Two-parameter Bifurcation Diagrams Research Articles

Two-parameter bifurcations analysis of a delayed high-temperature superconducting maglev model with guidance force.

A modified high-temperature superconducting maglev model is studied in this paper, mainly considering the influence of time delay on the dynamic properties of the system. For the original model without time delay, there are periodic equilibrium points. We investigate its stability and Hopf bifurcation and study the bifurcation properties by using the center manifold theorem and the normal form theory. For the delayed model, we mainly study the co-dimension two bifurcations (Bautin and Hopf-Hopf bifurcations) of the system. Specifically, we prove the existence of Bautin bifurcation and calculate the normal form of Hopf-Hopf bifurcation through the bifurcation theory of functional differential equations. Finally, we numerically simulate the abundant dynamic phenomena of the system. The two-parameter bifurcation diagram in the delayed model is given directly. Based on this, some nontrivial phenomena of the system, such as periodic coexistence and multistability, are well presented. Compared with the original ordinary differential equation system, the introduction of time delay makes the system appear chaotic behavior, and with the increase in delay, the variation law between displacement and velocity becomes more complex, which provides further insights into the dynamics of the high-temperature superconducting maglev model.

Complex Dynamics of a Three-Species Food Chain Model with Fear and Allee Effect

In ecology, predator–prey interactions are very complex in nature. Apart from direct killing, predator induces fear among their prey which affects the life-history, behavioral changes, and reproduction potential of the prey population. On the other hand, the Allee effect has a great impact on regulating the population size, community structure and population dynamics. In the present investigation, we modify the Hastings–Powell (HP) [1991] model by considering the cost of fear in middle predator and the Allee effect in top predator. The stability conditions for the biologically feasible equilibria are derived using linear stability analysis. Considering the cost of fear and the Allee effect as key parameters, the Hopf bifurcation analysis is carried out around the interior equilibrium. The direction of Hopf bifurcation and the stability of the bifurcating periodic solution are determined by applying the normal form theory and center manifold theorem. Our numerical results suggest that the fear effect can stabilize the system. It is observed that high levels of fear among middle predator decrease the population density of top predator. We also observe that if the Allee parameter is increased, then the system becomes stable from chaotic oscillations. However, further increase in the Allee parameter leads to population extinction. We have also drawn several one- and two-parameter bifurcation diagrams which explore rich dynamical behaviors.

Coexisting multiple firing behaviors of fractional-order memristor-coupled HR neuron considering synaptic crosstalk and its ARM-based implementation

The crosstalk between synapses has a far-reaching impact on the specificity of synaptic communication in human brain. Therefore, the influence of synaptic crosstalk on the dynamic behaviors of neural network cannot be ignored. In this paper, we investigate the dynamics of fractional-order (FO) memristor-coupled Hindmarsh-Rose (HR) neuron model considering synaptic crosstalk. Firstly, the equilibrium points and stability are investigated via qualitative analysis, from which it can be obtained that the number and stability of equilibrium points have diversity for different crosstalk strength parameters. Secondly, the global coexistence of multiple firing patterns is reflected by phase diagrams, Lyapunov exponent spectrums and bifurcation diagrams. The local attraction basins reveal the multi-stability phenomenon related to the initial value. The parameter related firing behaviors are described by the two-parameter bifurcation diagram. In particular, the model shows the offset boosting control method for a single variable. Then, spectral entropy (SE) and C0 complexity chaos diagram are used to observe the change of system complexity when two parameters change simultaneously. Finally, the digital implementations based on ARM are given to verify the consistency with the numerical simulation results.

Bifurcation analysis of mixed-mode oscillations and Farey trees in an extended Bonhoeffer–van der Pol oscillator

