Local Attractor Research Articles

Transition of patterns in the cell-chemotaxis system with proliferation source

Phase transition analysis is a fundamental component in understanding pattern-forming behavior of many biochemical systems, such as those found in developmental biology. The purpose of this work is to analyze the phase transition property of a diffusion–chemotaxis model with proliferation source, a macroscopic model of mobile species aggregation. Along the way, we will show that the system exhibits a very rich pattern-forming behavior, resulting in a competition between hexagonal, roll, and rectangular patterns. In particular, we show existence of regular hexagonal patterns in a confined spatial geometry where only two modes become unstable. It is also shown that they are either saddle points or attracting nodes. Moreover, we show that the competition happens inside a local attractor which consists of a finite number of stationary solutions and their connecting heteroclinic orbits. The structure of this attractor will be precisely determined as well.

Asymptotic dynamics of the Leslie-Gower competition system with Allee effects and stocking

We investigate the classical Leslie-Gower competition system where one of the two competing populations is subject to Allee effects and is also under constant stocking. The model can have either no interior steady state, a unique interior steady state, two interior steady states or three interior steady states depending on parameter values. Using the tools of monotone planar systems, we provide basins of attraction for the local attractors and for the non-hyperbolic steady states. It is concluded that stocking of the weaker competitor can promote the coexistence of both competing populations.

一类双约束单自由度碰振系统的擦边运动分析

本文分析了一类单自由度双侧约束碰振系统的擦边周期运动的稳定性。利用不连续映射的方法建立了擦边周期轨道的局部Poincare映射。在此基础上得到了双擦周期轨道的稳定性判据。根据此判据，可知系统在双擦周期轨道附近不存在局部吸引子，即，发生不连续擦边分岔。最后，用数值结果验证了理论方法的可行性。 The stability of grazing periodic motion in a single degree of freedom vibro-impact system with double constrains is analyzed. The Poincare mapping near the grazing trajectory is established by using the discontinuity mapping method. And the stability criterion of double grazing periodic motion is obtained. According to the criterion, it is demonstrated that local attractors do not exist near the double grazing trajectory, i.e., the grazing bifurcation is discontinuous. Finally, validity of the theoretical analysis is verified by the numerical results.

Computation of Lyapunov functions for systems with multiple local attractors

We present a novel method to compute Lyapunov functions for continuous-time systems with multiple local attractors. In the proposed method one first computes an outer approximation of the local attractors using a graph-theoretic approach.Then a candidate Lyapunov function is computed using a Massera-like construction adapted to multiple local attractors. In the final step this candidate Lyapunov function is interpolated over the simplices of a simplicialcomplex and, by checking certain inequalities at the vertices of the complex, we can identify the region in which the Lyapunov function is decreasing along system trajectories. The resulting Lyapunov functiongives information on the qualitative behavior of the dynamics, including lower bounds on the basins of attraction of the individual local attractors.We develop the theory in detail and present numerical examples demonstrating the applicability of our method.

Morse decomposition of global attractors with infinite components

In this paper we describe some dynamical properties of a Morse decomposition with a countable number of sets. In particular, we are able to prove that the gradient dynamics on Morse sets together with a separation assumption is equivalent to the existence of an ordered Lyapunov function associated to the Morse sets and also to the existence of a Morse decomposition -that is, the global attractor can be described as an increasing family of local attractors and their associated repellers.

Число локальных аттракторов безмасштабных сетей Хопфилда

Estimates of the number of local attractors for the Hopfield model of attractor neural network with continuous time and states where the neuron interaction graph has a scale-free structure are considered. The number of local attractors defines the network capacity, which is an important network characteristic. Numerous works were devoted to the problem of capacity estimations but mainly Boolean networks and the Hopfield models with symmetric interactions were studied. In the second case the capacity is proportional to the neuron number N. An estimation of the capacity via characteristics of the network interaction graph is found. This estimate implies that the capacity may increase as exp( cN a ), where c, a > 0. Furthermore, a formula, which connects the capacity and the number of strongly connected neurons (hubs) in the network has been found by computer simulations. We show that the logarithm of the capacity is proportional to the hub number and the hub number is proportional to the root of N. Results can have applications to associative memory neural models and morphogenesis modeling by genetic networks.

