A novel approach to generate attractors with a high number of scrolls
aa r X i v : . [ n li n . C D ] A p r A novel approach to generate attractors with a high number ofscrolls
J. L. Echenaus´ıa-Monroy , and G. Huerta-Cu´ellar , , , Dynamical Systems Laboratory, CULagos, Universidad de Guadalajara, Centro Universitario de losLagos, Enrique D´ıaz de Le´on 1144, Paseos de la Monta˜na, 47460, Lagos de Moreno, Jalisco, M´exico. Applied Mathematics Division, Instituto Potosino de Investigaci´on Cient´ıfica y Tecnol´ogica, Caminoa la Presa San Jos´e 2055, Col. Lomas 4ta. Secci´on, 78216, San Luis Potos´ı, S. L. P., M´exico. Department of Physics and Earth Sciences, St.Marys University, SanAntonio, TX78228, [email protected] , [email protected] Abstract
In this paper, it is presented a novel method for increasing the number of scrolls in a hybridnonlinear 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 thebehavior of the system measured by means of the coefficient of determination, denoted R , andpronounced “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 presentedprovides a new approach for the generation and control of a high number of scrolls in a hybridsystem. The results are verified for all the scenarios that the equations covers. Multiscroll attractors; High number of scrolls; Round Function; Unstable Dissipa-tive Systems; Coefficient of Determination.
The switched nonlinear systems are mostly associated to the generation of chaotic behaviors withthe presence of multiple scrolls in their phase spaces [1, 2, 3]. The study of this kind of systems haspresented a great interest in the last three decades of scientific development, due to the endless numberof possible applications that these systems can have in different areas of science. The term multiscrollit is referred to the generation of an attractor that has, at least, three scrolls in its state space, unlikethe dynamical systems of Lorenz [4] or Chua [5], which only have attractors of double-scroll. Suykensand Vanderwall [6], proposed this type of systems at the beginning of the 90’s, motivated by the ideaof obtaining a richer dynamical system, referring to the number of scrolls, than the system of Chua,on which they were based.There are several approaches for obtaining systems with multiscrolls, which can be classified byconsidering their constructions as follows: i) adding break points to the function of Chua [7, 8], ii)using hysteresis functions [9, 10], iii) implementing functions of sinusoidal type and, iv) applying Piece-Wise Linear functions (PWL), [11, 12, 13, 14, 15]. By means of applying a PWL, an hybrid system isgenerated, which is characterized by the coexistence of continuos dynamics, such as the state variableof the numercial model, and logical decision making [16, 17]. The biggest handicap lies when theexpected result entails obtaining a higher number of scrolls in the system.The generation of simple systems with a high number of scrolls is still an open problem, that’swhy it is proposed an hybrid system based in the use of Unstable Dissipative Systems (UDS) [1, 2, 3]with the implementation of the Round to Nearest Integer Function as commutation law [18, 19], forthe generation of attractors with a large number of scrolls, which can be controlled based on a growthequation dependent on the control parameters of the system, thus obtaining a system, numerically,easy to implement.This work is distributed as follows: the first section of the article contains a brief introduction to theproblem where the scientific background is described. Throughout the second section, the basic con-cepts for designing UDS systems, as well as the mechanism for generating attractors with multiscrollsusing PWL functions are described. The mathematical model used, as well as the description of themethodology implemented, are addressed in section number three. The results of the characterizationof the model and the construction of the equation that governs the growth in the number of scrolls, aredescribed in the fourth caption. The conclusions about the work are shown at the end of this paper. Corresponding Author Theory
It is well known that the generation of attractors with multiple scrolls depends both, on the stabilityof the generated equilibrium points, as well as on the type of the implemented switching function. Itis possible to analyze the stability of the equilibrium points through the theory of Unstable Dissipa-tive Systems (UDS), which describes a variety of three-dimensional systems showing dissipative andconservative components. The coexistence of both components causes the appearance of the so-calledattractors with multiscrolls.As in previous works [3, 15, 19], consider a system of three coupled autonomous differential equa-tions,(1) ˙ X = AX+Bf(x) , where X = [ x, y, z ] T ∈ R , is the state vector, B = [ b , b , b ] T ∈ R , is a constant position vector, A = [ a i,j ] ∈ R × , is a constant matrix, and f(x) is a nonlinear function. The kind of behaviorexhibited by the system, is defined by the eigenvalues of matrix A , which can generate a great varietyof combinations and, therefore, the same diversity of behaviors.Considering only the cases in which the system described by ec. (1) has saddle-node equilibriumpoints, since they have both, stable and unstable varieties, it is possible to characterize the model inthe following way: i) a system it is considered as UDS I, if their equilibrium points are hyperbolic-saddle-node, i.e., one eigenvalue is a negative real one, and the other two are complex conjugated withpositive real part, where the sum of the components must be less than zero. This last condition fulfillsthe dissipative conditions of the system [2, 20]. ii) By the other side, a UDS II system it is defined asthose which has a real positive eigenvalue, and two complex conjugated with negative real part, wherethe sum of its components it is also negative. If the linear operator of the system defined in ec. (1) , fullfil all the conditions to be defined as a UDSI, then it is possible to generate an attractor with multiple scrolls by means of the construction of acommutation law, in this case, a PWL function. The purpose of the commutation law is control thevisit in the different equilibrium points of the system, being achieved by means of the coexistence ofa large number of one-spiral unstable trajectories. To illustrate such behavior [1, 3, 15], consider thefollowing linear operator:(2) A = − . − . − . , which satisfies the conditions that define a UDS I system, having eigenvalues equal to λ = [ − . , . ± . i ] and P λ = − .
