Different Kinds Of Bifurcations Research Articles

Nonlinear Vibration Characteristics and Bifurcations of a Rotor System Subjected to Brush Seal Forces

In the paper, nonlinear vibration characteristics of a rotor system are investigated. Such a nonlinear rotor system is subjected to brush seal forces, which are obtained by integrating the bristle force along the entire ring. The nonlinear brush seal rotor system is constructed by merging a flexible rotor with nonlinear seal forces. The research is aimed at studying the nonlinear vibration characteristics and bifurcations of the motions under a variety of eccentricity circumstances. Different kinds of bifurcations are successfully obtained by mathematical discretization and mapping manipulation. Such a discrete mapping method successfully predicts the stable and unstable motions accurately. The period-doubling bifurcations and saddle node bifurcations of the rotor system are obtained. The sole unstable solutions are obtained, which are special, and a normal numerical integration method cannot solve this problem, which provides advantages in rotor design and motion control. According to the results, nonlinear resonances are found between the stable and unstable motions. The greater the eccentricity of the rotor, the greater the number of bifurcation points that occur during the rotor’s nonlinear motions, as well as the larger the ranges of speeds where the motions are unstable. Saddle node bifurcations generate unstable nonlinear motions and non-smooth motions, which may bring damage to the mechanical rotors. The period-doubling bifurcations produce the route from period-1 to period-2 motions in the nonlinear rotor system. The research provides a new perspective to study the bifurcations and stability of the nonlinear rotor systems.

Modeling the Effect of TV and Social Media Advertisements on the Dynamics of Vector-Borne Disease Malaria

Vector-borne disease malaria is transmitted to humans by arthropod vectors (mosquitoes) and contributes significantly to the global disease burden. TV and social media play a key role to disseminate awareness among people by broadcasting awareness programs. In this paper, a nonlinear model is formulated and analyzed in which cumulative number of advertisements through TV and social media is taken as dynamical variable that propagates awareness among people to control the prevalence of vector-borne disease. The human population is partitioned into susceptible, infected and aware classes, while the vector population is divided into susceptible and infected classes. Humans become infected and new cases arise when bitten by infected vectors (mosquitoes) and susceptible vectors get infected as they bite infected humans. The feasibility of equilibria is justified and their stability conditions are discussed. A crucial parameter, basic reproduction number, which measures the disease transmission potentiality is obtained. Bifurcation analysis is performed by varying the sensitive parameters, and it is found that the proposed system shows different kinds of bifurcations, such as transcritical bifurcation, saddle-node bifurcation and Hopf bifurcation, etc. The analysis of the model shows that reduction in vector population due to intervention of people of aware class would not efficiently reduce the infective cases, rather we have to minimize the transmission rates anyhow, to control the disease outbreak.

Stability Analysis of a Modified Leslie–Gower Predation Model With Weak Allee Effect in the Prey

In this manuscript, we study a Leslie–Gower predator-prey model with a hyperbolic functional response and weak Allee effect. The results reveal that the model supports coexistence and oscillation of both predator and prey populations. We also identify regions in the parameter space in which different kinds of bifurcations, such as saddle-node bifurcations, Hopf bifurcations and Bogdanov–Takens bifurcations.

Modeling the effect of time delay in the increment of number of hospital beds to control an infectious disease.

One of the key factors to control the spread of any infectious disease is the health care facilities, especially the number of hospital beds. To assess the impact of number of hospital beds and control of an emerged infectious disease, we have formulated a mathematical model by considering population (susceptible, infected, hospitalized) and newly created hospital beds as dynamic variables. In formulating the model, we have assumed that the number of hospital beds increases proportionally to the number of infected individuals. It is shown that on a slight change in parameter values, the model enters to different kinds of bifurcations, e.g., saddle-node, transcritical (backward and forward), and Hopf bifurcation. Also, the explicit conditions for these bifurcations are obtained. We have also shown the occurrence of Bogdanov-Takens (BT) bifurcation using the Normal form. To set up a new hospital bed takes time, and so we have also analyzed our proposed model by incorporating time delay in the increment of newly created hospital beds. It is observed that the incorporation of time delay destabilizes the system, and multiple stability switches arise through Hopf-bifurcation. To validate the results of the analytical analysis, we have carried out some numerical simulations.

