Advances in Pure Mathematics, 2013, 3, 190203 http://dx.doi.org/10.4236/apm.2013.31A027 Published Online January 2013 (http://www.scirp.org/journal/apm) More Compactification for Differential Systems Harry Gingold, Daniel Solomon Department of Mathematics, West Virginia University, Morgantown, USA Email: gingold@math.wvu.edu, solomon@math.wvu.edu Received October 29, 2012; revised November 30, 2012; accepted December 8, 2012 ABSTRACT This article is a review and promotion of the study of solutions of differential equations in the “neighborhood of infini ty” via a non traditional compactification. We define and compute critical points at infinity of polynomial autonomuos differential systems and develop an explicit formula for the leading asymptotic term of diverging solutions to critical points at infinity. Applications to problems of completeness and incompleteness (the existence and nonexistence re spectively of global solutions) of dynamical systems are provided. In particular a quadratic competing species model and the Lorentz equations are being used as arenas where our technique is applied. The study is also relevant to the Painlevé property and to questions of integrability of dynamical systems. Keywords: Nonlinear; Polynomial; Compactification; Ultra Extended Euclidean Space; Critical Point; Equilibrium Point; Critical Point at Infinity; Critical Direction at Infinity; Basin of Divergence; Basin of Convergence; Ideal Solutions; Asymptotic; Stability; Global; Globally Asymptotically Stable; Jacobian; Painleve Analysis, Competing Species; Model; Lorenz Equations; Periodic Surface; Differential Geometry; Attractor; Repeller 1. Introduction The projection of the real line on a circle is a form of compactification that was known to Greek mathematic cians before the commeon era. In 1881 Poincaré [1], stu died limit cycles “at infinity” of two dimensional poly nomial differential equations via compactification. Al though the paper contained errors that were addressed more than a hundred years later by Roeder [2], the origi nnal ideas had lasting impact. An early study of differ ential equations via compactification was carried out by Bendixson [3]; see Andronov et al. ([4], p. 216). Ben dixson used the stereographic projection that does not account for all directions at infinity. See e.g. Ahlfors [5] and Hille [6], for versions of the stereographic projection. The Poincaré compactification is adopted in various textbooks on differential equations. See also [710]. It is widely used to study critical points at infinity. Compare e.g. with the studies of Chicone and Sotomayor [11], Cima and Llibre [12], Schlomiuk and Vulpe [13], and their references. It is noteworthy that the stereographic projection is obtained by Y. Gingold and H. Gingold [14] as a degenerate limit of a family of compactifications that account for all directions at infinity. However, that com pactification is akin to the Poincaré compactification [1], and possesses radicals that prevent it being a tool for rational approximations. Compactification is an excellent mean to obtain global phase portraits of vector fields of dynamical systems, that include the neigborhood of in finity. [12] is a welcome global analysis and supplement to the analysis of Chen, Guang Qing and Liang, Zhao Jun [15]. This article is dedicated to the review to the exposition and to the promotion of the study of solutions of diffe rential systems and dynamical systems in the “neighbor hood of infinity”. This study promotes and utilizes a cer tain non traditional compactification. The treatment is based on a series of papers published in the span of the years 2004 to 2012. We describe the theoretical back ground necessary to define neighborhoods and critical points at infinity of solutions of differential systems. We develop an explicit formula for the leading asymptotic term of diverging solutions to critical points at infinity. Applications to problems of completeness and incom pleteness of dynamical systems are also brought to the fore. In particular quadratic systems and the Lorentz eq uations are being used as examples where new and old results are obtained. The quadratic system of competing species is utilized as an example of an incomplete system to which a main result, Theorem 14, applies. Theorem 14 is also related to the Painleve property and consequently is related to issues of integrability. The Lorenz system is used as a prototype of a nonlinear quadratic system that is complete for a much larger set of parameters known before. A simple bijection, , , that has a rich geometrical interpretation, plays a major 1 † 1yxx x ,k xy C opyright © 2013 SciRes. APM
