Complexity Of Dynamic Behavior Research Articles

Behavior Analysis of a Class of Discrete-Time Dynamical System with Capture Rate

In this paper, we study the stability and bifurcation analysis of a class of discrete-time dynamical system with capture rate. The local stability of the system at equilibrium points are discussed. By using the center manifold theorem and bifurcation theory, the conditions for the existence of flip bifurcation and Hopf bifurcation in the interior of R+2 are proved. The numerical simulations show that the capture rate not only affects the size of the equilibrium points, but also changes the bifurcation phenomenon. It was found that the discrete system not only has flip bifurcation and Hopf bifurcation, but also has chaotic orbital sets. The complexity of dynamic behavior is verified by numerical analysis of bifurcation, phase and maximum Lyapunov exponent diagram.

The diffusion-driven instability and complexity for a single-handed discrete Fisher equation

For a reaction diffusion system, it is well known that the diffusion coefficient of the inhibitor must be bigger than that of the activator when the Turing instability is considered. However, the diffusion-driven instability/Turing instability for a single-handed discrete Fisher equation with the Neumann boundary conditions may occur and a series of 2-periodic patterns have been observed. Motivated by these pattern formations, the existence of 2-periodic solutions is established. Naturally, the periodic double and the chaos phenomenon should be considered. To this end, a simplest two elements system will be further discussed, the flip bifurcation theorem will be obtained by computing the center manifold, and the bifurcation diagrams will be simulated by using the shooting method. It proves that the Turing instability and the complexity of dynamical behaviors can be completely driven by the diffusion term. Additionally, those effective methods of numerical simulations are valid for experiments of other patterns, thus, are also beneficial for some application scientists.

Discrete-time dynamics of a system with crowding effect and predator partially dependent on prey

In the present study, the stability and bifurcation analysis of discrete-time predator–prey system with predator partially dependent on prey and crowding effect of predator is examined. Global stability of the system at the fixed points has been discussed. The specific conditions for existence of flip bifurcation and Hopf bifurcation in the interior of R+2 have been derived by using a center manifold theorem and bifurcation theory. Numerical simulations have been carried out to show the complex dynamical behavior of the system and to justify our analytic results. In case of flip bifurcation, numerical simulations presented cascade of period-doubling bifurcation in the orbits of period 2, 4, 8, chaotic orbits and stable window of period 9 orbit; whereas in case of Hopf bifurcation, smooth invariant circle bifurcates from the fixed point. The complexity of dynamical behavior is confirmed by computation of Lyapunov exponents.

A zero-watermarking algorithm of stereoscopic image based on hyperchaotic system

In order to protect the copyright of stereoscopic images in the condition of the image quality, a zero-watermarking stereoscopic image algorithm is presented based on hyperchaotic system. In the proposed algorithm, disparity zero-watermark is constructed according to the stability of disparity between low frequency bands of wavelet decomposition of the left and the right views and direct coefficients of discrete cosine transform; in the process of zero-watermark construction, the zero-watermark position mapping according to the features of the sensitivity of hyperchaotic discrete system initial value, ampleness of parameter key space and complexity of dynamic behavior, enhances the security of algorithm. The relationship between security and watermark capacity is also analyzed. Experimental results show that the proposed algorithm is strong robust to noise, filtering, compression, and the shearing of asymmetrical and symmetrical attacks.