Coexisting Hidden and self-excited attractors in an economic system of integer or fractional order
CCoexisting Hidden and self-excited attractors in aneconomic system of integer or fractional order
Marius-F. DancaRomanian Institute of Science and Technology,Cluj-Napoca, Romaniaemail: [email protected] 22, 2020
Abstract
In this paper the dynamics of an economic system with foreign fi-nancing, of integer or fractional order, are analyzed. The symmetry ofthe system determines the existence of two pairs of coexisting attrac-tors. The integer-order version of the system proves to have severalcombinations of coexisting hidden attractors with self-excited attrac-tors. Because one of the system variables represents the foreign capitalinflow, the presence of hidden attractors could be of a real interest ineconomic models. The fractional-order variant presents another inter-esting coexistence of attractors in the fractional order space.
Keywords
Hidden attractor; self-excited attractor; coexisting attrac-tors; saddle; economic system
According to Shilnikov criteria, chaos emergence requires at least one unsta-ble equilibrium [Shilnikov(1965)]. So, if for some values of the key parameterthe equilibria are unstable and one tries to generate attractors with initialconditions near these equilibria, one obtains chaotic attractors and coulddeduce that the system evolves chaotically. On the other side, this con-clusion could be incomplete, or even false, since for initial conditions takenfrom an attraction basin which does not intersect with any small neighbor-hoods of unstable equilibria, the underlying trajectories could lead to someother attractor that might be a regular motion, but not a chaotic motion.Recently this problem has been managed by introducing the term of hid-den attractor (see e.g. [Leonov & Kuznetsov(2013)], one of the first workson this subject, or [Kuznetsov(2015)]). If the attraction basin intersectswith any open neighborhoods of an equilibrium the attractor is called self-excited, otherwise, it is called hidden. Usually, the self-excited attractors1 a r X i v : . [ q -f i n . GN ] S e p re numerically derived from unstable equilibria, while hidden attractors aredifficult to be localized, because their attraction basins have no relation withsmall neighborhoods of any equilibria. Hidden attractors can be found ina nonlinear system with one or more stable equilibria [Molaie et al.(2013),Wang et al.(2017), Cang et al.(2019), Qigui et al.(2019)], with a line of equi-libria (infinite equilibria) [Jafari & Sprott(2013)], with both unstable equi-libria and stable equilibria [Danca(2017)], with coexistence of various attrac-tors [Zhou et al.(2018)], or even without equilibria [Wei(2011)]. Note thatthere is a lack of correlation between hidden attractors and equilibria. More-over, the precise localization of hidden attractors seems to be an intractableissue and, to our best knowledge, there is still no general analytical way, butonly numerically with luck, to find hidden attractors (for the particular caseof Chua’s circuit, see [Leonov et al.(2011)]).To understand better the importance of hidden attractors consider thecrash of aircraft YF-22 Boeing in April 1992. The analysis of aircrafts andlaunchers control systems with saturation regards the linear stability andalso hidden oscillations which might occur [Andrievsky et al.(2013)]. Thedifficulties of rigorous analysis and design of nonlinear control systems withsaturation related to this case are presented in [Lauvdal et al.(1997)].Therefore, the identification of hidden attractors in a given system canhelp drive the system along the desired attractor, which can be a self-excitedattractor or a hidden attractor.The fractional order (FO) calculus is as old as the integer-order (IO)one, but its application was exclusive in mathematics. In the latter yearsFO systems where proved to describe, generally more accurately and in com-pact expressions, the behavior of real dynamical systems, compared to theIO models. This happens taking into account the nonlocal characteristic of“infinite memory”, i.e. the evolution of the system depends at each momenton the entire previous history. Basic aspects of the theory of FO can befound in [Oldham & Spanier(2006)]. While the definition of FO derivativefor continuous-time real functions has been formulated by Liouville, Grun-wald, Letnikov, and Riemann, in the late 19th century, the first definition ofa fractional difference operator was proposed in 1974 [Diaz & Olser(1974)].In this paper the FO derivative is considered in the sense of