Oscillatory Solutions Research Articles

Nonlinear model and characteristic analysis of fractional-order high frequency oscillator

Recently, fractional calculus theory has been widely used in various fields, including the design of oscillators. However, there are still some issues that need to be studied in depth. Firstly, it is necessary to classify the fractional-order oscillators according to the properties of the frequency selective network components, and analyze the influence of the fractional order of components on the characteristics of different types of fractional-order oscillators. Secondly, for fractional-order oscillators with nonlinear devices such as transistors, the traditional linear models cannot describe them accurately. Thirdly, due to the complexity of fractional differential equation, it is difficult to obtain the general analytical solution of the fractional-order oscillator. For these reasons, this paper takes the fractional-order capacitive feedback LC oscillator as an example, and establishes a nonlinear model of this oscillator based on the transistor characteristic mechanism. With the help of the equivalent small parameter method, the influence of the fractional-order characteristics of the components on the output waveform is analyzed. It is revealed that for the fractional-order capacitive feedback LC oscillator, the order of capacitance C1 has the most significant effect on frequency and amplitude, followed by the order of inductance L2, and the order of the feedback capacitor C2 has very little effect on the oscillator. In addition, although the introduction of fractional order increases the oscillation frequency, it will also lead to an increase in THD in some cases. Simulation and experimental results verify the correctness of the proposed nonlinear model and characteristic analysis of fractional-order high frequency oscillator.

Thermosolutal Convection of Natural and Anti-Natural Solutions Through an Angled Cavity Under Cross Gradients in Temperature and Concentration

In the current study, cross-temperature and concentration gradients are used to model the in a binary fluid contained in an angled square cavity. Using a method, the , , and conservation equations were numerically solved. The inclined cavity under equal solutal buoyancy and thermal forces was the subject of the study . Since the horizontal components of the thermal and singular volume forces were equal but opposed to one another, an equilibrium solution for this situation that corresponds to the rest state of the immobile fluid is feasible. However, this equilibrium solution becomes unstable above a specific critical value of the , leading to vertical density stratification inside the enclosure. The results are shown using the and as well as the and for the flow intensity. The existence of the commencement of convection is demonstrated in this work, and both natural and anti-natural flow solutions are obtained. Subcritical convection has also been seen for the natural solution when the is more or less than unity. For the start of supercritical and subcritical convection, the number's critical values are identified. As the climbed, so did the flow's intensity and the rates at which heat and mass were transferred. Reducing flow intensity and accelerating mass transfer are the results of raising the . Different flow patterns are shown for an aspect ratio of 4, and the existence interval of the oscillatory solutions is calculated.

Are Chebyshev-based stability analysis and Urabe’s error bound useful features for Harmonic Balance?

Harmonic Balance is one of the most popular methods for computing periodic solutions of nonlinear dynamical systems. In this work, we address two of its major shortcomings: First, we investigate to what extent the computational burden of stability analysis can be reduced by consistent use of Chebyshev polynomials. Second, we address the problem of a rigorous error bound, which, to the authors’ knowledge, has been ignored in all engineering applications so far. Here, we rely on Urabe’s error bound and, again, use Chebyshev polynomials for the computationally involved operations. We use the error estimate to automatically adjust the harmonic truncation order during numerical continuation, and confront the algorithm with a state-of-the-art adaptive Harmonic Balance implementation. Further, we rigorously prove, for the first time, the existence of some isolated periodic solutions of the forced-damped Duffing oscillator with softening characteristic. We find that the effort for obtaining a rigorous error bound, in its present form, may be too high to be useful for many engineering problems. Based on the results obtained for a sequence of numerical examples, we conclude that Chebyshev-based stability analysis indeed permits a substantial speedup. Like Harmonic Balance itself, however, this method becomes inefficient when an extremely high truncation order is needed as, e.g., in the presence of (sharply regularized) discontinuities.

