Multiple Bifurcations Research Articles

Dynamics and bifurcation characteristics of actively Q-switched Erbium-doped fiber ring laser based on EOM with large modulation index

Dynamic and bifurcation characteristics of actively Q-switched Erbium-doped fiber ring laser (EDFRL) with large modulation index are investigated theoretically and experimentally. Using pump power and modulation index as control parameters, the laser exhibits a pulse splitting dynamic response process. More pulse splitting states such as double pulse, triple pulse, and quadruple pulse can be achieved in the process of increasing pump power and modulation index, while chaotic outputs are present during the pulse splitting evolution. The dynamic response of the output due to modulation index and modulation frequency change is also discussed. At large modulation index, the formed multiple periodic route exhibits a reverse multiple periodic bifurcation after the maximum multiple period. The dynamic response of Q-switched EDFRL with large modulation index contributes to a more comprehensive understanding of the dynamic characteristics of EDFRL and its applications.

Sex-biased predation and predator intraspecific competition effects in a prey mating system

<abstract><p>In this work, we propose and investigate a predator-prey model where the prey population is structured by sex and the predators (unstructured) depredate based on sex-bias. We provide conditions for the existence of equilibrium points and perform local stability analysis on them. We derive global stability conditions for the extinction state. We show the possible occurrence of Hopf and saddle-node bifurcations. Multiple Hopf bifurcations are observed as the sex-biased predation rate is varied. This variation also shows the opposite consequences in the densities of the sex-structured prey. Our results show that sex-biased predation can cause both stabilizing and destabilizing effects for certain parameter choices. It can also cause an imbalanced sex-ratio, which has ecological consequences. Furthermore when intraspecific competition among predators is minimized, it can lead to the extinction of prey. We discuss the ecological implications and application of our results to the biocontrol of invasive species susceptible to sex-biased predation.</p></abstract>

Multiple solitons with bifurcations, lump waves, M-shaped and interaction solitons of three component generalized (3+1)-dimensional Breaking soliton system

<abstract><p>The generalized (3+1)-dimensional Breaking soliton system (gBSS) has numerous applications across various scientific fields. This manuscript presents a study on important exact solutions of the gBSS, with a focus on novel solutions. Using the Hirota bilinear technique, we derive the general solution of the proposed system and obtain the novel solutions by considering different types of auxiliary functions. Our analysis includes the study of multi-solitons, multiple bifurcation solitons, lump wave solutions, M-shaped solitons, and their interactions. We also observe several hybrid solitons, including tuning fork-shaped, X-Y shaped, and double Y shaped. Our results are presented through graphical representations.</p></abstract>

Multiple bifurcations of a discrete modified Leslie-Gower predator-prey model.

In this paper, we work on the discrete modified Leslie type predator-prey model with Holling type II functional response. The existence and local stability of the fixed points of this system are studied. According to bifurcation theory and normal forms, we investigate the codimension 1 and 2 bifurcations of positive fixed points, including the fold, 1:1 strong resonance, fold-flip and 1:2 strong resonance bifurcations. In particular, the discussion of discrete codimension 2 bifurcation is rare and difficult. Our work can be seen as an attempt to complement existing research on this topic. In addition, numerical analysis is used to demonstrate the correctness of the theoretical results. Our analysis of this discrete system revealed quite different dynamical behaviors than the continuous one.

Effect of a novel generalized incidence rate function in [formula omitted] model: Stability switches and bifurcations

