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

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.

[1]  Henry Leung,et al.  Experimental verification of multidirectional multiscroll chaotic attractors , 2006, IEEE Transactions on Circuits and Systems I: Regular Papers.

[3]  Johan A. K. Suykens,et al.  Quasilinear approach to nonlinear systems and the design of n-double scroll (n=1, 2, 3, 4, . . .) , 1991 .

[4]  R Femat,et al.  Multiscroll attractors by switching systems. , 2010, Chaos.

[5]  Henry Leung,et al.  Design and implementation of n-scroll chaotic attractors from a general jerk circuit , 2005, IEEE Transactions on Circuits and Systems I: Regular Papers.

[6]  Eric Campos-Cantón,et al.  Multistability in Piecewise Linear Systems versus Eigenspectra Variation and Round Function , 2016, Int. J. Bifurc. Chaos.

[7]  Eric Campos-Cantón,et al.  Chaotic attractors based on unstable dissipative systems via third-order differential equation , 2016 .

[8]  Xinghuo Yu,et al.  Generating 3-D multi-scroll chaotic attractors: A hysteresis series switching method , 2004, Autom..

[9]  E. Tlelo-Cuautle,et al.  Generating a 50-scroll chaotic attractor at 66 MHz by using FPGAs , 2016 .

[10]  E. Campos-Cant'on,et al.  Nonclassical point of view of the Brownian motion generation via fractional deterministic model , 2018, 1802.04382.

[11]  E. Ott.  Strange attractors and chaotic motions of dynamical systems , 1981 .

[12]  Johan A. K. Suykens,et al.  n-scroll chaos generators: a simple circuit model , 2001 .

[13]  Robert W. Newcomb,et al.  Chaos generation using binary hysteresis , 1986 .

[14]  Ricardo Femat,et al.  Attractors generated from switching unstable dissipative systems. , 2012, Chaos.

[15]  L. Chua,et al.  The double scroll family , 1986 .

[16]  Xinghuo Yu,et al.  Design and analysis of multiscroll chaotic attractors from saturated function series , 2004, IEEE Transactions on Circuits and Systems I: Regular Papers.

[17]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[18]  Johan A. K. Suykens,et al.  Families of scroll Grid attractors , 2002, Int. J. Bifurc. Chaos.

[19]  Eric Campos-Cantón,et al.  Analog Electronic Implementation of a Class of Hybrid Dissipative Dynamical System , 2016, Int. J. Bifurc. Chaos.

[20]  L. Chua.  The Genesis of Chua's circuit , 1992 .

[21]  Guanrong Chen,et al.  Generation of n-scroll attractors via sine function , 2001 .

[22]  J. Suykens,et al.  Generation of n-double scrolls (n=1, 2, 3, 4,...) , 1993 .

[23]  J. Suykens,et al.  Experimental confirmation of 3- and 5-scroll attractors from a generalized Chua's circuit , 2000 .

[24]  Leon O. Chua,et al.  A family of n-scroll attractors from a generalized Chua's circuit , 1997 .

[25]  F. Windmeijer,et al.  An R-squared measure of goodness of fit for some common nonlinear regression models , 1997 .

[26]  Alexander N. Pisarchik,et al.  An approach to generate deterministic Brownian motion , 2014, Commun. Nonlinear Sci. Numer. Simul..

[27]  Rider Jaimes-Reátegui,et al.  Family of Bistable Attractors Contained in an Unstable Dissipative Switching System Associated to a SNLF , 2018, Complex..

[28]  R. Sanfelice,et al.  Hybrid dynamical systems , 2009, IEEE Control Systems.