Narrow Results

We report an ultra-narrowband absorber with a dielectric grating and metal substrate. The simulation results show that we can achieve ultra-narrowband absorption with the absorption bandwidth less than 0.6 nm and the absorption rate more than 0.99 for TE-polarization (electric field is parallel to grating grooves). The simulation results also show the guide-mode resonance in the grating region and low power loss at the absorption peak. In addition, the ultra-narrowband absorption peak can be tuned by shrinking or enlarging the structure parameters. The figure of merit (FOM) is larger than 760 if this absorber is applied as a refractive index sensor.

We report a graphene-based tunable ultra-narrowband mid-infrared filter which can be tuned from 4.45122 $\mu$m to 4.44675 $\mu$m by tuning the Fermi level from 0.2 eV to 0.6 eV. Furthermore, the reflection bandwidth is less than 0.2 nm and the reflection rate is more than 0.55. The ultra-narrowband filter is designed based on the guided-mode resonance (GMR) effect. The shift of reflection peak is mainly caused by the change of the real part of the graphene’s permittivity. This tunable ultra-narrowband mid-infrared filter can be applied in the mid-infrared microscopy.

Ultra-narrowband light absorption is desired for many applications. A near-infrared TM-polarization (magnetic field is parallel to grating grooves) ultra-narrowband absorber with dielectric metamaterials has been reported theoretically in this paper. The simulation results show that we can achieve an ultra-narrowband absorption at the incident wavelength of $1.0646\mu \mathrm{\text{m}}$ with the absorption bandwidth less than 0.9 nm and the absorption rate more than 0.99 for TM polarization. At the same time, we find that the high absorption rate can only remain up to $5^{\circ }$, which means our absorber has high directivity. In addition, the field distribution at the resonance wavelength shows that the ultra-narrowband absorption in our absorber has originated from magnetic resonance effect. Our near-infrared TM-polarization ultra-narrowband absorber is a good candidate for laser stealth.

We report a tunable bandpass mid-infrared filter with microstructure and graphene, and the transmission peaks can be tuned from $4.5638\mu \mathrm{\text{m}}$ to $4.5461\mu \mathrm{\text{m}}$ when the graphene’s Fermi level increases from 0.2 eV to 1.0 eV. This bandpass mid-infrared filter is originated from the guided-mode resonance (GMR) effect, and the tunable mechanism is mainly attributed to the change of the refractive index of the graphene. This tunable mid-infrared filter can be applied in non-dispersive infrared analyzer.

In this paper, the effects of plasmas on the S-wave resonance states in the positron-potassium system are investigated. Two different plasma conditions are considered, namely weakly coupled classical plasma (WCCP) and dense quantum plasma (DQP). The effective interaction potential in WCCP and DQP has been described by the Debye–Hückel model (static screened Coulomb potential) and a modified version of Debye–Hückel model (exponential cosine screened potential). Resonance parameters are determined by calculating the energy-density within the framework of the stabilization method. For the plasma-free case, three states, lying below the Ps$(n=2)$ threshold, are identified. The energy and the width agree well with the existing results in the literature. A detailed study is carried out to explore the changes emerging in the resonance parameters due to screening effects of WCCP and DQP. It is found that the energies of the three states are gradually pushed toward the Ps$(n=2)$ threshold due to the increasing plasma screening strength, whereas widths of the states vary differently.

A model which allows a double impacting regime for a particle undergoing simple harmonic motion is considered in some detail. The behavior of the particle in the weak spring limit is considered. Symmetries of the motion are found and the extent of the resonant dynamical behavior is considered. Control equations are developed and strategies are described for both the preservation and the annihilation of experimental and analytical resonant periodic orbits.

Bifurcations of homoclinic orbit with orbit-flip and resonant eigenvalues corresponding to the tangent directions are investigated in a four-dimensional system. The existence, number, coexistence and incoexistence of 1-homoclinic orbit, 1-periodic orbit, 2n-homoclinic orbit and 2n-periodic orbit are given, and the bifurcation surfaces and the existence regions are also located.

We study a minimal integrate-and-fire based model of a "ghostbursting" neuron under periodic stimulation. These neurons are involved in sensory processing in weakly electric fish. There exist regions in parameter space in which the model neuron is mode-locked to the stimulation. We analyze this locked behavior and examine the bifurcations that define the boundaries of these regions. Due to the discontinuous nature of the flow, some of these bifurcations are non-smooth. This exact analysis is in excellent agreement with numerical simulations, and can be used to understand the response of such a model neuron to biologically realistic input.

