Two-parameter Bifurcation Diagrams Research Articles

Amplitude equations for wave bifurcations in reaction–diffusion systems

A wave bifurcation is the counterpart to a Turing instability in reaction–diffusion systems, but where the critical wavenumber corresponds to a pure imaginary pair rather than a zero temporal eigenvalue. Such bifurcations require at least three components and give rise to patterns that are periodic in both space and time. Depending on boundary conditions, these patterns can comprise either rotating or standing waves. Restricting to systems in one spatial dimension, complete formulae are derived for the evaluation of the coefficients of the weakly nonlinear normal form of the bifurcation up to order five, including those that determine the criticality of both rotating and standing waves. The formulae apply to arbitrary n-component systems ( n⩾3 ) and their evaluation is implemented in software which is made available as supplementary material. The theory is illustrated on two different versions of three-component reaction–diffusion models of excitable media that were previously shown to feature super- and subcritical wave instabilities and on a five-component model of two-layer chemical reaction. In each case, two-parameter bifurcation diagrams are produced to illustrate the connection between complex dispersion relations and different types of Hopf, Turing, and wave bifurcations, including the existence of several codimension-two bifurcations.

Fractional-order heterogeneous memristive Rulkov neuronal network and its medical image watermarking application

This article proposes a novel fractional heterogeneous neural network by coupling a Rulkov neuron with a Hopfield neural network (FRHNN), utilizing memristors for emulating neural synapses. The study firstly demonstrates the coexistence of multiple firing patterns through phase diagrams, Lyapunov exponents (LEs), and bifurcation diagrams. Secondly, the parameter related firing behaviors are described through two-parameter bifurcation diagrams. Subsequently, local attraction basins reveal multi-stability phenomena related to initial values. Moreover, the proposed model is implemented on a microcomputer-based ARM platform, and the experimental results correspond to the numerical simulations. Finally, the article explores the application of digital watermarking for medical images, illustrating its features of excellent imperceptibility, extensive key space, and robustness against attacks including noise and cropping.

Bifurcation mechanisms underlying the nested mixed-mode oscillations in a canard-generating driven Bonhoeffer–van der Pol oscillator

In previous works (Inaba and Kousaka, 2020; Inaba and Tsubone, 2020; Inaba et al, 2022) the bifurcation structures referred to as nested mixed-mode oscillations (MMOs) were discovered; these structures can be generated by a driven slow–fast Bonhoeffer–van der Pol (BVP) oscillator. It is known that this oscillator can generate canards in the absence of perturbations. In this work, we investigate nested MMOs in a canard-generating driven BVP oscillator near a supercritical Hopf bifurcation point subjected to weak periodic perturbations. The analysis of this system is undertaken using the techniques presented by Kawakami, (1984). These techniques include the use of a shooting algorithm to derive the parameters of the forced oscillators at which bifurcations occur by solving the simultaneous equations that identify the conditions in which the solution of the forced oscillators is periodic and the characteristic multiplier(s) with the largest absolute value of the periodic solution on the Poincaré section crosses the unit circle. We also obtain several two-parameter bifurcation diagrams that describe the system investigated here. One motivation for this study is to demonstrate that the bifurcation structures present in nested MMOs are likely to be a widespread phenomenon. We hypothesize here that nested MMOs represent a stable phenomenon; this is likely to be the case because they appear and disappear consistently and are bound by sequential saddle–node and period-doubling bifurcation curves; this hypothesis is supported by the observation that codimension-two bifurcation points or cusps do not appear in the region close to these bifurcation boundaries.

Localized pattern formation: semi-strong interaction asymptotic analysis for three components model

We investigate a three-component system involving the Belousov–Zhabotinsky reaction in water-in-oil microemulsions. Our goal is to investigate the connection between homoclinic snaking and semi-strength interaction in a three-variable reaction–diffusion system. A two-parameter bifurcation diagram of homogeneous, periodic and localized patterns is obtained numerically, and a natural asymptotic scaling for semi-strong interaction theory is found where an activator source term a = O ( δ 1 ) and b = O ( δ 1 ) , with δ 1 ≪ 1 being the diffusion ratio. Under this regime, singular perturbation techniques are used to construct localized steady-state patterns, and new types of non-local eigenvalue problems (NLEP) are derived to determine the stability of these patterns to O ( 1 ) time-scale instabilities. We extend the scope of the NLEP by considering a general scenario where both time-scaling parameters are non-zero. All analytical results are found to agree with numerics. Further numerical results are presented on the location of various types of breathing Hopf instability for localized patterns.

Bifurcation analysis of complex switching oscillations in a Kerr microring resonator.

