Cascade Of Period-doubling Bifurcations Research Articles

Evolving division of labor in a response threshold model

The response threshold model explains the emergence of division of labor (i.e., task specialization) in an unstructured population by assuming that the individuals have different propensities to work on different tasks. The incentive to attend to a particular task increases when the task is left unattended and decreases when individuals work on it. Here we derive mean-field equations for the stimulus dynamics and show that they exhibit complex attractors through period-doubling bifurcation cascades when the noise disrupting the thresholds is small. In addition, we show how the fixed threshold can be set to ensure specialization in both the transient and equilibrium regimes of the stimulus dynamics. However, a complete explanation of the emergence of division of labor requires that we address the question of where the threshold variation comes from, starting from a homogeneous population. We then study a structured population scenario, where the population is divided into a large number of independent groups of equal size, and the fitness of a group is proportional to the weighted mean work performed on the tasks during a fixed period of time. Using a winner-take-all strategy to model group competition and assuming an initial homogeneous metapopulation, we find that a substantial fraction of workers specialize in each task, without the need to penalize task switching.

Bursting activity in the reduced mean-field model of neuron-glial interaction

The study of neural activity synchronization in the brain is a central focus of modern neurobiology and neurodynamics. In a healthy brain, cognitive functions necessitate the accurate integration of neural activity at specific spatiotemporal scales. Some neurological and psychiatric disorders exhibit observable variations in patterns of synchronous population activity. One of the most significant and compelling synchronous population activity patterns is the bursting activity that participates in various informational and physiological processes, including epilepsy. Several mathematical models have been proposed to explain the mechanisms of formation of the bursting activity. The most intriguing and scientifically sound models enable incorporation of astrocytic modulation of neural activity [1]. In this paper, we present a novel phenomenological model for accurately replicating the bursting neural population activity. Our model builds upon the Tsodyks-Markram [2] model and incorporates vital aspects of neuron-glial interaction through a tripartite synapse. The presented model is a simplified version of the previously proposed model [3]. During astrocyte activity (which lasts seconds), we can set the probability of neurotransmitter release, u, which is determined in fractions of a second. This enables the proposed model to be written in the following manner: r=–E+α ln(1+exp((Ju(Y)xE+I0)/α)), =(1–x)/τD–u(Y)xE, =–y/τY+βσy(X). Here, E(t) represents the average neuronal activity of the excitatory population, while x(t) denotes the quantity of available neurotransmitter. Additionally, y(t) describes the concentration of gliotransmitters, which are released as a result of biochemical reactions during neuron-astrocyte interactions. The function describes the alteration in the likelihood of neurotransmitter release when gliotransmitters are present u(y)=u0+(∆u0)/(1+exp(–50(y–ythr)). Here u0 represents the probability of neurotransmitter release without astrocytic influence; ∆u0 indicates the change in the probability of neurotransmitter release caused by the interaction of the gliotransmitter with the presynaptic terminal, and ythr represents the threshold value that determines the change in the probability of neurotransmitter release due to the effect of a gliotransmitter. The function describes the influence of neurotransmitter on the concentration of the gliotransmitter σy(x)=1/(1+exp(–20(x–xthr)), where xthr is the astrocyte activation threshold. Note that the proposed model does not account for synaptic depression’s mechanism. Therefore, various dynamical regimes’ formation in the model is solely controlled by astrocytic dynamics. In the study, control parameters I0 and u0 were selected while the remaining parameters were held constant. Specifically, τ=0.013, τD=0.08, α=1.58, J=3.07. Changes in the concentration of neurotransmitters and gliotransmitters were characterized by parameters ∆u0=0.305, τν=3.3, β=0.3, xxthr=0.75, ythr=0.4. The research was conducted using computer modeling and numerical methods in nonlinear dynamics. The proposed model exhibits a diverse range of population activity patterns, such as spiking and bursting regular and chaotic activity. Chaotic dynamics zones are identified in the control parameter plane. An account is given of the mechanisms behind the emergence of these activity patterns through bifurcation analysis. It is shown that the occurrence of chaotic activity in the system can be associated both with the cascade of period doubling bifurcations and with the further development of chaos, as a result of which a homoclinic attractor appears in the system according to the Shilnikov scenario. It was also shown that multistability is observed in the system in a certain range of parameters. Note that the emergence of multistability and bursting activity in the model is independent of the intricacy of neuron and glial cell dynamics. These dynamics are instead influenced by the feedback loop between the presynaptic neuron and the glial cell. The effects of neuron-like dynamics and neural-glial interaction demonstrated are generally applicable, without specifying particular characteristics of neural-glial interaction, network architecture, or single-cell neuron dynamics. It should be noted that our proposed model solely concerns the potentiation of synaptic transmission by astrocytes. In summary, the proposed phenomenological model for population activity can replicate various patterns of neuron activity found in a wide range of dynamic memory and information processing studies. This research can potentially lead to the development of new, effective treatment methods for neurological diseases that involve neuron-glial interaction. Another potential application for these results is in the development of an optimized live chip that has preset functions. This requires a deeper understanding of rhythmogenesis in neural networks and brain function.

