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Consolidation of new information in memory occurs through the simultaneous occurrence of sharp-wave ripples (SWR) in the hippocampus network, fast–slow spindles in the thalamus network, and up and down oscillations in the cortex network during sleep. Previous studies have investigated the influential and active role of spindles and sharp-wave ripples in memory consolidation. However, a detailed investigation of the effect of membrane voltage of neurons and synaptic connections between neurons in the cortex, hippocampus, and thalamus networks to create spindle and SWR is required. This paper studies the dynamic behaviors of a hippocampal-thalamic-cortical network as a function of synaptic connection between excitatory neurons, inhibitory neurons (in the hippocampus and cortex), reticular neurons, and thalamocortical neurons (in the thalamic network). The bifurcation diagrams of the hippocampus, cortex, and thalamus networks are obtained by varying the strengths of different synaptic connections. The power diagrams for SWR and sleep spindles are shown accordingly. The results show that variations in synaptic self-connection (and inhibitory synaptic connection) of excitatory neurons in the CA3 region, as well as synaptic connection between excitatory neurons from CA1 region to excitatory neurons (and inhibitory neurons) in the cortex network have the most significant influence on dynamical behavior of the network. Furthermore, comparing diagrams for different synaptic connections shows that SWR is formed by excitatory neurons in CA3 region of the hippocampal network, passes through CA1 region, and enters cortex network. [ABSTRACT FROM AUTHOR]

In this paper, we propose a host–parasitoid model with a Holling type-II functional response and incorporate harvest effort. The Holling type-II response leads to saturation in parasitized hosts, creating a potential economic harvesting opportunity. To address overexploitation risks, we integrate a harvest effort function, determining an optimal threshold to prevent depletion. We explore model dynamics and bifurcations, including co-dimension one behaviors such as flip and Neimark–Sacker bifurcations, we provide numerical examples for validation. Our suggested difference-algebraic model, compared to continuous-time models, exhibits rich dynamics within the Nicholson–Bailey host–parasitoid framework. [ABSTRACT FROM AUTHOR]

This paper is concerned with a reaction–diffusion–advection chemostat model with two species growing and competing for a single-limited resource. By taking the growth rates of the two species as variable parameters, we study the effect of growth rates on the dynamics of this system. It is found that there exist several critical curves, which may classify the dynamics of this system into three scenarios: (1) extinction of both species; (2) competitive exclusion; (3) coexistence. Moreover, we take numerical approaches to further understand the potential behaviors of the above critical curves and observe that the bistable phenomenon can occur, besides competitive exclusion and coexistence. To further study the effect of advection and diffusion on the dynamics of this system, we present the bifurcation diagrams of positive equilibrium solutions of the single species model and the two-species model with the advection rates and the diffusion rates increasing, respectively. These numerical results indicate that advection and diffusion play a key role in determining the dynamics of two species competing in a flow reactor. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamical behavior of a one-dimensional piecewise map based on the logistic map, where generalized multistability can be observed. The proposed system has the unique property that the function is symmetric with respect to the origin but not its behavior, furthermore this system can display three types of multistability, and chaos for both, monostable and bistable behaviors. The stability analysis of the proposed system is presented. We describe the structure of bistable regions in the bifurcation diagram. Particular attention is paid to the chaotic regions. Corresponding to coexisting attractors, three scenarios of coexisting attractors, namely fixed points, periodic orbits, and chaotic attractors, can be found, which are unreported behaviors in discrete chaotic systems. The mechanism that leads to multistability phenomenon including pitchfork bifurcation, period-halving bifurcations, and the coexisting invariant sets is demonstrated. Furthermore, the Lyapunov exponent is analyzed with the type of multistability distinguished for a given set of parameters. [ABSTRACT FROM AUTHOR]

Social insect colonies' robust and efficient collective behaviors without any central control contribute greatly to their ecological success. Colony migration is a leading subject for studying collective decision-making in migration. In this paper, a general colony migration model with Hill functions in recruitment is proposed to investigate the underlying decision making mechanism and the related dynamical behaviors. Our analysis provides the existence and stability of equilibrium, and the global dynamical behavior of the system. To understand how piecewise functions and Hill functions in recruitment impact colony migration dynamics, the comparisons are performed in both analytic results and bifurcation analysis. Our theoretical results show that the dynamics of the migration system with Hill functions in recruitment differs from that of the migration system with piecewise functions in the following three aspects: (1) all population components in our colony migration model with Hill functions in recruitment are persistent; (2) the colony migration model with Hill functions in recruitment has saddle and saddle-node bifurcations, while the migration system with piecewise functions does not; (3) the system with Hill functions has only equilibrium dynamics, i.e. either has a global stability at one interior equilibrium or has bistablity among two locally stable interior equilibria. Bifurcation analysis shows that the geometrical shape of the Hill functions greatly impacts the dynamics: (1) the system with flatter Hill functions is less likely to exhibit bistability; (2) the system with steeper functions is prone to exhibit bistability, and the steady state of total active workers is closer to that of active workers in the system with piecewise function. [ABSTRACT FROM AUTHOR]

In this paper, bifurcation analysis of a three-dimensional discrete game model is provided. Possible codimension-one (codim-1) and codimension-two (codim-2) bifurcations of this model and its iterations are investigated under variation of one and two parameters, respectively. For each bifurcation, normal form coefficients are calculated through reduction of the system to the associated center manifold. The bifurcations detected in this paper include transcritical, fold, flip (period-doubling), Neimark–Sacker, period-doubling Neimark–Sacker, resonance 1:2, resonance 1:3, resonance 1:4 and fold-flip bifurcations. Moreover, we depict bifurcation diagrams corresponding to each bifurcation with the aid of numerical continuation method. These bifurcation curves not only confirm our analytical results, but also reveal a richer dynamics of the model especially in the higher iterations. [ABSTRACT FROM AUTHOR]

