A NEW CHAOTIC SYSTEM AND BEYOND: THE GENERALIZED LORENZ-LIKE SYSTEM

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1 Tutorials and Reviews International Journal of Bifurcation and Chaos, Vol. 14, No. 4) c World Scientific Pulishing Compan A NEW CHAOTIC SYSTEM AND BEYOND: THE GENERALIZED LORENZ-LIKE SYSTEM JINHU LÜ Institute of Sstems Science, Academ of Mathematics and Sstems Science, Chinese Academ of Sciences, Beijing 18, P. R. China jhlu@mail.iss.ac.cn GUANRONG CHEN Department of Electronic Engineering, Cit Universit of Hong Kong, Kowloon, Hong Kong, P. R. China gchen@ee.citu.edu.hk DAIZHAN CHENG Institute of Sstems Science, Academ of Mathematics and Sstems Science, Chinese Academ of Sciences, Beijing 18, P. R. China dcheng@iss.iss.ac.cn Received Januar 6, ; Revised Feruar 1, This article introduces a new chaotic sstem of three-dimensional quadratic autonomous ordinar differential equations, which can displa i) two 1-scroll chaotic attractors simultaneousl, with onl three equiliria, and ii) two -scroll chaotic attractors simultaneousl, with five equiliria. Several issues such as some asic dnamical ehaviors, routes to chaos, ifurcations, periodic windows, and the compound structure of the new chaotic sstem are then investigated, either analticall or numericall. Of particular interest is the fact that this chaotic sstem can generate a comple 4-scroll chaotic attractor or confine two attractors to a -scroll chaotic attractor under the control of a simple constant input. Furthermore, the concept of generalied Loren sstem is etended to a new class of generalied Loren-like sstems in a canonical form. Finall, the important prolems of classification and normal form of three-dimensional quadratic autonomous chaotic sstems are formulated and discussed. Kewords: Multi-scroll chaotic attractor; chaotification; three-dimensional quadratic autonomous sstem; Loren-like sstem; normal form. 1. Introduction Chaos as a ver interesting comple nonlinear phenomenon has een intensivel studied in the last four decades within the science, mathematics and engineering communities. Recentl, chaos has een found to e ver useful and has great potential in man technological disciplines such as in information and computer sciences, power sstems protection, iomedical sstems analsis, flow dnamics and liquid miing, encrption and communications, and so on [Chen & Dong, 1998; Chen, 1999; Lü et al., e; Chen & Lü,, and references therein]. Therefore, academic research on chaotic dnamics has evolved from the traditional trend of analing and 17

2 18 J. Lü et al. understanding chaos [Hao, 1984] to the new direction of controlling and utiliing it [Ott et al., 199; Chen, 199; Chen & Dong, 1998; Chen, 1999; Wang & Chen, ; Lü et al., e; Chen & Lü, ]. In a roader sense, chaos control can e divided into two categories: one is to suppress the chaotic dnamical ehavior when it is harmful [Ott et al., 199; Chen, 199], and the other is to create or enhance chaos when it is desirale known as chaotification or anticontrol of chaos [Chen & Lai, 1998; Chen & Dong, 1998; Wang & Chen, 1999; Wang et al., ; Lü et al., f; Chen & Lü, ; Lü et al., ]. Ver recentl, there has een increasing interest in eploiting chaotic dnamics in engineering applications, where some attention has een focused on effectivel creating chaos via simple phsical sstems such as electronic circuits [Tang et al., 1; Wang & Chen, ] and switching piecewise-linear controllers [Lü et al., f; Lü et al., ]. Chaotification is a ver attractive theoretical suject, which however is quite challenging technicall. This is ecause it involves creating some ver complicated ut well-organied dnamical ehaviors, which usuall also involves ifurcations or fractals. For a given sstem, which ma e linear or nonlinear and originall ma even e stale, the question is how to generate chaos out of it, using a simple and easil implementale controller such as a state or output feedack controller, or using an accessile parameter tuner. During the last few ears, a great deal of effort has een made toward this goal, not onl via computer simulations ut also development of rigorous mathematical theories. In the endeavor of chaotification, purposefull creating discrete chaos has gained great success [Chen & Lai, 1998; Wang & Chen, 1999]. At the same time, chaotification in continuous-time sstems is also developing rapidl. For eample, a simple linear partial state-feedack controller is ale to drive the Loren sstem [Loren, 196] from its nonchaotic state to chaotic state. This engineering design has led to the discover of the chaotic Chen sstem [Chen & Ueta, 1999], which is a dual of the Loren sstem [ Celikovský & Chen,, ], and led to the discover of a transition sstem etween the Loren sstem and the Chen sstem [Lü & Chen, ]. In 196, Loren discovered chaos in a simple sstem of three autonomous ordinar differential equations that has onl two quadratic nonlinearities, in order to descrie the simplified Raleigh Bénard prolem [Festa et al., ]. It is notale that the Loren sstem has seven terms on the right-hand side, two of which are nonlinear and ). In 1976, Rössler found a threedimensional quadratic autonomous chaotic sstem [Rössler, 1976], which also has seven terms on the right-hand side, ut with onl one quadratic nonlinearit ). Oviousl, the Rössler sstem has a simpler algeraic structure as compared to the Loren sstem. It was elieved that the Rössler sstem might e the simplest possile chaotic flow [Loren, 199], where the simplicit refers to the algeraic representation rather than the phsical process descried the equations or the topological structure of the strange attractor. It is therefore interesting to ask whether or not there are three-dimensional autonomous chaotic sstems with fewer than seven terms including onl one or two quadratic nonlinearities? The fact is that Rössler actuall had produced another even simpler chaotic sstem in 1979, which has onl si terms with a single quadratic nonlinearit ) [Rössler, 1979]. Thus, the question ecomes How complicated must a threedimensional autonomous sstem e in order to produce chaos? The well-known Poincaré Bendison theorem shows that chaos does not eist in a two-dimensional continuous-time autonomous sstem or a second-order equation) [Wiggins, 199]. Therefore, a necessar condition for a continuoustime autonomous sstem to e chaotic is to have three variales with at least one nonlinear term. As a side note, it is also known that there is a direct connection etween three-dimensional quadratic chaotic sstems and Lagrangian miing [Schmall et al., 199; Funakoshi, 1]. Lagrangian miing poses some interesting questions aout dnamical sstems; however, since realistic models are mainl eperimental and numerical, this suject is still in its earl involving phase of development. Three-dimensional quadratic autonomous sstems are ver important for studing ifurcations, limit ccles and chaotic flows. Recentl, it is proved [Zhang & Jack, 1997] that three-dimensional dissipative quadratic sstems of ordinar differential equations, with a total of four terms on the righthand side, cannot ehiit chaos. Ver recentl, this result was etended to three-dimensional conservative quadratic sstems [Jack & Zhang, 1999; Yang & Chen, ]. Later, it was known that autonomous chaotic flow could e produced