Microresonators are micron-scale optical systems that confine light using total internal reflection. These optical systems have gained interest in the past two decades due to their compact sizes, unprecedented measurement capabilities, and widespread applications. The increasingly high finesse (or Q factor) of such resonators means that nonlinear effects are unavoidable even for low power, making them attractive for nonlinear applications, including optical comb generation and second harmonic generation. In addition, light in these nonlinear resonators may exhibit chaotic behavior across wide parameter regions. Hence, it is necessary to understand how, where, and what types of such chaotic dynamics occur before they can be used in practical devices. We study here the underlying mathematical model that describes the interactions between the complex-valued electrical fields of two optical beams in a single-mode resonator with symmetric pumping. Recently, it was shown that this model exhibits a wide range of fascinating behaviors, including bistability, symmetry breaking, chaos, and self-switching oscillations. We employ here a dynamical system approach to perform a comprehensive theoretical study that allows us to identify, delimit, and explain the parameter regions where different behaviors can be observed. Specifically, we present a two-parameter bifurcation diagram that shows how (global) bifurcations organize the observable dynamics. Prominent features are curves of Shilnikov homoclinic bifurcations, which act as gluing bifurcations of pairs of periodic orbits or chaotic attractors, and a Belyakov transition point (where the stability of the homoclinic orbit changes). In this way, we identify and map out distinctive transitions between different kinds of chaotic self-switching behavior in this optical device.

Non-gradient-based simultaneous strategy for bifurcation parameter detection

This paper proposes a non-gradient-based simultaneous strategy for detecting bifurcation parameters in dynamical systems. The proposed method uses a single optimization algorithm with two interdependent objective functions, one for a periodic condition and the other for a bifurcation condition. In addition, a novel approach to easily detect a two-parameter bifurcation diagram is presented. A comparison between two population-based search algorithms, particle swarm optimization (PSO) and differential evolution (DE), shows that DE outperforms PSO in bifurcation parameter search. This work provides the first simultaneous strategy capable of directly detecting a two-parameter bifurcation diagram without requiring careful initialization, regardless of the stability of the periodic point. The proposed algorithm is simple, easy to understand, and computationally efficient, making it a powerful and widely applicable tool for accurate bifurcation parameter detection.

Dynamical analysis and circuit implementation of a memristive chaotic system with infinite coexisting attractors

In order to obtain a chaotic system with more complex dynamic characteristics and more suitable for engineering applications, this paper combines a general memristor containing a hyperbolic tangent function with a simple three-dimensional chaotic system to construct a four-dimensional memristive chaotic system with infinite coexisting attractors. The memristive chaotic system is thoroughly studied through numerical simulations of various nonlinear systems, including the Lyapunov exponent spectra, bifurcation diagram, C0 complexity, two-parameter bifurcation diagram and basins of attraction. The analysis reveals that this system has complex dynamical behavior. It includes not only periodic limit loops and chaotic attractors that depend on the variation of system parameters, but also the extreme multi-stability phenomenon of infinite coexisting attractors that depend on the variation of the initial conditions of the system. In addition, the chaos degradation and offset boosting control of the system are also studied and analyzed. Finally, the correctness and realizability of the memristive chaotic system are verified by circuit simulation and hardware circuit fabrication.The experimental results show that this memristive chaotic system can lay the foundation for practical engineering fields such as secure communication and image encryption.

Codimension-One and Codimension-Two Bifurcations of a Fractional-Order Cubic Autocatalator Chemical Reaction System

This article delves into an investigation of the dynamic behavior exhibited by a fractional order cubic autocatalator chemical reaction model. Specifically, our focus lies on exploring codimension-one bifurcations associated with period-doubling bifurcation and Neimark-Sacker bifurcation. Additionally, we undertake an analysis of codimension-two bifurcations linked to resonances of the types 1:2, 1:3, and 1:4. To achieve these outcomes, we employ the normal form method and bifurcation theory. The results are presented through comprehensive numerical simulations, encompassing visual representations such as phase portraits, two-parameter bifurcation diagrams, and maximum Lyapunov exponents diagrams. These simulations aptly examine the behavior of a system governed by two distinct parameters that vary within a three-dimensional space. Furthermore, the simulations effectively illustrate the theoretical findings while providing valuable insights into the underlying dynamics.

Chaos and Bistabilities in a Food-Chain Model with Allee Effect and Additional Food

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.

Bifurcation Analysis of a Wind Turbine Generator Drive System with Stochastic Excitation Under Both Displacement and Velocity Delayed Feedback

In this paper, a wind turbine generator drive system with stochastic excitation under both displacement and velocity delayed feedback is considered. Firstly, the center manifold method is used to approximate the delay term of the system, so that the Itô-stochastic differential equation can be obtained by random average method. Through the maximal Lyapunov exponential method, the local stochastic stability and random D-bifurcation conditions of the system are obtained. Secondly, it is verified that the increase of noise intensity and delay value induces the occurrence of random P-bifurcation of the system through Monte Carlo numerical simulations. In addition, the theoretical chaos threshold of the system is derived by the random Melnikov method. The results show that the chaos threshold decreases as the noise intensity increases, and the increase in time delay leads to a delay in the chaotic behavior of the system. Finally, the correctness and effectiveness of the chaos-theoretic analysis are verified based on the one-parameter bifurcation diagrams and the two-parameter bifurcation diagrams.