In this paper, we present a study on a mutual interference host-parasitoid model with Beverton–Holt growth. It is well known that, mutual interference of parasites has a stabilizing influence on the dynamics of the host-parasitoid model since the variance in searching efficiency, with parasite density, significantly depends on parasites' mutual interference. Thus, we have incorporated a mutual interference functional response into a host-parasitoid model to characterize such a phenomenon. The qualitative behaviors of the present model is investigated in this paper. Firstly, the existence and local stability of the model fixed points are discussed. Then, using perturbation method and normal form theory, we derived the emergence conditions of Neimark–Sacker bifurcation of the model. Furthermore, chaotic behavior of the model in the sense of Marotto is proved. In order to control chaotic behavior of the present model, we apply OGY feedback control strategy. Finally, numerical simulations are provided to support our theoretical discussion. [ABSTRACT FROM AUTHOR]

This paper thoroughly discusses the electric activities generated by ion communication between cells and their surrounding environment. Specifically, it focuses on the transients of firing activities of a four-dimensional pituitary model that evolves on three disparate timescales. To examine the impact of inward rectifying K + current and calcium concentration on the firing activities, a bifurcation analysis is conducted, categorizing three primary behaviors: resting, tonic spiking, and bursting. Each behavior is validated through their respective time courses, with the pituitary cells showing higher secretion rates of hormones and neurotransmitters during bursting than spiking. The geometric singular perturbed theory is applied to reveal hidden geometric features and the transient mechanisms associated with bursting, particularly mixed-mode oscillations (MMOs). Singular orbit construction performed in two-timescale separation with different viewpoints offers clarity on the underlying dynamic mechanisms. Canard-induced MMOs are observed in the context of 1 fast/3 slow and 2 fast/2 slow separations, facilitated by the presence of folded saddle-node and folded node, respectively. Additionally, the fast–slow analysis of the 3 fast/1 slow subsystem, which treats calcium concentration c as a parameter, in conjunction with the singular orbit constructions, effectively illustrates the system's complex dynamics. Furthermore, the information obtained in 1 fast/3 slow and 3 fast/1 slow discussions is interplayed in the context of three-timescale separation. The singular orbits identified within three-timescale framework offer a supplemental perspective to the delicate firing patterns observed in two-timescale analysis, enriching the overall understanding of the transient and long-term firing behaviors of the pituitary cells. This study presents valuable insights into the firing features in pituitary cells from the perspectives of dynamic systems. The singular perturbation analysis provides useful viewpoints for accessing firing patterns in multi-timescale systems. [ABSTRACT FROM AUTHOR]

Huanglongbing (HLB), primarily transmitted by the citrus psyllid, is the most devastating citrus disease worldwide. Planting disease-free citrus seedlings is one of the significant intervention strategies for HLB. In this paper, a compartment model is used to investigate the importation and local transmission of the disease. Due to the continuous input of the diseased citrus seedlings from outside, the traditional theoretical analysis methods are insufficient to describe and predict its dynamical behaviors based on the analysis determined by the basic reproductive ratio ℛ 0 . The criteria for the existence and local stability of disease-free and endemic equilibria are deduced. Then a complete theoretical, bifurcation, and sensitivity analyses are carried out on the simplified system. Based on our study, we have the following interesting findings: (a) the importation of diseased citrus seedlings in orchards plays a vital role in the transmission of HLB, and planting disease-free citrus seedlings will effectively inhibit the spread of the disease; (b) the importation of diseased citrus seedlings can affect the stability of the equilibrium of the full system; (c) the simplified system may exhibit some complex dynamic phenomena, such as saddle-node bifurcation, forward bifurcation, backward bifurcation, and bistability; (d) global sensitivity analysis shows that the mortality and biting rate of citrus psyllids have a greater impact on ℛ 0 than other parameters. These observations provide theoretical guidance for policymakers to determine the long-term viability of implemented HLB intervention strategies. [ABSTRACT FROM AUTHOR]

This paper attempts to find some interesting and unique properties in the dynamics of a permanent magnet synchronous motor (PMSM). When compared with the existing literature on the dynamics of a PMSM, we find some interesting and unique behaviors which have not been reported so far. These are (i) occurrence of multistability, (ii) existence of hidden attractors and (iii) equilibrium point with two stable node-foci and one saddle point index-1. The above-said unique behaviors in the dynamics of a PMSM are not found in the literature to the best of authors' knowledge. Three different cases with (a) v / f (voltage/frequency) control, (b) constant load torque and (c) constant direct and quadrature-axis voltage, and load torque are considered to show the multistability in the dynamics of a PMSM. The multistability is confirmed by using the bifurcation analysis. In another case, when the load torque is selected as a feedback of quadrature-axis voltage, the system depicts hidden attractors (point, periodic and transient chaotic). An adaptive sliding mode control is designed to control the hidden transient chaotic behavior of the system. The simulation results confirm the suppression of the transient chaotic attractors with smaller stabilization time and chattering free control input. [ABSTRACT FROM AUTHOR]