When a disease outbreaks in a population, the incidence rate function may increase initially with an increase of infective and then decrease due to inhibitory effects before getting saturated. For this purpose, we propose a new generalized incidence rate function and demonstrate the impact of this incidence rate function’s non-monotonic feature on the stability of endemic equilibria. In this paper, an SIR compartmental model with this new nonlinear incidence rate function and saturated treatment rate function is proposed. We obtain a disease free equilibrium which is locally stable for R0<1 and unstable otherwise. The system may have multiple endemic equilibria which can be either anti-saddle (node, focus or center) or saddle. We discover that when the incidence rate function is increasing at an anti-saddle endemic equilibrium, the endemic equilibrium may become either stable, unstable or switch stability via a parameter change. However, when the incidence rate function is non-monotonic and if the incidence rate decreases at an anti-saddle endemic equilibrium, the endemic equilibrium is always stable. Due to the limited treatment, we find forward (transcritical) and backward bifurcations. We also establish Hopf bifurcation and saddle–node bifurcation in the model system. Further, we incorporate the incubation delay in the model system. We obtain the same number of equilibria in the delay model. The disease free equilibrium is locally asymptotically stable for all delay lengths for R0<1. The endemic equilibrium switches its stability due to change in delay. Due to delay variation, the increasing incidence rate at an anti-saddle endemic equilibrium causes the multiple stability switches of the equilibrium from both stable and unstable initial states. Further, when the incidence rate decreases at the stable anti-saddle endemic equilibrium, it switches its stability once or multiple times and finally becomes unstable due to delay variation. We explicitly, show multiple stability switches of an endemic equilibrium due to existence of multiple Hopf bifurcations. When we further analyzed model system numerically, we could establish the existence of Hopf–Hopf bifurcation of codimension 2. In this case, the increase in the delay parameter causes the existence of two oscillatory solutions around the unstable endemic equilibrium. Hence, our results show that the delay system either destabilize developing the same frequency oscillations via Hopf bifurcation or two different frequency oscillations via Hopf–Hopf bifurcation of codimension 2. The existence and exhibition of these two frequency oscillations are novel and have not been explored much in epidemiological models.

Multiple bifurcation solitons, lumps and rogue waves solutions of a generalized perturbed KdV equation

Multiple bifurcation solitons, lumps and rogue waves solutions of a generalized perturbed KdV equation

Dialog modeling of domain specific cognitive assistants

Intelligent virtual assistants of software are becoming more common. Frequently, the communication interface is dialogue which is expressed in natural language. When an assistant consists of questions that are associated with a single answer, the abstraction of the dialogue is straightforward, however, when there are conversations with multiple bifurcations in the flow of communication, the representation and documentation is complex. In this sense, there is a lack of formalities that allow a dialogue to be accurately represented and based on them, to proceed with its programming in an industrial framework. In this work we propose the use of finite state machines to model complex dialogues. Thus, we introduce a formal model for the representation of dialogue and its mapping to a real assistant, using an industrial framework. We believe that our approach will make it easier to design and to program assistants with complex interaction, enabling better adoption of them.

Versatile benchmark model reproducing snap-through, asymmetric, symmetric unstable/stable, and multiple bifurcation including hill-top branching in structural instability

This paper proposes a simple yet innovative two degrees of freedom (2-DoFs) benchmark model for structural instability. The proposed rigid-body-spring model is versatile in that it reproduces various kinds of instability phenomena. Its parameter combinations comprehensively realize snap-through, asymmetric, symmetric unstable/stable, and multiple bifurcation including hill-top branching in structural stability problems. Governing equations including the total potential energy, nonlinear equilibrium equations, stiffness matrix (Jacobian), and stability information are all explicitly formulated. For post-buckling analysis, path branching is visualized for intuitive diagnosis. The versatile benchmark model is expected to yield a qualitative understanding of the instability mechanism depending on the model parameters. The benchmark model also contributes to the phenomenological understanding of stability design.

Coexistence of multiscroll chaotic attractors in two coupled inertial Hopfield neurons: numerical simulations and experiment

In this contribution, we consider the dynamics of a pair of coupled inertial neurons with hyperbolic tangent activation function. The two inertial neurons are coupled by adding to each one’s amplitude a perturbation proportional to the other one. The model is governed by a fourth-order autonomous system with hyperbolic tangent nonlinearities. The analysis of the coupled system yields nine equilibrium points some of which experience Hopf type bifurcation. When adjusting the coupling coefficients, striking nonlinear patterns are disclosed such as the coexistence of numerous bifurcation branches, merging crisis, multiple Hopf bifurcations, coexisting self-excited motions (e.g. two coexisting double-scroll chaos, four coexisting period-n cycles, four coexisting single-scroll chaos), and four-scroll chaotic attractors. These latter features are diagnosed with the help of classic numerical tools (e.g. 1D and 2D maximum Lyapunov exponent diagrams, 1D bifurcation diagrams of local peaks of variables, frequency spectrum plots, phase space trajectory plots, and attraction basins). The analogue electronic circuit design of the coupled inertial neurons system is carried out and simulated in PSpice to verify diverse types of features reported during the theoretical study. One of main achievements of the present article is that the coupling of inertial neurons can be regarded as an alternative scheme to obtain multiscroll chaotic signals.