Numerical study of chronic hepatitis B infection using Marchuk–Petrov model

In this work, we briefly describe our technology developed for computing periodic solutions of time-delay systems and discuss the results of computing periodic solutions for the Marchuk-Petrov model with parameter values, corresponding to hepatitis B infection. We identified the regions in the model parameter space in which an oscillatory dynamics in the form of periodic solutions exists. The respective solutions can be interpreted as active forms of chronic hepatitis B. The period and amplitude of oscillatory solutions were traced along the parameter determining the efficacy of antigen presentation by macrophages for T- and B-lymphocytes in the model.. The oscillatory regimes are characterized by enhanced destruction of hepatocytes as a consequence of immunopathology and temporal reduction of viral load to values which can be a prerequisite of spontaneous recovery observed in chronic HBV infection. Our study presents a first step in a systematic analysis of the chronic HBV infection using Marchuk-Petrov model of antiviral immune response.

Successive approximate solutions for nonlinear oscillation and improvement of the solution accuracy with efficient non-perturbative technique

In the present study, several successive approximate solutions of the nonlinear oscillator are derived by using the efficient frequency formula. A systematical analysis of the formulation of the nonlinear frequency helps to establish a general periodic solution. Each approximation represents, individually, the solution of the nonlinear oscillator. For the optimal design and accurate prediction of structural behavior, a new optimizer is demonstrated for efficient solutions. The classical Duffing frequency formula has been modified. The numerical calculations show high agreement with the exact frequency. The justifiability of the obtained solutions is confirmed by comparison with the numerical solution. It is shown that the enhanced solution is accurate for large amplitudes and is not restricted to oscillations that have small amplitudes. The new approach can provide a perfect approximation for the nonlinear oscillation.

On the Existence and Oscillatory Solutions of Multiple Delay Differential Equation

In this paper, we introduce new conditions to prove that the existence and boundedness of the solution by convergent sequences and convergent series. The theorem of Krasnoselskii, Lebesgue’s dominated convergence theorem and fixed point theorem are used to get some sufficient conditions for the existence of solutions. Furthermore, we get sufficient conditions to guarantee the oscillatory property for all solutions in this class of equations. An illustrative example is included as an application to the main results.

Allee effect in a diffusive predator–prey system with nonlocal prey competition

Nonlocal competition and Allee effect have extensively been considered in modeling population dynamics of species independently. This paper introduces two aspects, which prey has nonlocal competition and predator is subject to Allee effect, in a predator–prey system to investigate the effects of predation on the spatial distribution of prey. The conditions for the coexistence equilibrium point to remain stable and to undergo spatially inhomogeneous Hopf bifurcation and Turing bifurcation have been studied. In the absence of Allee effect, we find that the coexistence equilibrium point of the system is locally asymptotically stable independent of the nonlocal competition. In the presence of Allee effect, nonlocal prey competition can destabilize the coexistence equilibrium point. Numerical simulations are carried out to illustrate the theoretical results. The amplitude of oscillation solution for nonlocal prey competition system is larger than local prey competition system until oscillation solution evolves to periodic solution. Also, nonlocal prey competition term can drive a spatially inhomogeneous Hopf bifurcation, and the spatially inhomogeneous periodic solution emerges. Moreover, it is showed that when the habitat domain is larger, comparing with local prey competition system, the prey diffusion coefficient of system with nonlocal prey competition needs to be larger for two species coexistence in the spatially homogeneous form.

The existence of unbounded solutions of asymmetric oscillations in the degenerate resonant case

We prove the existence of unbounded solutions of the asymmetric oscillation in the case when each zero of the discriminative function is degenerate. This is the only case that has not been studied in the literature.

On One Type of Oscillating Solutions of a Second-Order Ordinary Differential Equation with a Three-Position Hysteresis Relay and a Perturbation

A second-order ordinary differential equation with a three-position hysteresis relay characteristic and a periodic perturbation function is considered. The existence theorem is proved for an oscillatory solution with a complete traversal of the characteristic with a possible exit into its saturation zones in some finite time and with a closed phase trajectory of an arbitrary shape. Sufficient conditions for the existence of periodic solutions with arbitrary and symmetric phase trajectories are established, as well as conditions for the nonexistence of a periodic solution with a symmetric phase trajectory. Numerical examples are given.