The two-variable Bonhoeffer–van der Pol oscillator, which is equivalent to the FitzHugh–Nagumo model, can be represented by a natural circuit, i.e., a circuit consisting only of simple two-terminal elements. An extended Bonhoeffer–van der Pol (BVP) oscillator is a circuit extended to a three-variable system from a two-variable BVP oscillator; this is obtained by adding an inductor–resistor branch to the original BVP oscillator. The extended BVP oscillator is known to generate mixed-mode oscillations. In this study, we classify the bifurcations of the simple sequences and their daughters, which constitute asymmetric Farey trees. Because the extended BVP oscillator is an extremely simple circuit, the parents–daughter processes can be quite precisely explained. We confirm that simple sequences 1s (s≥0) are born via saddle–node bifurcations and can be basic parents, which correspond to each stable fixed point of a Poincaré return map, where 1s represents one large excursion followed by a number s of small peaks. The two basic parents 1s and 1s+1 generate daughters [1s+1,1s×n] sequentially for the successive values of n via mixed-mode oscillation-incrementing bifurcations. The terms [1s+1,1s×n] indicate that 1s+1 is followed by 1sn-times. The daughters are born in a similar fashion to period-adding bifurcations generated by the circle map, and the parents–daughter processes satisfy Farey arithmetic and are terminated by a saddle–node bifurcation through which 1s appears. We create multiple two-parameter bifurcation diagrams and confirm that these phenomena are stably observed in these diagrams over broad ranges, i.e., complex bifurcations, such as codimension-two bifurcations or cusps, do not appear in the diagrams. Furthermore, the theoretical results are confirmed using laboratory measurements and experiments.

Stability and bifurcation analyses of p53 gene regulatory network with time delay

<abstract><p>In this paper, based on a p53 gene regulatory network regulated by Programmed Cell Death 5(PDCD5), a time delay in transcription and translation of Mdm2 gene expression is introduced into the network, the effects of the time delay on oscillation dynamics of p53 are investigated through stability and bifurcation analyses. The local stability of the positive equilibrium in the network is proved through analyzing the characteristic values of the corresponding linearized systems, which give the conditions on undergoing Hopf bifurcation without and with time delay, respectively. The theoretical results are verified through numerical simulations of time series, characteristic values and potential landscapes. Furthermore, combined effect of time delay and several typical parameters in the network on oscillation dynamics of p53 are explored through two-parameter bifurcation diagrams. The results show p53 reaches a lower stable steady state under smaller PDCD5 level, the production rates of p53 and Mdm2 while reaches a higher stable steady state under these larger ones. But the case is the opposite for the degradation rate of p53. Specially, p53 oscillates at a smaller Mdm2 degradation rate, but a larger one makes p53 reach a low stable steady state. Besides, moderate time delay can make the steady state switch from stable to unstable and induce p53 oscillation for moderate value of these parameters. Theses results reveal that time delay has a significant impact on p53 oscillation and may provide a useful insight into developing anti-cancer therapy.</p></abstract>

Unified framework for localized patterns in reaction–diffusion systems; the Gray–Scott and Gierer–Meinhardt cases

A recent study of canonical activator-inhibitor Schnakenberg-like models posed on an infinite line is extended to include models, such as Gray-Scott, with bistability of homogeneous equilibria. A homotopy is studied that takes a Schnakenberg-like glycolysis model to the Gray-Scott model. Numerical continuation is used to understand the complete sequence of transitions to two-parameter bifurcation diagrams within the localized pattern parameter regime as the homotopy parameter varies. Several distinct codimension-two bifurcations are discovered including cusp and quadruple zero points for homogeneous steady states, a degenerate heteroclinic connection and a change in connectedness of the homoclinic snaking structure. The analysis is repeated for the Gierer-Meinhardt system, which lies outside the canonical framework. Similar transitions are found under homotopy between bifurcation diagrams for the case where there is a constant feed in the active field, to it being in the inactive field. Wider implications of the results are discussed for other pattern-formation systems arising as models of natural phenomena. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Carryover of a Saddle-Node Bifurcation After Transforming a Parameter into a Variable

In this article, we introduce and study the carryover of a saddle-node bifurcation, a concept that describes how a saddle-node bifurcation of a dynamical system is carried over into an extended dynamical system obtained by transforming one of the parameters of the original system into a variable. We show that additional transversality and singularity conditions are needed to guarantee the carryover of a saddle-node bifurcation and provide a graphical methodology with a two-parameter bifurcation diagram to verify that such conditions are met. The results are applied to a gene activation model when the parameter describing the signal for activation is transformed into a variable, and to a cell cycle regulatory model when the parameter describing the cell mass is transformed into a variable. In both cases, we show that a saddle-node bifurcation carryover takes place.

