit, respectively. The dynamical characteristics of the one- and two-periodic waves are summarized by selecting different parameters. Furthermore, we obtain some new types of the quasiperiodic wave solutions of the variable coefficient (3 + 1)-dimensional generalized breaking soliton equation. These solutions present the dynamical behaviors of C-type, anti-C-type and Z-type periodic waves moving on the background of the periodic waves of bell type.

Central Configurations and Action Minimizing Orbits in Kite Four-Body Problem

In the current article, we study the kite four-body problems with the goal of identifying global regions in the mass parameter space which admits a corresponding central configuration of the four masses. We consider two different types of symmetrical configurations. In each of the two cases, the existence of a continuous family of central configurations for positive masses is shown. We address the dynamical aspect of periodic solutions in the settings considered and show that the minimizers of the classical action functional restricted to the homographic solutions are the Keplerian elliptical solutions. Finally, we provide numerical explorations via Poincaré cross-sections, to show the existence of periodic and quasiperiodic solutions within the broader dynamical context of the four-body problem.

Quasi-periodic solutions of fractional nonlinear Schrödinger equation systems

In this paper, the fractional nonlinear Schrödinger equation system with periodic boundary conditions is considered. I establish an abstract infinite-dimensional KAM theorem and apply it to fractional nonlinear Schrödinger equation system with real Fourier multiplier. By establishing a block diagonal normal form, we prove the existence of a class of Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite-dimensional invariant tori of an associated infinite-dimensional dynamic system.

Applications of the Moser’s Twist Theorem to Some Impulsive Differential Equations

By KAM theorem, we prove that all solutions of Duffing equations of periodic coefficients undergoing suitable impulses are bounded for all time and that there are many quasi-periodic solutions clustering at infinity.

Nonlinear vibration analysis of a servo controlled precision motion stage with friction isolator

Precision motion stages are used in advanced manufacturing, metrological applications, and semiconductor industries for high precision positioning with high speed. However, friction-induced vibration undermines the performance of a servo-controlled motion stage. Recently, it was found that passive isolation in the form of friction isolator is very effective to mitigate the undesirable effects of friction in precision motion stages. This work presents, for the first time, a detailed nonlinear analysis of the dynamics of motion stage with a friction isolator. We consider a lumped parameter model of the precision motion stage with PID and a friction isolator modeled as two degrees of freedom system. Linear analysis of the system in the space of integral gain and reference velocity reveals that the inclusion of friction isolator increases the local stability region of steady states. We further observe the sensitivity of the stability of steady states towards the internal resonance between the motion stage and friction isolator. The influence of friction isolator on the nonlinear response of the system is examined analytically using the method of multiple scales and harmonic balance. We observe that the inclusion of friction isolator does not change the nature of Hopf bifurcation for higher values of reference velocity, and it remains subcritical bifurcation with or without friction isolator. However, for lower values of reference velocity, the inclusion of friction isolator leads to change in bifurcation from supercritical to subcritical for the given values of parameters. This observation further implies that the inclusion of friction isolator increases the local stability of steady states, whereas the global stability of steady states depends on the interaction between friction isolator and operating parameters. Furthermore, a detailed numerical bifurcation analysis of the system reveals the existence of period-2, period-4, quasi-periodic, and chaotic solutions. Also, the stability of period-1 solutions near Hopf point is determined by Floquet theory, which further reveals the existence of period-doubling bifurcation.

A Holling Type II Discrete Switching Host-Parasitoid System with a Nonlinear Threshold Policy for Integrated Pest Management

A nonlinear discrete switching host-parasitoid model with Holling type II functional response function, in which the switch is guided by an economic threshold (ET), is proposed. Thus, if the weighted density of two generations of the host population increases and exceeds the ET, then integrated pest management (IPM) measures are enacted, i.e., biological and chemical measures are implemented together, assuming that the chemical immediately precedes the biological inputs to avoid pesticide-induced deaths of the natural enemies. First, the existence and local stability of the equilibria of two subsystems were studied, and the existence and coexistence of several types of equilibria of a nonlinear switching system were analysed. Next, the nonlinear switching system was investigated by numerical simulation, showing that the system exhibits quite complex dynamic behaviour. A two-dimensional bifurcation diagram revealed the existence and coexistence regions of different types of equilibria including regular and virtual equilibria. Moreover, period-adding bifurcations in two-dimensional parameter spaces were found. One-dimensional bifurcation diagrams revealed that the system has periodic, quasiperiodic, and chaotic solutions, Neimark–Sacker bifurcation, multiple coexisting attractors, period-doubling bifurcations, period-halving bifurcations, and so on. Finally, the initial densities of hosts and parasitoids associated with host outbreaks and their biological implications are discussed.