3 A New Chaotic Sstem and Beond: The Generalied Loren-Like Sstem 19 a three-dimensional quadratic autonomous sstem having five terms on the right-hand side, with at least one quadratic nonlinearit, or having si terms with a single quadratic nonlinearit. Latel, chaotic flow in an algeraicall simplest three-dimensional quadratic autonomous sstem was found using jerk functions [Sprott & Lin, ], which has onl five terms with a single quadratic nonlinearit ). In fact, this sstem is simpler than an others previousl found, regarding oth its jerk representation and its representation as a dnamical sstem. However, it is noticed that the simplicit of a sstem can e measured in various was. Algeraic simplicit of sstem s structure is one wa, and topological simplicit of chaotic attractor is another. Rössler s attractor and most of Sprott s eamples are topologicall simpler than the twoscroll Loren attractor [Sprott, 1994, 1997; Sprott & Lin, ]. In fact, Rössler attractor has a singlefold and structure. Furthermore, its one-scroll structure is the simplest topological structure for a three-dimensional quadratic autonomous chaotic sstem. Thus, it is interesting to ask whether or not there are three-dimensional quadratic autonomous chaotic sstems that can displa attractors with more comple topological structures than the two-scroll Loren attractor. That is, Is the twoscroll Loren attractor the most comple topological structure of this class of chaotic sstems? The answer is no. In fact, the recentl discovered Chen attractor and its associate transition attractor have more comple topological structures than the original Loren attractor [Ueta & Chen, ; Lü & Chen, ]. Nevertheless, these newl found attractors also have two-scrolls ut not more than that. Therefore, in comining these two points of views on simplicit or compleit) of a chaotic sstem, it would e trul interesting to seek for lower-dimensional chaotic sstems that have a simple algeraic sstem structure ut with a comple topological attractor structure. This is not just for theoretical interest; such chaotic sstems would e useful in some engineering applications such as secure communications. In the endeavor of finding three-dimensional quadratic autonomous chaotic sstems, other than luckil encountering chaos in unepected simulations or eperiments, there seems to e two sensile methods: one is Sprott s ehaustive searching via computer programming [Sprott, 1994], and the other is Chen s theoretical approach via chaotification [Chen & Lai, 1998; Wang & Chen, 1999; Chen & Ueta, 1999; Lü & Chen, ]. For nearl 4 ears, one of the classic icons of modern nonlinear dnamics has een the Loren attractor. In, Smale descried eighteen challenging mathematical prolems for the twent-first centur [Smale, ], in which the fourteenth prolem is aout the Loren attractor. In this regard, one concerned prolem has een: Does it reall eist? Onl ver recentl, the Loren attractor was mathematicall confirmed to eist [Tucker, 1999; Steward, ]. Another interesting question regarding chaotic sstems is: How comple can the topological structure of the chaotic attractor, if eists, of a three-dimensional quadratic autonomous sstem e? Here, it is noted that the compleit of the topological structure of a chaotic attractor ma e measured in two aspects: the numer of suattractors and the numer of parts scrolls or wings ) of the attractor. It has een well known that piecewise-linear functions can generate n-scroll attractors in Chua s circuit [Sukens & Vandewalle, 199; Elwakil et al., 1; Elwakil et al., ], and in a circuit with the asolute value as the onl nonlinearit, the can also create a comple n-scroll chaotic attractor [Tang et al., ]. Recentl, Liu and Chen [] found a simple three-dimensional quadratic autonomous chaotic sstem, which can displa a -scroll and also visuall) a 4-scroll attractor. Motivated these works, this article introduces one more simple three-dimensional quadratic autonomous sstem, which can generate two 1-scroll chaotic attractors simultaneousl, or two comple -scroll chaotic attractors simultaneousl. It is elieved that a three-dimensional quadratic autonomous chaotic sstem can have at most two chaotic attractors simultaneousl, and the sstem to e discussed here is one such simple ut interesting chaotic sstem. More precisel, the ojective of this article is to present and further stud a simple, interesting, and et comple three-dimensional quadratic autonomous chaotic sstem, which can displa two 1-scroll attractors simultaneousl or two comple -scroll chaotic attractors simultaneousl. This new chaotic sstem is introduced in Sec.. Section eplains how to find this new chaotic sstem, and Sec. 4 further investigates the dnamical ehaviors of this chaotic sstem, emploing some tools used

4 11 J. Lü et al. in [Ueta & Chen, ; Lü et al., ]. The compound structure of chaotic attractors [Elwakil & Kenned, 1; Lü et al., c, g] is analed for the two -scroll attractor of this new sstem in Sec.. Then, Secs. 6 and 7 eplore the relationship and connecting function for the -scroll attractor. The concept of generalied Loren sstem [ Celikovský & Chen,, ] is then etended, and a class of generalied Loren-like sstems are defined and discussed in Sec. 8. Furthermore, classification and normal form of three-dimensional quadratic autonomous chaotic sstems are studied in Sec. 9. Finall, conclusions and discussions are given in Sec. 1, with a rather detailed list of references provided at the end of the article.. A New Chaotic Sstem Consider the following simple three-dimensional quadratic autonomous sstem, which can displa two chaotic attractors simultaneousl: ẋ = a + c 1) ẏ = a + ż = +, where a,, c are real constants. This sstem is found to e chaotic in a wide parameter range and has man interesting comple dnamical ehaviors. For eample, it is chaotic for the parameters a = 1, = 4, and c < 19., and for a = 1, = 4, c = 18.1, it displas two 1-scroll chaotic attractors as shown in Fig. 1. The Lapunov eponent spectrum of sstem 1) is found to e λ 1 =., λ =, λ = , and the Lapunov dimension is d L =.1 for initial value 1, 1, 1). Oviousl, when a = 1, = 4, c = 18.1, the sstem has onl three equiliria: E 1, E 1 6.,, ), , ) , E 8 1, , 1 ) It is a strange phenomenon to have two coeisting chaotic attractors in a three-dimensional quadratic autonomous chaotic sstem with onl three equiliria. Furthermore, when a = 1, = 4, c =, this sstem can displa two comple -scroll chaotic attractors, as shown in Fig.. According to their geometric locations, these two coeisting attractors are called upper-attractor and lower-attractor here for convenience. For three-dimensional autonomous sstems, Vanĕ cek and Celikovský [1996] gave a classification in terms of a 1 a 1, where a 1, a 1 are the corresponding entries of the linear part of the sstem descried the constant matri A = [a ij ]. According to this condition, the Loren sstem satisfies the Chen sstem satisfies a 1 a 1 >, a 1 a 1 <, and sstem 1) here satisfies a 1 a 1 =. This is similar to the sstem studied in [Lü & Chen, ] and then further discussed in [ Celikovský & Chen, ]. This means that sstem 1) is also a transition sstem that ridges the gap etween the Loren and Chen sstems. However, sstem 1) has similar ut different dnamical ehaviors with the transition sstem found Lü and Chen [], investigated in [Lü et al., ]. B a suitale nonsingular transform, it is assumed without loss of generalit that A = [a ij ] = diag{a,, c}. In this setting, Liu and Chen [] derived another classification condition in terms of a + ac + c for the three-dimensional quadratic autonomous sstems. Using this condition, the 4-scroll sstem discovered in [Liu & Chen, ] satisfies a + ac + c, while sstem 1) here satisfies a + ac + c =. Therefore, sstem 1) is a new and particular sstem that satisfies two different classification conditions as descried aove. Moreover, this sstem has man interesting comple dnamical ehaviors, as will e seen elow.

