Dynamic Bifurcation Research Articles

Biological emergent properties in non-spiking neural networks

<abstract><p>A central goal of neuroscience is to understand the way nervous systems work to produce behavior. Experimental measurements in freely moving animals (<italic>e.g.</italic> in the <italic>C. elegans</italic> worm) suggest that ON- and OFF-states in non-spiking nervous tissues underlie many physiological behaviors. Such states are defined by the collective activity of non-spiking neurons with correlated up- and down-states of their membrane potentials. How these network states emerge from the intrinsic neuron dynamics and their couplings remains unclear. In this paper, we develop a rigorous mathematical framework for better understanding their emergence. To that end, we use a recent simple phenomenological model capable of reproducing the experimental behavior of non-spiking neurons. The analysis of the stationary points and the bifurcation dynamics of this model are performed. Then, we give mathematical conditions to monitor the impact of network activity on intrinsic neuron properties. From then on, we highlight that ON- and OFF-states in non-spiking coupled neurons could be a consequence of bistable synaptic inputs, and not of intrinsic neuron dynamics. In other words, the apparent up- and down-states in the neuron's bimodal voltage distribution do not necessarily result from an intrinsic bistability of the cell. Rather, these states could be driven by bistable presynaptic neurons, ubiquitous in non-spiking nervous tissues, which dictate their behaviors to their postsynaptic ones.</p></abstract>

Dynamic analysis of a new aquatic ecological model based on physical and ecological integrated control.

Within the framework of physical and ecological integrated control of cyanobacteria bloom, because the outbreak of cyanobacteria bloom can form cyanobacteria clustering phenomenon, so a new aquatic ecological model with clustering behavior is proposed to describe the dynamic relationship between cyanobacteria and potential grazers. The biggest advantage of the model is that it depicts physical spraying treatment technology into the existence pattern of cyanobacteria, then integrates the physical and ecological integrated control with the aggregation of cyanobacteria. Mathematical theory works mainly investigate some key threshold conditions to induce Transcritical bifurcation and Hopf bifurcation of the model (2.1), which can force cyanobacteria and potential grazers to form steady-state coexistence mode and periodic oscillation coexistence mode respectively. Numerical simulation works not only explore the influence of clustering on the dynamic relationship between cyanobacteria and potential grazers, but also dynamically show the evolution process of Transcritical bifurcation and Hopf bifurcation, which can be clearly seen that the density of cyanobacteria decreases gradually with the evolution of bifurcation dynamics. Furthermore, it should be worth explaining that the most important role of physical spraying treatment technology can break up clumps of cyanobacteria in the process of controlling cyanobacteria bloom, but cannot change the dynamic essential characteristics of cyanobacteria and potential grazers represented by the model (2.1), this result implies that the physical spraying treatment technology cannot fundamentally eliminate cyanobacteria bloom. In a word, it is hoped that the results of this paper can provide some theoretical support for the physical and ecological integrated control of cyanobacteria bloom.

Dynamic analysis of a modified algae and fish model with aggregation and Allee effect.

In the paper, under the stress of aggregation and reproduction mechanism of algae, we proposed a modified algae and fish model with aggregation and Allee effect, its main purpose was to further ascertain the dynamic relationship between algae and fish. Several critical conditions were investigated to guarantee the existence and stabilization of all possible equilibrium points, and ensure that the model could undergo transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation and B-T bifurcation. Numerical simulation results of related bifurcation dynamics were provided to verify the feasibility of theoretical derivation, and visually demonstrate the changing trend of the dynamic relationship. Our results generalized and improved some known results, and showed that the aggregation and Allee effect played a vital role in the dynamic relationship between algae and fish.

Dynamics Analysis of an Aquatic Ecological Model with Allee Effect

In this paper, based on the dynamic relationship between algae and protozoa, an aquatic ecological model with Allee effect was established to investigate how some ecological environment factors affect coexistence mode of algae and protozoa. Mathematical derivation works mainly gave some key conditions to ensure the existence and stability of all possible equilibrium points, and to induce the occurrence of transcritical bifurcation and Hopf bifurcation. The numerical simulation works mainly revealed ecological relationship change characteristics of algae and protozoa with the help of bifurcation dynamics evolution process. Furthermore, it was also worth emphasizing that Allee effect had a strong influence on the dynamic relationship between algae and protozoa. In a word, it was hoped that the research results could provide some theoretical support for algal bloom control, and also be conducive to the rapid development of aquatic ecological models.