Reducible KAM tori for higher dimensional wave equations under nonlocal and forced perturbation

In this paper, we prove an infinite dimensional Kolmogorov-Arnold-Moser theorem. As an application, it is shown that there are many small-amplitude linearly-stable quasi-periodic solutions for higher dimensional wave equations with a real Fourier multiplier, which are under nonlocal and forced perturbations with a special structure in space and short range property.

Expansions in the delay of quasi-periodic solutions for state dependent delay equations* * Supported in part by NSF Grant DMS 1800241.

We consider several models of state dependent delay differential equations (SDDEs), in which the delay is affected by a small parameter. This is a very singular perturbation since the nature of the equation changes. Under some conditions, we construct formal power series, which solve the SDDEs order by order. These series are quasi-periodic functions of time. This is very similar to the Lindstedt procedure in celestial mechanics.Truncations of these power series can be taken as input for a posteriori theorems, that show that near the approximate solutions there are true solutions. In this way, we hope that one can construct a catalogue of solutions for SDDEs, bypassing the need of a systematic theory of existence and uniqueness for all initial conditions.

The quasi-periodic solutions for the supersymmetric variable-coefficient KdV equation

The supersymmetric variable-coefficient KdV equation is presented and it admits Painlevé property by the standard singularity analysis. Based on Hirota bilinear method and Riemann theta function, one and two quasi-periodic wave solutions for the supersymmetric variable-coefficient KdV equation are studied. In addition, we give the asymptotic relations between quasi-periodic wave solutions and soliton solutions.

The continuation and stability analysis methods for quasi-periodic solutions of nonlinear systems

The continuation and stability analysis methods for quasi-periodic solutions of nonlinear systems

Nonstationary coherence-incoherence patterns in nonlocally coupled heterogeneous phase oscillators.

We consider a large ring of nonlocally coupled phase oscillators and show that apart from stationary chimera states, this system also supports nonstationary coherence-incoherence patterns (CIPs). For identical oscillators, these CIPs behave as breathing chimera states and are found in a relatively small parameter region only. It turns out that the stability region of these states enlarges dramatically if a certain amount of spatially uniform heterogeneity (e.g., Lorentzian distribution of natural frequencies) is introduced in the system. In this case, nonstationary CIPs can be studied as stable quasiperiodic solutions of a corresponding mean-field equation, formally describing the infinite system limit. Carrying out direct numerical simulations of the mean-field equation, we find different types of nonstationary CIPs with pulsing and/or alternating chimera-like behavior. Moreover, we reveal a complex bifurcation scenario underlying the transformation of these CIPs into each other. These theoretical predictions are confirmed by numerical simulations of the original coupled oscillator system.

Response Solutions for a Singularly Perturbed System Involving Reflection of the Argument

In this paper, the existence and uniqueness of response solutions, which has the same frequencyωwith the nonlinear terms, are investigated for a quasiperiodic singularly perturbed system involving reflection of the argument. Firstly, we prove that all quasiperiodic functions with the frequencyωform a Banach space. Then, we obtain the existence and uniqueness of quasiperiodic solutions by means of the fixed-point methods and theB-property of quasiperiodic functions.

Damped superlinear Duffing equation with strong singularity of repulsive type

A damped Duffing equation with a singularity is considered in this paper, where the elastic restoring force g has a singularity at origin and satisfies superlinear condition at infinity. By applying the twist theorem of nonarea-preserving map, we obtain the equation has at least one period-mT solution, where the minimal period is mT. It is also shown by bifurcation analysis that for a explicit form, the equation undergoes fold bifurcation, period doubling bifurcation and Hopf bifurcation, which leads to different solutions, including harmonic solutions, subharmonic solutions and quasiperiodic solutions. At last, we give the phase portraits and correspond Poincare section of the solutions.

Twist dynamics and Aubry-Mather sets around a periodically perturbed point-vortex

We consider the model of a point-vortex under a periodic perturbation and give sufficient conditions for the existence of generalized quasi-periodic solutions with rotation number. The proof relies on Aubry-Mather theory to obtain the existence of a family of minimal orbits of the Poincaré map associated to the system.