Caputo, be-cause it allows the choosing initial conditions for the IO systems. Oneof the most utilized numerical methods to integrate continuous systems ofFO is the predictor-corrector Adams-Bashforth-Moulton predictor-correctormethod for fractional-order differential equations [Diethelm et al. (2002)].There are many continuous but also discrete chaotic systems of IO, suchas the supply and demand systems which are among the oldest and simplesteconomic discrete models, having complicated dynamics [Lorenz(1993)]. It isproved that the economical models exhibit generally unstable steady statesand also fluctuations if the income distribution varies sufficiently and ifshareholders save more than workers (see e.g. [Miloslav(2001)] and refer-2nces therein).In this paper a continuous economic system is considered, and provednumerically that it presents coexisting attractors for both IO and FO vari-ants. In [Miloslav(2001)] the question of whether using and/or un-using of foreigninvestments can change the qualitative properties is addressed for a growthpath of a proposed dynamical economic system with three endogenous vari-ables and only one non-linear term, modeled by 3-dimensional autonomousdifferential equations, while in [Pribylova(2009)] the system is further an-alyzed via bifurcations routes including supercritical and subcritical Hopfbifurcation, and generalized Hopf bifurcation as well. It is proved that theexisting cycle exhibits period-doubling bifurcation as a route to chaos.The economic system considered in this paper, with a foreign financing,has the following form ˙ x = ax + bx ( c − x ) , ˙ x = d ( x + x ) , ˙ x = ex − f x , (1)where the variables x represents the savings of households, x the GrossDomestic Product (GDP), x the foreign capital inflow, and the variables a the variation of the marginal propensity to savings, b the ratio of cap-italized profit, c the potential GDP, d = 1 /ν , with ν the capital/outputratio, e the capital inflow/savings ratio and f the debt refund/output ratio[Lorenz(1993)]. As explained in [Lorenz(1993)], if x < c , the activities arenot constrained and high profits are derived from the sectors where the mar-kets are not yet saturated. The single non-linearity represents the qualityof a capitalization of the profits. If x > c , then the inflation is possible. Alack of new investment opportunities modifies it by savings.Note that the system can also be viewed as an extension of the van derPols system on the plane ( x , x ), i.e. when economically the system has noforeign investment ( x = 0)˙ x = ax + bx ( c − x ) , ˙ x = dx , (2)which, as a two-dimensional autonomous system, cannot display chaoticbehavior.In this paper one considers the 3-dimensional system (1) with a thebifurcation parameter and b = 0 . c = 1, d = 0 . e = 0 . f = 0 . )˙ x = ax + 0 . x (1 − x ) , ˙ x = 0 . x + x ) , ˙ x = 0 . x − . x . (3)While in [Miloslav(2001)] coexisting stable cycles and equilibria and alsothe existence of a chaotic attractor are studied analytically, in this paper itis shown numerically that for a < f the system presents richer dynamics,including coexisting hidden and self-excited chaotic attractors and stablecycles. This behavior could be extremely interesting from an economicalpoint of view.Because the right hand side of the system (1) is an odd function, thesystem presents symmetries in the bifurcation planes, and also in the phasespace.The equilibria, which are collinear (see Fig. 4), have the following ex-pression X ∗ = (0 , , , X ∗ , = (cid:32) ± (cid:114) bf + aefbe , ± (cid:115) bf + aefbf , ∓ (cid:114) bf + aefbe (cid:33) Note that X ∗ , with the considered 4 decimals, are approximations ofthe exact equilibria X ∗ , .The stability of equilibria and the occurrence of the Hopf bifurcation for a ≥ .
25 are studied in [Miloslav(2001)].In this paper, the bifurcation parameter a is considered, 0 ≤ a ≤ . Lemma 1. [Miloslav(2001), Pribylova(2009)] For a ∈ [0 , . , equilibria X ∗ , , are hyperbolic unstable saddles. Equilibrium X ∗ is attracting focus-saddle and X ∗ , repelling focus-saddle.Proof. The stability is proved numerically by analyzing the eigenvalues ofthe Jacobi matrix which has the following form J = . − . x a − . x x . . . − .
250 0 . Parameter d has been chosen as in [Miloslav(2001)], with six decimals, because round-ing to fewer decimals affects significantly the results.
4) The characteristic equation P ( λ ) = 0, λ ∈ R , where P ( λ ) is the char-acteristic polynomial, for X ∗ , has the following form P ( λ ) = λ − . λ + (0 . − . a ) λ − . a − . . Because, for a > . / . ≈ .
