Existence Of Periodic Oscillations Research Articles

Analysis of ballistic transport and resonance in the α-Fermi-Pasta-Ulam-Tsingou model.

Ballistic transport and resonance phenomena are elucidated in the one-dimensional α-Fermi-Pasta-Ulam-Tsingou (FPUT) model using an approach of computing thermal response functions. The existence of periodic oscillations in spatially sinusoidal temperature profiles seen in previous studies is confirmed. However, the results obtained using response functions enable a more complete understanding. In particular, it is shown that resonance involves beats between normal modes which tend to reinforce in a one-dimensional chain. Anharmonic scattering acts to destroy phase coherence across the statistical ensemble, and with increasing anharmonicity, transport is driven toward the diffusive regime. These results provide additional insight into anomalous heat transport in low-dimensional systems. Normal-mode scattering is also explored using time correlation functions. Interestingly, these calculations, in addition to demonstrating loss of phase coherence across an ensemble of simulations, appear to show evidence of so-called q-breathers in conditions of strong anharmonicity. Finally, we describe how the approach outlined here could be developed to include quantum statistics and also also first-principles estimates of phonon scattering rates to elucidate second sound and ballistic transport in realistic materials at low temperatures.

Spatial movement with diffusion and memory-based self-diffusion and cross-diffusion

Spatial memory has been considered significant in animal movement modeling. In this paper, we formulate a two-species interaction model by incorporating both random walk and spatial memory-based walk in their movement. The spatial memory-based walk, described by a chemotactic-like term, is derived by a modified Fick's law involving a directed movement toward the gradient of the density distribution function at a past time. For the proposed model, local stability and bifurcations are studied at constant steady states. Unlike a classical reaction-diffusion equation, we show that the accumulation points of eigenvalues for the model will locate at a vertical line in the complex plane, which will make the model generate spatially inhomogeneous time-periodic patterns through Hopf bifurcation. As illustrations, we apply these results to competition and cooperative models with memory-based diffusion. For the competition model, it turns out that the outcomes are far more complicated than those of classic Lotka-Volterra reaction-diffusion models, due to the consideration of memory-based diffusion. In particular, the existence of periodic oscillations is proved under weak competition. Similar conclusions hold for the cooperative model.

Stability and bifurcation analysis of a gene expression model with small RNAs and mixed delays

This paper investigates a gene expression model, which is mediated by sRNAs (small RNAs) and includes discrete and distributed delays. We take both the strong and weak kernel forms of distributed delay into consideration. The discrete time delay is chosen as the bifurcation parameter. By analyzing the distribution of characteristic values, we obtain the sufficient conditions of stability and examine the existence of periodic oscillations. When the discrete time delay is small and not greater than the threshold, the equilibrium of the gene expression model is asymptotically stable. When the bifurcation parameter exceeds the critical value, the model can produce limit cycles. Finally, numerical simulations are implemented to verify the correctness of our theoretical results.

Stability switches in a Cohen–Grossberg neural network with multi-delays

In this paper, we consider a Cohen–Grossberg neural network with three delays. Regarding time delays as a parameter, we investigate the effect of time delays on its dynamics. We show that there exist stability switches for time delays under certain conditions and the conditions for the existence of periodic oscillations are given by discussing the associated characteristic equation. Numerical simulations are given to illustrate the obtained results and interesting network behaviors are observed, such as multiple stability switches of the network equilibrium and synchronous (asynchronous) oscillations.

Periodic oscillatory behavior on a four-node neural network model with distributed delay

This paper investigates the existence of periodic oscillation for a four-node neural network with distributed delay. Two theorems are provided to determine the conditions for periodic oscillations of the model. The criteria for selecting the parameters in this network are derived. Some simulation examples are presented to illustrate the accuracy of the results.

Switches of oscillations in coupled networks with multiple time delays

This paper reveals the behaviors of a coupled network consisting of four sub-networks each with two neurons and multiple couplings between the sub-networks. Time delays are introduced into the internal connections within the sub-networks and the couplings between the sub-networks. The stability switches of the network equilibrium are analyzed and the conditions for the existence of periodic oscillations are given by discussing the associated characteristic equation. Numerical simulations are performed to illustrate the effectiveness of the obtained results and interesting network behaviors are observed, such as multiple switches of periodic oscillations.