Interactions Obtained from Basic Mechanistic Principles: Prey Herds and Predators

We investigate four predator–prey Rosenzweig–MacArthur models in which the prey exhibit herd behaviour and only the individuals on the edge of the herd are subjected to the predators’ attacks. The key concept is the herding index, i.e., the parameter defining the characteristic shape of the herd. We derive the population equations from the individual state transitions using the mechanistic approach and time scale separation method. We consider one predator and one prey species, linear and hyperbolic responses and the occurrence of predators’ intraspecific competition. For all models, we study the equilibria and their stability and we give the bifurcation analysis. We use standard numerical methods and the software Xppaut to obtain the one-parameter and two-parameter bifurcation diagrams.

A novel memristor-based chaotic system with line equilibria and its complex dynamics

Memristor, as a nonlinear element, provides many advantages thanks to its superior properties to design different chaotic circuits. Thus, a novel four-dimensional double-scroll chaotic system with line equilibria as well as two unstable equilibria based on the flux-memristor model is proposed in this paper. The effects of initial values and parameters on the dynamic characteristics of the system are studied in detail by means of phase diagrams, Lyapunov exponent spectrums, bifurcation diagrams, two-parameter bifurcation diagrams and basins of attraction. Besides, a series of complex phenomena in the system, such as sustained chaos, bistability, transient chaos and coexisting attractors are observed, proving that the chaotic system has rich dynamic characteristics. Also, spectral entropy (SE) complexity algorithm and [Formula: see text] complexity algorithm based on structure complexity are adopted to analyze the complexity of the system. Additionally, PSPICE circuit simulation and Micro-Controller Unit (MCU) hardware experiment are carried out to verify the correctness of theoretical analysis and numerical simulation. Finally, the pulse chaos synchronization is achieved from the perspective of maximum Lyapunov exponent, and numerical simulations demonstrate the occurrence of the proposed system and practicability of the pulse synchronization control.

The dynamics of a Leslie type predator\u2013prey model with fear and Allee effect

In this article, we discuss the dynamics of a Leslie–Gower ratio-dependent predator–prey model incorporating fear in the prey population. Moreover, the Allee effect in the predator growth is added into account from both biological and mathematical points of view. We explore the influence of the Allee and fear effect on the existence of all positive equilibria. Furthermore, the local stability properties and possible bifurcation behaviors of the proposed system about positive equilibria are discussed with the help of trace and determinant values of the Jacobian matrix. With the help of Sotomayor’s theorem, the conditions for existence of saddle-node bifurcation are derived. Also, we show that the proposed system admits limit cycle dynamics, and its stability is discussed with the value of first Lyapunov coefficient. Moreover, the numerical simulations including phase portrait, one- and two-parameter bifurcation diagrams are performed to validate our important findings.

Computational Analysis of Ca2+ Oscillatory Bio-Signals: Two-Parameter Bifurcation Diagrams.

Two types of bifurcation diagrams of cytosolic calcium nonlinear oscillatory systems are presented in rectangular areas determined by two slowly varying parameters. Verification of the periodic dynamics in the two-parameter areas requires solving the underlying model a few hundred thousand or a few million times, depending on the assumed resolution of the desired diagrams (color bifurcation figures). One type of diagram shows period-n oscillations, that is, periodic oscillations having n maximum values in one period. The second type of diagram shows frequency distributions in the rectangular areas. Each of those types of diagrams gives different information regarding the analyzed autonomous systems and they complement each other. In some parts of the considered rectangular areas, the analyzed systems may exhibit non-periodic steady-state solutions, i.e., constant (equilibrium points), oscillatory chaotic or unstable solutions. The identification process distinguishes the later types from the former one (periodic). Our bifurcation diagrams complement other possible two-parameter diagrams one may create for the same autonomous systems, for example, the diagrams of Lyapunov exponents, diagrams for mixed-mode oscillations or the 0–1 test for chaos and sample entropy diagrams. Computing our two-parameter bifurcation diagrams in practice and determining the areas of periodicity is based on using an appropriate numerical solver of the underlying mathematical model (system of differential equations) with an adaptive (or constant) step-size of integration, using parallel computations. The case presented in this paper is illustrated by the diagrams for an autonomous dynamical model for cytosolic calcium oscillations, an interesting nonlinear model with three dynamical variables, sixteen parameters and various nonlinear terms of polynomial and rational types. The identified frequency of oscillations may increase or decrease a few hundred times within the assumed range of parameters, which is a rather unusual property. Such a dynamical model of cytosolic calcium oscillations, with mitochondria included, is an important model in which control of the basic functions of cells is achieved through the signal regulation.