25, the coefficient 0 . − . a < P ( λ ) : + , − , − , − . Therefore, by Descartes rule, there exists a singlereal positive zero of P ( λ ), λ . For a ≤ .
2, the coefficient 0 . − . a > , − , + , − , i.e. three signchanges, which means that the polynomial P ( λ ) has either 3 or 1positive zeros. In order to understand better the characteristics of X ∗ ,in Fig. 1 (a) the three roots of P ( λ ), λ , , , are plotted as functionof a , for a ∈ [0 , λ is positivefor all considered values of a and the other two eigenvalues λ , arecomplex with (cid:60) ( λ , ) <
0. Therefore, X ∗ is spiral saddle of index 1 (orattracting focus saddle). This means that trajectories will leave thisequilibrium along the unstable manifold of dimension 1 (generatedby the real positive eigenvalue λ ), by spiralling, due to the stablemanifold of dimension 2 (generated by (cid:60) ( λ , ) < X ∗ , , the characteristic polynomial is P ( λ ) = λ + 0 . λ + (0 . − . a ) λ + 0 . a + 0 . . The proof of instability of X ∗ , , via Hurwitz stability criterion, canbe found in [Miloslav(2001)]. To verify it numerically, in Fig. 1 (b)the zeros of P ( λ ) are plotted as function of a for a ∈ [0 , a ∈ [0 , X ∗ , are spiralsaddle of index 2 (or repelling focus saddle). Trajectories near X ∗ , are attracted along the stable manifold of dimension 1 (generated bythe real negative eigenvalue λ ) and then are rejected spiraling on theunstable manifold of dimension 2 (due to the positiveness of the realpart of λ , , (cid:60) ( λ , ) > (cid:60) ( λ , ) (cid:54) = 0, a ∈ [0 , . X ∗ , , are hyperbolic.Fig. 1 (c) presents a chaotic attractor for a = 0 . X ∗ and X ∗ , i.e. it is a heteroclinic connection.5 Hidden and self-exited attractors
Hereafter attractors are denoted as follows:Self-Excited Cycles:
SEC ;Self-Excited Chaotic Attractors:
SECH ;Hidden Chaotic Attractors: H ;Hidden Cycles: HC .The numerical method utilized for the integration of the system (3) ofIO is the Matlab variant of RK method, ode − − − . If M LE is zero, then, after tran-sients removed, the trajectory is considered as regular (stable cycle), whileif
M LE > x , for a ∈ [0 , . a , the underlying attractors are sym-metric. Due to the symmetry, the coexistence of attractors features thissystem. Thus, depending on the initial condition x , there exists one oreven two pairs of symmetric attractors (blue-red and black-yellow in thebifurcation diagram). Beside the standard period-doubling cascade whichleads to chaos, the zoomed region in Fig. 2 (b) reveals an interesting co-existing window for a ∈ [ a , a ], with a = 0 . a = 0 . a = a , there exist both exterior and interior crisis.However the most interesting characteristic of the window presented inFig. 2 (b) represents the coexistence of self-excited chaotic attractors withhidden chaotic attractors.Because there exists no general analytical way to find the attractionbasin of hidden attractors of a given nonlinear system, in the case when theconsidered system admits unstable equilibria, an acceptable way is to searchrandomly initial conditions in neighborhoods of all these equilibria. This can6e done either in planar (usually horizontal) sections through each equilib-rium or/and analyzing three-dimensional (usually spherical) neighborhoodscentered on equilibria.For the considered system (3), one considers lattices B , = [ − , × [ − ,
5] of 400 ×
400 points ( x , x ) and zoomed regions, as horizontal sectionsthrough X ∗ (with x = 0) and X ∗ (with x the third coordinate of X ∗ ),respectively, and also spherical neighborhoods of unstable equilibria.Because of the symmetry, the case of the equilibrium X ∗ is similar to X ∗ . Therefore only X ∗ and X ∗ are analyzed. a = 0 . X ∗ (2 . , . , − . SECH , H and SECH , H re-spectively (see Fig. 3 (a) for H and H and Fig. 3 (b) for both pairsof hidden attractors and self-excited attractors). The extensive numericalexperiments revealed that initial conditions within small neighborhoods ofunstable equilibria X ∗ and X ∗ , launch trajectories to self-excited chaoticattractors SECH , (blue and light brown). Initial conditions outside theattraction basins of the self-excited attractors SECH , lead to the othertwo chaotic attractors, H , (red and black), which are hidden chaotic attrac-tors . To verify that, one can explore planar neighborhoods of each equilibriain lattices B , containing X ∗ and defined by x = 0, and B , containing X ∗ and defined by x = − .