H. GINGOLD, D. SOLOMON 191 role in our study. Details are provided when new results are derived or an important point of view is stressed. Otherwise, we defer for details to the original papers. Traditionally, various aspects of an article and comparisons with refer ences are provided in an introduction section. However, we prefer to postpone such comparisons and comments to subsequent sections with the hope that the delay will make the ideas more tangible. We are happy to ac knowledge the influence of the article by Elias and Gin gold [14] on the application of compactification methods to the study of differential systems of equations. We could not find in the text books on discrete dy namical systems the utilization of compactification tech niques. A modest attempt to fill up this large gap is given in H. Gingold [1618]. The order of presentation in this article runs as fol lows. In Section 2 we define what divergence to infinity means in and we extend to a larger set to be called the Ultra Extended . k k k In Section 3 we discuss properties of the compactifica tion , some of its geometrical interpreta tions and the metric induced by it in the Ultra Extended . 1 † 1yxx x k In Section 4 we discuss the new equations resulting from the transformation of an initial value problems of a polynomial differential system fy x under the compactification . 1 † 1yxx In Section 5 we define what a critical point p of fy is, and we prove a theorem that a polynomial differential system has at least one critical point in the Ultra Extended . k In Section 6 we derive an explicit formula for a Jaco bian associated with a critical point of p fy . We obtain an explicit leading asymptotic term of solu tions of fy that diverge to . From this ex plicit formula old and new results follow. p In Section 7 we discuss the ramification of the pre vious sections on a large family of quadratic systems. In Section 8 we show how the compactification tech niques shed new light on the completeness of the cele brated Lorenz system. We also identify an extension of the attractor. The utility of ideal solutions yt is brought to the fore. In Section 9 we study fields of Lorentz like systems near infinity. 2. Divergence in the Ultra Extended k Denote by y a column vector in . Let denote a row vector that is the trans pose of y. In particular let be the trans pose of the zero vector. Let k 00 † 12 ,,, k yyy y † ˆ,,0 † 12 :,,, k yfyfyfy k be a vector field in where y1, 2,,j, k are scalar polynomial functions. Denote by min max ,tt the maximal interval of existence of a solution of a differential equation d d y fy t . We say that yd d y t is a polyno mial differential system of degree L if the vector function y is given by 1, LL01 yfyfyfyfy (1) where j y, 0,1, 2,,jL are homogeneous poly nomials column vectors of degree j and L fy 0 for some . [Note the difference between k y fy and j y]. What does it mean that a sequence of points zn k ,t , n = 1, 2, ··· converges to infinity in k? What does it mean that a continuous vector function minmax di verges in the direction p to infinity in . There are at least two different definitions. yt Ctk Definition 1. We say that the sequence k zn , 0,1, 2,n diverges to infinity if † zn zn . nlim (2) However, This definition is too restrictive for various purposes; e.g., mathematical physics. It blurs the distinc tion between the different directions at infinity. A defini tion that distinguishes between all directions at infinity requires the following. Definition 2. We say that the unbounded sequence ,0n,1,2, pp zn diverges in the direction p to infinity or diverges to † ,1p , if (2) holds and 1 † zn znlimn. (3) zn p The continuous analog of the definition above is given by Definition 3. We say that the vector function min max ,tp yt Ct diverges in the direction p to infin ity, or diverges to and we denote max lim , tt yt p min tt yt ,or limp if we have † max max yt 1 lim lim l, tt tt † max , and imtt yt ty tytytp (4) or † min min yt 1 † lim lim . tt tt min , and limtt yt ty tytytp (5) Definition 1 is compatible with a common one in complex analysis that is associated with the extended Copyright © 2013 SciRes. APM
