Synchronization and Pattern Formation in a Memristive Diffusive Neuron Model

In this article, we construct an excitable memristive diffusive neuron model by considering a biophysical slow–fast bursting oscillator and study the effects of electromagnetic induction on the dynamics of the single model as well as the coupled systems. We explore various firing regimes such as tonic spiking, bursting, and mixed-mode oscillations depending on the bifurcation structure with different injected current stimuli, then perform a comparative analysis on the synchronization of the coupled oscillators by setting the model into two different network architectures. First, a diffusively coupled network is considered, and later a global network is constructed. The results suggest that the diffusively connected neurons show complete synchronization at higher couplings for bursting and tonic spiking regimes. Furthermore, we show that the extended spatial system can generate spiral-like patterns in the vicinity of a Hopf bifurcation point and observe the impact of Gaussian white noise to study its effects on pattern formation. These types of patterns are robust in the excitable model. Our results might contribute significantly to the dynamical studies of irregular neural computation.

Nonlinear Dynamics of Interband Cascade Laser Subjected to Optical Feedback

We present a theoretical study of the nonlinear dynamics of a long external cavity delayed optical feedback-induced interband cascade laser (ICL). Using the modified Lang–Kobayashi equations, we numerically investigate the effects of some key parameters on the first Hopf bifurcation point of ICL with optical feedback, such as the delay time (τf), pump current (I), linewidth enhancement factor (LEF), stage number (m) and feedback strength (fext). It is found that compared with τf, I, LEF and m have a significant effect on the stability of the ICL. Additionally, our results show that an ICL with few stage numbers subjected to external cavity optical feedback is more susceptible to exhibiting chaos. The chaos bandwidth dependences on m, I and fext are investigated, and 8 GHz bandwidth mid-infrared chaos is observed.

Turing-Hopf Bifurcation in the Predator-prey Model with Cross-diffusion Considering Two Different Prey Behaviours Transition

In this paper, we study the Turing-Hopf bifurcation in the predator-prey model with cross-diffusion considering the individual behaviour and herd behaviour transition of prey population subject to homogeneous Neumann boundary condition. Firstly, we study the non-negativity and boundedness of solutions corresponding to the temporal model, spatiotemporal model and the existence and priori boundedness of solutions corresponding to the spatiotemporal model without cross-diffusion. Then by analyzing the eigenvalues of characteristic equation associated with the linearized system at the positive constant equilibrium point, we investigate the stability and instability of the corresponding spatiotemporal model. Moreover, by computing and analyzing the normal form on the center manifold associated with the Turing-Hopf bifurcation, we investigate the dynamical classification near the Turing-Hopf bifurcation point in detail. At last, some numerical simulations results are given to support our analytic results.

Feedback Control for a Diffusive and Delayed Brusselator Model: Semi-Analytical Solutions

This paper describes the stability and Hopf bifurcation analysis of the Brusselator system with delayed feedback control in the single domain of a reaction–diffusion cell. The Galerkin analytical technique is used to present a system equation composed of ordinary differential equations. The condition able to determine the Hopf bifurcation point is found. Full maps of the Hopf bifurcation regions for the interacting chemical species are shown and discussed, indicating that the time delay, feedback control, and diffusion parameters can play a significant and important role in the stability dynamics of the two concentration reactants in the system. As a result, these parameters can be changed to destabilize the model. The results show that the Hopf bifurcation points for chemical control increase as the feedback parameters increase, whereas the Hopf bifurcation points decrease when the diffusion parameters increase. Bifurcation diagrams with examples of periodic oscillation and phase-plane maps are provided to confirm all the outcomes calculated in the model. The benefits and accuracy of this work show that there is excellent agreement between the analytical results and numerical simulation scheme for all the figures and examples that are illustrated.

Numerical analysis of a degenerate generalized Hopf bifurcation

In this paper, we present a numeric bifurcation analysis of the normal form of degenerate Hopf bifurcation truncated up to seventh order with an equilibrium point located at the origin. By applying the genericity nondegenerate conditions and normal form theory, we study the bifurcation analysis of the codimension-3 Takens–Hopf bifurcation for the difficult case, where a rich bifurcation scenario is displayed. The third Lyapunov coefficient is used to distinguish the different cases of a codimension-3 Takens–Hopf bifurcation point, which can be efficiently computed with the aid of a software program based on the symbolic package Maple, presented in Appendix A. The normal form analysis results can be used to depict the complete bifurcation diagrams and phase portraits. In order to investigate the mechanism of the transitions between equilibrium and limit cycles, the methods of two scales in frequency domain are employed to study the evolutions.