928 (Fig. 4 (a)). All points in B , are consideredas initial conditions, to see where the underlying trajectories go. As thenumerical analysis shows, the attraction basins of H and H on B do notcontain (intersect) the equilibrium X ∗ (see the black and red plot in Figs. 4(b), (c)). Similarly, these attraction basins do not intersect the equilibrium X ∗ on B (Figs. 4 (d), (e)). Therefore H , are hidden attractors.Note that X ∗ is on the separatrix between the attraction basins of SECH and SECH and, therefore, every (no matter how small) neigh-borhoods around X ∗ (Fig. 4 (c)) share initial conditions toward SECH or SECH . This situation is typical for this system and also for saddlesgenerally. Therefore, SECH , are self-excited.In addition to planar numerical analysis, a three-dimensional test around X ∗ and X ∗ is done by analyzing initial conditions within a sphere surround-ing these equilibria. For clarity, 50 random points as initial conditions arechosen. Figs. 5 (a), (b) presents the case of equilibrium X ∗ with the neigh-borhood V X ∗ . As can be seen, due to the fact that X ∗ is placed on theseparatrix, the neighborhood V X ∗ shares initial conditions for attractors SECH and SECH .In Figs. 5 (c), (d), within a spherical neighborhood V X ∗ , 50 initialconditions are all leading to SECH .7herefore, because the chaotic attractor H ∗ is not connected with anyequilibria X ∗ and X ∗ , it can be considered hidden.The self-excited attractors SECH , have M LE = 0 . H , , M LE = 0 . a = 0 .
05. Within the window a ∈ [ a , . a = 0 .
05 and X ∗ (2 . , . , − . SEC and SEC (red and black plot, Fig. 6 (a)) andof chaotic attractors SECH and SECH (blue and light-brown).Consider the lattices B (Figs. 6 (b) and (c)) and B (Figs. 6 (d) and(e)) in the planar sections through X ∗ and X ∗ . As for the case a = 0 . X ∗ belongs to a separatrix between attraction basins of SEC and SEC (Fig.6 (c)) and, therefore, SEC , are self-exited attractors . The zoom in Fig. 6(e) shows the fact that for this value of a , X ∗ also belongs to the separatrixof attraction basins of SEC and SECH (red and blue respectyively, Fig. 6(d), (e)), confers the status of self-excited characteristic of these attractors.Even by considering that attraction basins of SECH , do not intersect X ∗ (blue and light-brown in Fig. 6 (c)), they can be hidden, because of theposition of X ∗ , (red in Fig. 6 (e)) SECH , are actually not hidden.Note that with an extremely small perturbation of a , ∆ a = 2 e −
3, fromthe previous case of a = 0 . X ∗ , so that the hidden characteristic vanished.The attractors SEC , have M LE = − . e −
004 and, therefore,
SEC , are stable cycles. For H , , M LE = 0 . >
0, i.e. these attractors arechaotic. a = 0 . D defined for a ∈ [0 . , . options = odeset ( (cid:48) RelT ol (cid:48) , e −
8) is used (the defaultrelative tolerance
RelT ol of ode45 integrator, 0 . T max , the chaotic transients from thestable cycles).For a = 0 . options , the equilibria X ∗ , become X , ( ± . , . , ∓ . self-excited cycles SEC and SEC (red and black in Fig.7(a)) and hidden cycles HC and HC (blue and light-brown). The presumedstability can be deduced from the time series in Fig. 7 (b), while the typeof attractors can be deduced as before from the sections through equilibria X ∗ , the lattice B through X ∗ (Fig. 7 (c)) and B through X ∗ (Fig. 7 (d)).Again, a small perturbation of the bifurcation parameter with only ∆ a =9 e − a = 0 .
