CYBERNETICS AND PHYSICS, VOL. 8, NO. 3, 2019, 114–120

INTERMITTENCY INDUCED IN A BISTABLE MULTISCROLL 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 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 , 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 , 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] ∈ R is the state variable, B = tion is defined as follows: T 3 [b1, b2, b3] ∈ R stands for a real constant vector. The behavior of the system is defined by the eigenvalues of  3×3 2k if x > mh + 1, the matrix A ∈ 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)   α11 α12 α13 A = α21 α22 α23 . (2)  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 in eq. (7), with a switching law that creates a 9-scroll attractor, a 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. Next results are obtained by numerical simulations, Figure 1. (a) Bifurcation diagram that shows monostable and bistable by using eq. (9) in the bistable region (Figure 1(a), behavior for ζ < 0.0570, and ζ > 0.0570, respectively, correspond- red fringe), ζ = 0.0585, and fixed initial conditions ing to eq. (7). (b) Zoom of the interest region for ζ = 0.0585, red (x = 0.1, y = 0.1, z = 0.1), which generates the be- fringe corresponds to a bistable multiscroll attractor. (c) Phase space havior shown in Figure 1(c), dark blue color. Figure 2, (x, y), of the bistable attractor, negative part (dark blue) which cor- shows some temporal series of the observed behavior, responds the initial conditions [0.1, 0.1, 0.1], same ones used in this where it can be seen the temporal evolution of the sys- work. tem. When noise is increased, jumps between the two possible initial states appears, then it is possible to ob- serve that the residence time in each state comes smaller of the system through numerical simulation in a better and with similar probability. way. The system shown in eq. (7), can change behav- If the applied noise has a low amplitude, probability ior from 1, 3, 5, 7, and 9-scrolls monostable attractor, of jumping to the other state is low, but increasing the and generate six bistable 1-scroll attractors, by means of noise intensity the double-well potential is tilted sym- bifurcation parameter variation. metrically up and down, in this way periodically raising In this work, an additive random value is applied as and lowering the potential barrier. additive modulation of z state of eq. (7), which is gen- Figure 3 shows the (x, y) phase space of the corre- erated from the Box-Muller¨ method defined as follows sponding temporal series presented in Figure 2. Here it [Box & Muller,¨ 1958]: is possible to observe the initial attractor phase space, for a noise N = 0, and its evolution when noise is increased. √ The changes in the dynamical response presents a noisy ω = −2lnU cos(2πU ), 0 √ 1 2 (8) attractor that jumps between the commutation surfaces ω1 = −2lnU2cos(2πU1), of the 9-scrolls defined dynamic, from which the system is generated. for which U1 and U2 are random values from interval One statistical tool used to evaluate fluctuations of sys- of −1 to 1, ω0 and ω1 are independent variables with tems is the Detrended Fluctuation Analysis (DFA) [Peng standard deviation equal to 1, and changes its values for et al., 1994]. The DFA allows to measure a simple quan- each simulation step. titative parameter, the scaling exponent βν which char- acterizes a signal correlation properties. The main ad- CYBERNETICS AND PHYSICS, VOL. 8, NO. 3, 2019 117

2 15 a) b) with 1.23 6 N 6 2, showing a perturbations robustness 0 10 region, then the slope value decreases slowly with the -2 5 noise amplitude increment. -4 0

x (a.u.) -6 x (a.u.) From Figure 2((b)-(f)), it is possible to see an intermit-

-5 -8 tent behavior evolution with the noise increase. In Figure -10 -10 3 it can be seen that the obtained intermittency not only -12 -15 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 4 4 visits the commutation surfaces defined for the bistable Time (a.u.) ×10 Time (a.u.) ×10

15 15 attractor, but also visit and lives in all the switching sur- c) d)

10 10 faces that creates the natural attractor. As can be seen in

5 5 Figure 5(b), for ζ = 0.056, the obtained behavior cor-

0 0 responds to a 9-scrolls monostable attractor, for which x (a.u.) x (a.u.) -5 -5 each scroll have a equilibrium point, and in the case of -10 -10 ζ = 0.0585, the behavior corresponds to the bistable at-