A Ricker–type predator–prey system with hunting cooperation in discrete time

A predator–prey system in discrete-time for which the predators engage in cooperative hunting and the prey population is subject to scramble intra-specific competition is developed and analyzed. We prove that the number of positive equilibrium points depends not only on the predator’s basic reproduction number but also on the intensity of predator cooperation. The model exhibits a discrete Hopf bifurcation when the unique positive equilibrium point loses its stability. Two-parameter bifurcation diagrams using MATLAB conclude that hunting cooperation can increase the likelihood of predator persistence but may destabilize the predator–prey interaction and drive the predator to extinction.

Compound bursting behaviors in a forced Mathieu-van der Pol-Duffing system

• Compound “fold/fold-cycle” bursting. • Compound “subHopf/fold-cycle” bursting via “fold/fold” hysteresis loop. • Compound “supHopf/supHopf” bursting via “fold/fold” hysteresis loop. • Compound bursting behaviors are investigated by the slow-fast analysis method. • Dynamical evolutions among different bursting patterns are studied by two-parameter bifurcation diagram. The mechanism of compound bursting behaviors of a forced Mathieu-van der Pol-Duffing system is explored. Four compound bursting types and one typical relaxation oscillation, i.e., compound “Homoclinic/Homoclinic” bursting, compound “fold/fold-cycle” bursting, compound “subHopf/fold-cycle” bursting via “fold/fold” hysteresis loop, compound “supHopf/supHopf” bursting via “fold/fold” hysteresis loop and “fold/fold” bursting, are investigated. We find that different parameter values result in different repetitive spiking states, which lead to different bursting patterns. Our research provides more possible routes to compound bursting dynamics, and strengthens a better understanding of the compound bursting behaviors.

Deterministic and stochastic dynamics in a gene regulatory network mediated by miRNA

MicroRNAs are able to modulate gene expression at the posttranscriptional level and play an essential role in various biological processes. In this paper, we establish a general model of an miRNA-mediated gene regulatory network motif with extracellular stimulus. Dynamical properties of the regulatory motif without stimulus focus on diverse codimension-1 and codimension-2 bifurcation analyses of two important feedback strengths as the parameters. Under varying levels of stimulus, we further consider stochastic dynamics controlled by crucial behaviors of monostability, bistability, excitability and oscillation in four typical regions on different two-parameter bifurcation diagrams. Furthermore, we apply potential landscapes to characterize global dynamics and stability behind physical properties of the stochastic behaviors. More essentially, our model can qualitatively simulate experimental findings of long-term memory formation in Aplysia.

Spikes and localised patterns for a novel Schnakenberg model in the semi-strong interaction regime

An analysis is undertaken of the formation and stability of localised patterns in a 1D Schanckenberg model, with source terms in both the activator and inhibitor fields. The aim is to illustrate the connection between semi-strong asymptotic analysis and the theory of localised pattern formation within a pinning region created by a subcritical Turing bifurcation. A two-parameter bifurcation diagram of homogeneous, periodic and localised patterns is obtained numerically. A natural asymptotic scaling for semi-strong interaction theory is found where an activator source term \[a = O(\varepsilon )\] and the inhibitor source \[b = O({\varepsilon ^2})\], with ε2 being the diffusion ratio. The theory predicts a fold of spike solutions leading to onset of localised patterns upon increase of b from zero. Non-local eigenvalue arguments show that both branches emanating from the fold are unstable, with the higher intensity branch becoming stable through a Hopf bifurcation as b increases beyond the \[O(\varepsilon )\] regime. All analytical results are found to agree with numerics. In particular, the asymptotic expression for the fold is found to be accurate beyond its region of validity, and its extension into the pinning region is found to form the low b boundary of the so-called homoclinic snaking region. Further numerical results point to both sub and supercritical Hopf bifurcation and novel spikeinsertion dynamics.

Extreme multistability in a fractional-order thin magnetostrictive actuator (TMA)

In this paper, we consider and investigate the dynamics of a fractional-order thin magnetostrictive actuato with quintic nonlinearity. The energy balance method is used to establish the motion equation of the system. Numerical solutions of the system are studied using the Caputo method and Adams Bashforth Moulton scheme. In order to analyze the dynamical behaviors of the proposed system, some mathematical tools including bifurcation diagrams, Lyapunov exponents, two-parameter bifurcation diagrams, time series plots and phase portraits are exploited. The bifurcation diagrams show that the system under consideration exhibits rich and intriguing behaviors including period-doubling route to chaos, complex phase space trajectory and extreme multistability. One of the most novelty of the proposed system is that its exhibits extreme multistability, which was not yet reported in the existing thin magnetostrictive actuators discussed in the literature to the author’s knowledge. The results carried out in this work may help us in better understanding of the thin magnetostrictive actuators.