05 to a = 0 . X ∗ (compare Fig. 7 (d) with Fig. 6 (d)), sothat the hiddenness property appeared again.For both pairs of cycles, M LE = − . e − (3) Probably the most interesting characteristic of this system is the coexistenceof attractors in the space of the fractional-order systems in the sense thatfor some fractional order q and fixed parameter a , there exist two symmetricpairs of coexisting chaotic and stable cycles attractors.The commensurate FO variant of the system is D q ∗ x = ax + 0 . x (1 − x ) ,D q ∗ x = 0 . x + x ) ,D q ∗ x = 0 . x − . x , (4)where D q ∗ denotes the Caputo differential operator of order q ∈ (0 ,
1) withstarting point 0 (see e.g. [Podlubny(1999), Diethelm et al. (2002)]) D q ∗ = 1Γ( (cid:100) q (cid:101) − q ) (cid:90) t ( t − τ ) (cid:100) q (cid:101)− q − D (cid:100) q (cid:101) x ( τ ) dτ, where (cid:100)·(cid:101) denotes the ceiling function that rounds up to the next integer and D (cid:100) q (cid:101) represents the standard differential operator of order (cid:100) q (cid:101) ∈ N . Due tothe use of Caputo’s operator, the initial conditions can be taken as for theinteger-order case. Lemma 2.
For a ∈ [0 , . , equilibria X ∗ and X ∗ , are unstable.Proof. In order to study the stability of equilibria X ∗ and X ∗ , of the system(4), denote α min = min {| α i |} , for i = 0 , ,
2, where α i are the arguments ofthe eigenvalues. The stability theorem of equilibria of a FO system can bestated in the following practical form9 heorem 1. [Tavazoei & Haeri(2008), Danca(2017)] An equilibrium point X ∗ is asymptotically stable if and only if the instability measure ι = q − α min π is strict negative. If ι ≤ ι = 0 have geometricmultiplicity one, X ∗ is stable. The graphs of ι for all equilibria, with a ∈ [0 , . ι > a = 0 .
05. The bifurcation diagram with respect the frac-tional order q is presented in Fig. 9 (a). As can be seen, at the right ofthe bifurcation diagram, there exists a FO coexisting window where twosymmetric pairs of attractors coexist. Remark 3.
Generally for FO systems, attractors coexisting phenomenonis searched for a fixed fractional order q in the parameter bifurcation space,by searching different initial conditions for some fixed parameter value. Forthis system of FO the coexistence of attractors can be found also in the frac-tional order q space, by searching for initial conditions for a fixed parameterbifurcation. Note that, as for continuous integer-order systems and contrary to FOdifference systems, where the chaos strongness decreases for increasing q (seee.g. [Danca(2020)]), for this system of FO, the chaos strongness increaseswith q tending to 1.Because the inherent time history of the ABM method, a deep numericalanalysis to detect hidden attractors is tedious and time consuming. There-fore, the quality of the obtained attractors is determined by several empiricaltests. q = 0 . X ∗ , are the same as for the IO case, for a = 0 .
05, namely X ∗ (2 . , . , − . self-excited cycle , SEC or SEC (red and black respectively) and one hidden chaotic attractor H or H (blue and light-brown, respectively).Now, the cycles have M LE = 5 e − M LE = 0 . a , the FO system (4) admitsat least one FO coexisting window in the space of fractional order q , with The geometric multiplicity represents the dimension of the eigenspace of correspondingeigenvalues. x Attractor Figure0 .
052 ( ± . , , , ( ± , ± , ± ± . , , , ( ± , ± , ∓ H , SECH , Figs. 3,4 IO .
05 ( ± . , , , ( ± , ± , ± ± . , , , ( ± , ± , ∓ SEC , SECH , Fig. 60 . ± . , , , ( ± , ± , ± ± . , ,
0) ( ± , ± , ∓ SEC , HC , Fig. 7
F O .