1; this linear operator can be associated to a PWL function of three levels, ec. (3) . As can be seen in Figure 1(a), this conception of the function f ( x ) generates an attractor ofthree scrolls with equally distributed trajectories and equidistant equilibrium points.If the same linear operator as the one shown in ec. (2) is considered, and it is desired to obtain anattractor with a higher number of scrolls, a new commutation law must be built in, taking special carein obtaining equidistant equilibrium points, as well as the distribution of the trajectories. In order toobtain as many scrolls as segments have been introduced to the switching function, i.e., be f ( x ) theswitching function presented in ec. (4) , an attractor with seven scrolls in its phase space is obtaining,Figure 1(b). This same exercise can be done to build in larger switching laws ec. (5) , and therefore,generate attractors with the same number of scrolls, Figure 1(c).In general, the process to increase the number of scrolls in a system, is exemplified in the set ofequations ec. (3-5) in Table 2.2, the disadvantage lies when an attractor with many more scrolls mustbe implemented, i.e. more than a hundred scrolls, which obviously implies a not so simple task toaddress [21]. 2 a) (b)(c) (d) Figure 1: Attractors generated with the equations (a) ec. (2,3) , (b) ec. (2,4) , (c) ec. (2,5) , (d) ec.(2,6) .The task of adding a higher number of equilibrium points to a system, in a simpler way, it ispossible through the implementation of a switching function as in [18, 19], where the results have beenvalidated for the generation of attractors with multiscrolls. As an example, the Round to the NearestInteger Function (RNIF) is implemented, ec. (6) , which results in the attractor shown in Figure 1(d).
The multiscroll generator system here studied, is similar to those studied in [3, 19, 22], which iscomposed of three coupled differential equations, and implements a generalization of a PWL functionas an approach to obtain multiscroll attractors:(7) ˙ x = y, ˙ y = z, ˙ z = − α x − α y − α z + α ,α = C (cid:20) g (cid:18) xC (cid:19)(cid:21) , whre the descriptor system to analyze is composed of three state variables x, y, z . The values α , , are control parameters of the system that can modify its dynamics, and α is the switching functionassociated with the system. This work is focused on the operation region where the system responds3 o. of scrolls PWL3 f ( x ) = − , if x ≥ − , if − < x < , if x ≤ . (3) f ( x ) = − , if x ≥ − − , if − < x < − − , if − < x < − , if − < x < , if 1 < x < , if 3 < x < , if x ≤ . (4) f ( x ) = − , if x ≥ − − , if − < x < − − , if − < x < − − , if − < x < − , if − < x < , if 1 < x < , if 3 < x < , if 5 < x < , if x ≤ . (5) N f ( x ) = . (cid:16) round h x . i(cid:17) . (6)Table 1: PWL functions used for generate the attractors shown in Figure 1.4o the configuration of eigenvalues defined as UDS I; so the system is studied under the premise thatall control parameters are equal, α = α = α = α .Contemplating the previous statement, the behavior of the characteristic polynomial of the system, λ + α ( λ + λ +1) = 0, is analyzed under the modification for which the model responds to the definitionof a system cataloged as UDS I, resulting in a region of operation defined by 0 < α < α , C and C are real constants associated with the control parameters of the system. Thefunction g ( x ) responds as the RNIF, in this case, of the state variable x . Because this definition of theround function can be ambiguous for fractional values, the following consideration is adopted: g = (cid:26) Up round , by taking ⌊ x + 0 . ⌋ , Down round , by taking ⌈ x − . ⌉ . (8) The generalization of the nonlinear function, α , guarantees the generation of equidistant equi-librium points [19], and the amplitude of the jumps between the different switching surfaces. Thisconception is similar to those used in the set of equations described by ec. (3-5) (Table 2.2), and theresult is shown in Figure 1(d). The nonlinear function implemented in this system has a very simi-lar operation to a PWL function, conceived in order to simplify the construction of many switchingsurfaces as the control parameters allow. As it has been demonstrated in previous works [18, 19, 22], the use of the nonlinear function α , ec.(8) , presents a dependence on the control parameters of the system, and on the integration time. Thisis due that the function does not presents any type of limitation, either temporary or in the number ofswitching surfaces to visit, so it is defined from −∞ to ∞ . It is due to this, and the imperative needto generate systems with a higher number of scrolls, that this work is focused on the characterizationof the system proposed by ec. (7) , to obtain a system of easy numerical implementation, which is ableto generate as many scrolls as desired.According to the considerations raised by the theory that describes the unstable dissipative systems,it was established that the region where the analysis of the proposed model will be performed is definedfor values 0 < α <