Featured Cover

The cover image is based on the Research Article Gröbner bases plugged into graphical skills to solve a set of multiple bifurcation equations in structural compound stability problems by Masato Tanaka et al., https://doi.org/10.1002/nme.7088.

Multiple bifurcations in a predator–prey system of modified Holling and Leslie type with double Allee effect and nonlinear harvesting

In this paper, a modified Leslie-type predator–prey model with simplified Holling type IV functional response is established, in which double Allee effect on prey and nonlinear prey harvesting are considered. The analysis of the model shows that there exists a Bogdanov–Takens singularity (focus case) of codimension 4, and also multiple other nonhyperbolic and degenerate equilibria. Bifurcations are explored and it is found that transcritical bifurcation, saddle–node bifurcation, Bogdanov–Takens bifurcation of codimension 2, degenerate cusp type Bogdanov–Takens bifurcation of codimension 3, and degenerate focus type Bogdanov–Takens bifurcation of codimension 4 occur as parameters vary. The bifurcations result in complex dynamic behaviors, such as double limit cycle, triple limit cycle, quadruple limit cycle, cuspidal loop, (multiple) homoclinic loop, saddle–node loop, and limit cycle(s) simultaneously with homoclinic loop. We run numerical simulations to verify the theoretical results, and it is found that the system admits bistability, tristability, or even tetrastability.

On the mechanism of multiscroll chaos generation in coupled non-oscillatory rayleigh-duffing oscillators

We investigate the dynamics of a pair of coupled non oscillatory Rayleigh-Duffing oscillators (RDOs here after). The RDO serves as a model for a class of nonlinear oscillators including microwave Gunn oscillators [Guin et al Comm. in Nonlinear Sci. Numerical Simulat, 2017]. Here, the coupling between the two oscillators is obtained by superimposing to each one’s amplitude a perturbation proportional to the other one. We demonstrate that the coupling induces more equilibrium points and results in extremely complex nonlinear behaviors including multistability (up to six coexisting attractors), multiple Hopf bifurcations, multi-scroll chaos, and coexisting bifurcation trees. These phenomena are studied in detail by utilizing one-parametric bifurcation plots, bi-parametric Lyapunov exponent diagrams, phase space trajectory plots, and basins of attraction as well. Experimental results captured from an Arduino microcontroller-based realization of the coupled RDOs are included to support the observations made through numerical analysis. We would like to point out that the coupling scheme followed in this work may stimulate the research on multiscroll chaos generation based on coupled nonlinear oscillators.

Coupled non-oscillatory Duffing oscillators: Multistability, multiscroll chaos generation and circuit realization

We consider the dynamics of two coupled double-well non-oscillatory cubic Duffing oscillators. The coupling is realized by perturbing each one’s amplitude with a signal proportional to the amplitude of the other one. We prove that the coupling induces in the resulting fourth-order self-driven nonlinear system up to nine equilibrium points and engendered extremely complex dynamic features amongst which the coexistence of several bifurcation branches, multiple Hopf bifurcations (when a single parameter is considered), multistable behaviors (e.g. a pair of coexisting double scroll chaos, four coexisting mono-scroll chaos), and four-scroll chaotic attractor. A detailed investigation of these latter features is carried out by resorting to both analytical (i.e. Hopf bifurcation theorem, Routh–Hurwitz criterion) and numerical tools (e.g. Lyapunov exponent diagrams, 1D bifurcation diagrams, frequency spectrum, phase space trajectory plots, time series and attraction basins). An analog circuit implementing the coupled Duffing oscillators is performed by using simple electronic components (i.e. capacitors, resistors, quadruple operational amplifiers and analog multipliers chips). Experimentally captured phase portraits support the conclusions of the theoretical study. One of the important conclusions of this work is that the coupling of non-oscillatory Duffing type oscillators can be considered as an alternative approach to the generation of multiscroll chaos.

