Non-resonant Case Research Articles

Periodic attitude motions of an axisymmetric spacecraft in an elliptical orbit near the hyperbolic precession

This paper deals with the existence and stability of periodic attitude motion near hyperbolic precession (HP) for a dynamically symmetric rigid body (RB) with its center of mass moving along an elliptical orbit. We present the definitions of undisturbed and disturbed periodic attitude motions, and approximate analytical solutions for them are derived using the multiscale method under non-resonance, internal resonance, and combination resonance conditions. Based on the definition of Lyapunov stability, we demonstrate stability in the non-resonant case and establish the conditions require for stability in the internal resonance scenario and a numerical analysis supports the theoretical results. The findings of this study extend classical results on RB dynamics in central gravitational fields. These insights demonstrate that appropriate parameter selection can achieve stable periodic attitude motion for RBs is in elliptical orbits, offering practical implications for spacecraft attitude control and mission reliability.

Quantitative reducibility of ${\boldsymbol {C}^{\boldsymbol {k}}}$ quasi-periodic cocycles

Abstract This paper establishes an extreme $C^k$ reducibility theorem of quasi-periodic $SL(2, \mathbb {R})$ cocycles in the local perturbative region, revealing both the essence of Eliasson [Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation. Comm. Math. Phys.146 (1992), 447–482], and Hou and You [Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems. Invent. Math.190 (2012), 209–260] in respectively the non-resonant and resonant cases. By paralleling further the reducibility process with the almost reducibility, we are able to acquire the least initial regularity as well as the least loss of regularity for the whole Kolmogorov–Arnold–Moser (KAM) iterations. This, in return, makes various spectral applications of quasi-periodic Schrödinger operators wide open.

Nonlinear Dynamics of an Electromagnetically Actuated Cantilever Beam Under Harmonic External Excitation

The present work is devoted to the study of nonlinear vibrations of an electromagnetically actuated cantilever beam subject to harmonic external excitation. The soft actuator that controls the vibratory motion of such components of a robotic structure led to a strongly nonlinear governing differential equation, which was solved in this work by using a highly accurate technique, namely the Optimal Auxiliary Functions Method. Comparisons between the results obtained using our original approach with those of numerical integration show the efficiency and reliability of our procedure, which can be applied to give an explicit analytical approximate solution in two cases: the nonresonant case and the nearly primary resonance. Our technique is effective, simple, easy to use, and very accurate by means of only the first iteration. On the other hand, we present an analysis of the local stability of the model using Routh–Hurwitz criteria and the eigenvalues of the Jacobian matrix. Global stability is analyzed by means of Lyapunov’s direct method and LaSalle’s invariance principle. For the first time, the Lyapunov function depends on the approximate solution obtained using OAFM. Also, Pontryagin’s principle with respect to the control variable is applied in the construction of the Lyapunov function.

From molecules to materials: SHG-CD microscopy of structured chiral films

The interplay between molecular and structural chirality as a function of local sample morphology determines the nonlinear optical properties of many organic and hybrid organic–inorganic thin films. Here, we used second harmonic generation circular dichroism (SHG-CD) microscopy of thin molecular films of 1,1′-bi-2-naphthol (R-BINOL) as a research model. Our results show that the SHG signal measured at frequencies close to the electronic transition of BINOL molecules is resonantly enhanced by more than an order of magnitude compared to the non-resonant case. The extracted resonant SHG-CD signal is dominated by the chiral response of the molecule. In contrast, structural chirality determines the non-resonant SHG-CD images. We see clear evidence that the interference of the SHG signals caused by the molecular and structural chirality can lead to a decrease in the overall SHG intensity when both SHG signals are out of phase. These findings highlight the intricate relationship between molecular and structural chirality on the one hand and structural morphology on the other hand and pave the way for novel applications by exploiting the chiroptic properties of thin films.