A fractional-order discrete memristor neuron model: Nodal and network dynamics

<abstract><p>We discuss the dynamics of a fractional order discrete neuron model with electromagnetic flux coupling. The discussed neuron model is a simple one-dimensional map which is modified by considering flux coupling. We consider a discrete fractional order memristor to mimic the effects of electromagnetic flux on the neuron model. The bifurcation dynamics of the fractional order neuron map show an inverse period-doubling route to chaos as a function of control parameters, namely the fractional order of the map and the flux coupling coefficient. The bifurcation dynamics of the systems are derived both in the time and frequency domains. We present a two-parameter phase diagram using the Lyapunov exponent to categorize the various dynamics present in the system. In addition to the Lyapunov exponent, we use the entropy of the model to distinguish the various dynamics of the systems. To investigate the network behavior of the fractional order neuron map, a lattice array of $ N\times N $ nodes is constructed and external periodic stimuli are applied to the network. The formation of spiral waves in the network and the impact of various parameters, like the fractional order, flux coupling coefficient and the coupling strength on the wave propagation are also considered in our analysis.</p></abstract>

Dynamic properties for a stochastic food chain model

In this paper, Lyapunov exponents of ergodic invariant measures are used to study dynamic properties for a stochastic food chain model, which consists of two competing predators and one prey. Ayala’s experimental result, or rather, competitive coexistence is showed to be possible in this random case. Furthermore, the considered stochastic model have five points of dynamical bifurcation, which happen to be the thresholds (between survival and extinction) of the system or each species. In addition, the necessity of introducing environmental noise is verified by the fact that environmental driving force can drive the system towards extinction from partial extinction or coexistence. Moreover, all the theoretical results are well verified by numerical simulations. It is worth mentioning that we make a first attempt at using meshing method and statistical data to test Lyapunov exponents for two-dimensional boundary measures, and this is an innovation in the numerical methods.

Bifurcation Trees of (1:2)- Asymmetric Periodic Motions with Corresponding Infinite Homoclinic Orbits in the Lorenz System

In this paper, bifurcation dynamics and infinite homoclinic orbits of (1:2)- asymmetric periodic motions in the Lorenz systems are studied through the discrete mapping method. The infinite homoclinic orbits pertaining to the unstable periodic motions on the bifurcation trees of (1:2)-asymmetric periodic motions to chaos are determined. The stability and bifurcations of periodic motions are determined through the eigenvalue analysis. A bifurcation tree of (1:2) asymmetric periodic motions, varying with the Rayleigh number, is presented by discrete nodes, and the corresponding harmonic frequency-amplitude characteristics of periodic motions on the bifurcation tree are discussed through finite Fourier series analysis. A sequence of period-doubling bifurcations is observed on the bifurcation tree of the (1:2)- asymmetric periodic motion to chaos. The scenario of (1:2)- asymmetric period-1, period-2, period-4 motions to chaos scenario is illustrated. Homoclinic orbits are as appearance or vanishing of unstable periodic motions on the bifurcation tree. Illustrations of periodic motions and homoclinic orbits are completed for intuitive demonstrations of the corresponding topological structures. This study extends the initial study of the bifurcation tree from the (1:1)-symmetric periodic motions to asymmetric periodic motions then to chaos in the Lorenz system, and the corresponding infinite homoclinic orbits induced by all unstable periodic motions can be determined. Such results can enrich the understanding of bifurcation dynamics of asymmetric periodic motions to chaos in the Lorenz system.

Two-color optically addressed spatial light modulator as a generic spatiotemporal system.

Nonlinear spatiotemporal systems are the basis for countless physical phenomena in such diverse fields as ecology, optics, electronics, and neuroscience. The canonical approach to unify models originating from different fields is the normal form description, which determines the generic dynamical aspects and different bifurcation scenarios. Realizing different types of dynamical systems via one experimental platform that enables continuous transition between normal forms through tuning accessible system parameters is, therefore, highly relevant. Here, we show that a transmissive, optically addressed spatial light modulator under coherent optical illumination and optical feedback coupling allows tuning between pitchfork, transcritical, and saddle-node bifurcations of steady states. We demonstrate this by analytically deriving the system's dynamical equations in correspondence to the normal forms of the associated bifurcations and confirm these results via extensive numerical simulations. Our model describes a nematic liquid crystal device using nano-dimensional dichalcogenide (a-As 2S 3) glassy thin films as photo sensors and alignment layers, and we use device parameters obtained from experimental characterization. Optical coupling, for example, using diffraction, holography, or integrated unitary maps allows implementing a variety of system topologies of technological relevance for neural networks and potentially Ising or XY-Hamiltonian models with ultralow energy consumption.