Multiple bifurcated reconfiguration of double-loop metamorphic mechanisms with prismatic joints

The design and analysis of multi-loop metamorphic mechanisms are challenging in the research of metamorphic mechanisms. Inspired by the parallelogram mechanism, this paper proposes a metamorphic mechanism by adding a link to the parallelogram mechanism and changing the properties of some kinematic joints. The metamorphic mechanism has two loops and is composed of six links connected by six revolute joints and one prismatic joint. The topological structure of the metamorphic mechanism is represented by a directed graph, and the multiple bifurcation of the metamorphic mechanism is analyzed by using screw theory and high-order kinematics. The metamorphic mechanism has six bifurcation points and four different motion branches, including one parallelogram branch, one isosceles trapezoid branch and two oscillating block branches. Meanwhile, by combining these metamorphic mechanisms, some novel metamorphic mechanisms with variable sizes and different configurations can be obtained. The metamorphic mechanism proposed in this paper provides a fresh perspective on how to design new metamorphic mechanisms.

Gröbner bases plugged into graphical skills to solve a set of multiple bifurcation equations in structural compound stability problems

AbstractA core issue in structural multiple bifurcations (MB) in computational engineering is to identify all existing branching paths emanating from the MB point in compound stability problems. The governing MB equations (MBEs) will commonly result in a set of three (or occasionally two) polynomial equations in asymptotic stability theory when the singular stiffness matrix is subject to a rank deficiency of two (i.e., two null eigenvalues). However, no general solution strategy has been established to solve MBEs so far. This study proposes innovative graphical solution ideas to intuitively visualize multiple path branching in 2D‐ and 3D‐spaces of variables. Although the graphical skills display real solutions in specified search areas on a graphical monitor, it is not assured that “all” real roots are detected. The total number of identified real and complex roots of simultaneous equations must be generally consistent with that predicted algebraically to ensure that all real and complex roots are captured in MB. In computational algebra, Gröbner bases are employed to convert a set of polynomial equations into single recursively solvable equations and can be plugged into visualization steps. Therefore, Gröbner bases and graphical skills are complementary and can be applied to numerically solve a set of plate/shell structural MBEs.

Dynamics and stability of two predators–one prey mathematical model with fading memory in one predator

This paper concentrates on dynamics and stability analysis of two predators–one prey mathematical model with competition between predators and fading memory in one predator. The investigation of the constructed model shows that there exist five equilibria, e.g. trivial extinction state of all populations, extinction of both predators state, extinction of first or second predator state and coexisting state. Investigating the eigenvalues of characteristic polynomial, conditions for the local stability around each equilibrium are also determined depending on the parameter space. Analytical formulations are complemented with numerical simulations, where time simulations and single parameter numerical continuation of each variable are performed with respect to model parameters and multiple sub-and super-critical Hopf bifurcations, period doubling bifurcation and transcritical bifurcation are detected for different values of memory related parameter. Our results show that fading memory and competition between predators have substantial impact on the existence and dynamics of all three populations and may shed lights on further understanding of interacting species in ecology.

A class of periodic lattices for tuning elastic instabilities

With in mind microstructures exhibiting unconventional macroscopic mechanical behaviors, characterized by overall auxetic responses and strain localization due to local elastic instabilities, in this work we conceived a simple two-dimensional non-chiral architecture in the form of a periodic lattice, whose drawing is decided by varying the thickness ratios of the cells’ walls and in turn their slenderness. Inspired by nature, which often uses supposedly naive geometries that conceal complex functions, and focusing the attention on two limit geometrical configurations of main interest, we show how these non-chiral settings, as a function of the prescribed boundary conditions, might lead to symmetry breaking associated to a variety of non-trivial deformation modes, which in some cases also retrace and generalize chiral as well as auxetic responses already observed in literature in simpler microstructures. Interestingly, for both the above mentioned limit cases, we were able to idealize the mechanical response of the system through geometrically nonlinear beam-based models, in this way obtaining helpful analytical solutions and explicit formulas to estimate the tangent effective stiffness of the lattice as well as the critical load at the onset of instability. The post-buckling was instead analyzed and discussed in detail through parametric numerical finite element simulations and ad hoc laboratory experiments, performed by faithfully realizing 3D printed prototypes, constructed via additive manufacturing technologies and made of rubber-like material, to follow extreme deformation patterns, including multi-stable states, localization and compaction with possible self-contact/touching of the elements. At the end, we exploited the obtained closed-form solutions and some associated inequalities to derive the transition from the discrete lattice to its continuum limit, which – together with the multiple equilibrium bifurcation points exhibited by the proposed microstructure – could be used to broaden the spectrum of metamaterial geometries designed to exhibit complex tunable properties.