5 A New Chaotic Sstem and Beond: The Generalied Loren-Like Sstem a) ) Fig. 1. Two coeisting 1-scroll chaotic attractors. a) Initial value,, ) > ), ) initial value,, ) < ). a = 1, = 4, c = 18.1.) a) ) Fig.. Two coeisting -scroll chaotic attractors. a) Upper-attractor, ) lower-attractor. a = 1, = 4, c =.). The Upper-Attractor and Lower-Attractor This section provides a rief discussion of how the idea of chaotification leads to the discover of the sstem 1) that has two coeisting -scroll chaotic attractors. Start with the following general form of a threedimensional quadratic autonomous sstem: ẋ = a + c + d ) ẏ = a + e ż = + f, where a,, c, d, e, f are all real constant parameters. The sstem Jacoian, evaluated at,, ), is given a d d J,, ) = e a e. ) f f To find the equiliria of sstem ), let Clearl, if a ef d dt = d dt = d dt =. 4) a >, A = )df + c ef df a >,

6 11 J. Lü et al. and B = a )df c ef df a, then there eist three equiliria: ) )c S 1,, a S +, A, f ) A+ S +, A, f ) A+ ; if a/ef >, A, and B >, then sstem ) has another set of three equiliria: ) )c S 1,, a S 4, B, f ) B S, B, f ) B ; et if a/ef >, A >, and B >, then sstem ) has five equiliria: S 1, S, S, S 4, S ; ut if a/ef >, A <, and B <, or a/ef, then the sstem has a unique equilirium S 1, where ± = ± a/ef. In order to ensure sstem ) e chaotic, just like the tpical three-dimensional autonomous chaotic sstems such as the Loren sstem, it is required that sstem ) is dissipative, that is, V = ẋ/ )+ ẏ/ ) + ż/ ) = a/)) + = a + a + )/) < ; sstem ) has three or five equiliria, and all these equiliria are unstale. It is noticed that a three-dimensional conservative quadratic autonomous sstem can also displa chaos. Indeed, Sprott [1997] found an eample of three-dimensional conservative autonomous chaotic sstem. Here, onl the dissipative sstem case is discussed. Since a +a+ = a+1/)) + /4) >, one has <. When a/ef > and A, B are not simultaneousl ero, sstem ) has three or five equiliria. We ma choose d = 1, e = 1, f = 1 for the simplicit ecause the constants d, e, f determine the sign of the quadratic items. To have chaotic ehavior, these equiliria cannot e stale, that is, the Jacoian should have at least one unstale eigenvalue when it is evaluated at each of these equiliria. Thus, theoretical analsis and various trial tests reveal two sets of parameters: a <, <, c R, d = 1, e = 1, f = 1 a <, <, c R, d = 1, e = 1, f = 1, which ield the new chaotic sstem 1) and the following sstem: ẋ = a + c + ) ẏ = a ż =. It is noticed that sstems 1) and ) are equivalent under the transformation,, ),, ). In the following, therefore, onl sstem 1) is considered. Furthermore, for simplicit, assume c =, which ields the following sstem: ẋ = a 6) ẏ = a + ż = +, where a, are real constants. This is the sstem to e further analed in the rest of this article. First, Fig. shows the two coeisting -scroll chaotic attractors upper-attractor and lowerattractor of sstem 6). More detailed dnamical ehaviors of sstem 6) will e further investigated in the following sections. 4. Dnamical Behaviors of the New Chaotic Sstem 4.1. Some asic properties Sstem 6) shares various properties with some known three-dimensional quadratic autonomous sstems, such as the Loren sstem. These are listed in the following: Smmetr and invariance First, note the invariance of the sstem under the transforms,, ),, ),,, ),, ), and,, ),, ). That is, sstem 6) is smmetrical aout the three coordinate aes,,, respectivel. Furthermore, these smmetries persist for all values of the sstem parameters. Also, it is clear that the three coordinate aes,, themselves are solution orits of