Nonlinear dynamic response and bifurcation of asymmetric rotor system supported in axial-grooved gas-lubricated bearings

Dynamic characteristics of the asymmetric rotor system supported in axial-grooved gas-lubricated bearings are studied. In order to solve nonlinear dynamic response of rotor system effectively, a hybrid numerical model is established by coupling the motion equation of rotor with the rational function model of the gas film forces. The rational function model of the gas film forces of gas-lubricated bearing is established based on vector fitting theory. By using the hybrid numerical model, the repeated calculations of the unsteady Reynolds equation and gas film forces are avoided; the continuous rotor trajectory and the dynamic gas film forces can be calculated simultaneously; and for the rotor system supported in the same bearings, the computing cost can be saved effectively. The nonlinear dynamic responses of asymmetric rotor system supported in axial-grooved gas-lubricated bearings are investigated by trajectory diagrams, frequency spectrum, Poincaré maps, and time series. The bifurcations are analyzed by the bifurcation diagrams with different rotating speeds and mass eccentricities. The dynamic behaviors of the asymmetric rotor system appear complex nonlinear dynamic phenomenon and specific bifurcation characteristics.

Mathematical modelling, numerical and experimental analysis of one-degree-of-freedom oscillator with Duffing-type stiffness

The aim of the work is to develop and test simple mathematical models suitable for fast and realistic simulations of bifurcation dynamics of systems with a double potential well generated by a pair of repulsive magnets positioned transversely to the direction of motion, where there are motion resistances typical for rolling guides found in industry. Mathematical modelling of a harmonically forced one-degree-of-freedom oscillator with magnetically modified elasticity generating a double symmetrical minimum of potential was carried out. The system is composed of a cart moving along a linear rolling bearing with inertial excitation by means of a stepper motor with an unbalanced disc. The stiffness is realized by a pair of neodymium magnets of axes perpendicular to the direction of motion, in a position corresponding to the static equilibrium position of the oscillator, with additional mechanical linear springs, which generate a system similar to the Duffing system. Different models of the interaction force between cylindrical neodymium magnets were developed. The model parameters are estimated based on experimental data. Then the model is validated through additional experimental and numerical bifurcation analysis of the system. A very good agreement between numerical simulations and experimental data is obtained.

Dynamic transitions and bifurcations of 1D reaction–diffusion equations: The self‐adjoint case

This paper deals with the classification of transition phenomena in the most basic dissipative system possible, namely, the 1D reaction–diffusion equation. The emphasis is on the relation between the linear and nonlinear terms and the effect of the boundaries which influence the first transitions. We consider the cases where the linear part is self‐adjoint with second‐order and fourth‐order derivatives which is the case which most often arises in applications. We assume that the nonlinear term depends on the unknown function and its first derivative which is basically the semilinear case for the second‐order reaction–diffusion system. As for the boundary conditions, we consider the typical Dirichlet, Neumann, and periodic boundary settings. In all the cases, the equations admit a trivial steady state which loses stability at a critical parameter. We aim to classify all possible transitions and bifurcations that take place. Our analysis shows that these systems display all three types of transitions: continuous, jump and mixed. Moreover they exhibit transcritical, supercritical bifurcations with bifurcated states such as finitely many equilibria, circle of equilibria, and slowly rotating limit cycle. Many applications found in the literature are basically corollaries of our main results. We apply our results to classify the first transitions of the Chaffee–Infante equation, the Fisher‐KPP equation, the Kuramoto–Sivashinsky equation, and the Swift–Hohenberg equation.