Future Imaginings in Art and Artificial Intelligence

ABSTRACT In considering future imaginings of art and Artificial Intelligence (AI), we point out two key premises to our approach: the human-machine relation without opposition between the two terms; and the implications of artificial intelligence within practice-led art practices. According to Gilbert Simondon, there is a biological and technological evolution that does not separate nature and technics or culture and technics, since what is defined by human nature is already part of a technological system. Yuk Hui criticizes the idea of technological singularity supported by strong artificial intelligence which converges on itself. However, there is also the possibility of multiple bifurcations emerging from technodiverse practices and aesthetics. On algorithmic art, Lev Manovich points out that AI is increasingly culturally influencing our imagination, choices, and behaviours and that it has built its own imagery and cognitive patterns in society. Joanna Zylinska connects the dots between AI, creativity and innovation, which by bringing AI to research into practice-led art practices, opens the problematization of AI to the possibilities of reprogramming the human sensory-cognitive apparatus. In this paper, we extend these approaches to examine whether these computer systems that enable the processing of big data can or cannot program our imagination, desires, likes, sensations, perceptions and attitudes. Finally, in order to think the worlds built through these emerging AI technologies, we ask what kind of society we want to live in, and what roles we will allow ubiquitous computational technologies, such as AI, to play in our daily lives.

DYNAMIC ANALYSIS OF A CHAOTIC 3D QUADRATIC SYSTEM USING PLANAR PROJECTION.

The theory of dynamical systems is one of the most important theorems of scientific research because it relies heavily on most of the major fields of applied mathematics to give a sufficiently broad view of reality, but it still poses some problems, especially with regard to the modeling of certain physical phenomena. Since most of these systems are designed as continuous or discrete dynamic systems with large dimensions and multiple bifurcation parameters, researchers face major problems in qualitative study. In this paper, we propose a method to study bifurcations of continuous three-dimensional dynamic systems in general and chaotic systems in particular, which contains many bifurcation parameters. This method is mainly based on the projection on the plane and on the appropriate bifurcation parameter.

Multiple and complex codimension-2 bifurcations underlying post-inhibitory rebound spike and excitability transition

Neuron exhibits nonlinear dynamics such as excitability transition and post-inhibitory rebound (PIR) spike related to bifurcations, which are associated with information processing, locomotor modulation, or brain disease. PIR spike is evoked by inhibitory stimulation instead of excitatory stimulation, which presents a challenge to the threshold concept. In this paper, 7 codimension-2 or degenerate bifurcations related to 10 codimension-1 bifurcations are acquired in a neuronal model, which presents the bifurcations underlying the excitability transition and PIR spike. Type I excitability corresponds to saddle-node bifurcation on an invariant cycle (SNIC) bifurcation, and type II excitability to saddle-node (SN) bifurcation or sub-critical Hopf (SubH) bifurcation or sup-critical Hopf (SupH) bifurcation. The excitability transition from type I to II corresponds to the codimension-2 bifurcation, Saddle-Node Homoclinic orbit (SNHO) bifurcation, via which SNIC bifurcation terminates and meanwhile big homoclinic orbit (BHom) bifurcation and SN bifurcation emerge. A degenerate bifurcation via which BHom bifurcation terminates and limit point of cycle (LPC) bifurcation emerges is responsible for spiking transition from type I to II, and the roles of other codimension-2 bifurcations (Cusp, Bogdanov-Takens, and Bautin) are discussed. In addition, different from the widely accepted viewpoint that PIR spike is mainly evoked near Hopf bifurcation rather than SNIC bifurcation, PIR spike is identified to be induced near SNIC or BHom or LPC bifurcations, and threshold curves resemble that of Hopf bifurcation. The complex bifurcations present comprehensive and deep understandings of excitability transition and PIR spike, which are helpful for the modulation to neural firing activities and physiological functions.