Bifurcations in a Time-Delayed Birhythmic Biological System with Fractional Derivative and Lévy Noise

The birhythmic oscillation is of great significance in biology and engineering, and this paper presents a bifurcation analysis in a time-delayed birhythmic oscillator containing fractional derivative and Lévy noise. The numerical method is used to explore the influence of various parameters on the bifurcation of the birhythmic system, and the role of fractional derivative and Lévy noise in inducing or inhibiting birhythmicity in a time-delayed birhythmic biological system is examined in this work. First, we use a numerical method to calculate the fractional derivative, which has a fast calculation speed. Then the McCulloch algorithm is employed to generate Lévy random numbers. Finally, the stationary probability density function graph of the amplitude is obtained by Monte Carlo simulation. The results show that the fractional damping and Lévy noise can effectively control the characteristics of the birhythmic oscillator, and the change of the parameters (except the skewness parameter) can cause the system bifurcation. In addition, this article further discusses the interaction of fractional derivative and time delay in a birhythmic system with Lévy noise, proving that adjusting parameters of time delay can lead to abundant bifurcations. Our research may help to further explore the bifurcation phenomenon of birhythmic biological system, and has a practical significance.

New Traveling Wave Solutions and Interesting Bifurcation Phenomena of Generalized KdV-mKdV-Like Equation

Using the bifurcation method of dynamical systems, we investigate the nonlinear waves and their limit properties for the generalized KdV-mKdV-like equation. We obtain the following results: (i) three types of new explicit expressions of nonlinear waves are obtained. (ii) Under different parameter conditions, we point out these expressions represent different waves, such as the solitary waves, the 1-blow-up waves, and the 2-blow-up waves. (iii) We revealed a kind of new interesting bifurcation phenomenon. The phenomenon is that the 1-blow-up waves can be bifurcated from 2-blow-up waves. Also, we gain other interesting bifurcation phenomena. We also show that our expressions include existing results.

Bifurcation phenomenon in molecular dynamics model of coalescence/sintering on the nanoscale

Using the isothermal molecular dynamics, coalescence/sintering of Au nanoparticles (NPs) was simulated. We have found that the solid NP sintering scenario is switched to the coalescence scenario not at the NP melting temperature T m exactly but at a lower temperature T 0 ≈ 0.9T m interpreted as the critical temperature corresponding to a coalescence/sintering bifurcation phenomenon: in the temperature range from T 0 – 2 K to T 0 + 2 K to the resulting (daughter) NPs of the same size can have either liquid-like or crystalline structure after coalescence/sintering at the same fixed temperature.

Equilibria and Systemic Risk in Saturated Networks

We undertake a fundamental study of network equilibria modeled as solutions of fixed-point equations for monotone linear functions with saturation nonlinearities. The considered model extends one originally proposed to study systemic risk in networks of financial institutions interconnected by mutual obligations. It is one of the simplest continuous models accounting for shock propagation phenomena and cascading failure effects. This model also characterizes Nash equilibria of constrained quadratic network games with strategic complementarities. We first derive explicit expressions for network equilibria and prove necessary and sufficient conditions for their uniqueness, encompassing and generalizing results available in the literature. Then, we study jump discontinuities of the network equilibria when the exogenous flows cross certain regions of measure 0 representable as graphs of continuous functions. Finally, we discuss some implications of our results in the two main motivating applications. In financial networks, this bifurcation phenomenon is responsible for how small shocks in the assets of a few nodes can trigger major aggregate losses to the system and cause the default of several agents. In constrained quadratic network games, it induces a blow-up behavior of the sensitivity of Nash equilibria with respect to the individual benefits.

Bifurcation Analysis and H∞ Control of a Stochastic Competition Model with Time Delay

In this paper, we study a stochastic competition model with time delay and harvesting.We first simplify it through the stochastic center manifold reduction principle and stochastic averaging method as a one-dimensional Markov diffusion process. Singular boundary theory and an invariant measure are applied to analyze stochastic stability and bifurcation. The T-S fuzzy model of the system is constructed, and the H∞ fuzzy controller is designed to eliminate the bifurcation phenomenon through a linear matrix inequality approach. Numerical simulation is used to demonstrate our results.

Epidemic Model and Mathematical Study of Impact of Vaccination for the Control of Malware in Computer Network

Malware remains a significant threat to computer network. In this paper, we consideredthe problem which computer malware cause to personal computers with its control by proposing a compartmental model SVEIRS (Susceptible Vaccinated-Exposed-infected-Recovered-Susceptible) for malware transmission in computer network using nonlinear ordinary differential equation. Through the analysis of the model, the basic reproduction number were obtained, and the malware free equilibrium was proved to be locally asymptotical stable if is less than unity and globally asymptotically stable if Ro is less than some threshold using a Lyapunov function. Also, the unique endemic equilibrium exists under certain conditions and the model underwent backward bifurcation phenomenon. To illustrate our theoretical analysis, some numerical simulation of the system was performed with RungeKutta fourth order (KR4) method in Mathlab. This was used in analyzing the behavior of different compartments of the model and the results showed that vaccination and treatment is very essential for malware control.

Analysis of Autonomous Coordination Between Actuators in the Antagonist Musculoskeletal Model

The McKibben Pneumatic Actuator (MPA) is well-known as a type of soft actuator. As MPA generates tension only in the direction of compression, it is necessary to construct an antagonistic structure to drive a joint by MPAs and to coordinate antagonized MPAs. Similar to MPA, muscles in animals also generate tension only in the direction of contraction. Some studies have reported that animals utilize tension information to coordinate muscles for various autonomous movements. The purpose of this study is to realize autonomous cooperation between antagonized MPAs by applying tension feedback control and analyzing the mechanism of coordination. For this purpose, we verify the effect of tension feedback control on the 1-DOF pendulum model with antagonized MPAs. First, through numerical simulations, it is confirmed that the tension feedback generates various coordinated movements of antagonized MPAs, and the pendulum exhibits a bifurcation phenomenon based on the phase difference of the inputs of MPAs. Thereafter, we develop an actual experimental machine based on the model and confirm the autonomous cooperation between actual MPAs through verification experiments similar to the numerical simulations.

Shape Perturbation of Grushin Eigenvalues

We consider the spectral problem for the Grushin Laplacian subject to homogeneous Dirichlet boundary conditions on a bounded open subset of $${\mathbb {R}}^N$$ R N . We prove that the symmetric functions of the eigenvalues depend real analytically upon domain perturbations and we prove an Hadamard-type formula for their shape differential. In the case of perturbations depending on a single scalar parameter, we prove a Rellich–Nagy-type theorem which describes the bifurcation phenomenon of multiple eigenvalues. As corollaries, we characterize the critical shapes under isovolumetric and isoperimetric perturbations in terms of overdetermined problems and we deduce a new proof of the Rellich–Pohozaev identity for the Grushin eigenvalues.