Bifurcations and Dynamical Behavior of 2D Coupled Chaotic Sine Maps

The main characteristics of a dynamical system are determined by the bifurcation theory. In particular, in this paper we examined the properties of the discrete dynamical system of a two coupled maps, i.e. the maps with an invariant unidimensional submanifold. The study of coupled chaotic systems shows rich and complex dynamic behaviors, particularly through structures of bifurcations or chaotic synchronization. A bifurcation is a qualitative change of the system behavior under the influence of control parameters. This change may correspond to the disppearance or appearance of new singularities or a change in the nature of singularities. We can define different kinds of bifurcations for fixed points and period two cycles as, saddle-node, period doubling, transcritical or pitchfork bifurcations. The study of the sequence of bifurcations permits to understand the mechanisms that lead to chaos. The phenomena of synchronization and antisynchronization in coupled chaotic systems is very important because its applications in several areas, such as secure communication or biology. In this paper, we study bifurcation properties of a two-dimensional coupled map T with three parameters. The first objective is to locate the bifurcation curves and their evolution in the parametric plane (<i>a,b</i>), when a third parameter <i>c</i> varies. The equations of some bifurcation curves are given analytically; cusp points and co-dimension two points on these bifurcation curves are determined. The second is related to the study of the chaotic synchronization and antisynchronization in the phase space (<i>x,y</i>).

Evolutionary effects of non-compliant behavior in public procurement

The main goal of the present work is to analyze the evolution of non-compliant firms in public procurement. The dynamic setting is formalized by an evolutionary adaptation process which describes whether honest or dishonest behavior prevails in society at any given time. Firms will either behave in an honest or dishonest way depending on the payoffs and the type of firms they meet, through a word-of-mouth process. The honesty propensity assumption is introduced. By making use of both analytical tools and numerical techniques, we show that monomorphic configurations may represent stable or unstable equilibria, while coexistence of both groups in the steady state is possible and its asymptotic stability is sensitive to parameter modifications. Moreover, a rich variety of integrated dynamic behavior can be observed due to different kinds of bifurcations, smooth or border collision. A social planner can evaluate social and/or fiscal policies to manipulate preferences for honest behavior.

Bifurcations and multistability on the May-Holling-Tanner predation model considering alternative food for the predators.

In this paper a modified May-Holling-Tanner predator-prey model is analyzed, considering an alternative food for predators, when the quantity of prey i scarce. Our obtained results not only extend but also complement existing ones for this model, achieved in previous articles. The model presents rich dynamics for different sets of the parameter values; it is possible to prove the existence of: (i) a separatrix curve on the phase plane dividing the behavior of the trajectories, which can have different ω-limit; this implies that solutions nearest to that separatrix are highly sensitive to initial conditions, (ii) a homoclinic curve generated by the stable and unstable manifolds of a saddle point in the interior of the first quadrant, whose break generates a non-infinitesimal limit cycle, (iii) different kinds of bifurcations, such as: saddle-node, Hopf, Bogdanov-Takens, homoclinic and multiple Hopf bifurcations. (iv) up to two limit cycles surrounding a positive equilibrium point, which is locally asymptotically stable. Thus, the phenomenon of tri-stability can exist, since simultaneously can coexist a stable limit cycle, joint with two locally asymptotically stable equilibrium points, one of them over the y-axis and the other positive singularity. Numerical simulations supporting the main mathematical outcomes are shown and some of their ecological meanings are discussed.

A piecewise smooth model of evolutionary game for residential mobility and segregation.