EEG Analysis for Olfactory Perceptual-Ability Measurement Using a Recurrent Neural Classifier

A recurrent neural network model is designed to classify (pretrained) aromatic stimuli and discriminate noisy stimuli of both similar and different genres, using EEG analysis of the experimental subjects. The design involves determining the weights of the selected recurrent dynamics so that for a given base stimulus, the dynamics converges to one of several optima (local attractors) on the given Lyapunov energy surface. Experiments undertaken reveal that for small noise amplitude below a selected threshold, the dynamics essentially converges to fixed stable attractor. However, with a slight increase in noise amplitude above the selected threshold, the local attractor of the dynamics shifts in the neighborhood of the attractor obtained for the noise-free standard stimuli. The other important issues undertaken in this paper include a novel algorithm for evolutionary feature selection and data-point reduction from multiple experimental EEG trials using principal component analysis. The confusion matrices constructed from experimental results show a marked improvement in classification accuracy in the presence of data point reduction algorithm. Statistical tests undertaken indicate that the proposed recurrent classifier outperforms its competitors with classification accuracy as the comparator. The importance of this paper is illustrated with a tea-taster selection problem, where an olfactory perceptual-ability measure is used to rank the tasters.

Two species competitive model with the Allee effect

We consider the following system of difference equations: , , , where a, b, c, d are positive constants and are initial conditions. This system has interesting dynamics and can have up to nine equilibrium points. The most complex and perhaps most interesting case is the case of nine equilibrium points, four of which are local attractors, four of which are saddle points, and one of which is a repeller. Using recent results of Kulenović and Merino we are able to characterize the basins of attractions of all local attractors and thus to describe the global dynamics of this system. This case can be considered as a two-dimensional version of the Allee effect for competitive systems. MSC:39A10, 39A30, 37G35.

Subharmonic parametric resonance of simply supported buckled beams

This study presents the nonlinear vibrations of simply supported beams that are subjected to a subharmonic resonance excitation of order two. The applied axial force has a static component to buckle the beam and a dynamic component for parametric excitation. The equation governing the nonlinear transverse deformations is an integral partial differential equation. The static counter part of the governing equation represents the nonlinear buckling problem that is solved for the critical buckling loads and exact buckled configurations. A reduced-order model based on the Galerkin’s discretization is obtained where a multi-mode discretization is assumed. The significance of the number of modes retained in the discretization is examined. It is found out that for simply supported beams, the second and higher modes have no contribution on either the static or the dynamic response of the beam. To study the nonlinear dynamics of the problem, a shooting method is used to locate periodic orbits and long-time numerical integration using fifth-order Runge–Kutta method is used to find the fast Fourier transforms of the response. A forward sweep, where the amplitude of the force is increased, while the frequency is kept fixed, is used to drive the beam. We find local attractors coexisting with snapthrough motion. Nonlinear instabilities such as period-doubling leading to chaos, symmetry-breaking, cyclic-fold, period-demultiplying, and jumping are reported. A bifurcation diagram showing all possible attractors near and far from the equilibrium positions is presented. This diagram is a helpful tool to design beams for a desired response especially for energy harvesting applications where high-amplitude vibration is needed.

Regular and chaotic regimes in Saltzman model of glacial climate dynamics under the influence of additive and parametric noise

It is well-known that the climate system, due to its nonlinearity, can be sensitive to stochastic forcing. New types of dynamical regimes caused by the noise-induced transitions are revealed on the basis of the classical climate model previously developed by Saltzman with co-authors and Nicolis. A complete parametric classification of dynamical regimes of this deterministic model is carried out. On the basis of this analysis, the influence of additive and parametric noises is studied. For weak noise, the climate system is localized nearby deterministic attractors. A mixture of the small and large amplitude oscillations caused by noise-induced transitions between equilibria and cycle attraction basins arise with increasing the noise intensity. The portion of large amplitude oscillations is estimated too. The parametric noise introduced in two system parameters demonstrates quite different system dynamics. Namely, the noise introduced in one system parameter increases its dispersion whereas in the other one leads to the stabilization of the climatic system near its unstable equilibrium with transitions from order to chaos.

Place cell rate remapping by CA3 recurrent collaterals.