Mathematical Modeling of the Evolutionary Dynamics of a Planktonic Community Using a Discrete-Time Model

This study proposes a discrete-time eco-genetic model of a planktonic community that includes zooplankton and two competing phytoplankton haplotypes with and without a toxicity trait. The Holling type II response function describes predator consumption. We use the Ricker model to consider density limitation and regulation. The model is analytically and numerically studied. The loss of stability of fixed points occurs via the Neimark–Sacker scenario and a cascade of period-doubling bifurcations. The model reveals bistability and multistability. Therefore, the initial conditions can determine which of the coexisting dynamic modes will be attracted. If the competition of haplotypes is weaker than their self-regulation, then the variation in the current densities of community components can shift the observed dynamics, while the evolution direction remains unchanged. The ratio of haplotype fitnesses and predator pressure generally determines the asymptotic genetic composition of phytoplankton. If competition of haplotypes is higher than their self-regulation, then the bistability of monomorphic fixed points occurs when the displacement of one haplotype by another depends on initial conditions. The presence of predators can maintain the genetic polymorphism of the prey. This system shows dynamic modes similar to experimental dynamics: oscillation with delay, long-period antiphase fluctuations, and cryptic cycles emerging due to rapid evolution.

Routes from stationary dissipative solitons to chaos in a Mamyshev oscillator

Mamyshev oscillators (MOs), a new type of femtosecond fiber laser, can produce not only high-performance stationary dissipative solitons (SDSs), but also exhibit chaotic dynamics. Since chaotic behavior may lead to dramatic degradation of the performance, it is particularly important to investigate chaotic dynamics in MOs. Here, using numerical simulation and experimental research methods, we systematically investigate different routes from SDSs to chaos in an MO. Specifically, by simply increasing the gain saturation energy under different polarization states, we obtained different routes from SDSs to chaos, including via a cascade of period-doubling bifurcations (CPDB), via a quasi-cascade of period-doubling bifurcations, and via a double CPDB. In addition, the evolution from dissipative soliton molecules to chaos via CPDB was also observed by increasing the gain saturation energy with a small filter wavelength separation. This finding shows that the evolution from SDSs to chaos is a generic property of laser systems and can appear in various pulse patterns such as single pulse, dissipative soliton molecules, and harmonic mode-locking pulses, which may be beneficial for better understanding chaotic dynamics in MOs.

Effects of Competition Intensities and R&D Spillovers on a Cournot Duopoly Game of Digital Economies

In this paper, we introduce a Cournot duopoly game that can characterize fierce competition in digital economies and employ it to examine the effects of research and development (R&D) spillovers while considering various competition intensities. We obtain the analytical solution of the Nash equilibrium and the expression of commodity price, firm production, and variable profit under some key competition intensities. Furthermore, we analyze the local stability of the Nash equilibrium and derive that the equilibrium may lose its stability only through a 1:4 resonance bifurcation. Numerical simulations are conducted, through which we find that the Nash equilibrium transitions to complex dynamics through a cascade of period-doubling bifurcations. Phase portraits are also provided to illustrate more details of the dynamics, which confirm the previous theoretical finding that the Nash equilibrium loses its stability through a 1:4 resonance bifurcation.

