Luo Mao-Kang

Control of fractional chaotic and hyperchaotic systems based on a fractional order controller

We present a new fractional-order controller based on the Lyapunov stability theory and propose a control method which can control fractional chaotic and hyperchaotic systems whether systems are commensurate or incommensurate. The proposed control method is universal, simple, and theoretically rigorous. Numerical simulations are given for several fractional chaotic and hyperchaotic systems to verify the effectiveness and the universality of the proposed control method.

Lattice-valued Hahn-Dieudonné-Tong insertion theorem and stratification structure

Abstract In this paper, we shall generalize the Hahn-Dieudonne-Tong Insertion Theorem, a classical result on semicontinuous functions, to the case that the ranges are a certain kind of lattices L. We consider the family of all lattice-valued (lower) semicontinuous mappings as a topology, and then solve this problem via reducing it to a problem on a certain kind of separation property. Therefore, as a by-product, the following theorem is obtained for a suitable range L: An induced space is normal if and only if the underlying space is normal. The relative counterexamples show that the limitation for L is necessary. These results and counterexamples point out that normality of induced spaces and the existence of latticed-valued inserting mappings close relates to the internal characterization of the range L. The previous proofs of the classical insertion theorem are analytic and skillful; compared with them, the present method determining mappings via the stratifications is very natural and conceptual. The success of the method is based on the study in depth on the topological relations among the stratifications. We may hope that the method, which determines the mapping via inductively getting its stratifications, will be further applied in other aspects.

Weak Signal Frequency Detection Based on a Fractional-Order Bistable System

We apply spectral power amplification to quantifying and investigating the phenomenon of stochastic resonance in fractional-order systems. Moreover, the frequency of the weak mixed signal is detected by a stochastic resonance mechanism together with a power spectrum. The corresponding numerical results demonstrate the effectiveness of the proposed methods.

Weak Signal Frequency Detection Method Based on Generalized Duffing Oscillator

The sensitive characteristic to the initial value of chaos system sufficiently demonstrates the superiority in weak signal parameters detection. Analyzing the current chaos-based frequency detection method, a novel generalized Duffing equation is proposed to detect weak signal frequency. By choosing a suitable adjusting factor, when the outside driving force frequency is equal to that of the detected signal, the generalized Duffing oscillator is in great period state, which can obtain the frequency information of the detected signal. The simulation results indicate this method is rapidly convenient and shows better accuracy.

A Denoising Algorithm for Noisy Chaotic Signals Based on the Higher Order Threshold Function in Wavelet-Packet

Aiming at the shortage of conventional threshold function in wavelet noise reduction of chaotic signals, we propose a wavelet-packet noise reduction method of chaotic signals based on a new higher order threshold function. The method retains the useful high-frequency information, and the threshold function is continuous and derivable, therefore it is more consistent with the characteristics of the continuous signal. Contrast simulation experiment shows that the effect of noise reduction and the precision of noise reduction of chaotic signals both are improved.

Lattice-valued mappings, completely distributive law and induced spaces

The lattice-valued mappings whose value domain is a completely distributive lattice are investigated. We describe the analytic condition - the semicontinuity of lattice-valued mappings on a topological space X via the algebraic property - the completely distributive law and conversely, we use the semicontinuity to depict the completely distributive law. The family of all the upper semicontinuous lattice-valued mappings on a topological space X (or a part of them) constitutes a L-fuzzy topological space on X; a series of global properties of these lattice-valued mappings is presented. In particular, we investigate the impact of N-compactness on the semicontinuity. Using these results, we solve some problems on fuzzy Stone-Cech compactification.

Fractional backward Kolmogorov equations

This paper derives the fractional backward Kolmogorov equations in fractal space-time based on the construction of a model for dynamic trajectories. It shows that for the type of fractional backward Kolmogorov equation in the fractal time whose coefficient functions are independent of time, its solution is equal to the transfer probability density function of the subordinated process X(Sα(t)), the subordinator Sα(t) is termed as the inverse-time α-stable subordinator and the process X(τ) satisfies the corresponding time homogeneous Ito stochastic differential equation.

Stochastic Multi-Resonance in a Linear System Driven by Multiplicative Polynomial Dichotomous Noise

We investigate stochastic resonance in a linear system subjected to multiplicative noise that is a polynomial function of colored noise. Using the stochastic averaging method, the analytical expression of the output signal-to-noise ratio (SNR) is derived. Theoretical analysis and numerical results show that the output SNR is a nonmonotonic function of both the noise intensity and the correlation rate. Moreover, the phenomoenon of stochastic multi-resonance (SMR) is found, which is not observed in conventional linear systems driven by multiplicative noise with only a linear term.

T D property and spatial sublocales

As one of main backgrounds of locale theory, topologies have close connections with locales. But locales have other backgrounds such as algebra, mathematical logic, etc. So there are many differences between locales and topologies. Spatiality is an important localic property to investigate the connections between locales and topologies. TheT D property is a special separation property which plays an important role in this kind of investigations. Just as it will be proved in this paper, theT D property often appears as the lowest requirement for many topological spaces such that they can be described with localic properties and vice versa. In this paper, we show these special properties of theT D axiom and investigate some other interesting and important problems ofT D -spatiality of locales.

Lattice-valued semicontinuous mappings and induced topologies

Lattice-valued semicontinuous mappings play a basic and important role in solving the problems ofL-fuzzy compactification theory, and make the previous work on weakly induced spaces and induced spaces determinatively generalized and strengthened. Moreover, we can describe the complete distributivity of lattices with them as well. In this paper, we give the mutually descriptive relation between lattice-valued semicontinuous mappings and the complete distributivity of lattices, and the construction theorems of open sets and closed sets in lattice-valued fully stratified spaces, weakly induced spaces and induced spaces (they are calledS-spaces). Furthermore, we will investigate the structure of the co-topology ofS-space, solve a series of interesting problems on product,N-compactness and metrization ofS-spaces.