Concentration phenomena of solutions for critical perturbed Hénon problems

The main aim in this paper is to carry out a comprehensive research on the critical Hénon problem{−Δu=|x|αupα+ϵk(x)uqinΩ,u>0inΩ,u=0on∂Ω, where α>0, pα=N+2+2αN−2, q≥1, ϵ>0, k(x)∈C2(Ω¯), Ω is a smooth bounded domain containing the origin in RN, N≥3. Based on Lyapunov-Schmidt reduction argument, we provide some sufficient conditions for the existence of concentrating solutions without any condition on the Robin function. The main results depend on the non-resonant case that k(0)≠0andq≠N+2N−2 and the resonant case that k(0)=0orq=N+2N−2. The novelty in our study is significantly different from the case that α=0.

Wind Tunnel Test Research on the Aerodynamic Behavior of Concrete-Filled Double-Skin Steel (CFDST) Wind Turbine Towers

To explore the potential application of concrete-filled double-skin steel tubular (CFDST) structures in wind turbine towers, this study carried out wind tunnel tests to explore the aerodynamic behavior of CFDST tower-based wind turbine systems. Two scaled models including traditional steel tower-based and CFDST tower-based wind turbine systems were designed and tested in the field of typhoons. Then, the vibration characteristics in both the downwind and crosswind directions were systematically investigated, in terms of acceleration and displacement response, motion trajectory, dynamic characteristics, etc. The findings demonstrate that CFDST structures can have significantly improved performance against both blade harmonic excitation and external environmental excitation. Compared to traditional steel towers, CFDST towers exhibit a substantial reduction in aerodynamic response. In particular, the reduction in the RMS value can be over five times in the resonance case and 457.69% in the non-resonance case. The CFDST towers predominantly exhibited converged motion trajectory and concentrated on lower vibration modes. The energy dissipation capability was remarkably enhanced, with the damping ratio increasing up to 40.98%. Overall, it was experimentally demonstrated that CFDST towers can efficiently address the dynamic problems of large-scale wind turbine towers in engineering.

Bifurcations of a Laminated Circular Cylindrical Shell

In this paper, the stability and local bifurcations of a composite laminated circular cylindrical shell with radially pre-stretched membranes are explored by using analytical and numerical methods. On the basis of the four-dimensional averaged equation in the case of 1:1 internal resonance, three types of critical points are studied in detail. They are characterized as (1) one pair of purely imaginary eigenvalues and two negative eigenvalues; (2) a simple zero and one pair of purely imaginary eigenvalues; (3) two pairs of purely imaginary eigenvalues in nonresonant case. With the aid of normal form theory and Maple software, the steady-state solutions and the stability regions of the initial equilibrium solutions are obtained. The explicit expressions for the critical bifurcation curves leading to static bifurcation and Hopf bifurcation are also presented. The presence of Hopf bifurcation indicates that the circular cylindrical shell will flutter. The results contribute to the design of reasonable structure parameters to avoid flutter. Finally, numerical simulations are also presented to demonstrate the good agreement with the analytical predictions.

Shape stability of a microbubble in a power–law liquid

The onset of non-spherical oscillations of a microbubble in an unbounded power–law liquid, important for biomedical ultrasound applications, is studied. Two sets of evolution equations are obtained from the equation of motion: a Rayleigh Plesset-type equation for the spherical oscillations and an equation for the non-spherical oscillations. The non-spherical oscillations are modeled using the perturbation method via the Legendre polynomials. Two kinds of instabilities, namely parametric and Rayleigh-Taylor instabilities, are investigated. A higher power–law index causes the damping of the oscillations for both spherical and non-spherical oscillations. The power–law index damping effect depends on the ultrasonic drive frequency. At natural frequency, the amplitude of the perturbations is high compared to the non-resonant cases. At a low consistency index, the damping effect of the power–law index decreases. Unlike Newtonian liquids, the viscosity of power–law liquids is affected by the frequency of the acoustic field, thereby affecting Rayleigh-Taylor instability.