-15 -15 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 tractor which has been studied. In the case of the bistable Time (a.u.) ×104 Time (a.u.) ×104

15 15 attractor, each scroll have oscillations around two fixed e) f)

10 10 points, depending on the initial conditions. In Table 1,

5 5 the equilibrium location for the 9-scrolls attractor and

0 0 the bistable analyzed attractor is displayed.

x (a.u.) x (a.u.) -5 -5 As mention in section 2, the equilibrium points of the -10

-10 -15 analyzed system (when N = 0), behaves according ∗ −1 0 -15 -20 to the equation χ = −A [0, 0, αζf(x; k, h, p)] , and 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 Time (a.u.) ×104 Time (a.u.) ×104 have the property that whenever the state of the system ∗ ∗ start at χi , it will remain at χi for all future time, be- Figure 2. Temporal series obtained from the eq. (9) for different ing the system autonomous. When N > 0, the system noise amplitude: (a) N = 0, (b) N = 1, (c) N = 1.5, (d) N = 2.5, equilibrium respond to the following equation, χ∗ = (e) N = 3, and (f) N = 4. Noise amplitude values in arbitrary units. −A−1[0, 0, αζf(x; k, h, p)]0 − A−1[0, 0, Nη], where η is a time dependent function which varies in every iter- ation, becoming the system into a non-autonomous one, vantage of the DFA, over many other methods, is that al- lows the detection of long-range correlations of a signal

3 4 embedded in seemingly nonstationary time series, and a) b) 3 also avoids the spurious detection of apparent long-range 2 2 correlations that are an artifact of nonstationarity. Fluc- 1 1

tuation function F (ν; s) obeys the following power law y (a.u.) y (a.u.) 0 0 scaling relation: -1 -1 -2

-2 -3 -12 -10 -8 -6 -4 -2 0 2 -15 -10 -5 0 5 10 15 x (a.u.) x (a.u.) F (ν; s) ∼ sβν , (10) 5 6 c) d) for which the time series is segmented in s pieces with 4 2 length ν. When the scaling exponent βν > 0.5, three 0 0 distinct regimes can be defined as follows: y (a.u.) y (a.u.) -2

1. If βν ∼ 1, DFA defines 1/f noise. -4

-5 -6 2. If β > 1, DFA defines a non stationary or uncorre- -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 ν x (a.u.) x (a.u.)

lated behavior. 6 10 e) f) 8 3. If βν ∼ 1.5, DFA defines Brownian motion or Gaus- 4 6

sian noise. 2 4

2 0 In order to analyze the noise effects over the dynamical y (a.u.) y (a.u.) 0 -2 -2 changes of the bistable studied system, a noise amplitude -4 -4 -6 variation 0 N 4 in steps of ∆N = 0.001, was ap- 6 6 -6 -8 -15 -10 -5 0 5 10 15 -20 -15 -10 -5 0 5 10 15 plied. Figure 4 is made by considering the average of 50 x (a.u.) x (a.u.) temporal series for each noise amplitude. Figure 4 shows the evolution of the slope obtained by the DFA method, Figure 3. Phase spaces (x, y), corresponding to the temporal series where βν remains unchanged until a value N = 0.245. in Figure 2, obtained from the eq. (9) for different noise amplitude: It can be observed, from the mean line, that a change (a) N = 0, (b) N = 1, (c) N = 1.5, (d) N = 2.5, (e) N= N = 3, in the slope occurs after N > 0.245, and then it con- and (f) N = 4. Noise amplitude values in arbitrary units. tinues increasing until a laminar region for βν = 1.34, CYBERNETICS AND PHYSICS, VOL. 8, NO. 3, 2019 118