7 A New Chaotic Sstem and Beond: The Generalied Loren-Like Sstem 11 the sstem. Moreover, the orits on the -ais and -ais tend to the origin, while the orits on the -ais tend to infinit, as t. This chaotic sstem is roust to various small perturations due to its highl smmetric structure Dissipativit and eistence of attractor For sstem 6), it is noticed that V = ẋ + ẏ + ż = a + a + 1 ) + 4 =. So, when <, sstem 6) is dissipative, with an eponential contraction rate: dv dt = a + a + V. That is, a volume element V is contracted the flow into a volume element V e a +a+ a+ t in time t. This means that each volume containing the sstem trajector shrinks to ero as t at an eponential rate, a + a + )/). Therefore, all sstem orits are ultimatel confined to a specific suset of ero volume, and the asmptotic motion settles onto an attractor Diffeomorphism and topological equivalence Theorem 1. Sstem 1) is not diffeomorphic and, further, not topologicall equivalent with an known three-dimensional quadratic autonomous chaotic sstem. Proof. For simplicit of notation and discussion, denote an of the known three-dimensional quadratic autonomous chaotic sstem, e.g. the Loren sstem, ẋ = f), 7) and the new chaotic sstem 1), ẏ = g). 8) If the numers of equiliria for sstems 7) and 8) are not the same, then the are not diffeomorphic. For eample, the Loren sstem has three equiliria, while sstem 1) with two -scroll attractors has five equiliria. So the are not diffeomorphic. If the numers of suattractors for sstems 7) and 8) are not the same, then the are not diffeomorphic. For eample, the Loren sstem has a single -scroll chaotic attractor, while sstem 1) with three equiliria has two coeisting 1-scroll chaotic attractors. So the are not diffeomorphic. In the following, assume that the numers of equiliria and suattractors for sstems 7) and 8) are the same. If sstems 7) and 8) are diffeomorphic, then there would e a diffeomorphism, such that = h), 9) f) = M 1 )gh)), 1) where M) = dh)/d is the Jacoian of h) evaluated at. Let and = h ) e such equiliria and let A ) and B ) denote the corresponding Jacoians. Then, differentiating 1) would ield A ) = M 1 )B )M ). 11) Therefore, the characteristic polnomials for the matrices A ) and B ) should coincide. However, ecause the eigenvalues of the corresponding Jacoians are not equivalent, sstems 7) and 8) are not diffeomorphic. Furthermore, chaotic sstem 1) is not topologicall equivalent with an known three-dimensional quadratic autonomous chaotic sstem. Actuall, if the numers of equiliria or suattractors for sstems 7) and 8) are not the same, then the are not topologicall equivalent. Similarl, we can verif the topological equivalent rather tedious algeraical operation and is omitted here. The proof is thus completed. 4.. Equiliria and ifurcations It is easil verified that if a > then sstem 6) has five equiliria: S 1,, ) ) a, a S, a ) a, a S, a

8 114 J. Lü et al. S 4 a a,, a S a a,, a ) ) If a, then sstem 6) has a unique equilirium, S 1,, ). Pitchfork ifurcation of the null solution at a = or = ) can e oserved, if or a) is fied ut a or ) is varied. Moreover, an two nonero equiliria are smmetric aout one of the aes,,, that is, S and S, S 4 and S are smmetric with respect to the -ais. S and S 4, S and S are smmetric with respect to the -ais. S and S, S and S 4 are smmetric with respect to the -ais. It is also noticed that the origin is alwas a saddle point in the three-dimensional space for an nonero real numers a,. Lineariation of sstem 6) aout the null solution S 1 gives three eigenvalues: λ 1 = a/), λ = a, and λ =. Net, lineariing the sstem aout the nonero equiliria S, S, S 4, S ields the following same characteristic equation: fλ) = λ a + a + λ 4a =. 1) Theorem. Hopf ifurcation will not appear at an equilirium S i i = 1,..., ) of sstem 6). Proof. Since the characteristic equation for equilirium S 1,, ) has three eigenvalues: λ 1 = a/), λ = a, and λ =, one can see that there is no Hopf ifurcation at S 1. For equiliria S, S, S 4, S, the have the same characteristic equation 1), so onl the equilirium S is discussed. If Hopf ifurcation appears at equilirium S, then it ma e assumed that two eros are λ = ±ωi for some real ω, and the sum of the three eros of the cuic polnomial f is λ 1 + λ + λ = a + a + Hence, λ = a + a + )/). However a + a + ) f = 4a =,.. that is, a =. This contradicts with the condition a >. Therefore, there is no Hopf ifurcation at S. This completes the proof of the theorem. Remarks 1) It is quite a strange phenomenon to have no Hopf ifurcation in a three-dimensional quadratic autonomous chaotic sstem. All known three-dimensional quadratic autonomous chaotic sstems have Hopf ifurcations at some of their equiliria, such as in the Loren sstem [Loren, 196], Chen sstem [Chen & Ueta, 1999; Lü et al., g, h], transition sstem [Lü & Chen, ; Lü et al., ], unified chaotic sstem [Lü et al., a]. ) Since the coefficient of λ in Eq. 1) is ero, Eq. 1) does not have two conjugate imaginar roots for an parameters a,. However, the coefficient of λ in the characteristic equation is nonero for most known three-dimensional quadratic autonomous chaotic sstems such as those just mentioned. If sstem 6) is dissipative, then a+ <. Since a >, one has a <, <. Note that the coefficients of the cuic characteristic polnomial are all positive, so that fλ) > for all λ >. It is noted that instailit appears Reλ) > ) onl if there are two comple conjugate eros of f. In fact, Eq. 1) has one negative real root and two comple conjugate roots with positive real part. Consider Eq. 1). Denote A = a + a + >, B =, C = 4a > and assume that λ = A/) + Λ, then, Eq. 1) ecomes where and fλ) = Λ + P Λ + Q, P = A + B = A Q = A 7 AB + C = A 7 + C. This third-order polnomial in Λ can e solved using the Cardan formula, where one ma set

9 A New Chaotic Sstem and Beond: The Generalied Loren-Like Sstem 11 = 4P +7Q = 7C +4A C >. Consequentl, Eq. 1) has a unique real eigenvalue. λ 1 = A + Λ = A Q + 1 1P + 81Q P 18Q + 1, 1) 1P + 81Q along with two comple conjugate eigenvalues, λ, = A 1 18Q + 1 1P Q P + 18Q + 1 1P + 81Q ± i 1 18Q + 1 1P Q P + 18Q + 1, 14) 1P + 81Q when a <, <, λ 1 < and Reλ, ) >. Therefore, the equiliria S i i = 1,..., ) are unstale and, in fact, the are saddle-foci. 4.. Dnamical structure of the new chaotic sstem The dnamical ehaviors of sstem 6) are further investigated means of Poincaré mapping, parameter phase portraits, and calculated Lapunov eponents and power spectra. Figure shows some dnamical ehaviors in the a plane for sstem 6). In this figure, generall there eist three divisions in the plane, that is, limit ccles, a chaotic region, and then limit ccles again. The parameter values that detach these regions are calculated with Lapunov eponents and then analing the resulting phase portraits, Poincaré mapping and power spectra. Region A is a limit ccle region, where all orits converge to some limit ccles. Furthermore, when the initial point,, ) is aove the plane =, that is, for >, all trajectories converge to the upper limit ccle as shown in Fig. 4a). However, when the initial point,, ) is elow the plane =, that is, when <, all orits converge to the down limit ccle displaed in Fig. 4a). Region B has the onset of chaos, and it is a region with period-douling ifurcations. Similarl, if > then the period-douling ifurcations for the upper-attractor will appear, as displaed in Fig. 4); if <, then the period-douling ifurcations for the lower-attractor will appear, as shown in Fig. 4). The forming procedure for the A Limit Ccle B 4 W1 C 6 7 Chaos W W D E 8 W4 9 1 Limit Ccle a Fig.. Dnamical ehaviors of sstem 6).