In this paper, we propose a diffusive SIR model with general incidence rate, saturated treatment rate and spatially heterogeneous diffusion coefficients. We first prove the global existence of bounded solutions for the model and compute the basic reproduction number. We study the local and global stabilities of the disease-free equilibrium and the uniform persistence. In the case when the diffusion rate of infected individuals is constant, we carry out a bifurcation analysis of equilibria by considering the maximal treatment rate as the bifurcation parameter. Finally, we perform some numerical simulations, which show that the solutions to our model present periodic oscillations for certain values of the parameters. [ABSTRACT FROM AUTHOR]

This paper investigates an 11-dimensional slow–fast system of multiscale COVID-19 model with interval parameters. For the slow subsystem, the local dynamics of disease-free equilibrium (DFE) and endemic equilibrium (EE) are studied by using the fast–slow theory. Two criteria of the existence of forward and backward bifurcations are further obtained. When the basic reproduction number R 0 is less than unity, we show that bistabilities of DFE and EE for R c < R 0 < 1 can be caused by backward bifurcation. Thus, controlling R 0 below critical value R c is effective to eliminate endemic diseases. Meanwhile, the slow subsystem undergoes saddle-node bifurcation at R 0 = R c and undergoes pitchfork bifurcation at R 0 = 1 for EE. Moreover, the stability and bifurcation of the multiscale full system are established. As an application, the numerical simulations of real data of COVID-19 in Hong Kong are used to verify these results, which show that increasing vaccination rate, improving vaccine effectiveness and decreasing the fraction of individuals in risky state 2 are necessary to control the COVID-19. [ABSTRACT FROM AUTHOR]

Children frequently experience absence seizures, a form of seizure that is characterized by brief periods of unconsciousness and staring spells. While many studies have been conducted on absence seizures, there is still some uncertainty regarding the precise mechanisms causing absence seizures. The basal ganglia are believed to be essential in regulating thalamocortical network activity responsible for such seizures. Controlling or designing a treatment for this disorder requires an understanding of the contribution of the basal ganglia regions in the absence seizures. In this regard, efforts have been made to propose a mathematical model of brain neuronal substructures and their connections in the basal ganglia. The basal ganglia cortex-thalamus (BGCT) model is one of the most-studied mathematical models investigating absence seizures. However, this model has not been comprehensively studied from the viewpoint of dynamical behavior. Hence, to evaluate the BGCT model, this paper is devoted to studying a detailed and in-depth bifurcation analysis of the basal ganglia regions in the BGCT loop. Moreover, the 0–1 test for chaos is performed to confirm the results shown in the bifurcation diagrams. Our results suggest that the BGCT model can exhibit chaotic behavior in small regions of the coupling parameter, which is consistent with the complex nature of the brain neuronal network. [ABSTRACT FROM AUTHOR]

There is now much geological evidence that the Earth was fully glaciated during several periods in the geological past (about 700 Myr ago) and attained a so-called Snowball Earth (SBE) state. Additional support for this idea has come from climate models of varying complexity that show transitions to SBE states and undergo hysteresis under changes in solar radiation. In this paper, we apply large-scale bifurcation analyses to a novel, fully-implicit Earth System Model of Intermediate Complexity (I-EMIC) to study SBE transitions. The I-EMIC contains a primitive equation ocean model, a model for atmospheric heat and moisture transport, a sea ice component and formulations for the adjustment of albedo over snow and ice. With the I-EMIC, high-dimensional branches of the SBE bifurcation diagram are obtained through parameter continuation. We are able to identify stable and unstable equilibria and uncover an intricate bifurcation structure associated with the ice-albedo feedback. Moreover, large-scale linear stability analyses are performed near major bifurcations, revealing the spatial nature of destabilizing perturbations. [ABSTRACT FROM AUTHOR]

This paper reports the complex dynamics of a class of two-dimensional maps containing hidden attractors via linear augmentation. Firstly, the method of linear augmentation for continuous dynamical systems is generalized to discrete dynamical systems. Then three cases of a class of two-dimensional maps that exhibit hidden dynamics, the maps with no fixed point and the maps with one stable fixed point, are studied. Our numerical simulations show the effectiveness of the linear augmentation method. As the coupling strength of the controller increases or decreases, hidden attractor can be annihilated or altered to be self-excited, and multistability of the map can be controlled to being bistable or monostable. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics and optimal control of the Monod–Haldane predator–prey system with mixed harvesting that combines both continuous and impulsive harvestings. The periodic solution of the prey-free is studied and the local stability condition is obtained. The boundedness of solutions, the permanence of the system, and the existence of nontrivial periodic solution are studied. With the change of parameters, the system appears with a stable nontrivial periodic solution when the prey-free periodic solution loses stability. Numerical simulations show that the system has complex dynamical behaviors via bifurcation diagrams. Further, the maximum yield problem of the harvested system is studied, which is transformed into a nonlinear programming problem and solved by the method of combined multiple shooting and collocation. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of a vector-borne disease model with two transmission paths: direct transmission through contact and indirect transmission through vector. The direct transmission is considered to be a nonmonotone incidence function to describe the psychological effect of some severe diseases among the population when the number of infected hosts is large and/or the disease possesses high case fatality rate. The system has a disease-free equilibrium which is locally asymptotically stable when the basic reproduction number ( R 0 ) is less than unity and may have up to four endemic equilibria. Analytical expression representing the epidemic growth rate is obtained for the system. Sensitivity of the two transmission pathways were compared with respect to the epidemic growth rate. We numerically find that the direct transmission coefficient is more sensitive than the indirect transmission coefficient with respect to R 0 and the epidemic growth rate. Local stability of endemic equilibrium is studied. Further, the global asymptotic stability of the endemic equilibrium is proved using Li and Muldowney geometric approach. The explicit condition for which the system undergoes backward bifurcation is obtained. The basic model also exhibits the hysteresis phenomenon which implies diseases will persist even when R 0 < 1 although the system undergoes a forward bifurcation and this phenomenon is rarely observed in disease models. Consequently, our analysis suggests that the diseases with multiple transmission routes exhibit bistable dynamics. However, efficient application of temporary control in bistable regions will curb the disease to lower endemicity. Additionally, numerical simulations reveal that the equilibrium level of infected hosts decreases as psychological effect increases. [ABSTRACT FROM AUTHOR]