1.4 noise. It was observed that an intermittent attractor ap- max 1.3 mean min pears between the initial states, but with the noise am- 1.2 plitude increase, stochasticity is higher and domains are 1.1 visited with the absence of fixed points. A DFA analysis ν β 1 reveals that for noise amplitude N = 0.245, the inter- 0.9 mittent attractor appears, for 1.23 6 N 6 2, the fluctu- 0.8 ation analysis remains with a constant slope βν = 1.34, 0.7 and for N > 2 the behavior turns to uncorrelated, show- 0.6 0 1 2 3 4 ing an indistinguishable attractor in the phase space. In N (a.u.) the case of noise amplitude 1.23 6 N 6 2, the system shows a constant behavior between Brownian motion Figure 4. With the increasing of the applied noise amplitude it can be and uncorrelated fluctuations. There are several works seen the evolution of the slope βν with the maximum, minimum and that presents multiscrolls attractors useful to the design mean values. of systems with memory applications, in this case, the added noise could be applied to dynamically change the memory properties of the system, or in modeling sys- thus the system does not have equilibrium points [Khalil, tems immune to perturbations. 2002]. An important point to remark is that the vec- tor field associated to linear operator A oscillates in the range of x state, that depends of noise amplitude. Acknowledgements After the applied noise, the system jumps between all J.L.E.M. acknowledges CONACYT for the finan- the commutation surfaces unachievable for N = 0. As cial support (National Fellowship CVU-706850, No. seen in Figure 5, the noise amplitude 1.23 6 N 6 2 can offer the possibility to distinguish the natural attrac- 4 tor structure. Moreover, the induced stochastic dynamics a) between the nine domains in the system can not be ob- 3 served in Figure 3(d), (e) and (f), because of the noise 2 amplitude (N > 2). 1 0

In Figure 5, a comparison between the monostable at- y (a.u.) tractor (1(c), dark blue color), with added noise N = -1 -2 1.5, and the 9-scrolls natural attractor is shown. As men- -3 tion before the noise induce a visit of the nine domains -4 -15 -10 -5 0 5 10 15 in the system, for N w 1.5, shown in Figure 5(a). x (a.u.)

2 b) 1.5 Table 1. Equilibrium points location for the analyzed attractors, ∗ 1 (χi , 0, 0), N = 0. 0.5

0

Equilibrium points for the 9-scrolls attractor, ζ = 0.056 y (a.u.) -0.5 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ χ1 χ2 χ3 χ4 χ5 χ6 χ7 χ8 χ9 -1 -1.5

-2 −8 −6 −4 −2 0 2 4 6 8 -10 -5 0 5 10 x (a.u.) Equilibrium points for the bistable attractor, ζ = 0.0585

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ χ1 χ2 χ3 χ4 χ5 χ6 χ7 χ8 χ9

−8.3 −6.2 −4.1 −2.1 0 2.1 4.1 6.2 8.3

4 Conclusion In this work numerical simulations over a bistable at- Figure 5. Phase spaces (x, y), corresponding to temporal series, (a) tractor generated by means of a multiscroll system with shows intermittent behavior for a noise amplitude N = 1.5, (b) shows added noise are presented. Results are obtained over the natural attractor of 9-scrolls, and (c) shows that the intermittent a bistable region which is very close of the natural at- attractor visits the same commutation surfaces defined for the natural tractor of the system, modulated by means of Gaussian attractor. CYBERNETICS AND PHYSICS, VOL. 8, NO. 3, 2019 119

582124) and the University of Guadalajara, CULa- Perales, G., & Femat, R. (2010). Multiscroll attrac- gos (Mexico). G.H.C. acknowledges to University tors by switching systems. Chaos: An Interdisciplinary of Guadalajara for the suppot (project 241596, fund Journal of Nonlinear Science, 20(1), 013116. 1.1.9.24) to the realization of the sabbatical stay at St Lorenz, E. N. (1963). Deterministic nonperiodic flow. Mary’s University, and to E. Campos-Canton´ for fruc- Journal of the atmospheric sciences, 20(2), 130-141. tiferous discussions and the opportunity of realize simu- Chua, L. O. (1992). The genesis of Chua’s circuit. Berke- lations in the IPICYT. This work was supported by the ley, CA, USA: Electronics Research Laboratory, Col- University of Guadalajara under the project “Research lege of Engineering, University of California. Laboratory Equipment for Academic Groups in Opto- Arathi, S., Rajasekar, S., & Kurths, J. (2013). 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