H. GINGOLD, D. SOLOMON 192 complex plane. Just one geometrical point corresponding to the symbol augments . It is compatible with the compactification that is given by the stereographic projection. See e.g. [5,6]. 2 Definition 4. Denote by the union of and an ideal point k Ek P and call it the extended where k :,: kk. PE IP (6) However, compatible with Definition 3 we need a larger ideal set ID. Analogous to Y. Gingold and H. Gingold [14], we define below an Ultra Extended and produce an induced metric in Section 3. k Definition 5. Denote by the union of and a certain ideal set ID and call it the Ultra extended where k UEk k † :1,,: kkk . Dpppp UEID (7) As seen in the sequel there is good reason to introduce nonlinear transformations that will allow us to reduce the investigation of differential systems with unbounded solutions to the investigation of differential systems with apriori bounded solutions. 3. Compactification and a Metric In preparation to transforming the equation fy 1 ,0 k k y we need a diffeomorphism that will facilitate computa tions and will take the space into a bounded set. We sketch the main ideas. For more details see H. Gin gold [19]. We project the point 1 k ,,yy through the point on the surface (8) 0, ,0,1 12 22 11kk xx x (8) and single out as one of the two points of intersection of the parabolic surface (8) and the straight line connecting and . The determination of Z will be done by the determination of a certain branch of a multi valued function as given below. Then, all the points map onto a 12 1 ,,,, kk Zxx xx 1,,,0 k yy 1,,yy 0, ,0,1 ,0 k y parabolicbowl with coordinate 12 22 11 1 kk xx x , and all the points , map onto the “circle” with 1 11 ,,, k kk xxx † ,1ppp 12 22 11 1 kk xx x . Denote by U the unit ball and by its boundary. U 2 2 : : Ux xx Ux xx † † 1, 1. (9) Denote † ,:ryyyx xxR † . (10) The transformation 1 2 2 11 R yRxr R (11) is shown in [19] to be a bijection from onto the interior of U. It is also a bijection from the ideal set k :IDp † p p1 onto .U The inverse of † 1 y in 1R is defined by the branch x †2 2 ,. 114 114 y xR yy r 2r (12) The compactification (12) induces a metric in k UE in a natural manner. We consider two points y, . Denote their images under the above bijection by Z, k yUE ˆ ˆ respectively. Let denote the Euclidean norm. Define a positive definite function ˆ , yy by ˆ ˆ ,: . yyZ Z (13) Put, †† 1,, or,1, k yyy yppp †† 1 ˆˆ ˆˆˆˆˆ ,, or,1 k yyy yppp † 22 22 ˆ ˆˆˆ ˆ,, ˆ 114 114 ryyy rr , The next theorem borrowed from H. Gingold [19] shows how to make the a complete metric space. Then, divergence of solutions of dynamical systems is dealt with by convergence in the induced metric. k UE Theorem 6. The Ultra Extended is a complete metric space with respect to the chordal metric k ˆ ˆ ,: yyZ Z . It is given by the following. 2 2 2 ˆˆ 1 ˆ ˆˆ ,i ˆ k Myyy yyy ˆ f, , 2 †22 22 † ˆˆˆ ˆˆˆˆ ,12 1 ˆ if, 1,, k yyp yrr ypppy † †† ˆˆ ,21 ˆˆˆˆ if, 1,, 1, Myy pp ypppyppp Proof. See [19] for details. □ See e.g. Willard [20] for topics of compactification in general topolgy. We turn now to a set of new differential equations re sulting from compactifying the differential system d d y y t. 4. Compactifyng a Polynomial Differential System Put Copyright © 2013 SciRes. APM
H. GINGOLD, D. SOLOMON 193 1 22 2 1 22 01 21 22 21 ,1: 11 11 11 L LL LL ffxRRfRx RfxR fx Rf xRfxfx . L (14) Then the following proposition holds. Proposition 7. The compactification (11) takes the differential system fy into the differential system 2 1 22 12,1 d, d11 L RIxx fxR x tRR †2 (15) with †2 2 2 22 2,1 d1 . d11 L xf xR R tRR (16) Moreover, consider and as functions of a new independent variable where t , 22 2 0 d1,12,1 d 0, xRfxR xfxR xx †,x (17) 1 22 0minmax d11 ,0, d L tRRtttt , (18) 2 †2 2 00 d1 2,1 1 d 101. R2 , fx RR Rxx † (19) If then the initial value problems (17)(19), possess unique solutions on † 00 1xx such that . Rx † 21x Furthermore the Equation (18) generates a one to one mapping between the variable on and the variable t on . We have then, min max ,tt , 1 22 0011 d L ttRR (20) 1 22 max 00 1 022 0min 11d 11 d L L ttR R ttRR , . † (21) Furthermore, if 00 , then the initial value prob lems (17)(19), possess unique solutions on 1xx 2 10 such that , such that and such that . † xx 0 tt 1 R Proof. The proof is left as an exercise. Compare with derivations in Elias and Gingold and Gingold and Solo mon [21,22]. □ Also note: Remark 8. The compactified equations above, contain useful information that will become apparent in the se quel. The formulas (21), contains the following qualita tive information. The larger