Narrow Results

Discrete chaotic systems in simple algebraic forms show great potential in many fields due to their complex dynamics and coexisting behaviors. Recently, an analog circuit scheme for implementing discrete systems has been proposed, but this scheme cannot exhibit essential coexisting behaviors in discrete systems. To address this issue, this paper proposes a novel analog circuit scheme for experimentally observing the coexisting behaviors in discrete systems. A discrete memristor model and two discrete chaotic maps are taken as examples, and their analog circuits are designed and physically implemented. By appropriately setting the initial values, the pinched hysteresis loops related to the initial values of the discrete memristor model, the coexisting attractors of the discrete memristor map and the initial-offset-control coexisting behaviors of the discrete neuron model are simulated, respectively. In particular, the printed circuit board based on trigonometric chip, sample-and-hold device, operational amplifier chip, and other components is fabricated. The coexisting behaviors are observed experimentally, thus verifying the numerical simulations and circuit simulations. The proposed scheme provides a new framework for the physical realization of discrete systems.

Multiple chaotic attractors, implying several independent chaotic attractors generated simultaneously in a system from different initial values, are a very interesting and important nonlinear phenomenon, but there are few studies that have previously addressed it to our best knowledge. In this paper, we propose a polynomial function method for generating multiple chaotic attractors from the Sprott B system. The polynomial function extends the number of index-2 saddle foci, which determines the emergence of multiple chaotic attractors in the system. The analysis of the equilibria is presented. Two coexisting chaotic attractors, three coexisting chaotic attractors and four coexisting chaotic attractors are investigated for verifying the effectiveness of the method. The chaotic characteristics of the attractors are shown by bifurcation diagrams, Lyapunov exponent spectrum and phase portraits.

In the preceding paper (arXiv: 0710.2724 [quant-ph]) we have constructed the general solution for the master equation of quantum damped harmonic oscillator, which is given by the complicated infinite series in the operator algebra level. In this paper we give the explicit and compact forms to solutions (density operators) for some initial values. In particular, the compact one for the initial value based on a coherent state is given, which has not been given as far as we know. Moreover, some related problems are presented.

We study the well-formulation of the initial value problem of f(R)-gravity in the metric-affine formalism. The problem is discussed in vacuo and in presence of perfect-fluid matter, Klein–Gordon and Yang–Mills fields. Adopting Gaussian normal coordinates, it can be shown that the problem is always well-formulated. Our results refute some criticisms to the viability of f(R)-gravity recently appeared in literature.