Unperturbed System Research Articles

Cyclicity of a Class of Hamiltonian Systems under Perturbations of Piecewise Smooth Polynomials

In the study of the weakened Hilbert’s 16th problem, there usually appears period annulus surrounding more than one singular point, implying that there are often more than three generate functions in the Abelian integrals (or the first order Melnikov functions). It is also known that there are usually more than three generate functions in the study of the bifurcation of limit cycles in piecewise smooth Hamiltonian systems. This paper aims to explore a method so that we can estimate the upper bound of limit cycles bifurcating from the period annulus if the unperturbed system has some good properties such as Picard–Fuchs equations. This method is based on [Gavrilov & Iliev, 2009] and is useful for smooth and piecewise smooth polynomial vector fields of a general degree. Finally, we present an example to illustrate an application of the theoretical results.

On maintaining the stability of the equilibrium of nonlinear oscillators under conservative perturbations

Abstract. The problem of Yu.N. Bibikov on maintaining the stability of the equilibrium position of two interconnected nonlinear oscillators under the action of small, in a certain sense, conservative perturbing forces is considered. With different methods of reducing the system to the Hamiltonian form, some features are revealed for the case when the perturbing forces of the interaction of two oscillators are potential. The conditions for preserving the stability and instability of the equilibrium of two oscillators for the case of sufficiently small disturbing forces are obtained. The problem of maintaining the stability of the equilibrium under conservative perturbations is also considered in the more general situation of an arbitrary number of oscillators with power potentials with rational exponents, which leads to the case of a generalized homogeneous potential of an unperturbed system. The example given shows the applicability of the proposed approach in the case when the order of smallness of the perturbing forces coincides with the order of smallness of the unperturbed Hamiltonian.

Limit cycles appearing from a generalized heteroclinic loop with a cusp and a nilpotent saddle

This paper studies the limit cycle bifurcation problem of a class of piecewise smooth differential polynomial systems of degree n by perturbing a piecewise cubic polynomial system having a generalized heteroclinic loop with a cusp and a nilpotent saddle. First, we provide all possible phase portraits of the unperturbed system on the plane with crossing periodic orbits and obtain a condition for the appearance of a generalized heteroclinic loop with a cusp and a nilpotent saddle by qualitative theoretical knowledge. Then, we investigate the algebraic structure of the first order Melnikov function and give its asymptotic expansion near the generalized heteroclinic loop with the help of analytical skills. Finally, we employ the expansion together with its coefficients to obtain the existence of at least 3n−1 limit cycles.

Razumikhin qualitative analyses of Volterra integro-fractional delay differential equation with caputo derivatives

A non-linear system of Volterra integro-fractional delay differential equations with Caputo fractional derivatives is considered. New sufficient conditions for uniform stability, asymptotic stability, and Mittag-Leffler stability of the zero solution of the unperturbed system, and the boundedness of all solutions of the perturbed system, are presented. The technique of proof involves the Razumikhin method with an appropriate Lyapunov function. For illustrative purposes, two examples are provided.

Resonant amplification of enzymatic chemical oscillations by oscillating flow.

Using theory and simulation, we analyzed the resonant amplification of chemical oscillations that occur due to externally imposed oscillatory fluid flows. The chemical reactions are promoted by two enzyme-coated patches located sequentially on the inner surface of a pipe that transports the enclosed chemical solution. In the case of diffusion-limited systems, the period of oscillations in chemical reaction networks is determined by the rate of the chemical transport, which is diffusive in nature and, therefore, can be effectively accelerated by the imposed fluid flows. We first identify the natural frequencies of the chemical oscillations in the unperturbed reaction-diffusion system and, then, use the frequencies as a forcing input to drive the system to resonance. We demonstrate that flow-induced resonance can be used to amplify the amplitude of the chemical oscillations and to synchronize their frequency to the external forcing. In particular, we show that even 10% perturbations in the flow velocities can double the amplitude of the resulting chemical oscillations. Particularly, effective control can be achieved for the two-step chemical reactions where during the first half-period, the fluid flow accelerates the chemical flux toward the second catalytic patch, while during the second half-period, the flow amplifies the flux to the first patch. The results can provide design rules for regulating the dynamics of coupled reaction-diffusion processes and can facilitate the development of chemical reaction networks that act as chemical clocks.

Nonlinear density response from imaginary-time correlation functions: Ab initio path integral Monte Carlo simulations of the warm dense electron gas.