This paper studies the nonlinear equation for ventilation flows in a single opening enclosure. Periodic solutions, the global attractive region, periods and amplitudes of the periodic solutions, and quasi-periodic solutions in the nonlinear equation are investigated.

This paper is devoted to the numerical study of a two-dimensional nonautonomous ordinary differential equation with a strong cubic nonlinearity, submitted to an external periodical excitation of period τ = 2π/ω , having an amplitude E. In absence of this excitation the equation of Duffing type does not give rise to self oscillations. It may be a model of a series R-L-C electrical circuit with validity conditions in the parameter space. The purpose is essentially an analysis of the harmonics behavior of the period τ solutions according to points of the parameter plane (ω, E). Let r be the place occupied by a rank-m harmonic from an ordering based on line amplitudes of a frequency spectrum in descending order. Domains of the (ω, E) plane, for which the amplitude of the rank-m harmonic has the place r in the ordering mentioned above, are defined from properties of their boundaries. When r = 2 they are regions of predominance for the rank-m harmonic. When r = 1 they are regions of full predominance for the harmonic m, m = 2,3,4,…, which contain a set of points leading to a resonance for this harmonic. These regions fulfil the following important property: each of them is directly related to a fold bifurcation structure, called rank-mlip (two folds curves joining at two cusps), associated with a well-defined rank-m harmonic. The set of such structures, with m = 2,3,4,…, constitutes an isoordinal lips cascade. As an opening to Part II (to be published) period kτ solutions related to fractional harmonics behavior are also considered for k = 2,3.

We analyze an important class of engineering systems characterized by the discontinuous motion of a spring-mass constrained by the motion of a feedback-assisted actuator. We show that the combined effects of mechanical restitution coefficient and displacement feedback can be exactly represented by a single equivalent dissipation coefficient. We also show that the topological properties of the surfaces of section of orbits generated by impact oscillators which possess differing proportions of restitution and feedback levels, but whose equivalent dissipation coefficients are equal, are equivalent and universally scalable. The scaling law allows us to interchange the effects of restitution and feedback coefficients and so, effectively, eliminate one of these parameters from the equations of motion. Thus, the topological properties of dissipative feedback-assisted systems can be seen as scaled versions of either purely dissipative, or purely feedback-assisted, oscillators.

We provide an algorithm for studying invariant tori fulfilled by periodic orbits of a perturbed system which emerge from the set of periodic orbits of an unperturbed linear system in p : q resonance. We illustrate the algorithm with an application.

Duffing equation is a representative nonlinear equation in practical engineering systems, where the most existing research focuses on local solutions of weakly nonlinear systems. In this paper, we study global bifurcations and chaos of the standard Duffing system by employing the Arnold tongue, and use the basin of attraction to investigate the properties of the Arnold tongue overlap. Our results show that a resonance solution and chaos could coexist, when the parameters are on the Arnold tongue overlap. The phenomenon does not exist in a system described by a weak Duffing equation. Numerical solutions for these bifurcations and chaos are also provided to demonstrate the theoretical results.

We study the evolution of a conservative dynamical system with three degrees of freedom, where small nonconservative terms are added. The conservative part is a Hamiltonian system, describing the motion of a planetary system consisting of a star, with a large mass, and of two planets, with small but not negligible masses, that interact gravitationally. This is a special case of the three body problem, which is nonintegrable. We show that the evolution of the system follows the topology of the conservative part. This topology is critically determined by the families of periodic orbits and their stability. The evolution of the complete system follows the families of the conservative part and is finally trapped in the resonant orbits of the Hamiltonian system, in different types of attractors: chaotic attractors, limit cycles or fixed points.

A weakly nonlinear 2-DOF system, parametrically and externally excited, is studied. An intensive energy transfer between modes of vibrations is discovered. Multiple scales method is used for recognizing resonances occurring in the system. The amplitude response functions for some chosen cases of resonances are obtained and studied. Many of the engineering systems can be investigated in the presented way.

