Bifurcation Scenario Research Articles

Mathematical and Experimental Model of Neuronal Oscillator Based on Memristor-Based Nonlinearity

This article presents a mathematical and experimental model of a neuronal oscillator with memristor-based nonlinearity. The mathematical model describes the dynamics of an electronic circuit implementing the FitzHugh–Nagumo neuron model. A nonlinear component of this circuit is the Au/Zr/ZrO2(Y)/TiN/Ti memristive device. This device is fabricated on the oxidized silicon substrate using magnetron sputtering. The circuit with such nonlinearity is described by a three-dimensional ordinary differential equation system. The effect of the appearance of spontaneous self-oscillations is investigated. A bifurcation scenario based on supercritical Andronov–Hopf bifurcation is found. The dependence of the critical point on the system parameters, particularly on the size of the electrode area, is analyzed. The self-oscillating and excitable modes are experimentally demonstrated.

Coupled systems with quasi-periodic and chaotic dynamics

The interaction of a system with quasi-periodic autonomous dynamics and a chaotic Rössler system is studied. We have shown that with the growth of the coupling, regimes of two-frequency and three-frequency quasiperiodicity, a periodic regime and a regime of oscillation death sequentially arise. With a small coupling strength, doubling bifurcations of three-frequency tori are observed in the system. A chaotic regime, characterized by two additional zero Lyapunov exponents in spectrum, is revealed. Two-parameter Lyapunov exponent analysis and bifurcation analysis are presented. A new bifurcation scenario of transition from the regime of oscillation death to quasi-periodicity in coupled systems is described.

A Saddle-Node Bifurcation Scenario for Disappearance of Nonlinear Electrostatic Waves in an Ultra-Relativistic Degenerate Dense Plasma

The birth or destruction of electrostatic excitations is investigated in an ultra-relativistic degenerate dense dusty plasma by the saddle-node bifurcation scenario. An abstract form of bifurcation is derived which captures the main features of traveling wave solutions for various regions in the parameter space of plasma number density. We show that a sudden disappearance of solitary and supernonlinear periodic waves occurs at the critical point of saddle-node bifurcation. For a range of ion number density and dust concentration, we obtain the critical values of bifurcation and different existence domains of stability region. The numerical phase portrait analysis is also performed to examine the coexistence of solitary waves, nonlinear periodic, and supernonlinear periodic waves for different conditions. We further derive analytically the conditions with respect to the model parameters that give rise to both polarities of solitary waves. It is found that the amplitude of localized structures decreases with the increase in the plasma number density. Our results could be applicable for astrophysical objects with high density matter such as white dwarfs and neutron stars.

Semi-analytical method to study piecewise linear oscillators

This article proposes a semi-analytical method to investigate the dynamics and bifurcation scenarios of piecewise linear oscillators. The method is based on a mapping technique with a matrix structure that allows easy and rapid construction of any periodic orbit. When validated against direct numerical integration simulations, a good correlation and an accurate prediction of bifurcation phenomena were shown. The method is applied to analyse the nonlinear dynamic responses and bifurcations scenarios causes by changes of stiffness and viscous damping. A set of minimum conditions that the system must meet to present period doubling bifurcations and sub-harmonic orbits was given.

On the Bifurcation of Thresholds of the Essential Spectrum with a Spectral Singularity

We consider the Schrödinger operator on the plane with bounded potential, where is a real potential, and are compactly supported complex potentials, andis a small parameter, assuming that the lower part of the spectrum of the one-dimensional Schrödinger operator consists of a pair of isolated eigenvalues and the essential spectrum of the operator has a virtual level at its lower edge and a spectral singularity inside. Additionally, we assume that there is a certain superposition of eigenvalues of the operator with the virtual level and spectral singularity of the operator; this leads to the emergence of a special threshold in the essential spectrum of the perturbed operator, with the perturbation leading to a bifurcation of this threshold into eigenvalues and resonances with multiplicity doubling. The bifurcation scenario described in this paper is qualitatively different from the previously known ones.