Spatiotemporal dynamics for a diffusive mussel–algae model near a Hopf bifurcation point

In this paper, the Hopf bifurcation and Turing instability for a mussel–algae model are investigated. Through analysis of the corresponding kinetic system, the existence and stability conditions of the equilibrium and the type of Hopf bifurcation are studied. Via the center manifold and Hopf bifurcation theorem, sufficient conditions for Turing instability in equilibrium and limit cycles are obtained, respectively. In addition, we find that the strip patterns are mainly induced by Turing instability in equilibrium and spot patterns are mainly induced by Turing instability in limit cycles by numerical simulations. These provide a comprehension on the complex pattern formation of a mussel–algae system.

Bifurcation Analysis of a Railway Wheelset With Nonlinear Wheel-rail Contact

Stability is a key factor for the operation safety of railway vehicles, while current work employs linearized and simplified wheel/rail contact to study the bifurcation mechanism and assess the stability. To study the stability and bifurcation characters under real nonlinear wheel/rail contact, a fully parameterized nonlinear railway vehicle wheelset model is built. In modeling, the geometry nonlinearities of wheel and rail profiles come from field measurements, including the rolling radius, contact angle, and curvatures, etc. Firstly, four flange force models and their effects on the stability bifurcations are compared. It shows that an exponent fitting is more proper than a quintic polynomial one to simulate the flange, and works well without changing the Hopf bifurcation type. Then the effects of each term of the nonlinear geometry of wheel/rail contact on the Hopf bifurcation and Limit Circle bifurcation are discussed. Both the linear term and nonlinear term of rolling radius have a significant influence on Hopf bifurcation and Limit Point of Circle (LPC) bifurcation. The linear critical speed (Hopf bifurcation point) and the nonlinear critical speed (LPC bifurcation point) changes times while within the calculated range of the linear term of the rolling radius. Its nonlinear term changes the bifurcation type and the nonlinear critical speed almost by half. The linear term of contact angle, the radius of curvature of wheel, and rail profile should be taken into consideration since they can change both the bifurcation point and type, while the cubic term can be ignored. Furtherly, the field measured wheel profiles for several running mileages are employed to examine the real geometry nonlinearities and the according Hopf bifurcation behavior. The result shows that a larger suspension stiffness would increase the running stability under wheel wear.

Stability analysis for Selkov-Schnakenberg reaction-diffusion system

This study provides semi-analytical solutions to the Selkov-Schnakenberg reaction-diffusion system. The Galerkin method is applied to approximate the system of partial differential equations by a system of ordinary differential equations. The steady states of the system and the limit cycle solutions are delineated using bifurcation diagram analysis. The influence of the diffusion coefficients as they change, on the system stability is examined. Near the Hopf bifurcation point, the asymptotic analysis is developed for the oscillatory solution. The semi-analytical model solutions agree accurately with the numerical results.

Guaranteed Estimates for the Length of Branches of Periodic Orbits for Equivariant Hopf Bifurcation

Connected branches of periodic orbits originating at a Hopf bifurcation point of a differential system are considered. A computable estimate for the range of amplitudes of periodic orbits contained in the branch is provided under the assumption that the nonlinear terms satisfy a linear estimate in a ball. If the estimate is global, then the branch is unbounded. The results are formulated in an equivariant setting where the system can have multiple branches of periodic orbits characterized by different groups of symmetries. The nonlocal analysis is based on the equivariant degree method, which allows us to handle both generic and degenerate Hopf bifurcations. This is illustrated by examples.

An extension of the structured singular value to nonlinear systems with application to robust flutter analysis

The paper discusses an extension of $$\mu$$ μ (or structured singular value), a well-established technique from robust control for the study of linear systems subject to structured uncertainty, to nonlinear polynomial problems. Robustness is a multifaceted concept in the nonlinear context, and in this work the point of view of bifurcation theory is assumed. The latter is concerned with the study of qualitative changes of the steady-state solutions of a nonlinear system, so-called bifurcations. The practical goal motivating the work is to assess the effect of modeling uncertainties on flutter, a dynamic instability prompted by an adverse coupling between aerodynamic, elastic, and inertial forces, when considering the system as nonlinear. Specifically, the onset of flutter in nonlinear systems is generally associated with limit cycle oscillations emanating from a Hopf bifurcation point. Leveraging $$\mu$$ μ and its complementary modeling paradigm, namely linear fractional transformation, this work proposes an approach to compute margins to the occurrence of Hopf bifurcations for uncertain nonlinear systems. An application to the typical section case study with linear unsteady aerodynamic and hardening nonlinearities in the structural parameters will be presented to demonstrate the applicability of the approach.