Chaos Phenomenon in Power Systems: A Review

This review article puts forward the phenomena of chaotic oscillation in electrical power systems. The aim is to present some short summaries written by distinguished researchers in the field of chaotic oscillation in power systems. The reviewed papers are classified according to the phenomena that cause the chaotic oscillations in electrical power systems. Modern electrical power systems are evolving day by day from small networks toward large-scale grids. Electrical power systems are constituted of multiple inter-linked together elements, such as synchronous generators, transformers, transmission lines, linear and nonlinear loads, and many other devices. Most of these components are inherently nonlinear in nature rendering the whole electrical power system as a complex nonlinear network. Nonlinear systems can evolve very complex dynamics such as static and dynamic bifurcations and may also behave chaotically. Chaos in electrical power systems is very unwanted as it can drive system bus voltage to instability and can lead to voltage collapse and ultimately cause a general blackout.

Sliding Dynamics and Bifurcations of a Filippov System with Nonlinear Threshold Control

Considering the effectiveness of introducing the change rate of viral loads into the threshold setting policy for triggering interventions, we propose an immune-virus Filippov system with a nonlinear threshold. By developing new analytical and numerical methods, we systematically studied the rich dynamical behaviors and bifurcations of the proposed system, including the existence of three sliding segments and three pseudo-equilibria, boundary-center bifurcation, boundary-saddle bifurcation, pseudo-saddle-node bifurcation and tangency bifurcation. We further showed that the proposed system can exhibit virous structures in the coexistence of multiple steady states. Phenomena include bistability of two pseudo-equilibria, tristability and multiplestability of two pseudo-equilibria with regular equilibria or touching cycles. The modeling methods, as well as the analytical and numerical methods, can be widely applied to many other fields.

Global dynamic bifurcation of local semiflows and nonlinear evolution equations

In this work new global dynamic bifurcation theorems are established for local semiflows on complete metric spaces. These theorems are applied to the nonlinear evolution equation u t + A u = f λ ( u ) in a Banach space X , where A is a sectorial operator with compact resolvent, and f λ ( 0 ) = 0 for all λ ∈ R . In particular, if f λ takes the form f λ ( u ) = λ u + f ( u ) , it is shown that the global dynamic bifurcation branch of a bifurcation point ( 0 , λ 0 ) is necessarily unbounded without assuming the “crossing odd-multiplicity” condition. It has long been recognized that dynamical methods can be helpful in solving static problems. As an example, the bifurcation of elliptic equations on bounded domains associated with homogenous Dirichlet boundary condition is addressed. By considering the corresponding nonclassical parabolic flows and applying the global dynamic bifurcation theorems given here for local semiflows, some new results with global features are derived.

Nonlinear Vibration and Stability Analysis of Functionally Graded Nanobeam Subjected to External Parametric Excitation and Thermal Load

Analysis of vibration stability of simply supported Euler-Bernoulli functionally graded (FG) nanobeam embedded in viscous elastic medium with thermal effect under external parametric excitation is presented in this work. An attempt has been made for the first time is investigating the effect of thermal load on dynamic behavior, amplitude response, instability region and bifurcation points of functionally graded nanobeam. Thermal loads are supposed to be uniform, linear or nonlinear distribution along the thickness direction. Nonlocal continuum theory and the principle of the minimum total potential energy are applied to derive the governing equations. The partial differential equations (PDE) are transported to the ordinary differential equations (ODE) by using the Petrov-Galerkin method and the multiple time scales method are manipulated to solve the motion equation. To study the effect of external parametric excitation and thermal effect, different temperature distributions along the thickness such as uniform, linear, and nonlinear distribution are considered. Moreover, stable and unstable regions and bifurcation points are determined. It is obtained that the thermal load can affect the amplitude response of FG nanobeam. Also, it is observed that the instability of the system is affected by the detuning parameter and the parametric excitation amplitude plays great role in the instability of system. Nanobeams are used in many devices like nanoresonators, nanosensors and nanoswitches. This paper is helpful for designing and manufacturing nanoscale structures specially nanoresonators under different thermal loads.