Complex Dynamics Modes in a Simple Model of Prey-Predator Community: Bistability and Multistability

The paper proposes and studies a two-component discrete-time model of the prey-predator community considering zooplankton and fish interactions and their development features. Discrete-time systems of equations allow us to take into account naturally the rhythm of many processes occurring in marine and freshwater communities, which are subject to cyclical fluctuations due to the daily and seasonal cycle. We describe the dynamics of fish and zooplankton populations constituting the community by Ricker’s model, which is well-studied and widely used in population modeling. To consider the species interaction, we use the Holling-II type response function taking into account predator saturation. We carried out the study of the proposed model. The system is shown to have from one to three non-trivial equilibria, which gives the existence of the complete community. In addition to the saddle-node bifurcation, which generates bistability of stationary dynamics, a nontrivial equilibrium loses its stability according to the Neimark-Sacker scenario with an increase in the reproductive potential of both predator and prey species, as a result of which the community exhibits long-period oscillations similar to those observed in experiments. With the higher bifurcation parameter, the reverse Neimark-Sacker bifurcation is shown to occur followed by the closed invariant curve collapses, and dynamics of the population stabilizes, later losing stability through a cascade of period-doubling bifurcations. Multistability complicates the birth and disappearance of the invariant curve in the phase space scenario by the emergence of another irregular dynamics in the system with the single unstable nontrivial fixed point. At fixed values of the model parameters and different initial conditions, the system considered is shown to demonstrate various quasi-periodic oscillations. Despite extreme simplicity, the proposed discrete-time model of community dynamics demonstrates a wide variety and variability of dynamic modes. It shows that the influence of environmental conditions can change the type and nature of the observed dynamics.

Quasi-periodic structure in chaotic bursting attractor for a controlled jerk oscillator

The main purpose of the paper is to show that parts of the trajectory for chaotic bursting oscillations may oscillate according to different limit cycles, which implies orderly movement exists in chaos. For a typical jerk oscillator with a cubic nonlinearity, when a slow-varying controlling term is introduced, bursting oscillations may occur, appearing in the combination of large-amplitude and small-amplitude oscillations. Two coexisting stable equilibrium points may evolve to chaos via different cascades of period-doubling bifurcation, respectively, which means different types of stable attractors may coexist with the variation of slow-varying parameter, while several periodic windows exist in the chaotic region. It is found that the spiking oscillations of the full system may change according to different cycles not only for 2-D torus bursting oscillations but also for the chaotic bursting attractor. Furthermore, whether the influence of the short periodic window of the fast subsystem on the bursting oscillations appears or not depends on the size of the slow scale, since the inertia of the movement may cause the delay effect of the bifurcations at both two edges of the window. The findings suggest that coupling of multiple scales may be one of key factors for the diversity of real world.

Minimal Chaotic Networks of Linear Neuron-Like Elements with Single Rectification: Three Prototypes

Chaotic oscillations induced by single rectification in networks of linear neuron-like elements are examined on three prototype models: one nonautonomous system and two autonomous systems. The first is a system of coupled neurons with periodic input; the second is a system of three coupled neurons with six couplings; the third is a ring of four unidirectionally coupled neurons with one reverse coupling. In each system, the output function of one neuron is ramp and that of the others is linear. Each system is piecewise linear and the phase space is separated into two domains by a single border. Steady states, periodic solutions and homoclinic orbits are derived rigorously and their stability is evaluated with the eigenvalues of the Jacobian matrices. The bifurcation analysis of the three systems shows that chaotic attractors could be generated through cascades of period-doubling bifurcations of periodic solutions.

Scenarios for the creation of hyperchaotic attractors in 3D maps

We study bifurcation mechanisms for the appearance of hyperchaotic attractors in three-dimensional diffeomorphisms, i.e. such attractors whose orbits have two positive Lyapunov exponents in numerical experiments. In particular, periodic orbits belonging to the attractor should have two-dimensional unstable invariant manifolds. We discuss several bifurcation scenarios which create such periodic orbits inside the attractor. This includes cascades of supercritical period-doubling bifurcations of saddle periodic orbits and supercritical Neimark–Sacker bifurcations of stable periodic orbits, as well as various combinations of these cascades. These scenarios are illustrated by an example of the three-dimensional Mirá map.