The ab initio path integral Monte Carlo (PIMC) approach is one of the most successful methods in quantum many-body theory. A particular strength of this method is its straightforward access to imaginary-time correlation functions (ITCFs). For example, the well-known density-density ITCF F(q, τ) allows one to estimate the linear response of a given system for all wave vectors q from a single simulation of the unperturbed system. Moreover, it constitutes the basis for the reconstruction of the dynamic structure factor S(q, ω)-a key quantity in state-of-the-art scattering experiments. In this work, we present analogous relations between the nonlinear density response in the quadratic and cubic order of the perturbation strength and generalized ITCFs measuring correlations between up to four imaginary-time arguments. As a practical demonstration of our new approach, we carry out simulations of the warm dense electron gas and find excellent agreement with previous PIMC results that had been obtained with substantially larger computational effort. In addition, we give a relation between a cubic ITCF and the triple dynamic structure factor S(q1, ω1; q2, ω2), which evokes the enticing possibility to study dynamic three-body effects on an ab initio level.

Synchronization of quasiperiodic oscillations in nearly Hamiltonian systems: The degenerate case.

Quasiperiodic perturbations of two-dimensional nearly Hamiltonian systems with a limit cycle are considered. The behavior of solutions in a small neighborhood of a degenerate resonance is studied. Special attention is paid to the synchronization problem. Bifurcations of quasiperiodic solutions that arise when the limit cycle passes through the neighborhood of a resonance phase curve are investigated. The study is based on an analysis of an autonomous pendulum-type system, which is obtained by the method of averaging and determines the dynamics in the resonance zone. Two possible topological structures of the unperturbed averaged system are distinguished. For each case, the intervals of a control parameter that correspond to oscillatory synchronization are found. The results are applied to a Duffing-Van der Pol-type equation.

On the Stability Analysis of Linear Unperturbed Non-integer Differential Systems

In this paper, the stability of non-integer differential system is studied using Riemann-Liouville and Caputo derivatives. The stability notion for determining the stability/asymptotic stability or otherwise fractional differential system is given. Example is provided to demonstrate the effectiveness of the result.

Unified view of avalanche criticality in sheared glasses.

Plastic events in sheared glasses are considered an example of so-called avalanches, whose sizes obey a power-law probability distribution with the avalanche critical exponent τ. Although the so-called mean-field depinning (MFD) theory predicts a universal value of this exponent, τ_{MFD}=1.5, such a simplification is now known to connote qualitative disagreement with realistic systems. Numerically and experimentally, different values of τ have been reported depending on the literature. Moreover, in the elastic regime, it has been noted that the critical exponent can be different from that in the steady state, and even criticality itself is a matter of debate. Because these confusingly varying results have been reported under different setups, our knowledge of avalanche criticality in sheared glasses is greatly limited. To gain a unified understanding, in this work, we conduct a comprehensive numerical investigation of avalanches in Lennard-Jones glasses under athermal quasistatic shear. In particular, by excluding the ambiguity and arbitrariness that has crept into the conventional measurement schemes, we achieve high-precision measurement and demonstrate that the exponent τ in the steady state follows the prediction of MFD theory, τ_{MFD}=1.5. Our results also suggest that there are two qualitatively different avalanche events. This binariness leads to the nonuniversal behavior of the avalanche size distribution and is likely to be the cause of the varying values of τ reported thus far. To investigate the dependence of criticality and universality on applied shear, we further study the statistics of avalanches in the elastic regime and the ensemble of the first avalanche event in different samples, which provide information about the unperturbed system. We show that while the unperturbed system is indeed off-critical, criticality gradually develops as shear is applied. The degree of criticality is encoded in the fractal dimension of the avalanches, which starts from zero in the off-critical unperturbed state and saturates in the steady state. Moreover, the critical exponent τ is consistent with the prediction of the MFD τ_{MFD} universally, regardless of the amount of applied shear, once the system becomes critical.

Number of Limit Cycles from a Class of Perturbed Piecewise Polynomial Systems

In this paper, we study Poincaré bifurcation of a class of piecewise polynomial systems, whose unperturbed system has a period annulus together with two invariant lines. The main concerns are the number of zeros of the first order Melnikov function and the estimation of the number of limit cycles which bifurcate from the period annulus under piecewise polynomial perturbations of degree [Formula: see text].