10 116 J. Lü et al a) ) c) d) e) f) Fig. 4. Phase portraits of the stale attractors. a) =., ) =., c) = 4.48, d) =. 1, 1, 1), e) =. 1, 1, 1), f) =.7 1, 1, 1), g) =.7 1, 1, 1), h) = 6.

11 g) h) Fig. 4. Continued ) a) ) c) d) Fig.. Various projections of the upper-attractor and lower-attractor shown in Fig.. a) upper-attractor), ) lower-attractor), c), d). 117

12 118 J. Lü et al.. Lapunov Eponent Maimum Lapunov eponent a) ) Fig. 6. The Lapunov eponent spectra. a) a = 1, ) a =. upper-attractor and lower-attractor can e clearl seen from Fig. 4). Region C is a chaotic region, where there are three tpical periodic windows denoted W 1, W, W. Figure 4c) shows the tpical periodic orits in the periodic window W. When >, all orits will converge to the upper periodic orit, while if <, then the tend to the down periodic orit. In region C, if > then the sstem displas a chaotic attractor aove the plane =, as shown in Fig. 4d); if < then the sstem shows another chaotic attractor elow the plane =, as displaed in Fig. 4e). Region D is a region with period-douling ifurcations. When >, the period-douling ifurcations for the upper-attractor are shown in Fig. 4f); when <, the period-douling ifurcations for the lower-attractor are displaed in Fig. 4g). From Figs. 4f) and 4g), one can clearl see the forming mechanisms for the upper-attractor and lowerattractor. In addition, there is a periodic window W 4 in Region D a) ) Fig. 7. Poincaré mappings. a) = 1, ) = 4/7), c) = 1/7) 14, d) =.

13 A New Chaotic Sstem and Beond: The Generalied Loren-Like Sstem c) d) Fig. 7. Continued ) Region E is a limit cclic region, where all orits converge to one of the two limit ccles, as shown in Fig. 4h). Moreover, the trajectories converge to the upper limit ccle for >, ut to the lower limit ccle for <. Oviousl, there are two period-douling ifurcation regions, B, D, in the aove-descried diagram, that is, there are two routes to chaos. Especiall, these two routes are tpical of period- Tale 1. A summar of the parameter range for ehaviors of sstem 6) as determined oth theoretical analsis and numerical computation a = 1). For.4 < <, there are limit ccles; For.8.4, there are period-douling ifurcations; For =.4, there is a period- ifurcation point; For =.7, there is a period-4 ifurcation point; For.6 <.8, the sstem has two -scroll chaotic attractors, and one is aove the plane = while the other is elow the plane = ; For , there is a periodic window; For.8.78, there is a periodic window; For , there is a periodic window; For.9 <.6, there are period-douling ifurcations; For , there is a periodic window; For =.8, there is a period-4 ifurcation point; For =.9, there is a period- ifurcation point; For <.9, there are two period-1 limit ccles, and one is aove the plane = whilst the other is elow the plane =. douling ifurcations to chaos. However, the are different from man known three-dimensional quadratic autonomous chaotic sstems such as the Loren sstem. The Loren sstem has two routes to chaos, ut onl one route via the tpical perioddouling ifurcations. Moreover, the coeistence of period-douling ifurcations and intermittenc is oserved in Regions B and D here. Also, there are two limit ccle regions, A, E, in the diagram. Most of known three-dimensional quadratic autonomous chaotic sstems such as the Loren and Chen sstems have onl one limit ccle region together with a sink region. Notice also that in the diagram there are four tpical periodic windows, W 1, W, W, W 4, marked can, ellow, green, and lue, respectivel, in Fig.. These periodic windows pla an important role in the evolution of the dnamical ehaviors. Figure 4 displas the aundant and comple phenomena of sstem 6). It is ver interesting to oserve that the maimum periodic window W is located inside the chaotic regions. In order to investigate detailed dnamical ehaviors for sstem 6), several situations are further analed in the following. Case 1 a = 1). When a = 1, = 4, sstem 6) has five equiliria: S 1,, ), S 1, 4/7), 1/7) 14), S 1, 4/7), 1/7) 14), S 4 1, 4/7), 1/7) 14), S 1, 4/7), 1/7) 14).