A heuristic approach to the prediction of a periodic solution for a damping nonlinear oscillator with the non-perturbative technique

The present work attracts attention to obtaining a new result of the periodic solution of a damped nonlinear Duffing oscillator and a damped Klein–Gordon equation. It is known that the frequency response equation in the Duffing equation can be derived from the homotopy analysis method only in the absence of the damping force. We suggest a suitable new scheme successfully to produce a periodic solution without losing the damping coefficient. The novel strategy is centered on establishing an alternate equation apart from any difficulty in handling the influence of the linear damped term. This alternative equation was obtained with the rank upgrading technique. The periodic solution of the problem is presented using the non-perturbative method and validated by the modified homotopy perturbation technique. This technique is successful in obtaining new results toward a periodic solution, frequency equation, and the corresponding stability conditions. This methodology yields a more effective outcome of the damped nonlinear oscillators. With the help of this procedure, one can analyze many problems in the domain of physical engineering that involve oscillators and a linear damping influence. Moreover, this method can help all interested plasma authors for modeling different nonlinear acoustic oscillations in plasma.

Non-stationary nonzero mean probabilistic solutions of nonlinear stochastic oscillators subjected to both additive and multiplicative excitations

Non-stationary nonzero mean probabilistic solutions of nonlinear stochastic oscillators subjected to both additive and multiplicative excitations

Near-Inertial Surface Currents around Islands

Abstract Motivated by observations of enhanced near-inertial currents at the island chain of Palau, the modification of wind-generated near-inertial oscillations (NIOs) by the presence of an island is examined using the analytic solutions of Longuet-Higgins and a linear, inviscid, 1.5-layer reduced-gravity model. The analytic solution for oscillations at the inertial frequency f provides insights into flow adjustment near the island but excludes wave dynamics. To account for wave motion, the numerical model initially is forced by a large-scale wind field rotating at f, where the forcing is increased then decreased to zero. Numerical simulations are carried out over a range of island radii and the ocean response detailed. Near the island, wind energy in the frequency band near f can excite subinertial island-trapped waves and superinertial Poincaré waves. In the small-island limit, both the Poincaré waves and the island-trapped waves are very near f, and their sum resembles the Longuet-Higgins analytic solution but with increased amplitude near the island. The flow field can be viewed as primarily a far-field NIO locally deflected by the island plus an island-trapped contribution, leading to enhanced near-inertial currents near the island, on the scale of the island radius. As the island size is increased, the island-trapped wave frequency deviates further from f and its amplitude depends strongly on the frequency bandwidth and wavenumber structure of the wind forcing. In the large-island limit, the island-trapped wave resembles a Kelvin wave, and the sum of incident and reflected Poincaré waves suppresses the near-inertial current amplitude near the island. Significance Statement Strong, impulsive winds over the ocean excite currents that rotate in the opposite direction to Earth’s rotation. This work examines how these wind-generated currents, known as near-inertial oscillations (NIOs), are modified by the presence of an island. Around small islands, the primary response is locally enhanced near-inertial currents. Alternatively, around large islands, near-inertial currents are weaker. Understanding how these currents behave should provide insight into the physical processes that drive current variability near islands and spur local mixing.

An Insight into the Dynamical Behaviour of the Swing Equation

Motivated by the nonlinear dynamics of mathematical models encountered in power systems, an investigation into the dynamical behaviour of the swing equation is carried out. This paper examines analytically and numerically the development of oscillatory periodic solutions, whereby increases of the control parameter, lead to a cascade of period doubling bifurcations, before eventually loss in stability is exhibited and effective forerunners to chaos revealed. Gaining an understanding on the dynamical behaviour of the system can help to produce a deeper insight of the bifurcations entailed, with the appearance of the triggered sequence of the first period doubling’s acting as precursors of imminent danger and difficult operations of a practical system.

Improved Runge-Kutta Method for Oscillatory Problem Solution Using Trigonometric Fitting Approach

This paper provides a four-stage Trigonometrically Fitted Improved Runge-Kutta (TFIRK4) method of four orders to solve oscillatory problems, which contains an oscillatory character in the solutions. Compared to the traditional Runge-Kutta method, the Improved Runge-Kutta (IRK) method is a natural two-step method requiring fewer steps. The suggested method extends the fourth-order Improved Runge-Kutta (IRK4) method with trigonometric calculations. This approach is intended to integrate problems with particular initial value problems (IVPs) using the set functions and for trigonometrically fitted. To improve the method's accuracy, the problem primary frequency is used. The novel method is more accurate than the conventional Runge-Kutta method and IRK4. Several test problems for the system of first-order ordinary differential equations carry out numerically to demonstrate the effectiveness of this approach. The computational studies show that the TFIRK4 approach is more efficient than the existing Runge-Kutta methods.