The time-averaged spin-Hall conductivity in two-dimensional spin–orbit coupled systems: the role of spin dynamics

We investigate the spin-Hall conductivity (SHC) in the two-dimensional spin–orbit coupled systems by taking spin precession into account. The influence of spin precession on the spin-Hall current in the presence of disorder is investigated. Perturbation method leads to the result that the spin dynamics is composed of Larmor (periodic) and non-Larmor (non-periodic) precession. The non-Larmor component of spin is found to be the same result obtained from the Kubo formula in the short time limit. The Larmor component of spin, i.e., spin precession in the unperturbed system, was neglected in the previous studies, but it plays an important role in the linear response theory. The Larmor precession of spin should be zero after time average over several complete periods because there is no specific orientation perpendicular to the plane. By using the requirement of vanishing time-averaged Larmor precession of spin, we calculate the time-averaged non-Larmor SHC for k-cubic Rashba system by using the conventional definition of the spin current. Our result is 2.1(q/8π) which is very close to the experimental value 2.2(q/8π) by Wunderlich et al [2005 Phys. Rev. Lett. 94, 047204], where −q is the electric charge.

Chaos of a Single-Walled Carbon Nanotube Resulting from Periodic Parameter Perturbation

Carbon nanotubes (CNTs) are used in various nano-electromechanical systems (NEMS), and the parameters (including the system parameters and the excitation parameters) may result in chaos in these systems. Thus, understanding the mechanism of the chaos arising from NEMS is vital for CNT’s applications. Motivated by this need, the chaotic properties of a single-walled carbon nanotube system resulting from parametric excitation and external excitation are investigated in this paper. The criteria for the existence of the chaotic behavior in the system with periodic and quasi-periodic perturbations are obtained by the homoclinic Melnikov and the second-order average methods. Furthermore, in order to show the connection between periodic motion and complex behavior, the subharmonic periodic solutions, inside and outside the homoclinic loop, are analyzed. The global structure and the saddle-node bifurcation of the unperturbed averaged system are also considered. Finally, the Poincaré section and the transversal intersection of the unstable and stable manifolds are presented to verify the occurrence of chaos or subharmonic solution. The simulation results confirm the correctness of the theoretical analysis.

Damped Perturbations of Systems with Center-Saddle Bifurcation

An autonomous system of ordinary differential equations on the plane with a center-saddle bifurcation is considered. The influence of a class of time damped perturbations is investigated. The particular solutions tending to the fixed points of the limiting system are considered. The stability of these solutions is analyzed by Lyapunov function method when the bifurcation parameter of the unperturbed system takes critical and noncritical values. Conditions that ensure the persistence of the bifurcation in the perturbed system are described. When the bifurcation is broken, a pair of solutions tending to a degenerate fixed point of the limiting system appears in the critical case. It is shown that, depending on the structure and the parameters of the perturbations, one of these solutions can be stable, metastable or unstable, while the other solution is always unstable. The proposed theory is applied to the study of autoresonance capturing in systems with quadratically varying driving frequency.

A Canonical Transformation to Eliminate Resonant Perturbations. I.

Abstract We study dynamical systems that admit action-angle variables at leading order, which are subject to nearly resonant perturbations. If the frequencies characterizing the unperturbed system are not in resonance, the long-term dynamical evolution may be integrated by orbit-averaging over the high-frequency angles, thereby evolving the orbit-averaged effect of the perturbations. It is well known that such integrators may be constructed via a canonical transformation, which eliminates the high-frequency variables from the orbit-averaged quantities. An example of this algorithm in celestial mechanics is the von Zeipel transformation. However, if the perturbations are inside or close to a resonance, i.e., the frequencies of the unperturbed system are commensurate; these canonical transformations are subject to divergences. We introduce a canonical transformation that eliminates the high-frequency phase variables in the Hamiltonian without encountering divergences. This leads to a well-behaved symplectic integrator. We demonstrate the algorithm through two examples: a resonantly perturbed harmonic oscillator and the gravitational three-body problem in mean motion resonance.

The Admissible Control Correction Method in a Nonlinear Terminal Perturbed Problem

A solution of a nonlinear perturbed unconstrained point-to-point control problem, in which the unperturbed system is differentially flat, is considered in the paper. An admissible open-loop control in it is constructed using the covering method. The main part of the obtained admissible control correction in the limit problem is found by expanding the perturbed problem solution in series by the perturbation parameter. The first term of the expansion is determined by A.N. Tikhonov’s regularization of the Fredholm integral equation of the first kind. As shown by numerical experiments, the found structure of an admissible control allows one to find the final form of high precision point-to-point control based on the solution of an auxiliary variational problem in its neighborhood.