14 1 J. Lü et al. The characteristic roots for S 1 are λ 1 = /7, λ = 1, λ = 4, so S 1 is a saddle point. And the characteristic roots for the other equiliria are λ , λ, 1.9 ±.666i, thus the equiliria S, S, S 4, S are all saddle-foci. Figure further displas the projections of the upper-attractor and lower-attractor onto the, and planes, respectivel. Figure 6a) shows the corresponding Lapunov eponents versus the varing parameter. Figure 7 shows the Poincaré mapping on several sections, with several sheets of the attractors visualied. It is clear that some sheets are folded. Tale 1 lists the domains of the parameter, in which sstem 6) has comple dnamical ehaviors. Case a = ). Figure 6) shows the corresponding maimum Lapunov eponents versus the varing parameter. There are eleven periodic windows in the Lapunov eponent spectra: W 1 [ 4.981, 4.979], W [ 4.96, 4.946], W [ 6.84, 6.78], W 4 [ 6.7, 6.], W [ 9., 8.91], W 6 [ 9.44, 9.47], W 7 [ 9.1, 9.9], W 8 [ 9.97, 9.948], W 9 [ 1., 1.48], W 1 [ 1.81, 1.849], W 11 [ 11.41, 11.49]. These periodic windows pla an important role in the evolution of dnamical ehaviors of sstem 6). Other parameter ranges for ehaviors of sstem 6) can e similarl otained, which are however omitted here.. Compound Structure of the Upper-Attractor and Lower-Attractor It was found that the Chen attractor [Chen & Ueta, 1999] and transition attractor [Lü & Chen, ] oth have a compound structure [Lü et al., c, g]. That is, the can e otained merging together two simple attractors after performing a mirror operation. It is therefore interesting to ask if the upper-attractor and lower-attractor here also have a compound structure of two simpler attractors, respectivel. This section provides a positive answer to this question. In order to investigate the compound structures of the upper-attractor and the lower-attractor, a constant control term is added to the first equation of sstem 6), there otaining sstem 1), that is, ẋ = a + u 1) ẏ = a + ż = + where u = c. Assume that a = 1, = 4. When u = 9, one otains the right-attractors of the original upper-attractor and lower-attractor, and their projections on the plane are shown in Figs. 8a) and 8); while when u = 9, one has their mirror images, i.e. the two left-attractors and their projections on the plane, as shown in Figs. 8c) and 8d). Detailed theoretical as well as numerical analsis has confirmed that the upper-attractor and lower-attractor oth have a compound structure otained merging together the two corresponding simple left- and right-attractors via a mirror operation, respectivel. In order to stud the forming mechanisms of the upper-attractor and lower-attractor, and to clarif their topological structures, the dnamical ehaviors of the controlled sstem 1) is further studied here. It turns out to e possile to confine the chaotic dnamics to either one of the upperattractor and lower-attractor, changing the constant controller u, or to form two simple attractors which, when merged together, produces the upperattractor or lower-attractor. B tuning the parameter u, as listed in Tale, one can oserve different dnamical ehaviors of the controlled sstem. Remarks 1) It can e seen from Tale that i) when u is large enough, e.g. u > 1.4, sstem 1) has two period-1 limit ccles, as shown in Fig. 9a); ii) when u decreases graduall, there appear period-douling ifurcations [Fig. 9)] and then the two attractors evolve into onl one half of the original upper-attractor and lower-attractor; iii) when u is relativel small, the two -scroll attractors are ounded and form two partial chaotic attractors, as displaed in Fig. 9d); iv) when u is small enough, two complete -scroll attractors are otained.

15 A New Chaotic Sstem and Beond: The Generalied Loren-Like Sstem a) ) c) d) Fig. 8. The right- and left-forming attractors and their projections, for the upper-attractor and lower-attractor. a and ) u = 9; c and d) u = 9. Tale. A summar of the parameter domains for ehaviors of sstem 1), determined oth theoretical analsis and numerical computation. For u > 1.4, sstem 1) has two period-1 limit ccles [Fig. 9a)]; For 1. u 1.4, sstem 1) has two period- limit ccles; For 19. u < 1., there are period-douling ifurcations [Fig. 9)]; For 8.4 u < 19., sstem 1) has two left-attractors or right-attractors) Fig. 8); For 11.9 < u 14.7, there is a ig periodic window [Fig. 9c)]; For. u < 8.4, sstem 1) has two partial and ounded attractors [Fig. 9d)]; For u <, sstem 1) has two complete -scroll chaotic attractors Fig. ).

16 1 J. Lü et al a) ) c) d) Fig. 9. Phase portraits of sstem 1). a) u =, ) u = 19, c) u = 14.7, d) u = 7. Maimum Lapunov eponent u Fig. 1. The maimum Lapunov eponent spectrum for the controlled sstem 1). a = 1, = 4, initial value 1, 1, 1).) ) From Figs. 9a) 9d), one can see that the two -scroll attractors upper-attractor and lower-attractor oth have a compound structure, composed of two simple attractors respectivel, and each emerges from some simple limit ccles [Fig. 9c)]. Furthermore, the forming attractors of the upper-attractor and lowerattractor are otained simultaneousl tuning the simple controller u. ) There are two ig periodic windows, [11.9, 14.7] and [ 14.7, 11.9], in sstem 1), which pla a ke role in the forming of the two left-attractors or right-attractors. 4) u = ±1.4 are the period- ifurcation points and u = ±1. are the period-4 ifurcation points. ) The controller u can change the topological structures shapes) of the original chaotic attractors, rather than their qualitative

17 A New Chaotic Sstem and Beond: The Generalied Loren-Like Sstem 1 properties. Therefore, the controller is used here to provide a etter view of the topological structures of the sstem is chaotic attractors. According to Tale, one can see that the controlled sstem 1) is chaotic for u 19.. Figure 1 displas the maimum Lapunov eponent spectrum of sstem 1) for u 19.. It can e clearl seen that there are seven periodic windows in Fig. 1: W 1.79,.81), W 7.64, 7.71), W [8.78, 8.91], W 4 [11.1, 11.1], W [11.9, 14.77], W 6 [16.67, 16.78], W 7 [18., 18.4]. These periodic windows pla a ke role in the evolution of the dnamics for the controlled sstem 1). It is known that the numer of sstem equiliria and their stailities are ver important for the emergence of chaos. In the following, consider the equiliria of sstem 1). Theorem i) If u a/)) a, then sstem 1) has three equiliria: ) S 1 u,, a a, S a u a ), a a u ) ) a a, S a u a ), a a u ) ). a ii) If u a/)) a, then sstem 1) has three equiliria: ) S 1 u,, a S 4 a a, + u a ), a a + u ) ) a S a, a + u a ), a a + u a ) ). iii) If u < a/)) a, then sstem 1) has five equiliria: S 1 )/a)u,, ), S, S, S 4, S. Let a = 1, = 4. Then the equiliria of the controlled sstem 1) are otained as follows: When u 4/7) , sstem 1) has three equiliria: S 1 7u ),,, S S 1, u, u, 8 1 1, 7 + u, u ; a) ) Fig. 11. The plane projections of the two chaotic attractors of sstem 1). a) Initial value,, ) > ); ) initial value,, ) < ); a = 1, = 4, u = 18.1).