L the smaller 1 2 1 R could become and therefore the smaller may become. max0 0min ,tttt 5. Critical Points of the Compactified Equation The purpose of this section is to discuss a rigorous foot ing to the notion of a critical point of a dynamical system using the proposed compactification. If p limyp y q (22) holds, then p could be a candidate for a critical point of f y at infinity. Thanks to the definitions and the compactification above, we declare to be a critical point of p fy if p is a critical point of (17) as follows. Definition 9. We say that is a critical (equilib p rium) point of d d y fy t at infinity, or that p is a critical direction of fy at infinity, if there exist a unit vector p such that †. LL pf ppf p (23) If in addition we have then † 20pf p †† LL LLL fp ppfp fp pf p and we call p a generic critical point at infinity and we call p a generic direction at infinity. The set of initial points 0 such that the un bounded solutions of the initial value problem k y yfy , 00 ty , satisfies (4) or (5), is called the basin of di vergence of p or the basin of divergence in the p di rection. Notice that by this definition at least one value of 0 p must be included in the basin of convergence of p. This is a natural definition for a critical point at infini ty because of Proposition 10. If (4 ) or (5) holds then p must be a finite critical point of (17). Proof. Notice the identity 22 22 2 12 ,1 1,12,1 RIxx fxR RfxRxfxR x † †. (24) By virtue of (24), the relation † ,0 ,0fppfp p 0 then implies the nonlinear eigenvalue problem (23). Remark 11. Let †0 L pf p then max max 1 † lim lim tt tt tytyty tp Copyright © 2013 SciRes. APM
H. GINGOLD, D. SOLOMON 194 is impossible. Let then †0 L pf p min min 1 † lim lim tt tt tytyty tp is impossible. Hence, for (4) to hold we must have in forward time and we also must have in order for (5) to hold in backward time. †0 L pf p †0 L pf p Also notice: Remark 12. The set of critical points at infinity of a compactified and parametrized equation are not well de fined without a certain normalization that needs to be introduced or is implicitly assumed. In the above treat ment we “naturally” but arbitrarily defined a parametri zation (17). This determination causes the remaining Equations (17) and (19) to be uniquely determined. However, one may introduce spurious critical points as follows. Consider 22 †2 d1,1 d 2,1 xgxR fxR s xf xRx 1. (25) with 22 ,0 ,0 1 1 ,, j kk m jj jj j j gxx xmx Then the equation pertaining to t would be 1 22 d11 d L tgxRR . (26) The point 1,0 2,0,0 ,,, k xx can be made to be a spu rious critical point at infinity. It is noteworthy that the case differs from the case . For 1k1k1k we have (17) become 22 22 d12,1 d 21,1 . xRIxx fxR Rfx R † (27) Then, are the only two critical points of (17) so that are the only two critical points at infinity of a scalar polynomial differential equation. However, it seems desirable to choose for a different parame trization with 1x 1 1k 1 22 2 22 d 1,2,1 d d11 . d L x gxRfx R s tRR , This will eliminate the common factor of the right hand sides of 22 d2,1 1 d x xR R and 1 22 d11 d L tRR . Then, if , 0L 1x will not be critical points of 2 d2,1 d xfx R s . Must every polynomial differential system possess at least one critical point in the Ultra Extended ? The positive answer is given in: k Proposition 13. A polynomial differential system with L > 0 possesses at least one critical point in the . k UE Proof. If ˆ 0fy for some then we are done. Assume now without loss of generality that there does not exist such that . Consider the relation k y fy k yˆ 0 1 22 2 2 ,1 11 1. L L xRRf Rx Rfy It implies that for 2 10R also 2 ,1 0fxR r some x such . Two possibilities may occur. Ei that ther fo 2 1R0 we have that then the right hand side of ,1 20Rfx 2 d1 d xRI † 2 x 2 ,1fx R (28) vanishes and the result follows. If for 2 ,1 0fx R U then the mapping 1 † 22 :,1 ,1,1wxfxRfxRfxR 2 is a continuous mapping from U into U. By Brower’s fixed point theorem there exists U such that 1 † 22 2 ,1 ,1,1wxf xRf xRf xRx . By the definition of wx we have for all U that †1wxwx and therefore or that 21Rxx † † ˆˆˆ ,0,0,0 .fxfx fx x (29) Substitute (29) in the right hand side of (28) to obtain †† †† dˆˆ 2,0,0 d ˆˆˆ 2,0,0 xfxfxIxxx fx fxxxxx 0 and the result follows. 6. The Explicit Leading Asymptotic Term The purpose of this section is to produce conditions that guarantee the existence of solutions t that satisfy Copyright © 2013 SciRes. APM