Electromagnetic induction current sensed by the membrane potential in biological neurons can be characterized with a memristor synapse, which can be employed to demonstrate the real oscillating voltage patterns of Barnacle muscle fibers. This paper presents a 3D autonomous memristor synapse-based Morris–Lecar (abbreviated as m-ML) model, which is implemented through introducing a memristor synapse-based induction current to substitute the externally applied current in an existing 2D nonautonomous Morris–Lecar model. Making use of one- and two-parameter bifurcation plots and time-domain representations, diverse period-adding bifurcations as well as abundant periodic and chaotic burst firings are demonstrated. Through constructing the fold and Hopf bifurcation sets of fast spiking subsystem, bifurcation analyses of these chaotic and periodic burst firings are carried out. Moreover, the periodic and chaotic spiking firings and coexisting firing behaviors are illustrated by using one- and two-parameter bifurcation plots and local attraction basins. Finally, based on a field programmable gate array (FPGA) board, a compact digital electronic neuron is fabricated for the 3D m-ML model, from which periodic and chaotic bursting/spiking firings are experimentally measured to verify the results of the numerical simulations. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of immune response and autoimmunity with particular emphasis on the role of regulatory T cells (Tregs), T cells with different activation thresholds, and cytokines in mediating T cell activity. Analysis of the steady states yields parameter regions corresponding to regimes of normal clearance of viral infection, chronic infection, or autoimmune behavior, and the boundaries of stability and bifurcations of relevant steady states are found in terms of system parameters. Numerical simulations are performed to illustrate different dynamical scenarios, and to identify basins of attraction of different steady states and periodic solutions, highlighting the important role played by the initial conditions in determining the outcome of immune interactions. [ABSTRACT FROM AUTHOR]

In this paper, we address the free (uncontrolled) dynamics of a snakeboard consisting of two wheel pairs fastened to a platform. The snakeboard is one of the well-known sports vehicles on which the sportsman executes necessary body movements. From the theoretical point of view, this system is a direct generalization of the classical nonholonomic system of the Chaplygin sleigh. We carry out a topological and qualitative analysis of trajectories of this dynamical system. An important feature of the problem is that the common level set of first integrals is a compact two-dimensional surface of genus 5. We specify conditions under which the reaction forces infinitely increase during motion and the so-called phenomenon of nonholonomic jamming is observed. In this case, the nonholonomic model ceases to work and it is necessary to use more complex mechanical models incorporating sliding, elasticity, etc. [ABSTRACT FROM AUTHOR]

In this paper, we study an SIS epidemic model with nonlocal delay based on reaction–diffusion equation. The spatiotemporal distribution of the model solution is studied in detail, and sufficient conditions for the occurrence of the Turing pattern were obtained using the analysis of Turing instability. It was found that the delay not only prohibited the spread of infectious disease, but also had great effects on the spatial steady-state patterns. More specifically, the spatial average density of the infected populations will decrease, as well the width of the stripe pattern will increase as delay increases. When the delay increases to a certain value, the stripe pattern changes to the mixed pattern. The results in this work provide new theoretical guidance for the prevention and treatment of infectious diseases. [ABSTRACT FROM AUTHOR]

Since the bifurcation analysis remains the basic tool to inquire into nonlinear system dynamics, then, theoretically, the bifurcation analysis should be widely used to estimate the evolution of the climate dynamics at a certain geographic point over the last centenary. But it does not occur in practice. A viewpoint how to estimate such evolution is proposed and discussed in the paper. The viewpoint can be briefly formulated as the following: the extra-short timescale, plus the novel conceptual model, plus the modified bifurcation diagrams. So, first, only the data of instrumental measurements (hereafter observations) are used to build the bifurcation diagram, correspondingly, the timescale is restricted by the maximal duration of the reliable observations. In particular, 130-years time series of the land surface air temperature in Russia are analyzed in the paper. Second, the conceptual model (HDS-model) to describe the annual warming-cooling cycle dynamics similar to the dynamics of the hysteresis regulator with double synchronization (HDS-regulator) is introduced. Correspondingly, the local climate dynamics is considered as the dynamics of the system with a variable structure. The main advantage of HDS-model is the possibility to describe quite simply the rich variety of the observable temperature evolution based on the known nonlinear phenomena by the combinations of C-bifurcations (or, in other words, border-collision bifurcations) between three conjugate periodical processes with the same period but different orders of structural changes. Then HDS-hypothesis on the climate dynamics is proposed, where the local temperature evolution is considered under the competition between both amplitude and time quantizations within the Sun-Earth-Moon system. Processing the observations follows the SUC-logic conceptions on the experimental bifurcation analysis that allows to untangle the intricate nonlinear nonstationary behavior of a local climate system by using both special sections (S) and units (U) in a certain consequence (C). As a result, the modified bifurcation diagrams to show the temperature evolution in local climate systems are built under the variation of several bifurcation parameters. The diagrams exhibit intermittency phenomena of high activity in the European part of Russia at least. Generally speaking, HDS-hypothesis gives the good approximation to answer the question whether the observable changes in the local climate dynamics are qualitative and, if yes, in what meaning? [ABSTRACT FROM AUTHOR]