18 14 J. Lü et al. when u 4/7) 1, the sstem 1) has three equiliria: S 1 7u ),,, S , 7 u, 1 7 u, S 8 1 1, 7 u, u ; when u < 4/7) 1, sstem 1) has five equiliria: S 1, S, S, S 4, S. According to Tale, sstem 1) also has two -scroll chaotic attractors for 4/7) 1 u 19.. However, for u 4/7) 1, sstem 1) has onl three equiliria. This is quite a strange phenomenon in three-dimensional quadratic chaotic sstems. Assume that u = 18.1 > 4/7) 1, then sstem 1) has three equiliria: S 1 6.,, ), S 6.46, 4.789, 7.64), S 6.46, 4.789, 7.64). Figure 11 shows the plane projections of the two chaotic attractors for sstem 1). 6. Connecting the Upper-Attractor and Lower-Attractor As mentioned, sstem 6) has five equiliria: a) ) c) Fig. 1. The 4-scroll chaotic attractor of sstem 16) and its projections on various planes. a) D phase portrait, ) plane projection, c) plane projection, d) plane projection. a = 1, = 4, v =.) d)

19 A New Chaotic Sstem and Beond: The Generalied Loren-Like Sstem 1 S 1, S, S, S 4, S, in which S, S are aove the plane = while S 4, S are elow this plane. Furthermore, there is a close correlation etween the equiliria S 1, S, S and the upper-attractor. Also, there is a close correlation etween the equiliria S 1, S 4, S and the lower-attractor. Especiall, the upper-attractor and lowerattractor are smmetric. It is therefore interesting to ask if there is a simple controller that can connect the upper-attractor and lower-attractor. This section gives a positive answer to this question. In fact, a constant controller works well and can Tale. A summar of the parameter intervals for ehaviors of sstem 16), determined oth theoretical analsis and numerical computation. For u > 6., the trajectories of sstem 16) converge to one of the two points [Fig. 1a)]; For 4.1 u 6., the trajectories of sstem 16) converge to one of the two limit ccles upper-limit ccle and lower-limit ccle [Fig. 1)]; For.1 u < 4.1, the trajectories of sstem 16) converge to a limit ccle [Fig. 1c)]; For.7 u <.1, the sstem 16) has a 4-scroll chaotic attractor Fig. 1); For 11.7 u 1.1, there is a periodic window [Fig. 1d)]; For u <.7, sstem 16) has two -scroll attractors Fig. ) a) ) c) d) Fig. 1. Phase portraits of sstem 16). a) v =, ) v =, c) v = 4, d) v = 1. a = 1, = 4.)

20 16 J. Lü et al. Maimum Lapunov eponent iii) when v is relativel small, the two - scroll attractors upper-attractor and lowerattractor are connected and form a 4-scroll chaotic attractor, as shown in Fig. 1; iv) when v is small enough, sstem 16) has two -scroll attractors. ) There are two ig periodic windows, [11.7, 1.1] and [ 1.1, 11.7], in sstem 16), which pla an important role in forming the 4-scroll chaotic attractor. ) From Tale, one can see that the dnamical ehavior of the controlled sstem 16) is ver complicated. 1 1 v Fig. 14. The maimum Lapunov eponent spectrum for the controlled sstem 16). a = 1, = 4, initial value 1, 1, 1).) connect the upper-attractor and lower-attractor to form a 4-scroll chaotic attractor. In order to connect the upper-attractor and lower-attractor, a simple constant controller v is added to the second equation of sstem 6), giving ẋ = a 16) ẏ = a + + v ż = + Let a = 1, = 4. When v =, sstem 16) displas a 4-scroll chaotic attractor, as shown in Fig. 1. In order to stud the forming mechanism of the 4-scroll chaotic attractor and clarif its topological structure, the dnamical ehaviors of the controlled sstem 16) are further investigated here. B varing the control parameter v, as listed in Tale, one can otain different dnamical ehaviors. Remarks 1) One can see from Tale that i) when v is large enough, e.g. v > 6, the trajectories of sstem 16) converge to one of the two points, as shown in Fig. 1a); ii) when v decreases graduall, there appear two limit ccles upper-limit ccle and lower-limit ccle [Fig. 1)], and then the two limit ccles merge into one limit ccle, as displaed in Fig. 1c); Figure 14 displas the maimum Lapunov eponent spectrum of sstem 16) for u.1. It is clear that there are seven periodic windows in Fig. 14: W 1.1,.), W [11.7, 1.1], W 16.96, 16.99), W 4 [17., 17.7], W [18.79, 18.86], W 6 [19.49, 19.], W 7 [.1,.4]. These periodic windows pla a ke role in the evolution of the comple dnamics for the controlled sstem 16). To find the equiliria of sstem 16), let d/dt = d/dt = d/dt =. Theorem 4 i) If v a a/), then sstem 16) has three equiliria: S 1, v ) a, S S a + v a, a, a ) a + v a a + v a, a, a ) a + v ; a ii) If v a a/), then sstem 16) has

21 A New Chaotic Sstem and Beond: The Generalied Loren-Like Sstem 17 three equiliria:, v ) a, S 1 S 4 S a a v a,, a ) a v a a a v a,, a ) a v ; a iii) If v < a a/), then sstem 16) has five equiliria: S 1, v/a, ), S, S, S 4, S. Assume that a = 1, = 4. The equiliria of the controlled sstem 16) are as follows: when v 4 /7.86, sstem 16) has three equiliria: S 1 S S v ), 1,, v, 4 7, ) v, v, 4 7, ) v ; 7 when v 4 /7, the sstem has three equiliria: S 1 v ), 1,, S v, 4 7, ) v, 7 7 S 4 + v, 4 7, ) v ; 7 when v < 4 /7, the sstem has five equiliria: S 1, S, S, S 4, S. According to Tale, sstem 16) is chaotic for v <.1 <.86. However, for v <.86, the sstem has five equiliria and different dnamical ehaviors, including chaos, limit ccles, and sinks. 7. Controlling in etween the Upper-Attractor and Lower-Attractor As is known now, sstem 6) can displa two -scroll chaotic attractors upper-attractor and lowerattractor. It is therefore interesting to ask what is the relationship etween the upper-attractor and lower-attractor, and if the two -scroll chaotic attractors can e confined to either one of them via a simple constant control? This section provides a positive answer to this question a) ) Fig. 1. The -scroll chaotic attractors of sstem 17). a) m = 1, ) m = 1. a = 1, = 4.)