Biquadratic nontwist map: a model for shearless bifurcations

Area-preserving nontwist maps are used to describe a broad range of physical systems. In those systems, the violation of the twist condition leads to nontwist characteristic phenomena, such as reconnection–collision sequences and shearless invariant curves that act as transport barriers in the phase space. Although reported in numerical investigations, the shearless bifurcation, i.e., the emergence scenario of multiple shearless curves, is not well understood. In this work, we derive an area-preserving map as a local approximation of a particle transport model for confined plasmas. Multiple shearless curves are found in this area-preserving map, with the same shearless bifurcation scenario numerically observed in the original model. Due to its symmetry properties and simple functional form, this map is proposed as a model to study shearless bifurcations.

Rhythmogenesis in the mean field model of the neuron–glial network

Despite the fact that the phenomenon of bursting activity is important for functioning of living neural networks, the mechanisms of its origin are still not clear. In this paper, we propose a new phenomenological model that can explain the mechanisms of the formation of bursting activity based on short-term synaptic plasticity, recurrent connections, and neuron–glial interactions. We show that neuron–glial interactions can induce bursting activity. The bifurcation scenarios of emergence of bursting activity are in the focus of the paper. Proposed study is important for understanding the complex dynamics of neural networks.

The Dynamics of a Delayed Ecoepidemiological Model with Nonlinear Incidence Rate

In this paper, the general framework for calculating the stability of equilibria, Hopf bifurcation of a delayed prey-predator system with an SI type of disease in the prey population, is investigated. The impact of the incubation period delay on disease transmission utilizing a nonlinear incidence rate was taken into account. For the purpose of explaining the predation process, a modified Holling type II functional response was used. First, the existence, uniform boundedness, and positivity of the solutions of the considered model system, along with the behavior of equilibria and the existence of Hopf bifurcation, are studied. The critical values of the delay parameter for which stability switches and the nature of the Hopf bifurcation by using normal form theory and center manifold theorem are identified. Additionally, using numerical simulations and a hypothetical dataset, various dynamic characteristics are discovered, including stability switches, chaos, and Hopf bifurcation scenarios.

Toxicity-mediated regime shifts in a contaminated nutrient-plankton system.

In this article, we contemplate the dynamics of a three-tier system of nutrient, phytoplankton, and zooplankton with a gestation delay of discrete type and a distributed delay in nutrient recycling. Phytoplankton secretion-mediated alteration in the grazing pattern of zooplankton is encapsulated by a Monod-Haldane functional response. We carry out global sensitivity analysis for identifying the crucial model parameters having a significant impact on zooplankton density. The system potentially exhibits bistable configurations under identical ecological conditions by allowing different bifurcation scenarios, including multiple saddle-node and transcritical bifurcations with varying input rates of nutrients and inhibitory effects of phytoplankton against zooplankton. We observe that the gestation delay in zooplankton is responsible for the emergence of noxious bloom events. Interestingly, when the delay parameter crosses a threshold, the system experiences chaotic disorder, which prognosticates the onset of irregular bloom. Furthermore, by adding Gaussian white noise, we have extended the deterministic model to its stochastic counterpart. We found that white noise appears to regulate the survival and extinction of interacting populations. Comprehensive numerical simulations are consistent with mathematical results prognosticated by linear analysis.