Period-doubling bifurcations and islets of stability in two-degree-of-freedom Hamiltonian systems.

In this paper, we show that the destruction of the main Kolmogorov-Arnold-Moser (KAM) islands in two-degree-of-freedom Hamiltonian systems occurs through a cascade of period-doubling bifurcations. We calculate the corresponding Feigenbaum constant and the accumulation point of the period-doubling sequence. By means of a systematic grid search on exit basin diagrams, we find the existence of numerous very small KAM islands ("islets") for values below and above the aforementioned accumulation point. We study the bifurcations involving the formation of islets and we classify them in three different types. Finally, we show that the same types of islets appear in generic two-degree-of-freedom Hamiltonian systems and in area-preserving maps.

Stability of Periodic Orbits and Bifurcation Analysis of Ship Roll Oscillations in Regular Sea Waves

Response, stability, and bifurcation of roll oscillations of a biased ship under regular sea waves are investigated. The primary and subharmonic response branches are traced in the frequency domain employing the Incremental Harmonic Balance (IHB) method with a pseudo-arc-length continuation approach. The stability of periodic responses and bifurcation points are determined by monitoring the eigenvalues of the Floquet transition matrix. The primary and higher-order subharmonic responses experience a cascade of period-doubling bifurcations, eventually culminating in chaotic responses detected by numerical integration (NI) of the equation of motion. Bifurcation diagrams are obtained through the period-doubling route to chaos. Solutions are aided with phase portrait, Poincaré map, time history and Fourier spectrum for better clarity as and when required. Finally, the same ship model is investigated under variable excitation moments that may result from different wave heights in regular seas. The biased ship roll model exhibits primary and subharmonic responses, jump phenomena, coexistence of multiple responses, and chaotically modulated motion. The stable, periodic, and steady-state roll responses obtained by the IHB method are validated by the NI method. Results obtained by both methods are found to agree very well.

Correspondence to biophysical criteria of nonlinear effects in the occurrence of Feigenbaum bifurcation cascade in models of invasive processes

Introduction. The problem of creating a set of criteria for practically substantiated computational modeling of a number of complex staged biophysical processes with pronounced stages and critical transformations, for example, aggressive invasions, is discussed. Known models have a variety of behavior with the occurrence of bifurcations according to the same scenarios, the appearance of cycles, the coexistence of which is determined by Sharkovskii’s theorem. In the limit of complication of cyclic behavior in such models, they often encounter chaotization of the trajectory, but with the existence of an infinite number of periodicity windows. The conditions for an infinite cascade of bifurcations for iterations are determined by the fulfillment of the conditions of Singer’s theorem. The purpose of this work is to show that most of the nonlinear effects associated with chaotization scenarios do not have an ecological interpretation, but we will propose ways to exclude non-interpretable parametric ranges.Materials and methods. Using methods for estimating the stability of stationary states and cyclic trajectories using Singer’s theorem on the criterion for the occurrence of bifurcations for iterative models, we analyze interconnected nonlinear effects. The phenomena are considered on the example of cascades of the appearance of cycles of the period p = 2i + 1, i→∞ and a cascade of cycles p = 2i – 1, i→0 of “doubling” or “halfing” the period, which occur in ecological models often used to optimize fishing.Results. It is confirmed that the coexistence of nonlinear effects turns out to be contradictory if the simulation results are interpreted in the field of biocybernetics, on the basis of model and real examples. Iterative models generate unnecessary non-linear modes of behavior, when predicting the dynamics of invasions or harvesting bioresources, taking into account the regulatory impact, for example, in the case of the well-known Feigenbaum scenario. It has been established that bifurcations connected in one scenario have no explanation in ecological reality and are not reflected in the observed biophysical systems. These mathematical artifacts are common to several biophysical models that are very different in their theoretical foundations. Chaotization in real population dynamics has somewhat different properties than can be obtained in a cascade of period doubling bifurcations. The formation of a non-attractive chaotic set in the form of a strange repeller is more consistent with the dynamics of the development of fast invasions.Discussion and conclusions. It is shown that to describe the transformations of biosystemic processes with external influence, as the collapse of a commercial population, it is adequate to use models with the emergence of alternative attractors. These models correspond better to the transitions between the states of populations under the influence of fishing than models with the implementation of cascades of bifurcations of cycles, strange Cantor attractors and chaos regimes in the form of a continuum of unstable trajectories of all periods. The most promising are hybrid models of the life cycle with developmental stages for essential interpretation in ecology and forecasting of biosystems, as they allow to determine the parametric ranges of functioning and exclude unacceptable ranges of parameters where excessive nonlinear effects occur, which have no justification for population processes. The analysis of the adequacy criteria is based on degradation scenarios for a complexly structured sturgeon population in the Volga basin, cod off the coast of Canada, outbreaks of invasive insects, and the spread of the invasive ctenophore Mnemiopsis leidy in the Caspian Sea.