1. Considering future electronic implementations, this control parameter will beexplored 0 . ≤ α ≤ .
95, and with a variation in the increment equal to ∆ α = 0 . ec. (7) , will be analyzed for each of the values in the control parameter α , where the nonlinear function constants will be maintained as follows: C = 0 . , C = αC , whichguarantees the generation of equidistant equilibrium points. This system will be analyzed numericallythrough the implementation of an integrator type RK4, and a time scale τ = 0 .
1. For each of the α values analyzed, the integration time limit will be gradually increased, and the initial conditions ofthe system will be randomly changed. For each combination in the parameters α − t , the number ofscrolls in the model is calculated. An equation describing the growth in the number of scrolls will beapproximated, generating a control law to generate attractors with a high number of scrolls.In Figure 2, the obtained results from four different control parameters, α , and a sample of thetemporal values explored are shown. For example, consider α = 0 .
70 (green squares), the greendotted line indicates the average of scrolls obtained ( < N > ), with the same marking color, boththe maximum ( N Max) and minimum ( N Min) values are shown. The results are analogous for allthe control parameters shown in the figure; turning the graph into a similar one to a box-plot. Thetemporal spaces analyzed correspond to a value of 2 σ , where σ is the x axis of the graph. The behaviorof the different curves presents the same growth trend, except for the value α = 0 .
95, where the increasein the number of scrolls is slower, compared to the rest of the values.Each of the analyzed control parameters yields similar results to those shown in Figure 2, whichcan be summarized on the surface shown in FIG. 3a), where the increase in the number of averagescrolls is shown, ¯ N = < N > , obtained for each of the control parameters with respect to time. Onceobtained the results, these can be analyzed to construct an equation that approximates the generalbehavior of the system. In this case, the adjustment curve is constructed by linearizing the data [23],resulting in the equation shown in ec. (9) , such equation presents a dependency at two parameters,the simulation time given to the system, t = 2 σ , and the control parameter with which it is analyzed,5 Time, t σ , a.u. N , a . u . α =0.20 α =0.45 α =0.70 α =0.95 N MaxN Min
Figure 2: Behavior of the number of scrolls generated, along time, for differents α values. The dottedlines represents the average number of scrolls. α . The behavior of this equation, under the same scenario as those proposed in the numerical analysis,is shown in Figure 3(b).(9) ¯ N = e β + ln( t )2 , where β is defined as in ec. (10) , n ∆ α = α , responds to the value in the control parameter that isanalyzed, 1 ≤ n ≤
19, and t is the time to be simulated in the system, C = 0 . β = − ∆ α (cid:20) . n (cid:18) C n (cid:19)(cid:21) . Control parameter, α , a.u.Time, t σ , a.u.
500 11000 < N > , a . u . (a) Control parameter, α , a.u.Time, t σ , a.u.
500 11000 < N > , a . u . (b) Figure 3: Surface of behavior in the average number of scrolls generated, along time, for all the α values. (a) Obtained by means of the numerical time series analysis. (b) Obtained by means ofapplying the ec. (9) . 6 Control parameter, α , a.u. R , a . u . R
05 and α = 0 .
95. This lack of prediction in the model canbe understood from the point of view of the distribution of the recurrence points from the Poincar´e’ssection [15]. For very small α values, the disorder in the system is very high, so the visit to the differentdomains is presented in a highly random way, being more difficult to estimate the number of switchingsurfaces that will be visit. Similarly for values in the control parameter near to the threshold of theUDS I region, where the system has gained a significant amount of order, so the visit to the differentswitching surfaces is carried out with a probability that resembles to a Normal distribution, so theprediction in such scenario, also becomes difficult.7
15 -10 -5 0 5
Variable x, a.u. -0.4-0.3-0.2-0.100.10.20.30.4 V a r i ab l e y , a . u . (a) Time, t, a.u. × -15-10-505 V a r i ab l e x , a . u . (b) Figure 5: a) Attractor with N = 27 scrolls, and b) Time serie of the x variable, for the parameters α = 0 .