H. GINGOLD, D. SOLOMON Copyright © 2013 SciRes. APM 195 , p and p and is such that max limtt t p or min limtt t p and to de 2,as 0Vxp xp termine explicitly the leading asymptotic term in such solutions. (33) Let a, b, c be three k dimensional column vectors. Then, one can easily verify that the following (non asso ciative and non commutative) relations hold Theorem 14. Let be a critical point of the poly nomial differential system p fy and let . †0 L pf p † pfp †† †† .ab cba ccabcba (34) If , then the polynomial differential sys tem (17) possesses at least one parameter family of solu tions 0 L t such that Notice that 1 †1 max max 11 as . L L ytLpfpttp o tt † 2 1 † † 11 2 n i Rxppxp pxpxp p (35) (30) If , then the polynomial differential sys tem (17) possesses at least one parameter family of solu tions †0 L pf p t such that Hence, as p we have 1 †1 min min 11 as . L L ytLpfpttp o tt 2 † † 0 †2 1 12 . L L j j LL j j LL R Cpxpxp xp xp xpxp (36) (31) Proof. We first prove that the differential system (17) is equivalent to †† 1 † d, d :2 2 2 LL L xp AxpV Ipp Jfpfpp pf pI (32) C are of course the binomial coefficients. We now focus on the expansion of into a polynomial that depends on the variable p . Notice that 2, LL L fxfpJfp xp (37) where 2 is a polynomial of degree 2 in the vector variable p . where I denotes the k by k identity matrix, L fp is the Jacobian matrix of L x evaluated at p and V is a polynomial vector function of the vector variables We focus on the term in 1 2 1 1L Rfx . With the help of (35) we have 1 2 11 13 13 1()2 22 LLL LL Rfxpxpxpfpfx fp pxpf pf ppxp 11 , L † † †† where 2 3 p as p. In sum we have where 2 423 :,xp (39) 2 † 14 ,1 2 LLL fx R fpJfpfpp xp , (38) as p and 4 is a polynomial in the variable p . Notice that 22 1212222RIR IpxpxpIIpxpIxp xp †† †† . xxppxppppxpppxp xpxp †††† †† †††† 2 2 122222 22 2222 2 22222. RI xxIpxpIxpxpIppxpppxpxpxp pppxpIxpxpIxpppxpxpx p IpppxpIxpppx pxpxpxp † ††† †††† † ††† ††† ††† †††† † †
H. GINGOLD, D. SOLOMON 196 Hence, 22 2 2 1 2 1 d d12,1 dd 22 22 2 2022 2 22 . LLL LLL LL xp xRIxx fxR IpppxpIxpppx pxp fpJfpfppxpxp L ppfpxpfpxppfppx pfp Ipp Jfpfpp xpxp † ††† †† † ††† †† †† We analyze the above formula. Observe that by virtue of being a critical point p ††† LL Ippfppfp p ppp 0 Moreover, by virtue of (34) we have ††† 22 LL pxp fppxpf p 0 † †† † 22 22 LL LL pxp f pxppf p pxpf ppf pxp † † † . and the formula (32) follows. Next we observe that is a left eigenvector of A. † p ††† 1 †† †† 22 2 2 LL L L pApIppJfpfp p pf p pI pfp p because ††††††† 22 2pIpppppp pp 0 Thus, is an eigenvalue of A. Assume † 2L pf p that . Then, it is well known, see Hartman [25] that the differential system (17) possesses at least a one parameter family of solutions such that †0 L pf p 0 p is small and, such that and such that † 00 1xx lim p . Furthermore, because of the negative eigenvalue we have for every fixed p † 2L pf † 02 L pf p † exp2,as L xp pfp and consequently ††† ,exp2 as LL xf xvpfppfp, and 2? 1exp2 ,as L Rpfp The formula . d 1 22 max 0011 L ttR R guarantees that is finite because max t † max0 0exp2d . L tt pfp Our next aim is to determine the leading term of 2 1R as . Put in max tt 2 1vR †2 2 2 22 † 2 00 2,1 d1 , d11 101 0 L xf xR R tRR Rxx . Observe then that the numerator and denominator in † 2 ††† 2 2, d d2 2, 2 L LL L xf xv v tvv pf pxfxvpf p vv (40) preserve sign and establish a one to one correspondence among the variables ,,tv . Notice that max 0tt v . Therefore, there exist sets ,t 0max , 0 0,v and 0, such that for 0max ,tt , 0 0,vv and 0, vt the integration of (40) yields max 2 02d2 1 LL t vvvpfpot †d . or 111 max 21 21 LL L LvLvttpfpo † As max tt or 0v or we obtain from the above 1 max 1 21 max 1 11 LL L L vLttpfp RvLpfpt t . † † Thus we obtain as max tt or or 0v Copyright © 2013 SciRes. APM
H. GINGOLD, D. SOLOMON 197 1 2 1 1 max 1 1, L L ytR x Lpfpttpxp † and the result (30) follows. A similar analysis leads to the desired result if . †0 L pf p Remark 15. Our results are an improvement on Elias and Gingold and Gingold and Solomon [21,22] because; 1) we do not assume that all eigenvalues of A have nega tive real part as in [21], 2) in contrast to [21,22] we ex press explicitly an eigenvalue of A in terms of the critical direction p, 3) the coefficient of the leading term in the asymptotic formulas (30) and (31) is explicit and does not contain an unknown constant. Theorem 14 shows how the nonlinearity L of our dynamical system and the critical point determine precisely the asymptotic