Modeling the Spread and Control of Viral Infection in Damaged Aquatic System: Emergence of Patterns

In this paper, we propose an eco-epidemiological model for prey (free-living motile species, i.e., dinoflagellates) and predator in an aquatic system. The predator’s density is affected by both virus-induced prey and toxicity. The spread of viral infection is taken as simple law of mass, and interaction between prey and predator is described by the Holling type II functional response. Stability analysis of the model system has been studied for spatial and non-spatial systems, and the theoretical results are verified with numerical simulation. It is explored that the aquatic system is very sensitive to carrying capacity and able to generate chaotic phenomena. We have observed the Hopf and transcritical bifurcation scenario for different carrying capacity and transmission of virus contamination. The results show that high carrying capacity affects the aquatic system, and virus also influences the species dynamics. Further, the diffusion-driven Turing instability is analyzed and different hot and cold cluster-based Turing patterns are observed, emphasizing the effect of time variation and virus contamination. The results obtained in this study give a rich dynamics and show that the viruses with poor water quality, high nutrients, red and green algae, toxic substances, etc. are responsible for natural substances demises, aquatic habitats death. In large, it affects the environment and human health.

Asymptotic formulation of the nonlinear bifurcation scenarios in thermomechanically coupled plates

Asymptotic formulation of the nonlinear bifurcation scenarios in thermomechanically coupled plates

Comment on "A new class of out-gap discrete solitons in binary waveguide arrays" [Chaos 32, 073113 (2022)

Recent results [M. C. Tran and T. X. Tran, Chaos 32, 073113 (2022)] are put into the context of earlier work [A. V. Gorbach and M. Johansson, Eur. Phys. J. D 29, 77-93 (2004); M. Johansson and A. V. Gorbach, Phys. Rev. E 70, 057604 (2004)]. The two newly found families of "beyond-band discrete solitons" [M. C. Tran and T. X. Tran, Chaos 32, 073113 (2022)] are found to be smooth continuations of "on-top breathers" briefly mentioned by Gorbach and Johansson [Eur. Phys. J. D 29, 77-93 (2004)] and Johansson and Gorbach [Phys. Rev. E 70, 057604 (2004)] and the relevant bifurcation scenarios are described. One family is shown to be linearly unstable with respect to symmetry-breaking oscillations.

Allee effect-driven complexity in a spatiotemporal predator-prey system with fear factor.

In this paper, we propose a spatiotemporal prey-predator model with fear and Allee effects. We first establish the global existence of solution in time and provide some sufficient conditions for the existence of non-negative spatially homogeneous equilibria. Then, we study the stability and bifurcation for the non-negative equilibria and explore the bifurcation diagram, which revealed that the Allee effect and fear factor can induce complex bifurcation scenario. We discuss that large Allee effect-driven Turing instability and pattern transition for the considered system with the Holling-Ⅰ type functional response, and how small Allee effect stabilizes the system in nature. Finally, numerical simulations illustrate the effectiveness of theoretical results. The main contribution of this work is to discover that the Allee effect can induce both codimension-one bifurcations (transcritical, saddle-node, Hopf, Turing) and codimension-two bifurcations (cusp, Bogdanov-Takens and Turing-Hopf) in a spatiotemporal predator-prey model with a fear factor. In addition, we observe that the circular rings pattern loses its stability, and transitions to the coldspot and stripe pattern in Hopf region or the Turing-Hopf region for a special choice of initial condition.

Bifurcation phenomena in Taylor–Couette flow in a very short annulus with radial through-flow

In this study, the non-linear dynamics of Taylor–Couette flow in a very small-aspect-ratio wide-gap annulus in a counter-rotating regime under the influence of radial through-flow are investigated by solving its full three-dimensional Navier-Stokes equations. Depending on the intensity of the radial flow, either an axisymmetric (pure m=0 mode) pulsating flow structure or an axisymmetric axially propagating vortex will appear subcritical, i.e. below the centrifugal instability threshold of the circular Couette flow. We show that the propagating vortices can be stably existed in two separate parameter regions, which feature different underlying dynamics. Although in one regime, the flow appears only as a limit cycle solution upon which saddle-node-invariant-circle bifurcation occurs, but in the other regime, it shows more complex dynamics with richer Hopf bifurcation sequences. That is, by presence of incommensurate frequencies, it can be appeared as 1-, 2- and 3-torus solutions, which is known as the Ruelle–Takens–Newhouse route to chaos. Therefore, the observed bifurcation scenario is the Ruelle-Takens-Newhouse route to chaos and the period doubling bifurcation, which exhibit rich and complex dynamics.