05 ( ± , ± , ± ± , ± , ∓ SEC , H , Fig. 9
Table 1: Attractors datatwo symmetric pairs of coexisting attractors. Other potential FO-coexistingwindows, not examined here, are at the bifurcation points, e.q. at about a = 0 . q and parameter a , there could be two different systems. Choosing a suitable level of precision for computations is a real challengingtask. The numerical integrator for (3) in the IO case is the matlab ode45with implicit absolute and relative error tolerances. For the particular caseof a = 0 . options = odeset ( (cid:48) RelT ol (cid:48) , e − T max = 5000 − T max could lead to invalid results.To integrate numerically the system (4) of FO, the ABM method [Diethelm et al. (2002)]is utilized.The algorithms used to determine the finite time MLEs are the Benettin-Wolf algorithm for the IO case, and the algorithm presented in [Danca & Kuznetsov(2018)]for the FO case.The attractors data are presented in Table 1. Beside the initial conditionspresented in Table 1, points ( ± , ± , ±
1) and ( ± , ± , ∓
1) can be used asinitial conditions.Note that, because of the relatively large number of decimals of somesystem coefficients and also due the inherent numerical errors, the 4-digitscoordinates of X ∗ , (as approximations of the exact X ∗ , ) could also be takenas initial conditions, for cases with neighborhoods of X ∗ , .11 onclusion and Discussion In this paper several type of coexisting attractors of an economic systemof IO and FO are presented. Considering one of the several parametersas bifurcation parameter, the bifurcation diagram revealed windows wherehidden and self-excited attractors coexist.First, it is shown numerically that equilibria are unstable for both IO andFO cases, after which the exploration within neighborhoods of these pointsis investigated. The experiments, realized with the matlab routine ode45for the IO case and with the ABM method for the FO case, show that theconsidered attractors fit the definition of hidden or self-excited attractors.The characteristics of chaotic or regular attractors are revealed, aftertransients removed, by phase portraits, time series and maximal Lyapunovexponents.For the bifurcation diagram in Fig. 1, the four used initial conditionsare ( ± , ± , ±
1) and ± , ± , ∓ a close to a = 0 .
05, with relativesmall perturbations of a , of order of 1 e − e − a = 0 . a = 0 . x represents the foreign capital inflow [Miloslav(2001)],the presence of hidden attractors could represent an important phenomenonfrom the economic point of view. In fact under some circumstances suchas initial conditions, the dynamics of the system could cover some (hidden)behavior. Also, counterintuitively, due to the existence of hidden attractors,increasing (but also decreasing) the x variable is not necessarily relatedwith the system economical stability. References [Shilnikov(1965)] Shilnikov, L.P. [1965] “A case of the existence of a de-numerable set of periodic motions,”
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0) respectively;(b) Self-excited chaotic attractors
SECH (blue) and SECH (light brown)for x = ( ∓ . , ,
0) respectively. 19igure 4: Attraction basins of the chaotic attractors of the IO system, for a = 0 . H and H , respectively), while blue and light-brown correspond to self-excited at-tractors ( SECH and SECH , respectively); (a) Three-dimensional view oftwo planar horizontal sections through H and equilibrium X ∗ (plane B , x = 0) and also through equilibrium X ∗ (plane B , x = − . B , = [ − , × [ − , B ; (c)Rectangular zoomed region centered at X ∗ ; (d) View of B ; Rectangularzoomed region centered at X ∗ ; Points x represents initial conditions, whilescrolled arrows show to which attractors initial conditions tend.20igure 5: Three-dimensional spherical neighborhoods of equilibrium X ∗ and X ∗ of the IO system, for a = 0 . V X ∗ of X ∗ ; (b) Zoomof the neighborhood V X ∗ ; (c) Neighborhood V X ∗ of X ∗ ; (d) Zoom of theneighborhood V X ∗ . 21igure 6: Self-excited cycles SEC , and self-excited chaotic attractors SECH , of the IO system, for a = 0 .
05; (a) Three-dimensional phase plot;(b) View of lattice B of the planar section with plane x = 0 containingequilibrium X ∗ ; (c) Zoom of a neighborhood of X ∗ ; (d) View of lattice B of the planar section with plane x = − . X ∗ ;(e) Zoom of a neighborhood of X ∗ ; Points x represents initial conditions.22igure 7: Self-excited cycles SEC , (red and blue plot) and hidden cycles HC , (black and light-brown plot) of the IO system, for a = 0 . B of the planarsection with plane x = 0 containing equilibrium X ∗ and a zoomed regionof X ∗ ; (d) View of lattice B of the planar section with plane x = − . X ∗ and a zoomed region of X ∗ .23igure 8: The instability measure ι of equilibria X and X ∗ , , for the FOsystem with q = 0 . a .24igure 9: The FO system for a = 0 .
05 and q = 0 . q ∈ [0 . , SEC and hiddenchaotic attractor H ; (c) Self-excited stable cycle SEC and hidden chaoticattractor H ; (d)Time series for attractors SEC and H ; (e) Time seriesfor attractors SEC and H2