Compound bursting dynamics in a parametrically and externally excited mechanical system

This paper aims to report complex bursting dynamics in a parametrically and externally excited mechanical system. Typically, two compound bursting patterns, i.e., bursting of “supHopf/homoclinic-fold/supHopf” type and bursting of “supHopf/supHopf-fold/supHopf” type, composed of two different clusters during each period of bursting, can be observed in this system. To begin with, the generation of each of the two clusters in compound bursting of “supHopf/homoclinic-fold/supHopf” type is investigated by the fast-slow analysis. Then, a two-parameter bifurcation diagram is plotted to investigate the transition of the compound bursting. We show that there are two originally disunited active areas, which may inosculate as a whole by homoclinic bifurcations if the system parameter varies properly. Based on this, a different type of compound bursting, i.e., bursting of “supHopf/supHopf-fold/supHopf” type, is revealed.

Two-Parameter 0-1 Test for Chaos and Sample Entropy Bifurcation Diagrams for Nonlinear Oscillating Systems

This paper presents and compares two-parameter bifurcation diagrams obtained from the 0-1 test for chaos and sample entropy methods for nonlinear oscillating systems. Based on the computed diagrams it is often possible to establish a unique correspondence between the two types of diagrams. Comparison of the diagrams are presented for typical chaotic systems, including those of Lorenz, Rössler, electric arc circuits and chemical oscillating systems. The two types of bifurcation diagrams may also be related to the period-n and frequency distribution diagrams. The main goal is to address the issue of obtaining similar results from the 0-1 test for chaos and sample entropy methods and discuss difficulty in properly selecting parameters needed in the two methods. A possible link between the 0-1 test for chaos and surrogate data methods is also mentioned.

Exploring dynamical complexity in a time-delayed tumor-immune model.

The analysis of dynamical complexity in nonlinear phenomena is an effective tool to quantify the level of their structural disorder. In particular, a mathematical model of tumor-immune interactions can provide insight into cancer biology. Here, we present and explore the aspects of dynamical complexity, exhibited by a time-delayed tumor-immune model that describes the proliferation and survival of tumor cells under immune surveillance, governed by activated immune-effector cells, host cells, and concentrated interleukin-2. We show that by employing bifurcation analyses in different parametric regimes and the 0-1 test for chaoticity, the onset of chaos in the system can be predicted and also manifested by the emergence of multi-periodicity. This is further verified by studying one- and two-parameter bifurcation diagrams for different dynamical regimes of the system. Furthermore, we quantify the asymptotic behavior of the system by means of weighted recurrence entropy. This helps us to identify a resemblance between its dynamics and emergence of complexity. We find that the complexity in the model might indicate the phenomena of long-term cancer relapse, which provides evidence that incorporating time-delay in the effect of interleukin in the tumor model enhances remarkably the dynamical complexity of the tumor-immune interplay.

The Yamada Model for a Self-Pulsing Laser: Bifurcation Structure for Nonidentical Decay Times of Gain and Absorber

We consider self-pulsing in lasers with a gain section and an absorber section via a mechanism known as [Formula: see text]-switching, as described mathematically by the Yamada ordinary differential equation model for the gain, the absorber and the laser intensity. More specifically, we are interested in the case that gain and absorber decay on different time-scales. We present an overall bifurcation structure by showing how the two-parameter bifurcation diagram in the plane of pump strength versus decay rate of the gain changes with the ratio between the two decay rates. In total, there are ten cases BI to BX of qualitatively different two-parameter bifurcation diagrams, which we present with an explanation of the transitions between them. Moroever, we show for each of the associated eleven cases of structurally stable phase portraits (in open regions of the parameter space) a three-dimensional representation of the organization of phase space by the two-dimensional manifolds of saddle equilibria and saddle periodic orbits. The overall bifurcation structure provides a comprehensive picture of the observable dynamics, including multistability and excitability, which we expect to be of relevance for experimental work on [Formula: see text]-switching lasers with different kinds of saturable absorbers.