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.

Dynamical Behavior of Discrete Ratio-Dependent Predator-Prey System

In this paper, we propose a discrete ratio-dependent predator-prey system. The stability of the fixed points of this model is studied. At the same time, it is shown that the discrete model undergoes fold bifurcation and flip bifurcation by using bifurcation theory and the method of approximation by a flow. Numerical simulations are presented not only to demonstrate the consistence with our theoretical analyses, but also to exhibit the complex dynamical behaviors, such as the cascade of period-doubling bifurcation in period-2 and the chaotic sets. The Maximum Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors. These results show that the direct discrete method has more rich dynamic behaviors than the discrete model obtained by Euler method.

Comparative Dynamics Analysis of Simple Mathematical Models of the Plankton Communities Considering Various Types of Response Function

The paper proposes a two-component discrete-time model of the plankton community, taking into account features of the development and interaction of phytoplankton and zooplankton. To describe the interaction between these species and to compare the system dynamics, we use the following set of response functions: type II and III Holling function and the Arditi–Ginzburg response function, each of which describes trophic interactions between phytoplankton and zooplankton. An analytical and numerical study of the model proposed is made. The analysis shows that the variation of trophic functions does not change the dynamic behavior of the model fundamentally. The stability loss of nontrivial fixed point corresponding to the coexistence of phytoplankton and zooplankton can occur through a cascade of period-doubling bifurcations and according to the Neimark–Saker scenario, which allows us to observe the appearance of long-period oscillations representing the alternation of peaks and reduction in the number of species as a result of the predator-prey interaction. As well, the model has multistability areas, where a variation in initial conditions with the unchanged values of all model parameters can result in a shift of the current dynamic mode. Each of the models is shown to demonstrate conditional coexistence when a variation of the current community structure can lead to the extinction of the entire community or its part. Considering the characteristics of the species composition, the model with the type II Holling function seems a more suitable for describing the dynamics of the plankton community. Such a system is consistent with the idea that phytoplankton is a fast variable and predator dynamics is slow; thus, long-period fluctuations occur at high phytoplankton growth rates and low zooplankton ones. The model with the Arditi–Ginzburg functional response demonstrates quasi-periodic fluctuations in a narrow parametric aria with a high predator growth rate and low prey growth rate. The quasi-periodic dynamics regions in the model with the Holling type III functional response correspond to the conception of fast and slow variables, however in this case, the stability of the system increases, and the Neimark-Sacker bifurcation occurs even at a higher growth rate of zooplankton.

Ultradian Rhythmicity and Chaotic Behaviors in the Integrate-and-Fire Model of a Neuroendocrine System

We consider a mathematical model for pulsatility in a neuroendocrine regulation system. The impulsive neural activity is modeled using an integrate-and-fire mechanism, applied to ensembles of hypothalamic neurons. The system generates a discrete impulse-to-impulse mapping that can be used to detect periodic and chaotic behaviors. The numerical study of this mapping is illustrated with bifurcation diagrams. In a wide range of the model’s parameters, transition to chaos through cascades of period-doubling bifurcations was found. Merging and expansion bifurcations were also revealed in the system.