The massive straight segment, rotating around the axis perpendicular to its center, is a simple alternative for more precise and sophisticated models that take into account the shape and mass density of natural asteroids. In this article, we give numerical results for the families of periodic orbits that emanate from the five libration points of massive straight segment, and for some secondary bifurcating families. Possible application and extension of the results to research on the motion of satellites near small irregular-shaped celestial bodies are also discussed. The numerical continuation and bifurcation algorithms we use in our study are based on boundary value techniques, as implemented in AUTO. An extensive set of python scripts for running AUTO will be made available at http://users.encs.concordia.ca/~doedel. These scripts can be used to recompute the results in this paper, and to do a wide selection of related calculations. [ABSTRACT FROM AUTHOR]

This work is concerned with the bifurcational analysis of nonlinear dissipative systems affected by time delay. This issue typically arises when testing highly nonlinear energy dissipation devices, commonly used in vibration control of civil structures, and carried out experimentally via a hybrid technique known as Real-Time Dynamic Substructuring (RTDS) simulation. Unfortunately, the RTDS simulation is affected by time delay in the control feedback loop due to the actuator response, sensor reading and numerical processing. In essence, this paper focuses on studying the nonlinear dynamics induced by the interaction of a dynamical system with the nonlinear damper affected by the presence of time delay. Given the complexity of the system, numerical analysis is carried out in the context of bifurcational behavior, and bifurcation diagrams are computed using a continuation method. The bifurcational analysis presented here, provides a characterization of delay-induced nonlinear phenomena created by the interaction of the dynamical system with a delayed nonlinear response of the dissipation device. Nonlinear dynamics are also identified and characterized for different damper types when varying the damper model parameters, leading to the identification of system conditions at which the testing arrangement and test specimens can exhibit undesired dynamics. [ABSTRACT FROM AUTHOR]

Computer simulation of mathematical models of mammalian ventricular myocytes has become a very promising tool for understanding the underlying mechanisms of cardiac arrhythmias and may provide useful predictions of treatment in order to prevent fatal arrhythmias. In this paper, we employ the Luo-Rudy (LR) model of membrane action potential of the mammalian ventricular cell. Simulation is performed with the aid of a direct method for locating the equilibria and by an adaptive grid method for constructing bifurcation diagrams. Two-parameter bifurcation diagrams are constructed to show the parameter sets in which (a) spontaneous oscillations exist, (b) the equilibrium corresponding to the resting potential is stable, or (c) the equilibrium with the most depolarized potential is stable. Multiple attractors are detected near the boundary of these parameter sets. The generation of normal and abnormal pacemaking activities is elucidated via bifurcation analysis. Spontaneous oscillations appear and disappear via saddle-node homoclinic bifurcations, Hopf bifurcations, homoclinic bifurcations, or fold bifurcations of limit cycles. Limit cycles may become chaotic via period-adding cascades as the extracellular potassium concentration [K]o decreases. Pacemaking activities may also be generated when the time-independent potassium current is blocked. [ABSTRACT FROM AUTHOR]

Through the analysis of a mathematical model we construct in this paper the full catalog of possible love stories among standard (i.e. secure and non-synergic) individuals. For this, we first distinguish between robust and fragile couples, i.e. couples which have one or two stable equilibria. We also distinguish between high and low quality relationships by looking at the sign of the feelings among the partners. Finally, we split fragile couples into two distinct classes, namely those that start from the state of indifference to evolve toward their most desirable equilibrium, as opposed to those that have an unfavorable evolution. The boundaries separating the various classes in any two-dimensional parameter space are either bifurcation curves or special curves characterized by the annihilation of a suitably defined indicator associated to the equilibria or to their basins of attraction. The analysis is performed in part analytically and in part numerically (through continuation methods), and is mainly focused on the role of the appeals of the individuals. The results are in agreement with common wisdom and observations and show, in particular, that love stories can evolve positively even if one of the two individuals has negative feelings in the initial phase of the relationship. [ABSTRACT FROM AUTHOR]

In this tutorial the chaotic behavior of the fractional-order Chua's circuit is investigated from the time-domain point of view. The objective is achieved using the Adomian decomposition method, which enables the solution of the fractional differential equations to be found in closed form. By exploiting the capabilities offered by the decomposition method, the paper presents two remarkable findings. The first result is that a novel bifurcation parameter is identified, that is, the fractional-order q of the derivative. The second result is that chaos exists in the fractional Chua's circuit with order q = 1.05, which is the lowest order reported in literature for such circuits. Finally, a reliable and efficient binary test for chaos (called "0–1 test") is utilized to detect the presence of chaotic attractors in the system dynamics. [ABSTRACT FROM AUTHOR]

This paper shows how the appearance and disappearance of high-frequency oscillations in a batch reactor can be predicted by means of a minimal model describing the fast dynamics of the reaction. The analysis is based on the bifurcations of the model with respect to the slowly varying species, namely product and reactants. The method of analysis can, in principle, be applied to any slow-fast system provided the evolution of the slow variables is monotonic. [ABSTRACT FROM AUTHOR]