Hidden attractors in Chua circuit: mathematical theory meets physical experiments

After the discovery in early 1960s by E. Lorenz and Y. Ueda of the first example of a chaotic attractor in numerical simulation of a real physical process, a new scientific direction of analysis of chaotic behavior in dynamical systems arose. Despite the key role of this first discovery, later on a number of works have appeared supposing that chaotic attractors of the considered dynamical models are rather artificial, computer-induced objects, i.e., they are generated not due to the physical nature of the process, but only by errors arising from the application of approximate numerical methods and finite-precision computations. Further justification for the possibility of a real existence of chaos in the study of a physical system developed in two directions. Within the first direction, effective analytic-numerical methods were invented providing the so-called computer-assisted proof of the existence of a chaotic attractor. In the framework of the second direction, attempts were made to detect chaotic behavior directly in a physical experiment, by designing a proper experimental setup. The first remarkable result in this direction is the experiment of L. Chua, in which he designed a simple RLC circuit (Chua circuit) containing a nonlinear element (Chua diode), and managed to demonstrate the real evidence of chaotic behavior in this circuit on the screen of oscilloscope. The mathematical model of the Chua circuit (further, Chua system) is also known to be the first example of a system in which the existence of a chaotic hidden attractor was discovered and the bifurcation scenario of its birth was described. Despite the nontriviality of this discovery and cogency of the procedure for hidden attractor localization, the question of detecting this type of attractor in a physical experiment remained open. This article aims to give an exhaustive answer to this question, demonstrating both a detailed formulation of a radiophysical experiment on the localization of a hidden attractor in the Chua circuit, as well as a thorough description of the relationship between a physical experiment, mathematical modeling, and computer simulation.

Mathematical Analysis of the Transmission Dynamics of Skin Cancer Caused by UV Radiation

Nowadays, skin cancer is a worldwide panic. It is related to ultraviolet radiation. In this paper, we have formulated a SIRS type mathematical model to show the effects of ultraviolet radiation on skin cancer. At first, we have showed the boundedness and positivity of the model solutions to verify the model’s existence and uniqueness. The boundedness and positivity gave the solutions of our model bounded and positive, which was very important for real-world situation because in real world, population cannot be negative. Then, we have popped out all the equilibrium points of our model and verified the stability of the equilibrium points. This stability test expressed some physical situation of our model. The disease-free equilibrium point is locally asymptotically stable if R 0 < 1 and if R 0 > 1 , then it is unstable. Again, the endemic equilibrium point is stable, if R 0 > 1 and unstable if R 0 < 1 . In order to understand the dynamical behavior of the model’s equilibrium points, we examined the phase portrait. We also have observed the sensitivity of the model parameters. After this, we have investigated the different scenarios of bifurcations of the model’s parameters. At the set of Hopf bifurcation parameters when infection rate due to UV rays is less than α 1 = 0.01 , proper control may eradicate the existence of disease. From transcritical bifurcation, we can say when recovery rate greater than 1.9, then the disease of skin cancer can be eliminated and when recovery rate less than 1.9 then the disease of skin cancer cannot be eradicated. Finally, numerical analysis is done to justify our analytical findings.

Complex nonlinear dynamics of polarization and transverse modes in a broad-area VCSEL

Lasers can exhibit nonlinear and chaotic dynamics driven by the interaction of multiple lasing modes, and investigating the different scenarios of mode competition and bifurcations of their dynamics is of great interest on a fundamental level as well as in view of applications. We study the dynamics of a broad-area vertical-cavity surface-emitting laser (VCSEL) in solitary continuous-wave operation with a comprehensive investigation of its polarization state, lasing spectra, near-field distributions, and temporal dynamics. Fluctuations at the frequency of birefringence splitting and other frequency components develop in a series of bifurcations. The bifurcations coincide with changes of the transverse lasing modes and/or the polarization state, demonstrating the importance of both the spatial and polarization degrees of freedom for mode competition. As a consequence, the inherent nonlinear dynamics of broad-area VCSELs is significantly more complex than the dynamics of VCSELs with a single spatial mode.

