CYBERNETICS AND PHYSICS, VOL. 8, NO. 3, 2019, 114–120 INTERMITTENCY INDUCED IN A BISTABLE MULTISCROLL ATTRACTOR BY MEANS OF STOCHASTIC MODULATION Jose´ L. Echenaus´ıa-Monroy Guillermo Huerta-Cuellar∗ Laboratory of Dynamical Systems Laboratory of Dynamical Systems CULagos, Universidad de Guadalajara CULagos, Universidad de Guadalajara Mexico´ Mexico´ [email protected] ∗Corresponding author: [email protected] Rider Jaimes-Reategui´ Juan H. Garc´ıa-Lopez´ Hector´ E. Gilardi-Velazquez´ Laboratory of Dynamical Systems Laboratory of Dynamical Systems Facultad de Ingenier´ıa CULagos, Universidad de CULagos, Universidad de Universidad Panamericana Guadalajara, Mexico´ Guadalajara, Mexico´ Mexico´ [email protected] [email protected] [email protected] Article history: Received 16.10.2019, Accepted 26.11.2019 Abstract inverse period-doubling bifurcations, respectively, and In this work, numerical results of a nonlinear switch- crisis-induced intermittency with a crisis of chaotic at- ing system that presents bistable attractors subjected to tractors [Manneville & Pomeau, 1979; Hirsh, Nauen- stochastic modulation are shown. The system exhibits a berg, & Scalpino , 1982; Hirsh, Huberman, & Scalpino; dynamical modification of the bistable attractor, giving Hu, & Rudnick]. For systems that show multistable be- rise to an intermit behavior, which depends of modula- havior, noise presence can be useful to influence inter- tion strength. The resulting attractor converge to an in- esting dynamics as hopping attractor [Kraut, & Feudel; termittent double-scroll, for low amplitude modulation, Huerta-Cuellar et. al.; Pisarchik et. al.], as physical and and a 9-scroll attractor for a higher applied noise ampli- natural phenomenons [Huerta-Cuellar et. al.; Pisarchik tude. A Detrended Fluctuation Analysis (DFA) applied et. al; Gelens et. al.]. Also, noise can induce on-off inter- to the x state variable, shows a perturbations robust- mittency in those systems that exhibit bistable behavior ness region, since the increase of noise does not present [Campos-Mejia, 2013]. A system with periodic poten- changes. Due to the applied noise, the final obtained tial in the high frequency regime could shows the occur- system has higher randomness, compared with the orig- rence of intermittency as the case of a pendulum, where inal one. The understanding of the dynamical behavior the linear-response theory yields maximum frequency- of multiscrolls systems is highly important for advanc- dependent mobility as noise strength function [Saikia et ing technology in communications, as well in memory al., 2011]. systems applications. In the case of systems that generates intermittent ac- tivity, an analysis and characterization of the equilib- Key words rium points number in a Chua multiscroll system by ap- plied noise was shown by [Arathi, Rajasekar, & Kurths]. Multiscroll, Intermittency, Dynamics, Bistability. In that sense, the generation of systems with scrolls in their phase space, such as the Lorenz and Chua systems 1 Introduction [Lorenz, 1963; Chua, 1992], have been extensively stud- In many nonlinear dynamical systems, intermittency is ied from a dynamical point of view, being the Lorenz at- a common behavior, characterized by irregular burst that tractor a particular case with intermittent behavior [Man- modify regular behavior [Manneville & Pomeau, 1979]. neville & Pomeau, 1979]. Over the past few years, Different types of intermittency can be mentioned, e.g., the design and control of systems with multiple scrolls type I and on-off intermittency are related with saddle- have been a subject of interest for the scientific com- node bifurcations, type II and type III with Hopf and munity [Echenaus´ıa-Monroy, & Huerta-Cuellar],´ hav- CYBERNETICS AND PHYSICS, VOL. 8, NO. 3, 2019 115 ing a great impact in their application, such as secure UDS type II is defined by equilibrium points with a real communication systems, neural modeling, generation positive eigenvalue with two complex conjugated with of pseudo-random systems, and deterministic Brownian negative real part, with Σ(λ1;2;3) < 0 [Campos-Canton´ motion [Kwon et al., 2011; Yalcin et al., 2004; Huerta- et al., 2010]. Cuellar et al., 2014]. Recently a bistable multiscroll fam- Multiscroll generation by means of series of saturated ily has been presented and characterized [Echenaus´ıa et functions results in a better way to control the generated al., 2018]. In such system it is possible to obtain different fixed points, than with Piece Wise Linear (PWL) systems behaviors by means of bifurcation parameter variation [Lu¨ et al., 2004; Lu¨ et al., 2006]. This paper is based on (ζ). By considering the results shown in [Echenaus´ıa et a multiscroll generator system aimed by the jerk equa- al., 