Episodic-like memory is thought to be supported by attractor dynamics in the hippocampus. A possible neural substrate for this memory mechanism is rate remapping, in which the spatial map of place cells encodes contextual information through firing rate variability. To test whether memories are stored as multimodal attractors in populations of place cells, recent experiments morphed one familiar context into another while observing the responses of CA3 cell ensembles. Average population activity in CA3 was reported to transition gradually rather than abruptly from one familiar context to the next, suggesting a lack of attractive forces associated with the two stored representations. On the other hand, individual CA3 cells showed a mix of gradual and abrupt transitions at different points along the morph sequence, and some displayed hysteresis which is a signature of attractor dynamics. To understand whether these seemingly conflicting results are commensurate with attractor network theory, we developed a neural network model of the CA3 with attractors for both position and discrete contexts. We found that for memories stored in overlapping neural ensembles within a single spatial map, position-dependent context attractors made transitions at different points along the morph sequence. Smooth transition curves arose from averaging across the population, while a heterogeneous set of responses was observed on the single unit level. In contrast, orthogonal memories led to abrupt and coherent transitions on both population and single unit levels as experimentally observed when remapping between two independent spatial maps. Strong recurrent feedback entailed a hysteretic effect on the network which diminished with the amount of overlap in the stored memories. These results suggest that context-dependent memory can be supported by overlapping local attractors within a spatial map of CA3 place cells. Similar mechanisms for context-dependent memory may also be found in other regions of the cerebral cortex.

Allee effects in a Ricker-type predator–prey system

We study a discrete host–parasitoid system where the host population follows the classical Ricker functional form and is also subject to Allee effects. We determine basins of attraction of the local attractors of the single population model when the host intrinsic growth rate is not large. In this situation, existence and local stability of the interior steady states for the host–parasitoid interaction are completely analysed. If the host's intrinsic growth rate is large, then the interaction may support multiple interior steady states. Linear stability of these steady states is provided.

A pesca e o conhecimento ecológico local dos pescadores de acará-disco (Symphysodon aequifasciatus, Pellegrin 1904: Cichlidae) na Reserva de Desenvolvimento Sustentável Piagaçu-Purus, baixo rio Purus, Brasil

O acará-disco (Symphysodon aequifasciatus, Pellegrin 1904), peixe endêmico da bacia amazônica, é uma das espécies ornamentais mais conhecidas do mundo. Apesar disso, informações sobre a explotação desse animal em vida livre são quase inexistentes na literatura científica. Nesse artigo, a pesca de pequena escala do acará-disco na Reserva de Desenvolvimento Sustentável Piagaçu-Purus (RDS-PP), baixo rio Purus, Amazonas, é caracterizada com base em entrevistas estruturadas e abertas com pescadores e habitantes, bem como em observações de campo. A pesca do acará-disco é sazonal, ocorrendo durante o período de seca. É realizada por pescadores locais, utilizando atratores de pesca e uma técnica de detecção e estimativa de abundância por mergulho em apneia. O armazenamento local é feito em tanques de madeira e os peixes são transportados em recipientes plásticos para distribuição nos mercados regionais por meio de barcos dos próprios pescadores, barcos de linha ou jangadas. Os peixes são vendidos para empresas exportadoras especializadas de Manaus. Neste estudo, descrevemos o conhecimento ecológico local dos pescadores de acará-disco na RDS-PP e discutimos sua importância para a gestão da pesca nessa unidade de conservação.

Self-similar erbium-doped fiber laser with large normal dispersion

We report a large normal dispersion erbium-doped fiber laser with self-similar pulse evolution in the gain fiber. The cavity is stabilized by the local nonlinear attractor in the gain fiber through the use of a narrow filter. Experimental results are accounted for by numerical simulations. This laser produces 3.5 nJ pulses, which can be dechirped to 70 fs with an external grating pair.

Particle Swarm Optimization Based on Local Attractors of Ordinary Differential Equation System

Particle swarm optimization (PSO) is inspired by sociological behavior. In this paper, we interpret PSO as a finite difference scheme for solving a system of stochastic ordinary differential equations (SODE). In this framework, the position points of the swarm converge to an equilibrium point of the SODE and the local attractors, which are easily defined by the present position points, also converge to the global attractor. Inspired by this observation, we propose a class of modified PSO iteration methods (MPSO) based on local attractors of the SODE. The idea of MPSO is to choose the next update state near the present local attractor, rather than the present position point as in the original PSO, according to a given probability density function. In particular, the quantum-behaved particle swarm optimization method turns out to be a special case of MPSO by taking a special probability density function. The MPSO methods with six different probability density functions are tested on a few benchmark problems. These MPSO methods behave differently for different problems. Thus, our framework not only gives an interpretation for the ordinary PSO but also, more importantly, provides a warehouse of PSO-like methods to choose from for solving different practical problems.

