Spatiotemporal Dynamics Induced by Nonlocal Competition in a Diffusion Predator-Prey System with Habitat Complexity

In this paper, we study a delayed diffusive predator-prey model with nonlocal competition in prey and habitat complexity. The local stability of coexisting equilibrium are studied by analyzing the eigenvalue spectrum. Time delay inducing Hopf bifurcation is investigated by using time delay as bifurcation parameter. We give some conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution by utilizing the normal form method and center manifold theorem. Our results suggest that only nonlocal competition and diffusion together can induce stably spatial inhomogeneous bifurcating periodic solutions.

Delayed Feedback Control of Hidden Chaos in the Unified Chaotic System between the Sprott C System and Yang System

In this paper, the influence of delayed feedback on the unified chaotic system from the Sprott C system and Yang system is studied. The Hopf bifurcation and dynamic behavior of the system are fully studied by using the central manifold theorem and bifurcation theory. The explicit formula, bifurcation direction, and stability of the periodic solution of bifurcation are given correspondingly. The Hopf bifurcation diagram and chaotic phenomenon are also analyzed by numerical simulation to prove the correctness of the theory. It shows that this delay control can only be applied to the hidden chaos with two stable equilibria.

Regular Versus Singular Solutions in a Quasilinear Indefinite Problem with an Asymptotically Linear Potential

The aim of this paper is analyzing the positive solutions of the quasilinear problem-\bigl{(}u^{\prime}/\sqrt{1+(u^{\prime})^{2}}\big{)}^{\prime}=\lambda a(x)f(u)% \quad\text{in }(0,1),\qquad u^{\prime}(0)=0,\quad u^{\prime}(1)=0,where {\lambda\in\mathbb{R}} is a parameter, {a\in L^{\infty}(0,1)} changes sign once in {(0,1)} and satisfies {\int_{0}^{1}a(x)\,dx<0}, and {f\in\mathcal{C}^{1}(\mathbb{R})} is positive and increasing in {(0,+\infty)} with a potential, {F(s)=\int_{0}^{s}f(t)\,dt}, quadratic at zero and linear at {+\infty}. The main result of this paper establishes that this problem possesses a component of positive bounded variation solutions, {\mathscr{C}_{\lambda_{0}}^{+}}, bifurcating from {(\lambda,0)} at some {\lambda_{0}>0} and from {(\lambda,\infty)} at some {\lambda_{\infty}>0}. It also establishes that {\mathscr{C}_{\lambda_{0}}^{+}} consists of regular solutions if and only if\int_{0}^{z}\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}dx=+\infty\quad\text% {or}\quad\int_{z}^{1}\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}dx=+\infty.Equivalently, the small positive regular solutions of {\mathscr{C}_{\lambda_{0}}^{+}} become singular as they are sufficiently large if and only if \Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}\in L^{1}(0,z)\quad\text{and}% \quad\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}\in L^{1}(z,1).This is achieved by providing a very sharp description of the asymptotic profile, as {\lambda\to\lambda_{\infty}}, of the solutions. According to the mutual positions of {\lambda_{0}} and {\lambda_{\infty}}, as well as the bifurcation direction, the occurrence of multiple solutions can also be detected.

Bifurcation Phenomena in a Lotka–Volterra Model with Cross-Diffusion and Delay Effect

This paper focuses on a Lotka–Volterra model with delay and cross-diffusion. By using Lyapunov–Schmidt reduction, we investigate the existence, multiplicity, stability and Hopf bifurcation of spatially nonhomogeneous steady-state solutions. Furthermore, we obtain some criteria to determine the bifurcation direction and stability of Hopf bifurcating periodic orbits by using Lyapunov–Schmidt reduction.

Generalized Hopf Bifurcation for Neutral Functional Differential Equations

Here we employ the Lyapunov–Schmidt procedure to investigate bifurcations in a general neutral functional differential equation (NFDE) when the infinitesimal generator has, for a critical value of the parameter, a pair of nonsemisimple purely imaginary eigenvalues with multiplicity [Formula: see text]. We derive criteria, explicitly in terms of the system's parameter values, for the existence of two branches of bifurcating periodic solutions and for the description of the bifurcation direction of these branches. The general result is illustrated by a detailed case study of an oscillator.

Bifurcation Analysis of a Generic Reaction–Diffusion Turing Model

A generic Turing type reaction–diffusion system derived from the Taylor expansion near a constant equilibrium is analyzed. The existence of Hopf bifurcations and steady state bifurcations is obtained. The bifurcation direction and the stability of the bifurcating periodic obits are calculated. Numerical simulations are included to show the rich spatiotemporal dynamics.

Stability and Bifurcation Analysis of a Nonlinear Discrete Logistic Model with Delay

We consider a nonlinear discrete logistic model with delay. The characteristic equation of the linearized system at the positive equilibrium is a polynomial equation involving high order terms. We obtain the conditions ensuring the asymptotic stability of the positive equilibrium and the existence of Neimark-Sacker bifurcation, with respect to the parameter of the model. Based on the bifurcation theory, we discuss Neimark-Sacker bifurcation direction and the stability of bifurcated solutions. Finally, some numerical simulations are performed to illustrate the theoretical results.

Stability and Bifurcation Analysis of a Three-Dimensional Recurrent Neural Network with Time Delay

We consider the nonlinear dynamical behavior of a three-dimensional recurrent neural network with time delay. By choosing the time delay as a bifurcation parameter, we prove that Hopf bifurcation occurs when the delay passes through a sequence of critical values. Applying the nor- mal form method and center manifold theory, we obtain some local bifurcation results and derive formulas for determining the bifurcation direction and the stability of the bifurcated periodic solution. Some numerical examples are also presented to verify the theoretical analysis.

HOPF BIFURCATION ANALYSIS OF TWO SUNFLOWER EQUATIONS

In this paper, two sunflower equations are considered. Using delay τ as a parameter and applying the global Hopf bifurcation theorem, we investigate the existence of global Hopf bifurcation for the sunflower equation. Furthermore, we analyze the local Hopf bifurcation of the modified equation with nonlinear relation about stem's increase, including the occurrence, the bifurcation direction, the stability and the approximation expression of the bifurcating periodic solution using the theory of normal form and center manifold. Finally, the obtained results of these two equations are compared, which finds that the result about the period of their bifurcating periodic solutions is obviously different, while the bifurcation direction and stability are identical.

A NEW FAMILY OF FIRST-ORDER TIME-DELAYED CHAOTIC SYSTEMS

A general method for formulating first-order time-delayed chaotic systems with simple linear time-delayed term is proposed. The formulated systems are realized with electronic circuit experiments. In order to determine the unknown coefficients in a general delayed differential equations for having chaotic solutions, we follow the route of period-doubling bifurcation to chaos. Firstly, the conditions for a time-delayed system having a stable periodic solution, generating from a destablized steady state, is analyzed with Hopf bifurcation theory. Then the delay time parameter is changed according to the bifurcation direction to search the chaotic state, which is identified by the Lyapunov exponents spectra. The theoretical analysis is well confirmed by numerical simulations and circuit experiments.