An enhanced FitzHugh–Nagumo neuron circuit, microcontroller-based hardware implementation: Light illumination and magnetic field effects on information patterns

This contribution introduced and investigated an improved photosensitive memristive FitzHugh–Nagumo (FHN) neural circuit. The mathematical equations of the model have been derived from Kirchhoff’s electrical circuit law. Based on Helmholtz’s theorem, the Hamilton statistical function that enables us to obtain the physical energy of the neuron necessary to sustain electrical activity has been established. Therefore, the energy dependency on the light intensity and the magnetic field has been provided. Using two-parameter diagrams, bifurcation diagrams, Lyapunov exponents, and time series, the window of parameters in which regular and irregular patterns of the neuron occur have been characterized. Besides, the proposed FHN neuron model is implemented using a simple microcontroller platform and various firing patterns are acquired experimentally to validate the numerical results. Furthermore, light and field-induced pattern formation in a chain regular lattice network made of 100 photosensitive memristive neurons is being studied. Numerical experiments are carried out to identify the effects of photocurrent and magnetic flux parameters on information coding patterns translated by the membrane potential from voltage output. The spatiotemporal patterns revealed that the network presents localized wave patterns with some features of synchronization, sensitive to parameter change. This confirms that information coding and other collective behaviors of the network could be efficiently controlled by taming the intensity of light illumination and magnetic field exposure.

Bifurcation Stability Analysis of the Synchronverter in a Microgrid

Synchronized converters are being studied as a viable alternative to address the transition from synchronous generation to power-electronics-based generation systems. One of the important features that make the synchronous generator an unrivaled alternative for power generation is its stability properties and inherent inertial response. This work presents a stability analysis of a synchronverter-based system conducted through the bifurcation theory to expose its stability regions in a grid-connected configuration with an aggregate load model conformed by a ZIP model and an induction motor model. One and two-parameter bifurcation diagrams on the gain, load, and Thévenin equivalent plane are computed and analyzed. All the results confirm the strong stability properties of the syncronverter. Some relevant findings are that the reduction in a droop gain or time constant results in Hopf bifurcations and inertia reduction, but the increase in the time constant leads to decoupling between the reactive and active power loops. It is also found that the increment of a specific time constant (τf>0.02 s) increases the stability region on the droop gains plane to all positive values. It is also found that a low lagging power factor reduces the feasible operating and stable operating regions. For a lagging power factor above 0.755, subcritical Hopf bifurcation disappears, and also, the feasible operating solution overlaps the stability region. Finally, it is also found how the Thévenin equivalent affects the stability and that the stability boundary is delimited by Hopf bifurcations. The bifurcation diagrams are numerically computed using XPP Auto software.

Hyperchaotic System with Application to Image Encryption

This paper introduces a novel hyperchaotic circuit system that is extremely sensitive to circuit parameters and system initial values. The typical dynamical behaviors of the hyperchaotic system such as one-parameter bifurcation diagrams, two-parameter bifurcation diagrams and Lyapunov exponent spectrum are discussed in detail. Then an image encryption algorithm is designed by using the hyperchaotic sequences. In this algorithm, Zigzag algorithm is used to scramble pixel positions, and backward diffusion and forward diffusion algorithms are used to diffuse pixel values. The results indicate that the proposed algorithm can improve the security of the image encryption algorithm. In addition, some circuit simulation experiments are performed to verify the sensitivity of the hyperchaotic circuit to parameters by Multisim. Finally, the hyperchaotic circuit is implemented by using physical elements. The simulation results are consistent with the experimental results of hardware circuit, which effectively verifies the existence of hyperchaotic attractors.

Bifurcation and Path-Following Analysis of Periodic Orbits of a Fermi Oscillator Model

In this research, we offer a bifurcation analysis to describe impacting, stick, and grazing between a particle and the piston to better understand the nonlinear dynamics of a Fermi oscillator. The principles of hybrid dynamical systems will be utilized to explain the moving process in such a Fermi oscillator. The motion complexity and stick mechanism of such an oscillator are demonstrated using periodic and chaotic motions. We introduce the hybrid dynamical system and numerical continuation detection tool COCO. Physical and mathematical models are used to construct the obtained bounded mathematical model. The frequency, amplitude in oscillating base displacement, and the gap between the stationary boundary and the piston’s equilibrium position are the major parameters. We employ path-following analysis to illustrate the bifurcation that leads to solution destabilization. From the viewpoint of eigenvalue analysis, the essence of period-doubling and Fold bifurcation is revealed. Numerical continuation methods are used to perform a complete one-parameter bifurcation analysis of the Fermi oscillator. The presence of codimension-1 bifurcations of limit cycles, like period-doubling, fold, and grazing bifurcations, is shown in this work. Then, the one-parameter continuation analysis is tracked in two-parameter bifurcation diagrams that include a second important factor of interest. Bifurcations cause all solutions to lose stability, according to our findings. The acquired results provide a better understanding of the Fermi oscillator mechanism and demonstrate that we may control the system dynamics by modifying the parameters.

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'.