22 18 J. Lü et al a) ) Fig. 16. Phase portraits of sstem 17). a) m = 1, ) m = 6.4. a = 1, = 4.) To do so, a constant controller m is added to the third equation of sstem 6), [Shilnikov, 199] so as to otain ẋ = a ẏ = a + ż = + + m. 17) When a = 1, = 4, m = 1, sstem 17) displas a -scroll chaotic attractor upper-attractor as shown in Fig. 1a); when m = 1, the sstem shows another -scroll chaotic attractor lowerattractor as displaed in Fig. 1). Therefore, the original upper-attractor and lower-attractor of sstem 6) are confined to onl one, either the upper-attractor or the lower-attractor, via a simple constant control. In fact, sstem 17) estalishes a kind of connection [Lü et al., d] etween the upperattractor and the lower-attractor. When the controller m is large, e.g. m = 1, sstem 17) onl displas the upper-attractor; when the controller m is small, e.g. m = 1, the sstem onl shows the lower-attractor; when m =, the sstem displas oth. In order to stud the forming mechanism of the -scroll chaotic attractor and clarif its topological structure, the dnamical ehaviors of the controlled sstem 17) are further studied here. B changing the control parameter m, as listed in Tale 4, one can otain different dnamical ehaviors of the sstem. Remarks 1) It can e seen from Tale 4 that i) when m is large enough, e.g. m > 1., the trajectories of sstem 17) converge to a point, as shown in Fig. 16a); ii) when m decreases graduall, there appear period-douling ifurcations, as displaed in Fig. 16); iii) when m is relativel small, the upper-attractor and lowerattractor are confined to a chaotic attractor, as shown in Fig. 1; iv) when m is small enough, sstem 17) also has two -scroll attractors. ) For m > 1., the trajectories of sstem 17) converge to a point aove the plane =, while when m < 1., to a point elow the plane =. Similarl, for.1 < m < 6.4, sstem 17) is confined to an upper attractor whilst when 6.4 < m <.1, to a lower attractor. ) From Tale 4, one can see that the dnamical ehaviors of the controlled sstem 17) is relativel simple. Tale 4. A summar of the parameter domains for ehaviors of sstem 17), determined oth theoretical analsis and numerical computation. For m > 1., the trajectories of sstem 17) converge to a point [Fig. 16a)]; For m = 1.1, there are two period- ifurcation points; For 6.4 m 1., there are period-douling ifurcations [Fig. 16)]; For.1 < m < 6.4, sstem 17) is confined to a -scroll attractor Fig. 1); For m.1, sstem 17) has two -scroll chaotic attractors Fig. ).

23 Maimum Lapunov eponent A New Chaotic Sstem and Beond: The Generalied Loren-Like Sstem 19 a m, ) a m, a a m, ) a m, a ; S 4 S m Fig. 17. The maimum Lapunov eponent spectrum for the controlled sstem 17). a = 1, = 4, initial value 1, 1, 1).) Figure 17 shows the maimum Lapunov eponent spectrum of sstem 17) for u 6.4. Oviousl, there is onl one periodic window, W [.8,.4], in Fig. 17. However, this periodic window is ver important in the evolution of the comple dnamics for the controlled sstem 17). Consider the equiliria of sstem 17). Theorem. i) If m a /), then sstem 17 ) has three equiliria: S 1,, m ) S S a + m ) a + m, a a + m, ) a + m, a ; ii) If m a /), then sstem 17 ) has three equiliria: S 1,, m ), iii) If m < a /), then sstem 17 ) has five equiliria: S 1,, m/), S, S, S 4, S. Assume that a = 1, = 4. The equiliria of the controlled sstem 17) are as follows: when m 4 /7 1.89, sstem 17) has three equiliria: S 1 S S,, m ), m, ) m, 1, m, ) m, 1 ; 7 7 when m 4 /7, the sstem has three equiliria: S 1,, m ), 4 S m, ) 7 4 m, 1, 7 7 S m, ) m, 1 ; 7 7 when m < 4 /7, the sstem has five equiliria: S 1, S, S, S 4, S.

24 1 J. Lü et al. According to Tale 4, sstem 17) is chaotic for m 6.4 < Furthermore, for m < 1.89, the sstem has five equiliria with different dnamical ehaviors, including chaos, limit ccles and sinks. 8. The Generalied Loren-Like Sstem The Loren sstem is a ver important model for studing lower-dimensional chaotic sstems, especiall for three-dimensional quadratic autonomous chaotic sstems [Deng, 199]. For eample, Lorenlike sstems and classical dnamical equations with memor forcing were recentl studied in detail [Festa et al., ]. Vanĕ cek and Celikovský [1996] introduced the so-called generalied Loren sstem. More recentl, some similar ut different chaotic sstems were discovered [Chen & Ueta, 1999; Lü & Chen, ; Chen & Lü, ]. In order to further investigate the dnamical ehaviors of these chaotic sstems and the relationships among them, Celikovský and Chen [, ] introduced a generalied Loren canonical form of chaotic sstems and a hperolic-tpe generalied Loren sstem and its canonical form, which cover a ver large class of three-dimensional quadratic autonomous chaotic sstems. Yet, there are still some similar sstems such as the one discussed in [Rucklidge, 199] not elonging to these generalied Loren sstems. For this reason, the concept of generalied Loren sstem is eing further etended here, giving rise to what is called generalied Loren-like sstem in this section. First, recall the concept of generalied Loren sstem: Definition 1. The nonlinear sstem of ordinar differential equations in R of the following form is called a generalied Loren sstem: ) A ẋ = + 1 1, 18) λ 1 where = [ 1 ] T, λ R, and A is a real matri: ) a11 a A = 1, 19) a 1 a with eigenvalues λ 1, λ R such that λ > λ 1 > λ >. ) Moreover, the generalied Loren sstem is said to e nontrivial if it has at least one solution that goes neither to ero nor to infinit or a limit ccle. The following result can e found in [ Celikovský & Chen, ]. Lemma 1. For the nontrivial generalied Loren sstem 18) ), there eists a nonsingular linear change of coordinates, = T, which takes 18) into the following generalied canonical form: λ 1 ż = λ λ 1 + 1, 1, ) 1, 1) 1 τ where = [ 1 ] T and parameter τ 1, ). Note that the first condition in the definition of the geometric Loren model is that the flow should have an equilirium point O,, ). If F denotes the vector field of the flow, then the derivative DF O) should have one positive eigenvalue λ 1 > and two negative eigenvalues λ < λ <. Furthermore, the epanding eigenvalue should dominate the weakest contracting one, i.e. λ 1 > λ. In fact, this is easil understood in view of the familiar Shilnikov s criterion [Shilnikov, 199; Shilnikov et al., 199; Shilnikov, 199; Shilnikov et al., 1]. The eigenvalues condition ) is retained here in etending the concept of generalied Loren sstem: Definition. The nonlinear sstem of ordinar differential equations in R of the following form is called a generalied Loren-like sstem: a 11 a 1 a ẋ= a 1 a a a 1 a a 1 c 11 c 1 c 1 d 11 d 1 d 1 + c 1 c c + d 1 d d, c 1 c c d 1 d d ) where = [ 1 ] T, A = a ij ), B = ij ), C = c ij ) are all real matries, i = c i1, i = d i1, c i = d i i = 1,, ), and A has eigenvalues λ 1, λ, λ R such that λ > λ 1 > λ >. )