leading term. We could not find in the following sample of textbooks on differential equations, [4,79,2328], the above explicit asymptotic formulas. Notice that the coef ficient matrix A depends on p 1L y and not only on the highest degree term L y of the vector field y. This is counter intuitive. Corolla ry 16. Let be a critical point of the poly nomial differential system p fy . Let †0 L pfp and let . Then the system 2L fy does not have the Painleve property. Namely, not all of its moving singularities are simple poles. We turn to the completeness issue. Definition 17. A differential system fy is called complete if the solutions to all initial value prob lems 0 ,k yfyyt w exist on . Otherwise the system is called incom plete. , Studies of “completeness'” questions in nonlinear dy namical systems include [11,29,30] and references therein. The fact that compactification is central to un derstanding completeness as well as incompleteness is seen from the following theorem. We cannot see how the Gronwall lemma can be used to prove incompleteness. Theorem 18. Given the polynomial system fy 0p, let be a critical point such that L p† pf Then, fy is incomplete. A necessary condition for a polynomial dynamical system to be complete is that the real part of the eigenvalues of the Jacobians about all critical points be purely imaginary. p Proof. Use Theorem 14. Remark 19. It appears that in spite of a voluminous li terature on dynamical systems, at least two important outstanding questions remain unresolved. What is the actual interval of existence of their solutions as a func tion of the initial conditions? For which range of the pa rameters can we assert that fy possesses solu tions that exist on the semi infinite interval 0,t ? The analysis of non isolated critical points also needs more illumination. Observe: Remark 20. A leading asymptotic term of singular so lutions may be written in the form where Vt Vt is an unbounded vector function at a singular point say 0. A powerful technique of asymptotic analysis assumes a form t 0 VtSt t where the power is ob tained first by so called “balancing”. It is only afterwards that a constant vector S is determined. S is to be derived as a solution of a nonlinear system of algebraic equations that could be difficult to solve and is yet to become ex plicit. This technique of asymptotic analysis was ex tended refined and applied by various authors. Compare e.g. with [28,3147]. Applications of this technique to partial differential equations may be found in e.g. Ablo witz and Segur, [31]. This article pursues a different or der of operations in the determination of the leading term Vt. Compatification coupled with the identification of equilibrium points of a dynamical system helps first de termine the constant vector S from a definite explicit system of nonlinear algebraic equations. Namely, L pfp , †L pf p , and †1pp †L p pf S. p 7. An Application to the Competing Species Model The competing species model f y d d y y t is a poly nomial differential system of degree 2 where the vector function y is given by 12 , yfyfy (41) where ,1,2 j fyj, are the following homogeneous polynomials column vectors of degree j. 2 112 2 2 ,, cyy y 1 12 212 12 d. d by ay fy fy gy my yq yfy fy t (42) In this section I denotes the 2 by 2 identity matrix. The competing species model has attracted much at tention. Coppel, [48], attests to the large number of qua dratic differential systems that model various natural phenomenon, from fluid mechanics to stellar constella tions. They share similar features with the competing species model. Compare e.g. with [7,38,44,48,49]. For a partial glimpse into the immense literature on quadratic systems see Artes et al., Dumortier et al., Hua et al., Ince, Rein [4852]. Copyright © 2013 SciRes. APM
H. GINGOLD, D. SOLOMON 198 The reader should have no difficulty recognizing (43) below as a special case of formula (30) with 2L . The theorem below is part of a detailed analysis that can be found in H. Gingold [53]. The theorem reads Theorem 21. Given (42). Assume that a, b, c, g, m, q . Then, 1) With one exception all critical points are generic, namely, ; 2) The basin of divergence of every generic critical point , contains at least a one dimensional manifold; 3) A solution of p † 20pf pp fy , diverges to where is generic if and only if p p max † 2max as . p ytt t pfp tt (43) 4) In the exceptional case where , the real valued solutions may exist on the entire real line