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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.

Consider the parametric dynamical system

The study of stability with a polynomial approach had an important impulse when Kharitonov’s theorem was published in 1978. This theorem gives conditions for the stability of an interval family of polynomials (see [

However, stability is not always required. For example, there is a class of chaotic dynamical systems based on unstable equilibria. Several times a structural change is given by one

Consider an asymptotically stable linear system given by

If

In Section

Let

Let us recall that a set of points is convex if it contains, with any two points

The abscissa of stability

Consider the polynomial

If

Let

Consider the polynomial

Consider

Let

Theorem

Let

If

If

If

For the polynomial

Consider

Consider

In fact, another way to obtain

Consider the following polynomial

Let

For a family of Hurwitz polynomials of degree

Consider the family of Hurwitz polynomials

if

are lower bounds for the abscissa of stability of the family of polynomials.

From item (b) of Theorem

For item (b) of Theorem

Note that for every interval family of Hurwitz polynomials we give the lower bound

In Theorem

Consider the family of Hurwitz polynomials

Since

Consider the family of Hurwitz polynomials

In the study of multiscroll attractors different aspects are interesting and one of them is when the multiscroll attractor exists for a particular set of system’s parameters; then the interest is about robustness against parametric perturbation. For instance, we would like to know the variation of the values of parameters of a given system in order to preserve the multiscroll attractor. In this direction a polynomial approach has been used to find the maximal robust dynamics [

We have the following system:

This work is based on UDS-I, so a generalization of the above definition for UDS-I with dimension greater than three can be given as follows.

The system given by (

Due to the relation between the linear system like (

Let

The proof of (i) is obvious. We will focus on the proof of (ii). Firstly, it is not too hard to see that if the root

The previous lemma provides an upper bound for dissipativity. However, it may happen that

Given the fact that a Hurwitz polynomial

Consider the Hurwitz polynomial

If

In order to generate multiscroll attractors, let us consider the control system

Let

The linear part of the system must satisfy the dissipative condition

The affine vector

The commuting system given by (

Consider a system given by (

We exemplify the theory by presenting a case in

Consider system (

Note that the closed-loop system (

A system satisfying the previous theorem is candidate to generate multiscroll attractors emerging from its equilibria with a suitable step function

The controlled system is

Projections of the solution of system (

The abscissa of

Projections of the attractor onto the planes: (a)

The equilibria of the system for

In this paper we use the Gauss–Lucas theorem for obtaining an inequality between the abscissas of stability of a Hurwitz polynomial and its derivative. Then we use such inequality for getting a lower bound for the abscissa of a polynomial and for an interval family of polynomials. We have compared the lower bounds obtained with other works and we can say that the obtained bounds in this paper are easy to calculate and sometimes are better that others. Based on the aforementioned results, an approach to generate multiscroll attractors was presented. We consider that this result is important to help in understanding the emergence of chaos in stable systems. Using the abscissa of stability we can generate multiscroll attractors from a Hurwitz polynomial. One interesting aspect is that we can generate multiscroll attractor with the change of only one parameter.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The first author wishes to thank CONACYT for its Ph.D. scholarship support. Jorge Antonio López-Rentería also wishes to thank CONACYT for the postdoctoral grant (Grant no. 290941-UIA) and the Iberoamerican University for the support in the realization of this paper. C. A. Loredo-Villalobos also wishes to acknowledge the support of CONACYT through the postdoctoral fellowship. Eric Campos-Cantón acknowledges the CONACYT financial support for sabbatical. He would also like to thank the University of Houston for his sabbatical support and Professor Matthew Nicol for allowing him to work with him and his valuable discussions on dynamic systems.