A novel approach to generate attractors with a high number of scrolls

Abstract

In this paper, it is presented a novel method for increasing the number of scrolls in a hybrid nonlinear switching system. Using the definition of the "Round to the Nearest Integer Function", as a generalization of a PWL function, which is capable of generating up to a thousand of scrolls. An equation that characterizes the grown in the number of scrolls is calculated, which fits to the behavior of the system measured by means of the coefficient of determination, denoted R 2 , and pronounced "R squared". The proposed equation is based on obtaining as many scrolls as desired, based on the control parameters of the linear operator of the system. The work here presented provides a new approach for the generation and control of a high number of scrolls in a hybrid system. The results are verified for all the scenarios that the equations covers.

Electronic implementation dataset to monoparametric control the number of scrolls generated

“…The dynamical system implemented in the electronic circuit is governed by the set of equations shown in Eq. (1) , better described in [ 1 , [3] , [4] , [5] , [6] ]. …”

“…The parameter ρ controls the generation of attractors with a different number of scrolls, meanwhile, the α value is a control parameter associated with the stability of the equilibrium points in the system. Since it is locking for hyperbolic-saddle-node equilibrium points [3] , [4] , [5] , [6] , the control parameter is delimited by α ∈ (0, 1).…”

Multistability route in a PWL multi-scroll system through fractional-order derivatives

“…Consider a PWL multi-scroll system based on the jerk equation [12,[42][43][44][45], which satisfies all conditions to be considered as UDS I for the control parameter space α ∈ (0 − 1), since it has a real positive eigenvalue, a pair of complex conjugate eigenvalues with negative real component, where the eigenvalue sum function is negative.…”

“…The control parameter α is a positive constant defining the eigenvalues of the system, where α ∈ R. It is possible to generate an attractor with multiple scrolls by using a commutation law f (x), in this case a Piece-Wise Linear (PWL) function, as shown in Eq. ( 10) [23,44], whose purpose is to control the visitation in the different domains containing the equilibrium points of the system, which is achieved by the coexistence of a large number of unstable one-spiral trajectories.…”

Predicting Tipping Points in Chaotic Maps with Period‐Doubling Bifurcations

“…Two types of systems can show chaos: continuous systems (flows) and discrete systems (maps) [3][4][5]. Chaos is still a challenge, and there are many unknown mysteries about it in both continuous and discrete chaotic systems [6][7][8]. Studying various dynamics of discrete and continuous systems has been a hot topic [9][10][11].…”