Geographical variation in the periodicity of gypsy moth outbreaks

The existence of periodic oscillations in populations of forest Lepidoptera is well known. While information exists on how the periods of oscillations vary among different species, there is little prior evidence of variation in periodicity within the range of a single Lepidopteran species. The exotic gypsy moth is an introduced foliage‐feeding insect in North America. Its populations are characterized by high amplitude oscillations between innocuously low densities and outbreak levels during which large regions of forests are defoliated. These outbreaks are recognized to arise periodically with considerable synchrony across much of the gypsy moth's North American range. Our analysis indicates that gypsy moth outbreaks in North America are periodic but they exhibit two dominant periodicities: a primary period of 8–10 yr (as previously reported) and a secondary period of 4–5 yr (a new finding in this study). The outbreak periodicity varied geographically and this variation was associated with forest type. We found that in the most susceptible forest types, those on xeric sites where oak is often mixed with pines, outbreak periodicity had a more dominant 5‐yr period while in forest types characteristic of more mesic sites where oak was mixed with maples and other species, cycles were clearly operating on a 10‐yr period.

Nonlinear Oscillations in a System with Dry Friction

The case is examined where the right-hand side of the equations of motion is discontinuous. Attraction only in the stick domain ensures existence of periodic oscillations. Sufficient stability conditions for the periodic solution of a nonlinear system with dry friction are established

Periodic coexistence in the chemostat with three species competing for three essential resources

A chemostat model of three species of microorganisms competing for three essential, growth-limiting nutrients is considered. J. Husiman and F.J. Weissing [Nature 402 (1999) 407] show numerically that this model can generate periodic oscillations. The present contribution is concerned with rigorous analysis regarding the existence of periodic oscillations in this model. Our analysis is based on the following observation made by Huisman and Weissing: there is a cyclic replacement of species, if each species becomes limited by the resource for which it is the intermediate competitor. Using a permanence theory, an index theory, and a Poincaré–Bendixson theory for three-dimensional competitive systems, we analytically succeed to give sufficient conditions for the existence of periodic orbits in the limit sets in this model. The results in this paper suggest that with a wide range of parameter values, sustained periodic oscillations of species abundances for the model are possible, without involving external disturbances. Our results also suggest that competition is not necessarily destructive, i.e., in the case of existence of sustained periodic oscillations, if one of three competitors is absent, one of the other two rivals cannot survive.

Nonlinear semigroups and the existence and stability of solutionsof semilinear nonautonomous evolution equations

This paper is concerned with the existence and stability of solutions of a class of semilinear nonautonomous evolution equations. A procedure is discussed which associates to each nonautonomous equation the so‐called evolution semigroup of (possibly nonlinear) operators. Sufficient conditions for the existence and stability of solutions and the existence of periodic oscillations are given in terms of the accretiveness of the corresponding infinitesimal generator. Furthermore, through the existence of integral manifolds for abstract evolutionary processes we obtain a reduction principle for stability questions of mild solutions. The results are applied to a class of partial functional differential equations.

Dynamic behaviour of an isothermal fixed-bed reactor with narrow activity region catalyst

The dynamic behaviour of an isothermal axial dispersion fixed-bed reactor packed by catalyst pellets with Dirac delta distributed activity in the case of bimolecular Langmuir—Hinshelwood kinetics is studied. The existence of periodic oscillations is numerically established. The influence of the active point location, Thiele modulus and Damköhler number on the dynamic behaviour is discussed.

Analytical theory of non-linear oscillations

In this paper, we improve on the results of two previous papers (8, 9) by establishing a general existence theorem (section 1·3, below) for a class of periodic oscillations of a wide class of non-linear differential equations of the second order in the real domain which are perturbations of the autonomous differential equationwhere g(x) is strictly non-linear. We then, by way of illustrating the power of the theorem, apply it to the problems which Morris (section 2·2 below), Shimuzu (section 2·3 below) and Loud (section 2·5 below) set themselves on the existence of periodic oscillations of certain differential equations which are perturbations of equations of the form (1·1·1).

Periodic oscillations in integral pulse frequency-modulated feedback systems†

This paper presents a systematic method of investigating the existence of periodic oscillations in feedback systems with integral pulse frequency modulation. The method is based on the non-linear discrete equivalence of the system. Extension of results to neural pulse frequency modulation systems is discussed.

Periodic Oscillations in Feedback Systems with Neural Pulse Frequency Modulation

In this paper a study of effect of neural pulse frequency modulation (NPFM) on feedback systems is attempted. The overall system is discretized and a non-linear discrete equivalence of a NPFM feedback system is obtained. It is also shown that NPFM systems exhibit periodic oscillations and conditions for the existence of periodic oscillations in NPFM systems are obtained using the non-linear discrete equivalence. A method of implementation of a neural pulse frequency modulator as well as a few experimental results are presented.