The paper proposes an evolutionary version of a Schelling-type dynamic system to model the patterns of residential segregation when two groups of people are involved. The payoff functions of agents are the individual preferences for integration which are empirically grounded. Differently from Schelling's model, where the limited levels of tolerance are the driving force of segregation, in the current setup agents benefit from integration. Despite the differences, the evolutionary model shows a dynamics of segregation that is qualitatively similar to the one of the classical Schelling's model: segregation is always a stable equilibrium, while equilibria of integration exist only for peculiar configurations of the payoff functions and their asymptotic stability is highly sensitive to parameter variations. Moreover, a rich variety of integrated dynamic behaviors can be observed. In particular, the dynamics of the evolutionary game is regulated by a one-dimensional piecewise smooth map with two kink points that is rigorously analyzed using techniques recently developed for piecewise smooth dynamical systems. The investigation reveals that when a stable internal equilibrium exists, the bimodal shape of the map leads to several different kinds of bifurcations, smooth, and border collision, in a complicated interplay. Our global analysis can give intuitions to be used by a social planner to maximize integration through social policies that manipulate people's preferences for integration.

Harmonic balance method to analyze bifurcations in memristor oscillatory circuits

SummaryThe paper studies nonlinear dynamics and bifurcations of a class of memristor oscillatory circuits obtained by replacing the nonlinear resistor of a Chua's oscillator with a flux‐controlled memristor. A recently developed technique, named flux‐charge analysis method, has shown that the state space of such circuits can be decomposed in invariant manifolds, where each manifold is characterized by a different dynamics and different attractors. Goal of the paper is to investigate the use of the harmonic balance method in combination with flux‐charge analysis method in order to study the different kinds of bifurcations generated by changing the circuit parameters on a fixed manifold, changing manifold for a fixed parameter set (bifurcations without parameters), or changing simultaneously circuit parameters and manifolds. The main result is that the harmonic balance method is quite simple to apply in this rich bifurcation context and is effective to detect Hopf and to accurately predict period‐doubling bifurcations of all these different kinds. Copyright © 2017 John Wiley &amp; Sons, Ltd.

Dynamical phase transitions in a two-species bosonic Josephson junction

We investigate dynamical phase transitions that are induced by interspecies interaction in a two-species bosonic Josephson junctions (BJJ), based on semi-classical theory. In zero-phase mode, similar to the case of a single-species BJJ, we observe the well-known dynamical phase transition from Josephson oscillation to self-trapping, which can be induced by both enhanced repulsive and attractive interspecies interactions. In π phase mode, dynamical phase transitions are even more interesting and counter-intuitive. We characterize a dynamical phase transition with the merging of two separate phase space domains into one, which is induced by increasing repulsive interspecies interaction. On the other hand, we find that by increasing attractive interspecies interaction, a phase separation of two formally overlapped phase space domains will occur. At last, we reveal that these intriguing dynamical phase transitions are caused by different kinds of bifurcations.

Nonlinear characteristics of an autoparametric vibration system

The nonlinear characteristics of an autoparametric vibration system are investigated. This system consists of a base structure and a cantilever beam with a tip mass. The dynamic equations for the system are derived using the extended Hamilton's principle. The method of multiple scales (MMS) is used to determine an approximate analytical solution of the nonlinear governing equations and, hence, analyze the stability and bifurcation of the system. Compared with the numerical simulation, the first-order MMS is not sufficient. A Lagrangian-based approach is proposed to perform a second-order analysis, which is applicable to a large class of nonlinear systems. The effects of the amplitude and frequency of the external force, damping and frequency of the attached cantilever beam, and the tip mass on the nonlinear responses of the autoparametric vibration system are determined. The results show that this system exhibits many interesting nonlinear phenomena including saturation, jumps, hysteresis and different kinds of bifurcations, such as saddle-node, supercritical pitchfork and subcritical pitchfork bifurcations. Power spectra, phase portraits and Poincare maps are employed to analyze the unstable behavior and the associated Hopf bifurcation and chaos. Depending on the application of such a system, its dynamical behaviors could be exploited or avoided.

Time optimal control for the deployment of a tethered satellite allowing for a massive tether

The time optimal deployment of a satellite from a space ship is studied. In order to take the mass and lateral oscillations of the tether into account, a discretized model of the tether is build. Applying Pontryagin’s Maximum Principle a time-optimal deployment from a trivial downhanging configuration close to the space ship to another one farther away is computed. It is found, that the obtained solution displays a difficult switching pattern and during the variation of the initial length different kinds of bifurcations occur, leading to discontinuous variations of the optimal solution candidates.