Non-resonant anomaly detection with background extrapolation

Complete anomaly detection strategies that are both signal sensitive and compatible with background estimation have largely focused on resonant signals. Non-resonant new physics scenarios are relatively under-explored and may arise from off-shell effects or final states with significant missing energy. In this paper, we extend a class of weakly supervised anomaly detection strategies developed for resonant physics to the non-resonant case. Machine learning models are trained to reweight, generate, or morph the background, extrapolated from a control region. A classifier is then trained in a signal region to distinguish the estimated background from the data. The new methods are demonstrated using a semi-visible jet signature as a benchmark signal model, and are shown to automatically identify the anomalous events without specifying the signal ahead of time.

Branching patterns of wave trains in mass-in-mass lattices

We investigate the existence and branching patterns of wave trains in the mass-in-mass (MiM) lattice, which is a variant of the Fermi–Pasta–Ulam (FPU) lattice. In contrast to FPU lattice, we have to solve coupled advance-delay differential equations, which are reduced to a finite-dimensional bifurcation equation with an inherited Hamiltonian structure by applying a Lyapunov–Schmidt reduction and invariant theory. We establish a link between the MiM lattice and the monatomic FPU lattice. That is, the monochromatic and bichromatic wave trains persist near $\mu =0$ in the nonresonance case and in the resonance case $p:q$ where $q$ is not an integer multiple of $p$ . Furthermore, we obtain the multiplicity of bichromatic wave trains in $p:q$ resonance where $q$ is an integer multiple of $p$ , based on the singular theorem.

Probing dynamical sensitivity of a non-Kolmogorov-Arnold-Moser system through out-of-time-order correlators.

Non-Kolmogorov-Arnold-Moser (KAM) systems, when perturbed by weak time-dependent fields, offer a fast route to classical chaos through an abrupt breaking of invariant phase-space tori. In this work, we employ out-of-time-order correlators (OTOCs) to study the dynamical sensitivity of a perturbed non-KAM system in the quantum limit as the parameter that characterizes the resonance condition is slowly varied. For this purpose, we consider a quantized kicked harmonic oscillator (KHO) model, which displays stochastic webs resembling Arnold's diffusion that facilitate large-scale diffusion in the phase space. Although the Lyapunov exponent of the KHO at resonances remains close to zero in the weak perturbative regime, making the system weakly chaotic in the conventional sense, the classical phase space undergoes significant structural changes. Motivated by this, we study the OTOCs when the system is in resonance and contrast the results with the nonresonant case. At resonances, we observe that the long-time dynamics of the OTOCs are sensitive to these structural changes, where they grow quadratically as opposed to linear or stagnant growth at nonresonances. On the other hand, our findings suggest that the short-time dynamics remain relatively more stable and show the exponential growth found in the literature for unstable fixed points. The numerical results are backed by analytical expressions derived for a few special cases. We will then extend our findings concerning the nonresonant cases to a broad class of near-integrable KAM systems.

Study of the dynamic process in a nonlinear mathematical model of the transverse oscillations of a moving beam under perturbed boundary conditions

The study of transverse oscillations of systems moving along their axis is a very difficult, but at the same time a very important task. Mathematical models of nonlinear transverse oscillations of a beam moving along its axis are analyzed in this paper work, both for non-resonant and resonant cases. The task becomes even more complicated if we additionally take into account the method of fastening the ends of the beam or the perturbation at its ends. We have obtained dependencies that can be used in construction, transport, industry, mechanical engineering and other domains of technology, ensuring the stability and safety of the operation of such mechanical systems. Mathematical models have been obtained for structural engineers to determine the amplitude–frequency response of relevant structures. These mathematical models are key to researching the dynamics of moving media. The obtained results allow considering not only the influence of kinematic and physical-mechanical parameters on the amplitude–amplitude frequency response of the medium, but also the fastening method. In addition, the correlations obtained in the paper make it possible to study not only the influence of the moving medium parameters on the nature of changes in the frequency and amplitude of oscillations, but also to consider the movement at the points of support of the medium. Namely, even at the stage of designing a pipeline for a liquid flowing at a certain speed, it is possible to consider the influence of the oscillation of the supports or their fastening method on the dynamics of the oscillatory process. The resulting dependencies allow designers to consider the influence of the characteristics given in the paper with a high level of accuracy and predict dynamic phenomena in them. In engineering calculations of various mechanical systems, the resulting dependencies can be used to optimize parameters to avoid negative destructive phenomena during operation.