Oscillations in gas-grain astrochemical kinetics

ABSTRACT We have studied gas-grain chemical models of interstellar clouds to search for non-linear dynamical evolution. A prescription is given for producing oscillatory solutions when a bistable solution exists in the gas-phase chemistry and we demonstrate the existence of limit cycle and relaxation oscillation solutions. As the autocatalytic chemical processes underlying these solutions are common to all models of interstellar chemistry, the occurrence of these solutions should be widespread. We briefly discuss the implications for interpreting molecular cloud composition with time-dependent models and some future directions for this approach.

An Accuracy-preserving Block Hybrid Algorithm for the Integration of Second-order Physical Systems with Oscillatory Solutions

It is a known fact that in most cases, to integrate an oscillatory problem, higher order A-stable methods are often needed. This is because such problems are characterized by stiffness, chaos and damping, thus making them tedious to solve. However, in this research, an accuracy-preserving relatively lower order Block Hybrid Algorithm (BHA) is proposed for solution of second-order physical systems with oscillatory solutions. The sixth order algorithm was derived using interpolation and collocation of power series within a single step interval [tn; tn+1]. In order to circumvent the Dahlquist-barrier and also obtain an accuracy-preserving algorithm, four o-step points were incorporated within the single step interval. A number of special cases of oscillatory problems were solved using the proposed method and the results obtained clearly showed that it outperformed other existing methods we compared our results with even though the BHA is of lower order relative to such methods. Some of the second-order physical systems considered were the Kepler, Bessel and damped problems. Some important properties of the BHA were also analyzed and the results of the analysis showed that it is consistent, zero-stable and convergent

Stellar instability from parametric resonance

In this paper we examine the stability of stellar configurations in which the interior solution is described by a closed Friedmann-Lemaître-Robertson-Walker (FLRW) geometry sourced with a charged pressureless fluid and radiation. An interacting vacuum component and a conformally coupled massive scalar field are also included. Given a simple factor for the energy transfer between the pressureless fluid and the vacuum component we obtain bounded interior oscillatory solutions. We show that in proper domains of the parameter space the interior dynamics is highly unstable so that the break of the Kolmogorov–Arnold–Moser (KAM) tori leads to a disruptive ejection of mass. For such configurations the interior solution asymptotically matches an exterior Reissner–Nordström–de Sitter spacetime.

Classical and quantum dynamics of over-damped non linear systems

Overdamping is a regime in which friction is sufficiently large that the motion either decays to its equilibrium position or it crosses the equilibrium position exactly once before returning monotonically towards the equilibrium position. The phenomena of overdamping has been studied classically and quantum mechanically only for the case of the linear damped harmonic oscillator. Here we study the classical and quantum dynamics of a family of over-damped non linear systems. The main objective of this paper is to find a Lagrangian and Hamiltonian framework to study over-damped non linear systems and to show that a quantum mechanical description can be developed in the momentum representation. Our results reduce to the well known solution of the linear damped harmonic oscillator when the non linear part is set to zero.

Linearized Learning with Multiscale Deep Neural Networks for Stationary Navier-Stokes Equations with Oscillatory Solutions

Linearized Learning with Multiscale Deep Neural Networks for Stationary Navier-Stokes Equations with Oscillatory Solutions

Exact solution and coherent states of an asymmetric oscillator with position-dependent mass

We revisit the problem of the deformed oscillator with position-dependent mass [da Costa et al., J. Math. Phys. 62, 092101 (2021)] in the classical and quantum formalisms by introducing the effect of the mass function in both kinetic and potential energies. The resulting Hamiltonian is mapped into a Morse oscillator by means of a point canonical transformation from the usual phase space (x, p) to a deformed one (xγ, Πγ). Similar to the Morse potential, the deformed oscillator presents bound trajectories in phase space corresponding to an anharmonic oscillatory motion in classical formalism and, therefore, bound states with a discrete spectrum in quantum formalism. On the other hand, open trajectories in phase space are associated with scattering states and continuous energy spectrum. Employing the factorization method, we investigate the properties of the coherent states, such as the time evolution and their uncertainties. A fast localization, classical and quantum, is reported for the coherent states due to the asymmetrical position-dependent mass. An oscillation of the time evolution of the uncertainty relationship is also observed, whose amplitude increases as the deformation increases.