Conjoined Lorenz twins-a new pseudohyperbolic attractor in three-dimensional maps and flows.

We describe new types of Lorenz-like attractors for three-dimensional flows and maps with symmetries. We give an example of a three-dimensional system of differential equations, which is centrally symmetric and mirror symmetric. We show that the system has a Lorenz-like attractor, which contains three saddle equilibrium states and consists of two mirror-symmetric components that are adjacent at the symmetry plane. We also found a discrete-time analog of this "conjoined-twins" attractor in a cubic three-dimensional Hénon map with a central symmetry. We show numerically that both attractors are pseudohyperbolic, which guarantees that each orbit of the attractor has a positive maximal Lyapunov exponent, and this property is preserved under small perturbations. We also describe bifurcation scenarios for the emergence of the attractors in one-parameter families of three-dimensional flows and maps possessing the symmetries.

Frequency entrainment and suppression of stick–slip vibrations in a 3 DoF discontinuous disc brake model

Friction-induced stick–slip vibration has been identified as the main reason for various types of disc brake noises like creep groan, squeak and creak. In this work, we investigate the suppression of stick–slip vibrations in disc brakes due to the application of a small-amplitude normal harmonic force on the brake pad. The discontinuity induced bifurcations that lead to such suppression are studied in detail. The frequency range of the normal force in which suppression of stick–slip vibration occur is identified. A 3 DoF discontinuous model of the disc brake is considered. The discontinuous oscillator is modelled as a Filippov system with smooth vector fields separated by a discontinuity surface. It is shown that the normal harmonic force entrains the non-smooth limit cycle even for very small forcing amplitudes. Different frequency intervals exhibiting harmonic, super-harmonic, and sub-harmonic entrainments are identified. It is demonstrated that the initiation and termination of these entrainment intervals are associated with Neimark–Sacker bifurcations. It is also shown that discontinuity induced grazing-sliding mechanism brings about these bifurcations. Switching-sliding bifurcations happening in the entrainment region give rise to a sub-interval which is associated with suppression of stick–slip vibrations. We also report interesting discontinuity induced bifurcation scenarios in higher order entrainment regimes like grazing-sliding involving different orbit loops at lower and upper limits of the interval and splitting–merging of the sliding surface. • Stick–slip vibration suppression in disc brakes by using normal force is studied. • Various entrainment regions are detected and studied by Filippov’s method. • Discontinuity induced bifurcations causing these entrainments are analysed. • Entrainment sub-intervals with suppressed stick–slip vibration are identified.

Full bifurcation scenarios and pattern formation of laminar electroconvection in a cavity

This study numerically investigates the flow structures and bifurcation scenarios of three-dimensional (3D) laminar electroconvection (EC). An efficient parallel lattice Boltzmann model is undertaken to numerically solve the model problem. The results present three steady flow patterns and three pitchfork bifurcations. These three patterns each have one, two, or four charge void cells. The three critical values of electric Rayleigh number Tc are 242, 545, and 665, respectively. There are also two hysteresis loops whose nonlinear criteria Tf are 157 and 435, respectively. An unexpected flow pattern, which has two prism-shaped primary vortex structures, demonstrates the significance of 3D analysis. In addition, we find that the 3D flow in the cavity is more stable by studying the correlation between the 3D and two-dimensional laminar EC. Using dynamic mode decomposition for the flow structures, we reveal that the novel feature is the result of competition between the EC flow structure and the limitation of geometry.