or may possess singularities with an asymptotic leading term similar to the leading asymptotic term in a Laurent series expansion s with a pole of order one or two. 0mcqb It is interesting to compare the results obtained by Hille, [39,40], where psi series representations for solu tions of (42) are obtained for a special range of the para meters. Naturally, these psi series provide explicitly the desired leading asymptotic terms of singular solutions of (42). However, our approach covers numerous cases where the results in [39,40] do not apply. A detailed analysis of the competing species model of complex valued solutions of x being the independent complex variable, was undertaken by Garnier, [38]. It is not impossible to derive the leading asymptotic terms of singular solutions by the methods presented in [38]. However, this would entails the extraction of the leading asymptotic term of singular solutions of (42) from a my riad of transformations. Another indirect method that could lead to (43) requires the reduction of (42) to a cer tain pair of second order differential equations satisfied by each component of the vector y. The techniques of Bureau, [33,34], may be then applied. It is noteworthy that a more general quadratic system than (42) is not amenable to the results of [34]. This is so because then each component of y could satisfy a second order diffe rential equation where ,,uhuux ,,huux is not a rational function of u and . The references men tioned in this paragraph are part of a voluminous litera ture that deals with an outstanding question that origi nated with Fuchs, (1884). It stimulated a large amount of work on nonlinear differential equations of the form , where u ,,huux ,,uhuux is a scalar rational function of u and that possesses coefficients that are analytic functions of the independent variable x. The out standing question is: which equations of the form possess solutions that have fixed singu larities at certain fixed values of x. Thus, mimicking a property of linear non autonomous differential equations. (These singularities also called by a large school of au thors critical points and are not to be confused with the critical points of dynamical systems that are synonymous with equilibrium points of dynamical systems). Other related works include [3436,43]. A detailed account that lead to the Painleve transcendentals can be found in [26]. Applications to soliton theory may be found in [31]. It is noteworthy that a successful application of the technique in [32] that pursues “closed form” solutions of (42), re quires knowledge of the properties of one non constant vector solution of (42). u ,,uhuux 8. The Lorenz Completeness, a New Repeller, and an Extension of the Attractor In this section we discuss a result whose corollary shows that the Lorenz system is complete for all its real parame ters. This completeness property is shared by a larger fam ily of nonautonomous quadratic systems that is denoted below by . Then we show that the Lorenz system has a repeller at , a corollary of which is the existence of an attractor for the Lorenz system for 0 . By a Lorenz system [54] we mean a system satisfying 121 2121 3312 , yyy yyyy yyyy 3 y (44) with 0 , 0 , 0 . Note that most authors deal only with Lorenz systems with positive parameters, in which realm there is a global attractor. The existence of an attractor for is a corollary of our first result. Definition 22. Let CB d boun be the family of scalar functions continuous anded on . Let 2, ty be a column vector in k whose compents a dratic forms: onre qua † 22 ,n n ftyf ty, with each y 2n t a lower trianes in gular ma entritrix with . LetCB 11 ,, tyf ty where 1 t is a kk with matrix entries in 0 CB , and let 0 ft column vec tor in k tries in be a with en CB . Then (Non Autonomus Lorenzlike) is the class of syste oms † ,,,with,yfty fty ftyfty 210 2 0. (45) The completeness of descr is given in [22], in which cludes a more detailed iption of the structure of (autonomous Lorenzlike systems) that could explain orthogonality property in (45) as a source of the completeness. Obviously, the Lorenz system is in the for all real values of its parameters. Theorem 23. All systems in are complete. al va It is shown in [21] that given initial data, the initi lue problems (17)(19) possess unique solutions on such that 1x . In particular, it is easy (19) that they sphere 1R is inva riant. Thus we may consider the flow on oundary. to see from boundar the b Copyright © 2013 SciRes. APM