2018], in this work a study on noise-induced inter- tion, with the implementation of a Saturated Non Linear mittency between coexisting regimes of 1-scroll behav- Function (SNLF), as switching law, resulting in the sys- ior, under the influence of external Gaussian noise, is tem described by eq. (3). carried out. The observed intermitent dynamics may be associated with on-off intermittency. The interest in in- termittent fluctuations arises from its usefulness for both x_ = y; technical applications and fundamental research. There y_ = z; (3) are some works that presents useful memory applica- z_ = −α31x − α32y − α33z + α31f(x; k; h; p); tions based on multiscrolls [Itoh & Chua, 2008; Pham et al., 2019], in this case the applied noise could be imple- been f(x; k; h; p) the SNLF, defined as: mented to dynamically change the system memory prop- erties. p X f(x; k; h; p) = fm(x; k; h); (4) 2 Theory m=−p Consider a set of deterministic nonlinear differential equations, with chaotic behavior, defined as in [Campos- for which k > 0 is the slope of the saturated function Canton´ et al., 2010]. series, h > 2 is the saturated delay time of the saturated function, defined by the op-amp switching speed, p is a χ_ = Aχ + B; (1) positive integer, m = 1; 2; : : : ; n, where n defines the number of scrolls to generate. The function segmenta- T 3 where χ = [x1; x2; x3] 2 R is the state variable, B = tion is defined as follows: T 3 [b1; b2; b3] 2 R stands for a real constant vector. The behavior of the system is defined by the eigenvalues of 8 3×3 2k if x > mh + 1; the matrix A 2 R which is given as a linear operator < as follows: fm(x; k; h) = k(x − mh) + k; if x − mh ≤ 1; : 0 if x < mh − 1; (5) 0 1 α11 α12 α13 A = @α21 α22 α23A : (2) 8 0 if x > mh ± 1; α31 α32 α33 < f−m(x; k; h) = k(x ± mh) − k; if x ± mh ≤ 1; Discarding any combination of eigenvalues that is not : −2k if x < mh − 1; characteristic of hyperbolic-saddle-node points, that is, (6) the system is consistent with the Unstable Dissipative The SNLF contemplated for this study is constructed System type I (UDS I), defined in [Campos-Canton´ et based on [Echenaus´ıa et al., 2018], studied with a bifur- el., 2012; Ontanon˜ et al., 2012]. cation parameter ζ. This ζ parameter works as an indi- A switched system is implemented, constituted by a vidual control gain for the nonlinear function. This al- set of equations in form of eq. (1), in order to change lows the possibility to analyze the bifurcation diagrams the dynamical behavior which is governed by a switch- for a defined set of parameters in the model and gener- ing law, Si with i = 1; :::; n, and n ≥ 3. Each system ates more than one single attractor. The modified system 3 Si has a domain Di ⊂ R , containing the equilibrium is described by: ∗ −1 χi = −Ai Bi. Then, the switching law governs the ∗ ∗ SW dynamics by changing the equilibria from χi to χj , t 3 x_ = y; i 6= j, when the flow Φ : Di −! R crosses from the i − th to the j − th domain. y_ = z; (7) In order to generate multiscrolls dynamics, eq. (1) z_ = −α[x + y + z − ζf(x; k; h; p)]; must to accomplishes with the UDS I definition. A UDS 0 < α < 1 type I corresponds to equilibrium points with a real neg- where is a system parameter that modifies the equilibria stability, and ζ is the bifurcation param- ative eigenvalue (λ1) with two complex conjugated with positive real part (λ2, and λ3), with Σ(λ1;2;3) < 0, and a eter. This modification allows to analyze the behavior CYBERNETICS AND PHYSICS, VOL. 8, NO. 3, 2019 116 3 Methodology and Results From the dynamical system in eq. (7), with a switching law that creates a 9-scroll attractor, a bifurcation diagram constructed, generated by a gradual change of the bifur- cation parameter ζ, and a fixed parameter α = 0:45, is shown in Figure 1(a). A value of α parameter higher than 0:45 causes a more restrictive dynamical behavior of the system [Echenaus´ıa et al., 2018]. Figure 1(b), shows a zoom of the region to analyze with ζ = 0:0585, which generates a bistable behavior. Noise addition to the eq. (7) can stimulate jumps be- tween each of the bistable states. This bistable region forms a double potential well, for which the system re- sponse only have one state each time. With the noise amplitude increasing is possible to change the average residence times of the system until get an equilibrium of jumps between the two states. Without loss of generality, the equation system with the added noise is as follows: x_ = y; y_ = z; (9) z_ = −α[x + y + z − ζf(x; k; h; p)] + Nη; where N is the noise amplitude, and η represents the added noise, generated by the Box-Muller¨ method. The noise addition is made for each itteration of the numeri- cal system, in order to have a perturbation for every sim- ulation time.