Nonlinear stability of triangular equilibrium points in non-resonance case with perturbations

Nonlinear stability of triangular equilibrium points in non-resonance case with perturbations

An efficient compact difference-proper orthogonal decomposition algorithm for fractional viscoelastic plate vibration model

In this article, we aim to prove the solvability for the fractional viscoelastic plate vibration model through the Galerkin argument in Sobolev spaces as a compensation for the absence of its mathematical theory, then propose an efficient compact difference-proper orthogonal decomposition (CD-POD) algorithm. Rigorous numerical analyses are conducted to show that the CD-POD algorithm inherits the merits of the compact difference and the proper orthogonal decomposition technique, that is, it is unconditionally stable, possesses the spatial and temporal convergence rates of fourth order and second order, respectively, and reduces greatly the storage and computing cost, and thus may adapts for long time simulation for the plate vibration model. As a bridge for the plate vibration model and the CD-POD algorithm, a compact difference scheme is also proposed and analyzed previously. Numerical experiments, one of which is for resonance case, indicate that the CPU time of the CD-POD algorithm is at least 20 times less than that of the compact difference scheme, for resonance case or for non-resonance case, and thus verifies again that the CD-POD algorithm is a reasonable computation candidate for long time simulation of the viscoelastic plate vibration model.

Standing waves for Schrödinger equations with Kato–Rellich potentials

We show the existence of standing waves for the nonlinear Schrödinger equation with Kato-Rellich type potential. We consider both resonant with the nonlinearity satisfying one of Landesman–Lazer type or sign conditions and non-resonant case where the linearization at infinity has zero kernel. The approach relies on the geometric and topological analysis of the parabolic semiflow associated to the involved elliptic problem. Tail estimates techniques and spectral theory of unbounded linear operators are used to exploit subtle compactness properties necessary for use of the Conley index theory due to Rybakowski. The obtained results extend those by Prizzi (2003) for the non-resonant case, resolve the existence problem at resonance and complete those from A. Ćwiszewski and W. Kryszewski (2019) where the bifurcation of stationary solutions from infinity was studied.

AVERAGED CIRCULAR SPATIAL RESTRICTED THREE-BODY PROBLEM: INTERNAL CASE, NEW RESULTS

We investigate the spatial restricted circular three-body problem in the nonresonant case. Namely, we apply Gaussian averaging to obtain averaged equations in terms of osculating elements and then investigate them. Keplerian ellipse with a focus in the main body (the Sun) is taken as an unperturbed orbit assuming the semi-major axis of the ellipse to be less than the radius of the orbit of the outer planet (internal problem). Using the Parseval formula we have derived the twice-averaged perturbed force function of the problem in the form of an explicit analytical series with coefficients expressed in terms of the Gauss and Clausen hypergeometric functions. An investigation of averaged force function along it’s curves of non-analyticity showed that the series are asymptotic by Poincaré. For a reduced system with one degree of freedom, phase portraits of oscillations in the plane of the Keplerian elements \(e\), \(\omega \) are constructed in the second and fourth approximations. It is shown the topology of the phase portrait is more complicated in fourth approximation then in first and second approximations provided that the constant \({{c}_{1}}\) of the Lidov–Kozai integral belongs to the interval (0, 0.382).