A drillbit-rock vibro-impact system with a linear rock contact law is considered in this paper. This system is a simplified model to study the drillbit oscillation in a percussive drilling system. The main purpose of this work is to explore the optimized vibration condition of the drillbit to improve the drilling efficiency in practical drilling applications. In order to eliminate the influences of drilling progression on the dynamic analysis of drillbit oscillation, the drillbit oscillation is separated from its drift via oscillation reconstruction. The obtained bounded mathematical model is analyzed by numerical integration and numerical continuation. Based on the codimension-one bifurcations detected by TC-HAT during one-parameter continuation, the two-parameter bifurcation curves are traced by two-parameter continuation. These explored two-parameter bifurcation curves constitute the parameter boundaries for the period-one one-impact oscillation of drillbit, which is verified as the optimal oscillation with the highest drilling efficiency. [ABSTRACT FROM AUTHOR]

Generic results for degenerate Chenciner (generalized Neimark–Sacker) bifurcation are obtained in the present work. The bifurcation arises from two-dimensional discrete-time systems with two independent parameters. We define in this work a new transformation of parameters, which enables the study of the bifurcation when degeneracy occurs. By the four bifurcation diagrams we obtained, new behaviors hidden by the degeneracy are brought to light. [ABSTRACT FROM AUTHOR]

In the present paper, we perform a complete bifurcation analysis of a two-stage population model, in which the per capita birth rate and stage transition rate from juveniles to adults are density dependent and take the general Beverton-Holt functions. Our study reveals a rich bifurcation structure including codimension-one bifurcations such as saddle-node, Hopf, homoclinic bifurcations, and codimension-two bifurcations such as Bogdanov-Takens (BT), Bautin bifurcations, etc. Moreover, by employing the polynomial analysis and approximation techniques, the existences of equilibria, Hopf and BT bifurcations as well as the formulas for calculating their bifurcation sets have been provided. Finally, the complete bifurcation diagrams and associate phase portraits are obtained not only analytically but also confirmed and extended numerically. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of the paradigmatic hyperchaotic oscillator with gyrators introduced by Tamasevicius and co-workers (referred to as the TCMNL oscillator hereafter) is considered. This well known hyperchaotic oscillator with active RC realization of inductors is suitable for integrated circuit implementation. Unlike previous literature based on piecewise-linear approximation methods, I derive a new (smooth) mathematical model based on the Shockley diode equation to explore the dynamics of the oscillator. Various tools for detecting chaos including bifurcation diagrams, Lyapunov exponents, frequency spectra, phase portraits and Poincaré sections are exploited to establish the connection between the system parameters and various complex dynamic regimes (e.g. hyperchaos, period-3 doubling bifurcation, coexistence of attractors, transient chaos) of the hyperchaotic oscillator. One of the most interesting and striking features of this oscillator discovered/revealed in this work is the coexistence of a hyperchaotic attractor with a chaotic one over a broad range of system parameters. This phenomenon was not reported previously and therefore represents a meaningful contribution to the understanding of the behavior of nonlinear dynamical systems in general. A close agreement is observed between theoretical and experimental analyses. [ABSTRACT FROM AUTHOR]

Photovoltaic (PV) systems with a battery back-up form an integral part of distributed generation systems and therefore have recently attracted a lot of interest. In this paper, we consider a system of charging a battery from a PV panel through a current mode controlled boost dc-dc converter. We analyze its complete nonlinear/nonsmooth dynamics, using a piecewise model of the converter and realistic nonlinear v-i characteristics of the PV panel. Through this study, it is revealed that system design without taking into account the nonsmooth dynamics of the converter combined with the nonlinear v-i characteristics of the PV panel can lead to unpredictable responses of the overall system with high current ripple and other undesirable phenomena. This analysis can lead to better designed converters that can operate under a wide variation of the solar irradiation and the battery's state of charge. We show that the v-i characteristics of the PV panel combined with the battery's output voltage variation can increase or decrease the converter's robustness, both under peak current mode control and average current mode control. We justify the observation in terms of the change in the discrete-time map caused by the nonlinear v-i characteristics of the PV panel. The theoretical results are validated experimentally. [ABSTRACT FROM AUTHOR]

Inclined cables are essential structural elements that are used most prominently in cable-stayed bridges. When the bridge deck oscillates due to an external force, such as passing traffic, cable vibrations can arise not only in the plane of excitation, but also in the perpendicular plane. This undesirable phenomenon can be modeled as an auto-parametric resonance between the in-plane and out-of-plane modes of vibration of the cable. In this paper, we consider a three-mode model, capturing the second in-plane, and first and second out-of-plane modes, and use it to study the response of an inclined cable that is vertically excited at its lower (deck) support at a frequency close to the second natural frequency of the cable. Averaging is applied to the model and then the solutions and bifurcations of the resulting averaged differential equations are investigated and mapped out with numerical continuation. In this way, we present a detailed bifurcation study of the different possible responses of the cable. We first consider the equilibria of the averaged model, of which there are four types that are distinguished by whether each of the two out-of-plane modes is present or not in the cable response. Each type of equilibrium is computed and represented as a surface over the plane of amplitude and frequency of the forcing. The stability of the equilibria changes and different surfaces meet along curves of bifurcations, which are continued directly. Overall, we present a comprehensive geometric picture of the two-parameter bifurcation diagram of the constant-amplitude coupled-mode response of the cable. We then focused on bifurcating periodic orbits, which correspond to cable dynamics with varying amplitudes of the participating second in-plane and second out-of-plane modes. The range of excitation amplitude and frequency is determined where such whirling cable motion can occur. Further bifurcations - period-doubling cascades and a Shilnikov homoclinic bifurcation - are found that lead to a chaotic cable response. Whirling and chaotic cable dynamics are confirmed by time-step simulations of the full three-mode model. The different cable responses are characterized, and can be distinguished clearly, by their motion at the quarter-span and by their frequency spectra. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the complex dynamical behaviors of a biological network that is derived from innate immune responses and that couples positive and negative feedback loops. The stability conditions of the non-negative equilibrium points (EPs) of the system are obtained, using the theory of dynamical systems, and we deduce that no more than three stable EPs exist in this system. Through bifurcation analysis and numerical simulations, we find that the system presents rich dynamical behaviors, such as monostability, bistability and oscillations. These results reveal how positive and negative feedback cooperatively regulate the dynamical behavior of the system. [ABSTRACT FROM AUTHOR]