Qualitative Analysis of a Leslie-Gower Predator-Prey System with Nonlinear Harvesting in Predator

This paper deals with the study of the stability and the bifurcation analysis of a Leslie-Gower predator-prey model with Michaelis-Menten type predator harvesting. It is shown that the proposed model exhibits the bistability for certain parametric conditions. Dulac’s criterion has been adopted to obtain the sufficient conditions for the global stability of the model. Moreover, the model exhibits different kinds of bifurcations (e.g., the saddle-node bifurcation, the subcritical and supercritical Hopf bifurcations, Bogdanov-Takens bifurcation, and the homoclinic bifurcation) whenever the values of parameters of the model vary. The analytical findings and numerical simulations reveal far richer and complex dynamics in comparison to the models with no harvesting and with constant-yield predator harvesting.

A simple method for precisely evaluating the stability regions of a multi-mode laser

We recently introduced a heuristic method for delimiting the stability regions of a three-mode class A laser with three freedom degrees. Here this method is extended to cover the stability conditions of a three-mode class B laser with five freedom degrees. The diagonal arrays of the stability coefficient determinant are taken equal to zero until the determinant vanishes. It then turns out that the diagonal roots correspond to the laser stability boundaries. They form the different kinds of bifurcations that segregate the above-threshold state, with three simultaneous oscillating modes, from the bistability and below-threshold states, with two and no oscillating modes, respectively. The fast optical switches and high optical memories have been designed by exploiting the bistability properties of bifurcations as reported by Perez et al (2007 Opt. Express 15 12941–8).

Complexity analysis of a Cournot–Bertrand duopoly game with different expectations

In this paper, we consider a Cournot–Bertrand mixed duopoly model with different expectations, where the market has linear demand and the firms have fixed marginal cost functions. Two firms choose output and price as decision variables, respectively, under the assumption that there is a certain degree of differentiation between the products offered by firms to avoid the whole market is occupied by the one that applies a lower price. The two players are considered to have bounded rational and static expectations. The existence and local stability condition of Nash equilibrium is investigated. We find the stability region of Cournot–Bertrand system is bigger than that of Cournot or Bertrand system under the same conditions. Furthermore, there are two different kinds of bifurcations when the parameters pass through the different boundary curves of the stability region, which is different from the Cournot or Bertrand model. Numerical simulation method is used to display the dynamic behaviors of the dynamical system, such as periodic cycles, bifurcation diagrams and strange attractors of the systems. The economic explanations of the complex dynamic behaviors are also given.

Bifurcations in granular media: macro- and micro-mechanics approaches

Various failure modes related to different kinds of bifurcations occur in nonassociated elastoplastic materials such as geomaterials. After presenting experimental evidence, we study this question by means of phenomenological constitutive relations and direct numerical simulations based on the discrete element method. The second-order work criterion related to diffuse failure modes is particularly considered within the framework of continuum and discrete mechanics. The equations of the bifurcation domain boundary and unstable stress direction cones are established. Diffuse failure is simulated numerically by perturbing bifurcation states. To cite this article: F. Darve et al., C. R. Mecanique 335 (2007).

Vibrational time recurrences in a model of acetylene C212H2

The time-dependent intramolecular vibrational dynamics of acetylene is investigated with the vibrogram method for the effective Hamiltonian obtained by Abbouti Temsamani and Herman [J. Chem. Phys. 102, 6371 (1995)]. The quantum recurrences evidenced by the vibrogram are shown to be in correspondence with emerging periodic orbits, especially, of the bending type. In this model, we find different kinds of bifurcations and, in particular, a birth of bending local modes as well as further bifurcations causing the destabilization of bending periodic orbits. This destabilization leads to a faster damping of the corresponding recurrences and to an increase in the intramolecular vibrational relaxation. A comparison with monodeuterated acetylene is also carried out.

Nonlinear behavior of a reaction-diffusion system of the photochemistry within the mesopause region

The photochemistry of the mesopause region entails a driven chemical oscillator enforced by solar short-wave irradiation. Zero-dimensional calculations show that this oscillator is able to produce nonlinear reactions like cascades of subharmonics or chaos. We investigate what will happen if this system is subjected to atmospheric diffusion. We discuss the system response and introduce different kinds of bifurcations.