Plasmonic waveguides from Coulomb-engineered two-dimensional metals

Coulomb interactions play an essential role in atomically-thin materials. On one hand, they are strong and long-ranged in layered systems due to the lack of environmental screening. On the other hand, they can be efficiently tuned by means of surrounding dielectric materials. Thus all physical properties which decisively depend on the exact structure of the electronic interactions can be in principle efficiently controlled and manipulated from the outside via Coulomb engineering. Here, we show how this concept can be used to create novel plasmonic waveguides in metallic layered materials. We discuss how dielectrically structured environments can be utilized to non-invasively confine plasmonic excitations in an unperturbed homogeneous metallic two-dimensional system by modifications of its many-body interactions. We define optimal energy ranges for this mechanism and demonstrate plasmonic confinement within several nanometers. In contrast to conventional functionalization mechanisms, this scheme relies on a purely many-body concept and does not involve any direct modifications to the active material itself.

Qualitative Analyses of Differential Systems with Time-Varying Delays via Lyapunov–Krasovskiĭ Approach

In this paper, a class of systems of linear and non-linear delay differential equations (DDEs) of first order with time-varying delay is considered. We obtain new sufficient conditions for uniform asymptotic stability of zero solution, integrability of solutions of an unperturbed system and boundedness of solutions of a perturbed system. We construct two appropriate Lyapunov–Krasovskiĭ functionals (LKFs) as the main tools in proofs. The technique of the proofs depends upon the Lyapunov–Krasovskiĭ method. For illustration, two examples are provided in particular cases. An advantage of the new LKFs used here is that they allow to eliminate using Gronwall’s inequality. When we compare our results with recent results in the literature, the established conditions are more general, less restrictive and optimal for applications.

Superresolved polarization-enhanced second-harmonic generation for direct imaging of nanoscale changes in collagen architecture.

Superresolution (SR) optical microscopy has allowed the investigation of many biological structures below the diffraction limit; however, most of the techniques are hampered by the need for fluorescent labels. Nonlinear label-free techniques such as second-harmonic generation (SHG) provide structurally specific contrast without the addition of exogenous labels, allowing observation of unperturbed biological systems. We use the photonic nanojet (PNJ) phenomena to achieve SR-SHG. A resolution of with respect to the fundamental wavelength, that is, a -fold improvement over conventional or diffraction-limited SHG under the same imaging conditions is achieved. Crucially we find that the polarization properties of excitation are maintained in a PNJ. This is observed in experiment and simulations. This may have widespread implications to increase sensitivity by detection of polarization-resolved SHG by observing anisotropy in signals. These new, to the best of our knowledge, findings allowed us to visualize biological SHG-active structures such as collagen at an unprecedented and previously unresolvable spatial scale. Moreover, we demonstrate that the use of an array of self-assembled high-index spheres overcomes the issue of a limited field of view for such a method, allowing PNJ-assisted SR-SHG to be used over a large area. Dysregulation of collagen at the nanoscale occurs in many diseases and is an underlying cause in diseases such as lung fibrosis. Here we demonstrate that pSR-SHG allows unprecedented observation of changes at the nanoscale that are invisible by conventional diffraction-limited SHG imaging. The ability to nondestructively image SHG-active biological structures without labels at the nanoscale with a relatively simple optical method heralds the promise of a new tool to understand biological phenomena and drive drug discovery.

Chaos and subharmonic bifurcation of a composite laminated buckled beam with a lumped mass

With analytical and numerical methods, dynamic response of a composite laminated buckled beam with a lumped mass and an axial excitation is investigated in this paper. The dynamic equation of the first mode for this model, which is a perturbation of a Hamilton system with both quadratic and cubic nonlinearities as well as external and parametric excitations, is obtained with the Galerkin method. The distribution of the equilibriums, homoclinic orbits and periodic orbits for the unperturbed system under different system parameters are obtained analytically. Chaotic vibrations arising from homoclinic intersections are studied with Melnikov method. Chaotic feature on the location of the lumped mass is investigated in detail. It is presented that there exist controllable frequencies decreasing monotonously with the location of the lumped mass for this system. Subharmonic bifurcation is also studied with the subharmonic Melnikov method. It is numerically proved that the system may undergo chaotic motions through finite subharmonic bifurcations. Numerical simulations including the phase portraits, time histories, Poincare´ sections, Lyapunov spectrums and bifurcation diagram are given, which confirm the analytical results.

Geometric speed limit of neutrino oscillation

We investigate geometric quantum speed limit of neutrino oscillations in a presence of matter and CP-violation. We show that periodicity in the speed limit present in an unperturbed system becomes damped by interaction with a normal matter and decoherence. We also show that (hypothetical) CP-violation causes enhancement of periodicity and increases amplitude of an oscillating quantum speed limit and can quantify CP-violation.