In this paper, the analytical dynamics for singularity, switchability, and bifurcations of a 2-DOF friction-induced oscillator is investigated. The analytical conditions of the domain flow switchability at the boundaries and edges are developed from the theory of discontinuous dynamical systems, and the switchability conditions of boundary flows from domain and edge flows are presented. From the singularity and switchability of flow to the boundary, grazing, sliding and edge bifurcations are obtained. For a better understanding of the motion complexity of such a frictional oscillator, switching sets and mappings are introduced, and mapping structures for periodic motions are adopted. Using an eigenvalue analysis, the stability and bifurcation analysis of periodic motions in the friction-induced system is carried out. Analytical predictions and parameter maps of periodic motions are performed. Illustrations of periodic motions and the analytical conditions are completed. The analytical conditions and methodology can be applied to the multi-degrees-of-freedom frictional oscillators in the same fashion. [ABSTRACT FROM AUTHOR]

The nonlinear stability analysis of boiling water nuclear reactors (BWRs) is conducted with the aid of so-called advanced, well validated, system codes and an advanced reduced order model to build a detailed mathematical understanding of the BWR behavior in the practical relevant parameter space. In the last years, the existence of Hopf-bifurcation points was confirmed by some researchers. In the framework of this paper, a parameter region was analyzed in which the coexistence of different stability states is realized. As a novel result, we found a parameter region in which stable fixed points, unstable limit cycles and stable limit cycles coexist. This system behavior can be explained by a saddle-node bifurcation of cycles (turning point). The existence of this solution type in a BWR system indicates the possibility of large amplitude limit cycle oscillations in the linear stable region. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical response of a coupled oscillator is investigated, taking in consideration the nonlinear behavior of a SMA spring coupling the two oscillators. Due to the nonlinear coupling terms, the system exhibits both regular and chaotic motions. The Poincaré sections for different sets of coupling parameters are verified. [ABSTRACT FROM AUTHOR]

Bifurcation analysis has been applied to many power electronics circuits. Literature abounds with results regarding the various ways in which such circuits lose stability under variation of some selected parameters, e.g. via period-doubling bifurcation, Hopf bifurcation, border collision, etc. The current status of research in the identification of bifurcation behavior in power electronics has reached a stage where the salient types of bifurcation behavior, their underlying causes and the theoretical parameters affecting them have been well understood. Currently, the emphasis of research in this field has gradually shifted toward applications that are of direct relevance to practical design of power electronics. One direction is to apply some of the available research results in bifurcation behavior to the design of practical power electronics circuits. The main difficulty is that the abstract mathematical presentations of the available results are not directly applicable to practical design problems. In this paper we will discuss how research efforts may be directed to bridge this gap. [ABSTRACT FROM AUTHOR]

This paper concerns the bifurcation analysis of a recently proposed second-order impact model for forest fire prediction. The analysis is carried out with respect to four pairs of system parameters and is based on a one-dimensional piecewise-smooth map naturally induced by the two-dimensional flow. The results of the analysis confirm those reported in the literature and extend them, pointing out the different roles played by several system parameters in influencing the qualitative behavior of fire regimes. [ABSTRACT FROM AUTHOR]

This tutorial investigates bifurcation and chaos in the fractional-order Chen system from the time-domain point of view. The objective is achieved using the Adomian decomposition method, which allows the solution of the fractional differential equations to be written in closed form. By taking advantage of the capabilities given by the decomposition method, the paper illustrates two remarkable findings: (i) chaos exists in the fractional Chen system with order as low as 0.24, which represents the smallest value ever reported in literature for any chaotic system studied so far; (ii) it is feasible to show the occurrence of pitchfork bifurcations and period-doubling routes to chaos in the fractional Chen system, by virtue of a systematic time-domain analysis of its dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we extend the two-dimensional discrete time model of the virtual fly, developed by Böddeker and Egelhaaf [2003], into three space dimensions, and introduce its continuous time analog. Like real blowflies, the virtual counterparts exhibit a dichotomous chasing behavior: depending on size, velocity and the course of the targets; they catch the targets or pursue them at constant distance. Here, we analyze this behavioral characteristic with respect to the course of the target, in particular, we choose trajectories, spiraling upwards. After setting up the three-dimensional model, we transform it into the local coordinates of the pursuer, using equivariance properties. Then bifurcation tools apply, and it turns out that depending on the gradient and the curvature of the spiraling trajectory, a fixed point in the transformed system can lose or gain stability. A stable fixed point corresponds in the original system to a trajectory, on which the virtual fly follows the target at a constant distance. In this way, we explain the dichotomous behavior through the occurrence of bifurcations. [ABSTRACT FROM AUTHOR]