An instrumental insight for a periodic solution of a fractal Mathieu–Duffing equation

The primary goal of the present study is to investigate how to obtain a periodic solution for a fractal Mathieu–Duffing oscillator. To achieve this, the fractal oscillator in the fractal space has been transformed into a damping Mathieu–Duffing equation in the continuous space by employing a new modification of He’s definition of the fractal derivative. The required analytical periodic solution has been based on the rank upgrade technique (RUT) presented. The RUT successfully generates a periodic solution without sacrificing the damping coefficient by creating an alternate equation, aside from any challenges in managing the impact of the linear damping component. The homotopy perturbation method (HPM) has been used to find the required periodic solution for the alternate equation. A comparison of the numerical solutions of the original equation and the alternative equation showed good agreement. The stability behavior in the non-resonance case as well as in the sub-harmonic resonance case has also been discussed. Further, another method, “the non-perturbative approach”, that deals with the obtained equation has been introduced.

Protecting the fidelity of a pulsed-driven harmonic oscillator

The temporal fidelity of a non-dissipative quantised harmonic oscillator (QHO) driven by two types of pulsed laser; namely, rectangular and train of n-chirped Gaussian pulses is investigated. The computational results are presented for two initial states of the HO, namely, the superposition of coherent states (Schrödinger cat states type) and squeezed coherent state. Quantum interference between the two component states |α〉 and |−α〉 of the initial cat state in the non-resonance case induces oscillatory behaviour in the fidelity. The case of initial squeezed coherent state shows profound maximum bounds of the fidelity or lesser maximum peaks, depending on the pulse shape and other system parameters (squeezing, chirp and pulse number parameters).

Fucik spectrum with weights and existence of solutions for nonlinear elliptic equations with nonlinear boundary conditions

We consider the boundary value problem $$\displaylines{ - \Delta u + c(x) u = \alpha m(x) u^+ - \beta m(x) u^- +f(x,u), \quad x \in \Omega, \cr \frac{\partial u}{\partial \eta} + \sigma (x) u =\alpha \rho (x) u^+- \beta \rho (x) u^- +g(x,u), \quad x \in \partial \Omega, }$$ where \((\alpha, \beta) \in \mathbb{R}^2\), \(c, m \in L^\infty (\Omega)\), \(\sigma, \rho \in L^\infty (\partial\Omega)\), and the nonlinearities f and g are bounded continuous functions. We study the asymmetric (Fucik) spectrum with weights, and prove existence theorems for nonlinear perturbations of this spectrum for both the resonance and non-resonance cases. For the resonance case, we provide a sufficient condition, the so-called generalized Landesman-Lazer condition, for the solvability. The proofs are based on variational methods and rely strongly on the variational characterization of the spectrum. See also https://ejde.math.txstate.edu/special/02/m2/abstr.html

Analytical and numerical study of a vibrating magnetic inverted pendulum

The current study investigates the stability structure of the base periodic motion of an inverted pendulum (IP). A uniform magnetic field affects the motion in the direction of the plane configuration. Furthermore, a non-conservative force as one that dampens air is considered. Its underlying equation of motion is derived from traditional analytical mechanics. The mathematical analysis is made simpler by substituting the Taylor theory in order to expand the restoring forces. The modified Homotopy perturbation method (HPM) is employed to achieve a roughly adequate regular result. To support the prior result, a numerical method based on the fourth-order Runge-Kutta method (RK4) is employed. The graphs for both the analytic and numerical solutions are highly consistent with one another, which indicates that the perturbation strategy is accurate. The solution time history curve exhibits a decaying performance and indicates that it is steady and without chaos. The resonance and non-resonance cases are found through the stability study by using the time scale method. In all perturbation approaches, the methodology of multiple time scales is actually regarded as a further standard approach. The time history is used to create a collection of graphs. Some graphical representations are used to illustrate how the typical physical values affect the behavior of the discovered solution. It has been discovered that the statically unstable IP can have its instability reduced by raising the spring torsional constant stiffness as well as the damped coefficient. Moreover, the magnetic field has a significant role in the stability configuration, which explains that at higher values of this field, the decaying waves take much more time than the smaller values of this field. Accordingly, it can be employed in various engineering devices that need a certain period of time to be more stable.