5, and t = 70000 units. To demostrate the correct operation of the proposed equation, ec. (9) , the following results arepresented, Table 4.1, that cover the three possible scenarios: i) when α and t are known values,determine the number of scrolls to be obtained ec. (9) , ii) when is desired to obtain an attractor of N scrolls for a determined α value, determine the simulation time that takes the system to reach thevisit of the N switching surfaces ec. (13) , and iii) when both values, N and t , are set parameters,determine the α value that is required to obtain the desired attractor ec. (14) .Considering the first case proposed by ec. (9) , suppose that the control parameter is known, aswell as the simulation time to be used, α = 0 . , t = 70000 u . , and it is desired to know the number ofscrolls to be obtained, this to determine if the attractor is large enough for the purposes, or a highernumber of scrolls is required. With this information, it is possible to estimate the number of scrollsthat would be obtained by substituting in ec. (9) , resulting an average number of scrolls ¯ N = 35 . α = 0 . t = 70000 units. If this scenario is simulated, the systempresents the dynamics shown in Figure. 5, where an attractor with 27 scrolls is obtained, which is notso different from the value dessired.In the same way as in the previous case, it is possible to perform the following exercise: an attractorwith a hundred of scrolls is desired, associated to a control parameter α = 0 .
35, so the simulation timeneeded to obtain such attractor is the parameter to calculate. Substituting these values in ec. (13) ,¯ t = 404470 u . is obtained. This amount of time may be considered as the minimum time that the systemneeds to visit all the switching surfaces desired. If the system is simulated with these parameters, anattractor with 108 scrolls is obtained, Figure. 6.As a last proof of the right prediction performed by the equation that governs the number of scrollsto be obtained, assume a simulation time t = 23000 u . , and the desired number of scrolls is N = 13, sothe control parameter to be used is unknown. Substituting the information in ec. (14) , ¯ n = 19 . n = 19, and corresponds to α = 0 .
95. Carring out the simultaion, thesystem presents an attractor with only 5 scrolls. As mention before, the system is hard to predict inthis α value, because of the order that the dynamics in the system has gain. Considering this fact, thesimulation it is performed for the α previous value ( α = 0 . nown Parameters Estimated Parameter α, t ¯ N = e β + ln( t )2 . (9) α, N ¯ t = e Ne β . (13) t, N ¯ n = − (3 C + 28 . − ln Ne ln( t )2 (cid:18) α (cid:19) . (14)Table 2: Combinations of the equation for control the growth of the number of multiscrolls. (a) Time, t, a.u. × -40-30-20-100102030 V a r i ab l e x , a . u . (b) Figure 6: a) Attractor with N = 108 scrolls, and b) Time serie of the x variable, for the parameters α = 0 .
35, and t = 404470 units. 9 Variable x, a.u. -0.4-0.3-0.2-0.100.10.20.30.4 V a r i ab l e y , a . u . (a) Time, t, a.u. × -4-20246 V a r i ab l e x , a . u . (b) Figure 7: a) Attractor with N = 12 scrolls, and b) Time serie of the x variable, for the parameters α = 0 .
9, and t = 23000 units. With this research work, a hybrid system of three differential equations that implements the Roundto the Nearest Integer Function, as a generalization of a PWL function, for obtaining systems withmultiscrolls based on UDS models, has been analyzed. Based on the implementation of the RNIF, acharacterization of the behavior of the system was carried out, with which an equation is obtained,dependent on the control parameters, that facilitates the implementation of systems with a largenumber of scrolls. This equation was validated in all the scenarios that it covers, based on the coefficientof determination ( R ), guaranteeing a good prediction in the increase of the number of scrolls in thesystem.The model here described, furthermore of facilitate the obtention of systems with a high numberof scrolls, is capable of generating attractors with a higher level of disorder, based on the resultsshown in [15]. The limitations in the basins of attraction for UDS I systems are a major factorwhen designing multiscroll attractors. The implementation of the results here described, may beenhelpful in the analysis of systems like those studied in [19], developing the analysis of time series withthe minimum amount of data to reproduce the phenomenon. Contemplating all these factors, it isconsidered that the technological applications of the model are even more attractive, having potentialuse in fields of science like in neural systems, secure communication systems, electric motors withvariable torque, pseudo-random number generators, among others. The improvement in the equationand corresponding analogical implementation, it is proposed as future work. Acknowledgments
J.L.E.M. acknowledges CONACYT for the financial support (National Fellowship CVU-706850, No.582124), to the University of Guadalajara, CULagos (M´exico), and to E. Campos-Cant´on for fruc-tifer discussions and the opportunity of realize a research-stay in his investigation group. This workwas supported by the University of Guadalajara under the project “Research Laboratory Equipmentfor Academic Groups in Optoelectronics from CULAGOS”, R-0138/2016, Agreement RG/019/2016-UdeG, Mexico.