Simulation of Heartbeat Dynamics: A Nonlinear Model

The mathematical modeling of biological systems has proven to be a valuable tool by allowing experiments which would otherwise be unfeasible in a real situation. In this work we propose a system of nonlinear differential equations describing the macroscopic behavior of the cardiac conduction system. The model describes the interactoin between the SinoAtrial and AtrioVentricular node. Its very simple structure consists of two nonlinear oscillators resistively coupled. The numerical analysis detects different kinds of bifurcations whose pathophysiological meanings are discussed. Moreover, the model is able to classify different pathologies, such as several classes of arrhythmic events, as well as to suggest hypothesis on the mechanisms that induce them. These results also show that the mechanisms generating the heartbeat obey complex laws. The model provides a wuite complete description of different pathological phenomena and its simplicity can be exploited for further studies on the control of cardiac dynamics.

Dynamic behavior of a neural network model of locomotor control in the lamprey.

1. Experimental studies have shown that a central pattern generator in the spinal cord of the lamprey can produce the basic rhythm for locomotion. This pattern generator interacts with the reticular neurons forming a spinoreticulospinal loop. To better understand and investigate the mechanisms for locomotor pattern generation in the lamprey, we examine the dynamic behavior of a simplified neural network model representing a unit spinal pattern generator (uPG) and its interaction with the reticular system. We use the techniques of bifurcation analysis and specifically examine the effects on the dynamic behavior of the system of 1) changing tonic drives to the different neurons of the uPG; 2) altering inhibitory and excitatory interconnection strengths among the uPG neurons; and 3) feedforward-feedback interactions between the uPG and the reticular neurons. 2. The model analyzed is a qualitative left-right symmetric network based on proposed functional architecture with one class of phasic reticular neurons and three classes of uPG neurons: excitatory (E), lateral (L), and crossed (C) interneurons. In the model each class is represented by one left and one right neuron. Each neuron has basic passive properties akin to biophysical neurons and receives tonic synaptic drive and weighted synaptic input from other connecting neurons. The neuron's output as a function of voltage is given by a nonlinear function with a strict threshold and saturation. 3. With an appropriate set of parameter values, the voltage of each neuron can oscillate periodically with phase relationships among the different neurons that are qualitatively similar to those observed experimentally. The uPG alone can also oscillate, as observed experimentally in isolated lamprey spinal cords. Varying the parameters can, however, profoundly change the state of the system via different kinds of bifurcations. Change in a single parameter can move the system from nonoscillatory to oscillatory states via different kinds of bifurcations. For some parameter values the system can also exhibit multistable behavior (e.g., an oscillatory state and a nonoscillatory state). The analysis also shows us how the amplitudes of the oscillations vary and the periods of limit cycles change as different bifurcation points are approached. 4. Altering tonic drive to just one class of uPG neurons (without altering the interconnections) can change the state of the system by altering the stability of fixed points, converting fixed points to oscillations, single oscillations to two stable oscillations, etc. Two-parameter bifurcation diagrams show the critical regions in which a balance between the tonic drives is necessary to maintain stable oscillations. A minimum tonic drive is necessary to obtain stable oscillatory output. With appropriate changes in the tonic drives to the L and C neurons, stable oscillatory output can be obtained even after eliminating the E neurons. Indeed, the presence of active E neurons in the biological system does not prove they play a functional role in the system, because tonic drive from other sources can substitute for them. On the other hand, very high excitation of any one class of neurons can terminate oscillations. Appropriate balance of tonic drives to different neuron classes can help sustain stable oscillations for larger tonic drives. Published experimental results concerning changes in amplitude and swimming frequency with increased tonic drives are mimicked by the model's responses to increased tonic drive. 5. Interconnectivity among the neurons plays a crucial role. The analysis indicates that the C and L classes of neurons are essential components of the model network. Sufficient inhibition from the L to C neurons as well as mutual inhibition between the left and right halves is necessary to obtain stable oscillatory output. When the E neurons are present in the model network, they must receive appropriate tonic drive and provide appropriate excitation