[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

Fractional Differential Equations

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

The computational brain: Patricia S. Churchland and Terrence J. Sejnowski (MIT Press, Cambridge, MA, 1992); xi, 544 pages, $39.95

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Stochastic Differential Equations And Applications

Mittag-Leffler stability of fractional-order Hopfield neural networks

Global stability analysis of fractional-order Hopfield neural networks with time delay

In this paper, an adaptive backstepping design is proposed to synchronize two uncertain chaos systems. This method can be applied to a variety of chaos systems which can be transformed into the so-called general strict-feedback form no matter whether it contains external excitation or not. Rossler system and Duffing oscillator are used as examples for detailed description. Numerical simulations show the effectiveness and feasibility of the method.

Adaptive backstepping synchronization of uncertain chaotic system

The dynamic behaviors of fractional-order differential systems have received increasing attention in recent decades. But many results about fractional-order chaotic systems are attained only by using analytic and numerical methods. Based on the qualitative theory, the existence and uniqueness of solutions for a class of fractional-order Lorenz chaotic systems are investigated theoretically in this paper. The stability of the corresponding equilibria is also argued similarly to the integer-order counterpart. According to the obtained results, the bifurcation conditions of these two systems are significantly different. Numerical solutions, together with simulations, finally verify the correctness of our analysis.

Dynamic analysis of a fractional-order Lorenz chaotic system

Fractional-order neural networks play a vital role in modeling the information processing of neuronal interactions. It is still an open and necessary topic for fractional-order neural networks to investigate their global stability. This paper proposes some simplified linear matrix inequality (LMI) stability conditions for fractional-order linear and nonlinear systems. Then, the global stability analysis of fractional-order neural networks employs the results from the obtained LMI conditions. In the LMI form, the obtained results include the existence and uniqueness of equilibrium point and its global stability, which simplify and extend some previous work on the stability analysis of the fractional-order neural networks. Moreover, a generalized projective synchronization method between such neural systems is given, along with its corresponding LMI condition. Finally, two numerical examples are provided to illustrate the effectiveness of the established LMI conditions.

Yongguang Yu

Papers

Mittag-Leffler stability of fractional-order Hopfield neural networks

Global stability analysis of fractional-order Hopfield neural networks with time delay

Adaptive backstepping synchronization of uncertain chaotic system

Dynamic analysis of a fractional-order Lorenz chaotic system

LMI Conditions for Global Stability of Fractional-Order Neural Networks