Stochastic bifurcation in single-species model induced by α-stable Lévy noise

Bifurcation analysis has many applications in different scientific fields, such as electronics, biology, ecology, and economics. In population biology, deterministic methods of bifurcation are commonly used. In contrast, stochastic bifurcation techniques are infrequently employed. Here we establish stochastic P-bifurcation behavior of (i) a growth model with state dependent birth rate and constant death rate, and (ii) a logistic growth model with state dependent carrying capacity, both of which are driven by multiplicative symmetric stable Lévy noise. Transcritical bifurcation occurs in the deterministic counterpart of the first model, while saddle-node bifurcation takes place in the logistic growth model. We focus on the impact of the variations of the growth rate, the per capita daily adult mortality rate, the stability index, and the noise intensity on the stationary probability density functions of the associated non-local Fokker–Planck equation. Implications of these bifurcations in population dynamics are discussed. In the first model the bifurcation parameter is the ratio of the population birth rate to the population death rate. In the second model the bifurcation parameter corresponds to the sensitivity of carrying capacity to change in the size of the population near equilibrium. In each case we show that as the value of the bifurcation parameter increases, the shape of the steady-state probability density function changes and that both stochastic models exhibit stochastic P-bifurcation. The unimodal density functions become more peaked around deterministic equilibrium points as the stability index increases, while an increase in any one of the other parameter has an effect on the stationary probability density function. That means the geometry of the density function changes from unimodal to flat, and its peak appears in the middle of the domain, which means a transition occurs.

Dynamical phenomena connected with stability loss of equilibria and periodic trajectories

This is a study of a dynamical system depending on a parameter . Under the assumption that the system has a family of equilibrium positions or periodic trajectories smoothly depending on , the focus is on details of stability loss through various bifurcations (Poincaré–Andronov– Hopf, period-doubling, and so on). Two basic formulations of the problem are considered. In the first, is constant and the subject of the analysis is the phenomenon of a soft or hard loss of stability. In the second, varies slowly with time (the case of a dynamic bifurcation). In the simplest situation , where is a small parameter. More generally, may be a solution of a slow differential equation. In the case of a dynamic bifurcation the analysis is mainly focused around the phenomenon of stability loss delay. Bibliography: 88 titles.

The Dynamics of Axon Bifurcation Development in the Cerebral Cortex of Typical and Acallosal Mice

The corpus callosum (CC) is a major interhemispheric commissure of placental mammals. Early steps of CC formation rely on guidance strategies, such as axonal branching and collateralization. Here we analyze the time-course dynamics of axonal bifurcation during typical cortical development or in a CC dysgenesis mouse model. We use Swiss mice as a typical CC mouse model and find that axonal bifurcation rates rise in the cerebral cortex from embryonic day (E)17 and are reduced by postnatal day (P)9. Since callosal neurons populate deep and superficial cortical layers, we compare the axon bifurcation ratio between those neurons by electroporating ex vivo brains at E13 and E15, using eGFP reporter to label the newborn neurons on organotypic slices. Our results suggest that deep layer neurons bifurcate 32% more than superficial ones. To investigate axonal bifurcation in CC dysgenesis, we use BALB/c mice as a spontaneous CC dysgenesis model. BALB/c mice present a typical layer distribution of SATB2 callosal cells, despite the occurrence of callosal anomalies. However, using anterograde DiI tracing, we find that BALB/c mice display increased rates of axonal bifurcations during early and late cortical development in the medial frontal cortex. Midline guidepost cells adjacent to the medial frontal cortex are significant reduced in the CC dysgenesis mouse model. Altogether these data suggest that callosal collateral axonal exuberance is maintained in the absence of midline guidepost signaling and might facilitate aberrant connections in the CC dysgenesis mouse model.

Dynamic Analysis and Hopf Bifurcation of a Lengyel–Epstein System with Two Delays

In this paper, a Lengyel–Epstein model with two delays is proposed and considered. By choosing the different delay as a parameter, the stability and Hopf bifurcation of the system under different situations are investigated in detail by using the linear stability method. Furthermore, the sufficient conditions for the stability of the equilibrium and the Hopf conditions are obtained. In addition, the explicit formula determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained with the normal form theory and the center manifold theorem to delay differential equations. Some numerical examples and simulation results are also conducted at the end of this paper to validate the developed theories.

Bifurcation processes in pulse voltage regulator

Analysis of a pulse voltage regulator from the point of view of transition to an unstable position and possible bifurcation states. Calculation of the stability condition when changing the parameters of the system. Obtaining the pulse voltage regulator stability surface. This voltage stabilizer can have the basic intrinsic parameters of the system, such as, for example, the load resistance, the value of which can change over time according to unknown laws. In the course of the work, the stability limits for static and dynamic bifurcations were obtained with respect to many parameters of the system, including with respect to the value of the load conductivity. To obtain numerical results, a system with the following main parameters was used: U = 4,1 V, d0 = 0,4, C = 600 F, L = 2 mH, f1 = 0,8 Ohm, G = 0,5 V1, RL = 2 Ohm.