Narrow Results

A class of delayed ratio-dependent Gause-type predator–prey model is considered. We study the eigenvalue problem for the linearized system at the coexisting equilibrium. For a critical case when the characteristic equation has a single zero root and a simple pair of pure imaginary roots, a complete bifurcation analysis is presented by employing the center manifold reduction and the normal form method. We analyzed the influence of the time delay on the Hopf–Fold bifurcation and showed the occurrence of quasi-periodic motion and bursting behavior. This phenomenon is in line with the seasonal variation law of the population.

We investigate the dynamics of two types of nonautonomous ordinary differential equations with quasi-periodic time-varying coefficients and nonlinear terms. The vector fields for the nonautonomous systems are written as $ẇ=a{\mathrm{\Phi }}_1(w)+b{\mathrm{\Phi }}_2(t,w)$, $w\in {ℝ}^3$, where $a{\mathrm{\Phi }}_1$ is the spacial part and $b{\mathrm{\Phi }}_2$ is the time-varying part, and $a$ and $b$ are real parameters. The first type has a polynomial as the nonlinear term, another type has a continuous periodic function as the nonlinear term. The polynomials and periodic functions have simple zeros. Several examples with numerical experiments are given. It is found by numerical calculation that there might exist only one attractor for the systems with polynomials as nonlinear terms and $\left|b\right|\gg max\{1,|a|\}$, and there might exist infinitely many attractors for systems with periodic functions as nonlinear terms and $\left|b\right|\gg max\{1,|a|\}$. For $\left|b\right|$ sufficiently small, the parameter regions for $(a,b)$ are roughly divided into three parts: the spacial region ($\left|a\right|\gg \left|b\right|$), the balance region ($\left|a\right|\approx \left|b\right|$), and the time-varying region ($\left|a\right|\ll \left|b\right|$); (i) for $\left|a\right|\gg \left|b\right|$, the orbits approach some planes depending on the zeros of the polynomials or the periodic functions; (ii) for $\left|a\right|\approx \left|b\right|$, there exist attractors with the number no less than the number of zeros of the polynomials or the periodic functions, implying the existence of infinitely many attractors for systems with periodic functions as nonlinear terms; (iii) for $\left|a\right|\ll \left|b\right|$, the orbits wind around some region depending on the choice of the initial position. The shape of the attractors might be strange or regular for different parameters, and we obtain the existence of ball-like (regular) attractors, two-wings (strange) attractors, and other attractors with different shapes. The Lyapunov exponents are negative. These results reveal an intrinsic relationship between the existence of attractors (or strange dynamics) and the parameters $a$ and $b$ for nonautonomous systems with quasi-periodic coefficients. These results will be very useful in the understanding of the dynamics of general nonautonomous systems, nonautonomous control theory and other related fields.

In the present paper, an eco-epidemiological model consisting of susceptible prey, infected prey and predator has been proposed and analyzed. We have obtained conditions for the existence and persistence of all the three populations. To study the global dynamics of the system, numerical simulations have been performed. Our simulation results show that the system enters into quasi-periodic solutions or chaotic depending upon the choice of system parameters. To confirm the chaotic behavior of the system, we have calculated Lyapunov exponent and constructed Poincare section. Our analysis reveals that the infection and predation rates specially on the infected prey population are the key parameters that play crucial roles for controlling the chaotic dynamics of the system.

The reflection properties of light wave propagation in one-dimensional quasi-periodic metallic photonic crystal (PC) are comprehensively analyzed by transfer matrix method. In this work, we form a Fibonacci sequence quasi-periodic PC composed of metal and dielectric. The results demonstrate that the reflection stop band is strongly dependent on the periodic structure, metal thickness and incident angle. For this structure, the reflection stop band ranges from the visible light region to near-infrared region. Compared with the periodic metallic PC, the reflection stop bandwidth of our structure is wider. When the metal thickness increases, the reflection stop band is significantly enlarged. Furthermore, the reflection stop bandwidth slowly gets narrow and shifts to short wavelength region with the increase of incidence angle. Considering TE and TM wave at all incident angles, there is an omnidirectional reflection bandgap with width of 241nm for our investigated quasi-periodic metal PC.

The aim of this paper is to introduce the new hyperchaotic complex Lü system. This system has complex nonlinear behavior which is studied and investigated in this work. Numerically the range of parameters values of the system at which hyperchaotic attractors exist is calculated. This new system has a whole circle of equilibria and three isolated fixed points, while the real counterpart has only three isolated ones. The stability analysis of the trivial fixed point is studied. Its dynamics is more rich in the sense that our system exhibits both chaotic and hyperchaotic attractors as well as periodic and quasi-periodic solutions and solutions that approach fixed points.