Stability and Multiscroll Attractors of Control Systems via the Abscissa

Abstract

We present an approach to generate multiscroll attractors via destabilization of piecewise linear systems based on Hurwitz matrix in this paper. First we present some results about the abscissa of stability of characteristic polynomials from linear differential equations systems; that is, we consider Hurwitz polynomials. The starting point is the Gauss-Lucas theorem, we provide lower bounds for Hurwitz polynomials, and by successively decreasing the order of the derivative of the Hurwitz polynomial one obtains a sequence of lower bounds. The results are extended in a straightforward way to interval polynomials; then we apply the abscissa as a measure to destabilize Hurwitz polynomial for the generation of a family of multiscroll attractors based on a class of unstable dissipative systems (UDS) of affine linear type.

Design of a Nonhomogeneous Nonlinear Synchronizer and Its Implementation in Reconfigurable Hardware

“…For the implementation of the synchronization scheme, the selected piecewise UDS given in [43] presents a four scrolls attractor and it is defined aṡ…”

Output feedback controller for polytopic systems exploiting the direct searching of the design space

“…Therefore, the stability of all polynomials inside the polynomial set ( ) is equivalent to the stability of all its exposed edges due to Lemma 4. Obviously, this implies condition (17).…”

“…Remark 1. If one of the model conditions of Lemma 1 satisfies, condition (17) in Theorem 2 will be sufficient and necessary condition for the TS property of the pre-defined convex space. Generally, if the characteristic polynomial of the closed-loop model affinely depends on the uncertain variables , then Theorem 2 equivalently investigates the TS property.…”

“…There exists approaches that focus on the problem of stability check of different types of polynomial spaces. [10][11][12][13][14][15][16][17][18] One of the important theorems in this topic is the boundary crossing theorem that briefly states that, if there exist stable and unstable polynomials in a path-connected polynomial spaces, then there exists a polynomial with pure imaginary roots. López-Renteria et al generalize this theorem in order to discuss about the number of the pure imaginary roots of these critical polynomials.…”

A New Megastable Chaotic Oscillator with Blinking Oscillation terms

“…Among features related to the topology of attractors, ones which have multiscrolls are interesting for researchers [30]. Assessing the stability of multiscrolls attractors [31] and finding methods to preserve multiscrolls [32] are topics that grab much attention. Besides, some methods have been introduced to use multiscroll attractors such as switches in systems [33].…”