Bifurcations and Chaos in Three-Coupled Ramp-Type Neurons

The bifurcations and chaos in autonomous systems of two- and three-coupled ramp-type neurons are considered. An asymmetric piecewise linear function is employed for the output function of neurons in order to examine changes in the bifurcations from a sigmoid output function to a ramp output function. Steady solutions in the systems are obtained exactly and they undergo discontinuous bifurcations because the systems are piecewise linear. Periodic solutions and homoclinic/heteroclinic orbits in the systems are obtained by connecting local solutions in linear domains at borders and solving transcendental equations. The bifurcations of the periodic solutions are calculated with the Poincaré maps and the Jacobian matrices, which are also derived rigorously. A stable periodic solution in a two-neuron oscillator of the Wilson–Cowan type with three couplings remains in the case of a ramp neuron. A chaotic attractor of Rössler type emerges in a network of three ramp neurons with six couplings, which is due to two saddle-focuses. The network consists of the two-neuron oscillator and one bypass neuron connected through three couplings. One-dimensional Poincaré maps show the generation of the chaotic attractor through a cascade of period-doubling bifurcations. Further, multiple homoclinic orbits based at a saddle are generated from the destabilization of two focuses when asymmetry in the output function is large. This homoclinicity causes qualitative change in the bifurcations of the periodic solutions as the output function of neurons changes from sigmoid to ramp.

Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting

Abstract In this paper, a discrete Leslie-Gower predator-prey system with Michaelis-Menten type harvesting is studied. Conditions on the existence and stability of fixed points are obtained. It is shown that the system can undergo fold bifurcation, flip bifurcation, and Neimark-Sacker bifurcation by using the center manifold theorem and bifurcation theory. Numerical simulations are presented to illustrate the main theoretical results. Compared to the continuous analog, the discrete system here possesses much richer dynamical behaviors including orbits of period-16, 21, 35, 49, 54, invariant cycles, cascades of period-doubling bifurcation in orbits of period-2, 4, 8, and chaotic sets.

Dynamics of pendulum-based systems under human arm rotational movements

Wearable devices have applications in health monitoring and patient assistance devices. Harvesting energy from human motion kinetic energy by the pendulum-based electromagnetic generator is a promising way of extending the battery life of these devices. In this paper, the oscillatory and rotary dynamical response of pendulum-based systems under an open-circuit condition is investigated for two different models considering the lower arm’s rotation. The arm pitching model (in-plane excitation) represents the swing of the lower arm during walking. The arm rolling rotation model (out-of-plane excitation) represents the twisting type of motion of the lower arm movement associated with a specific kind of disorder, for example, tremors experienced by Parkinson’s patients. The governing equations of motion around stable equilibria are derived using Lagrange formulation, and their nondimensional forms are introduced for the two models. A special case of these models is developed for experimental validation and system identification, where the system parameters are identified by fitting frequency response functions on the experimental data. The system’s nonlinear dynamical response shows various bifurcations, including limit point bifurcation (i.e., saddle-node bifurcation), pitchfork bifurcation, a cascade of period-doubling bifurcations under amplitude or frequency sweeps. When the system shows a bi-stable response, the rotary response usually has a higher angular velocity amplitude than the oscillatory response. Basins of attraction plots for different operating conditions are presented to show the effect of initial conditions on the system’s type of steady-state response: oscillatory or rotary.

Nonlinear dynamics of the dissipative anisotropic two-photon Dicke model

We study the semiclassical limit of the anisotropic two-photon Dicke model with a dissipative bosonic field and describe its rich nonlinear dynamics. Besides normal and ‘superradiant’-like phases, the presence of localized fixed points reflects the spectral collapse of the closed-system Hamiltonian. Through Hopf bifurcations of superradiant and normal fixed points, limit cycles are formed in certain regions of parameters. We also identify a pole-flip transition induced by anisotropy and a region of chaotic dynamics, which appears from a cascade of period-doubling bifurcations. In the chaotic region, collision and fragmentation of symmetric attractors take place. Throughout the phase diagram we find several examples of phase coexistence, leading to the segmentation of phase space into distinct basins of attraction.