In this paper, a general and systematic scheme is provided to research strong and weak resonances derived from delay-induced various double Hopf bifurcations in delayed differential systems. The method of multiple scales is extended to obtain a common complex amplitude equation when the double Hopf bifurcation with frequency ratio k₁:k₂ occurs in the systems under consideration. By analyzing the complex amplitude equation, we give the conditions of the strong and weak resonances respectively in the analytical expressions. The weak resonances correspond to the codimension-two double Hopf bifurcations since the amplitudes and the phases may be decoupled, but the strong resonances to the codimension-three double Hopf bifurcations in the system. It is seen that the weak resonances happen in the system even for a lower-order ratio, i.e. k₁ + k₂ ≤ 4. As applications, two examples are displayed. Three cases of the delay-induced resonance with 1:2, 1:3, 1:5 and are discussed in detail and the corresponding normals are represented. Thus, the relative dynamical behaviors can be easily classified in the physical parameter space in terms of nonlinear dynamics. The results show the provided conditions may be used to determine that a resonance is strong or weak.

Interactions involving conjugate coupling and feedback have been known to induce rhythmogenesis in Chua’s circuits exhibiting fixed point behavior [Singla et al., 2011; Mandal et al., 2013]. It was reported, in these works, that beyond a threshold value of coupling/feedback strength, interaction term involving difference of conjugate variables can induce oscillations in Chua’s circuit. Furthermore, it has been observed that the parametric thresholds, in nonlinear oscillators, can be regulated by suitable forcing techniques [Parmananda et al., 2001; Mahara et al., 2005]. In the present work, a conjunction of these two ideas has been entertained i.e. we study the rhythmogenesis in Chua’s circuit when the conjugate coupling/feedback terms are modulated sinusoidally. It was noted that, when the forcing frequency lies in the vicinity of the natural frequency of Chua’s circuit, maximum lowering of the threshold coupling/feedback constant occurred. Appearance of Arnold’s tongue was also observed as a result of these perturbations. Numerical results have been complimented with the experimental observations.

In a neighborhood of stable equilibrium, we consider the dynamics for at least three degrees-of-freedom (dof) Hamiltonian systems (2 dof systems are not ergodic in this case). A complication is that the recurrence properties depend strongly on the resonances of the corresponding linearized system and on quasi-trapping. In contrast to the classical FPU-chain, the inhomogeneous FPU-chain shows nearly all the principal resonances. Using this fact, we construct a periodic FPU-chain of low dimension, called a FPU-cell. Such a cell can be used as a building block for a chain of FPU-cells, called a cell-chain. Recurrence phenomena depend strongly on the physical assumptions producing specific Hamiltonians; we demonstrate this for the $1:2:5$ resonance, both general and for the FPU case; this resonance shows dynamics on different timescales. In addition we will study the relations and recurrence differences between several FPU-cells and a few cell-chains in the case of the classical near-integrable FPU-cell and of chaotic cells in $3:2:1$ resonance.

The phenomenon of delay-induced resonance implies that in a nonlinear system a time-delay term may be used as an effective enhancer of the oscillations caused by an external forcing maintaining the same frequency. This is possible for the parameters for which the time-delay induces sustained oscillations. Here, we study this type of resonance in the overdamped and underdamped time-delayed Duffing oscillators, and we explore some new features. One of them is the conjugate phenomenon: the oscillations caused by the time-delay may be enhanced by means of the forcing without modifying their frequency. The resonance takes place when the frequency of the oscillations induced by the time-delay matches the ones caused by the forcing and vice versa. This is an interesting result as the nature of both perturbations is different. Even for the parameters for which the time-delay does not induce sustained oscillations, we show that a resonance may appear following a different mechanism.

In this paper, bifurcation analysis of a three-dimensional discrete game model is provided. Possible codimension-one (codim-1) and codimension-two (codim-2) bifurcations of this model and its iterations are investigated under variation of one and two parameters, respectively. For each bifurcation, normal form coefficients are calculated through reduction of the system to the associated center manifold. The bifurcations detected in this paper include transcritical, fold, flip (period-doubling), Neimark–Sacker, period-doubling Neimark–Sacker, resonance 1:2, resonance 1:3, resonance 1:4 and fold-flip bifurcations. Moreover, we depict bifurcation diagrams corresponding to each bifurcation with the aid of numerical continuation method. These bifurcation curves not only confirm our analytical results, but also reveal a richer dynamics of the model especially in the higher iterations.