In this paper a bifurcation analysis of a piecewise-affine discrete-time dynamical system is carried out. Such a system derives from a well-known map which has good features from its circuit implementation point of view and good statistical properties in the generation of pseudo-random sequences. The considered map is a generalization of it and the bifurcation parameters take into account some common circuit implementation nonidealities or mismatches. It will be shown that several different dynamic situations may arise, which will be completely characterized as a function of three parameters. In particular, it will be shown that chaotic intervals may coexist, may be cyclical, and may undergo several global bifurcations. All the global bifurcation curves and surfaces will be obtained either analytically or numerically by studying the critical points of the map (i.e. extremum points and discontinuity points) and their iterates. In view of a robust design of the map, this bifurcation analysis should come before a statistical analysis, to find a set of parameters ensuring both robust chaotic dynamics and robust statistical properties. [ABSTRACT FROM AUTHOR]

In much of the previous study of switched dynamical systems, it has been assumed that switching occurs at a common border between two regions in the same space as the system trajectory crosses the border. However, models arising from this consideration cannot cover systems whose trajectories do not actually "cross" the border. A typical example is the current-mode controlled boost converter whose trajectory is "reflected" at the border. In this paper, we propose a general method to model switched dynamical systems. Also, we suggest an analytical procedure to determine periodic solutions and their stability. The method is developed in terms of solution flows, and no solution has to be explicitly written. Most practical switched dynamical systems can be modeled and analyzed by this method. [ABSTRACT FROM AUTHOR]

This paper presents a study of the behavior of a special class of 3D dynamic systems (i.e. RHS of the third-order equation is a cubic polynomial), using Lyapunov–Andronov's theory. Considering the general case, we find a new analytical formula for the first Lyapunov's value at the boundary of stability. It enables one to study in detail the bifurcation behavior (and the route to chaos, in particular) of dynamic systems of the above type. We check the validity of our analytical results on the first Lyapunov's value by studying the route to chaos of two 3D dynamic systems with proved chaotic behavior. These are Chua's and Rucklidge's systems. Considering their route to chaos, we find new results. [ABSTRACT FROM AUTHOR]

This is the second of two papers introducing a new dynamical phenomenon, strongly related to the problems of synchronization and control of chaotic dynamical systems, and presenting the corresponding mathematical analysis, conducted both experimentally and theoretically. In particular, it is shown that different dynamical models (ordinary differential equations) admitting chaotic behavior organized by a homoclinic bifurcation to a saddle-focus (Shil'nikov-like chaos) tend to have a particular selective property when externally perturbed. Namely, these systems settle on a very narrow chaotic behavior, which is strongly correlated to the forcing signal, when they are slightly perturbed with an external signal which is similar to their corresponding generating cycle. Here, the "generating cycle" is understood to be the saddle cycle colliding with the equilibrium at the homoclinic bifurcation. On the other hand, when they are slightly perturbed with a generic signal, which has no particular correlation with their generating cycle, their chaotic behavior is reinforced. This peculiar behavior has been called qualitative resonance underlining the fact that such chaotic systems tend to resonate with signals that are qualitatively similar to an observable of their corresponding generating cycle. Here, a detailed mathematical analysis of the qualitative resonance phenomenon is presented, confirming the intuitions given by the geometrical model discussed in Part I. [ABSTRACT FROM AUTHOR]

In this paper, the theory of Lyapunov–Andronov is applied to investigate the route to chaos in Rossler system. On the base of a new analytical formula for the first Lyapunov value at the boundary of stability region, we make a detailed bifurcation analysis of this system. From the obtained results the following new conclusions are made: Transition to chaos in the Rossler's system takes place at soft stability loss in the form of a cascade of periodic self-oscillations. Then the occurrence of chaotic self-oscillations in this system takes place under hard stability loss. [ABSTRACT FROM AUTHOR]

A tritrophic "prey-intermediate predator-top predator" population system with disease in the intermediate predator is considered. For this 4D-model, the bifurcation analysis is performed. In this analysis, the rate of the disease transmission is used as a bifurcation parameter. A variety of mono-, bi- and tri-stable behaviors with regular and chaotic attractors are analyzed. It is shown how random disturbances of the bifurcation parameter cause multistage stochastic transformations, noise-induced excitement, and stochastic transitions from order to chaos and reversely. These noise-induced effects are studied in terms of stochastic P- and D-bifurcations. [ABSTRACT FROM AUTHOR]

In this research article, a three-species food chain model with Allee effect and additional food is proposed and analyzed. The Allee effect and additional food are introduced to the top predator population. The dynamical behavior of the system is studied analytically and numerically. We have performed equilibrium analysis and local stability analysis around the non-negative equilibria. We have also explored different bifurcations in the system. We have drawn several one- and two-parameter bifurcation diagrams to explore complex dynamical behaviors. We observe that top predator goes to extinction when Allee parameter crosses a threshold value, whereas additional food enhances the stability and persistence of the system. [ABSTRACT FROM AUTHOR]

This paper proposes a bifurcation point detection strategy based on nested layer particle swarm optimization (NLPSO). The NLPSO is performed by two particle swarm optimization (PSO) algorithms with a nesting structure. The proposed method is tested in detection experiments of period doubling bifurcation points in discrete-time dynamical systems. The proposed method directly detects the parameters of period doubling bifurcation regardless of the stability of the periodic point, but require no careful initialization, exact calculation or Lyapunov exponents. Moreover, the proposed method is an effective detection technique in terms of accuracy, robustness usability, and convergence speed. [ABSTRACT FROM AUTHOR]