Multistability and localized attractors in a dissipative discrete NLS equation

We consider a finite discrete nonlinear Schrodinger equation with localized forcing, damping, and nonautonomous perturbations. In the autonomous case these systems are shown numerically to have multiple attracting spatially localized solutions. In the nonautonomous case we study analytically some properties of the pullback attractor of the system, assuming that the origin of the corresponding autonomous system is hyberbolic. We also see numerically the persistence of multiple localized attracting states under different types of nonautonomous perturbations.

Variations of characteristics of the mean atmospheric energy level in the area of the Franz Josef Land archipelago in 1935–2012

Proposed is a method for computing the average temperature of the vertical column of the atmosphere (the temperature of the average energy level) based on some features of energy characteristics of the atmosphere and using the radiosonde data within the mid-troposphere. The modem database is supplemented with the data of radio sounding carried out at Russian upper-air stations in 1934–1959. Variations of average annual values of temperature of the mean atmospheric energy level are observed with the period of several decades and with the amplitude of 4°C in 1935–2012. Intensive decrease in the mean annual values of height-integrated temperature has been registered in recent years. Long-period variations of its average seasonal values of the same nature are registered. They are most pronounced in winter and transition seasons and are significantly reduced in summer. The observed oscillations indicate the existence of disturbance sources of long-term scale that is typical of the evolution of the anomalies of the sea surface temperature in the North Atlantic. The nature of long-term changes in the temperature of the mean energy level enables to assume the existence of a local attractor in atmospheric changes near the Franz Josef Land archipelago associated with the features of the thermal state of the North European basin and with the ice regime, first of all, in the Barents Sea. The temperature of the mean energy level depends weakly on local greenhouse effects that allows distinguishing natural (nonantropogenic) causes of atmospheric disturbances in a more explicit form.

Mirror Transform to Conquer Deception for Particle Swarm Optimization

Particle swarm optimization meets difficulties when handling deceptive problems, because the local optimal attractors misguide the particles and pull them away from the global optimum. Function transform that changes the shape to make it easier for the algorithm to optimize is an efficient way to change the original attraction, but the existing strategies cannot conquer deception. Therefore we propose the mirror transform by simply reversing the original attraction. Experimental results validate the efficiency of our strategy.

Chemical and physical insight on the local properties of the phosphides XSiP2 (X = Be, Mg, Cd, Zn and Hg) under pressure: from first principles calculations

Local properties of the XSiP2 (X = Be, Mg, Cd, Zn and Hg) compounds are revisited through the partition of static thermodynamic properties under pressure. We pay attention to the metallization that occurs when the investigated compounds undergo a phase transition from chalcopyrite to the NaCl structure. Electron localization function analysis shows that the local valence basin attractors values decrease as a function of pressure. As the pressure increases, the tetragonal distortion (c/a) diminishes while the degree of ionicity enhances. In addition, by means of atom in molecule approach, atomic-like local compressibility and pressures are analyzed. We found that the basins volumes of the investigated compounds in the NaCl phase have lower compressibilities than those in the chalcopyrite phase. According to the predicted core-valence basins, the phosphorus cation is found to be the more affected by the hydrostatic pressure.

Local Behavior Near Quasi-Periodic Solutions of Conformally Symplectic Systems

We study the behaviour of conformally symplectic systems near rotational Lagrangian tori. We recall that conformally symplectic systems appear for example in mechanical models including a friction proportional to the velocity. We show that in a neighborhood of these Lagrangian, invariant, rotational tori, one can always find a smooth symplectic change of variables in which the time evolution becomes just a rotation in some direction and a linear contraction in others. In particular quasi-periodic solutions with $$n$$ independent frequencies of contractive (expansive) diffeomorphisms are always local attractors (repellors). We present results when the systems are analytic, $$C^r$$ or $$C^\infty $$ . We emphasize that the results presented here are non-perturbative and apply to systems that are far from integrable; moreover, we do not require any assumption on the frequency and in particular we do not assume any non-resonance condition. We also show that the system of coordinates can be computed rather explicitly and we provide iterative algorithms, which allow to generalize the notion of “isochrones”. We conclude by showing that the above results apply to quasi-periodic conformally symplectic flows.