H. GINGOLD, D. SOLOMON 199 Setting 1R in (17) reduces it to † 21312 220, , fxxxx , which t † 1,0,0 and the entire circle 10x. The nonconstan are circles in x2 and x3, with x1 fixed: is readily solved. There ical points a are crit t solutions os2 in2 a a aa 2 2 1c , 1s xa (46) where ˆ 1a, and is related to the starting point 2 a 2 ,1s,1 sinaaco † . Note that the critical poiiting cases of tnts are limhe circles as 1a or 0. For ease of visualization let us orient the ahat x1 “points up”. Then the periodic orbits on the unit sphere may be viewed as circles of constant latitude. Note that the period is xes so t πa so the motion is very slow near the equator, and thquator full of critical points is a limit ing case. If viewed looking down (that is, in along the positive x1axis), orbits in the upper hemisphere rotate counterclockwise, and those in the lower hemisphere rotate clockwise. Since e e ˆ x 1, ˆ y to c doesond under the cofication to not corresp mpacti anthing known in the Lorenz system. However, these orbits could be interpreted to correspond to ideal solutions yt that belong to the ultra ex tended 3 . In facider large t, ons as ,y we restrour attention to the highest order te solve the approximate Lorenz system 0y ict rms and 3 (47) whose solution is easily seen to be (large) circles in y2 (48) where C, C and 1 21 312 , yy y yyy and y3, with y1 constant: 1 21 21 ˆcos , sin C yt CCt CCt 1 2 define the starting point † 12 2 ˆ0,cossin.yCCC, The limits of these cir cles as y do not exist in 3, but they can be understobits in the ideal sD, which bounds 3. Let 22 12 CC be large. Then these periodic vector tions pcertain enigma. They cannot be inter preted as natural approximations to solutions of the Lo renz system on an infinite time interval, because all solu tions must enter a certain ellipsoid in forward time [63]. We choose 1 Cra and od as or ose a et I solu 2 21Cra for 01a . Then as r transcompactification to a circleit sphere with constant first coordi nate. Choosing instead any finite C1 leads to a family of circles, all of which transform to the equator. Similarly, choosing a finite C2 leads to a family of circles which transform to the poles. Definition 24. We say , on the u ˆ y n forms un that a surface in is a peri od der 3 icity surface for the system fy i is the un ion of periodic orbits including critical points, and it is the maximal such object in some neighborhood of itself. The discussion above may be summarized by: f it Proposition 25. The ideal set ID is the preimage of the boundary sphere ,U which is a periodicity surface of the compactified Lorenz system (17). The periodic orbits are circles that are limit cycles when restricted to any of the planes with x1 fixed, 1 01x . Remark 26. There is great Hilinterest inbert’s 16th pr the boundary sphere can be shown to at oblem asking for the number of limit cycles in planar polynomial differential systems [5562]. Poincaré is cre dited with the discovery of limit cycles at infinity of pla nar polynomial systems [1,2], which are not part of the official count of total limit cycles in the original Hilbert’s 16th problem. It is natural now to view the set ID as a periodicity surface of the Lorenz system at infinity and to ask which dynamical systems possess a periodicity sur face at infinity. If the circles on tract nearby orbits (from inside the unit ball) in back wards time , it should be possible to say something about asymptotic behavior (in backwards time t) of the Lorenz equation. This suggests limit cycles at infinity. It is easy enough to show; see e.g., [63], that all trajectories eventually enter a compact set and do not leave it. So it seems plausible that in some sense is a global repel ler. On the other hand, 22 1d 22 12 312 2d yy y yy t takes both positive and negative values even for large . Similarly, if the invariant circles on the boundary ere are to be seen as repelling, we might hope that R decreases along orbits, at least near the boundary sphere. It does not in general since S takes both positive and negative values. However, we h sph ave in [64] proven via a Poincaré map ar gument on the compactified system Theorem 27. The ideal set at infinity D is a global repeller in the following sense: If 11 rt t is large enough, then there exists a 21 tt sucy h that 12 rt rt. . EvRemark 28 en thouy eigenvalue of the Ja co hat th gh ever bian at every critical point on the boundary sphere of the compactified Lorenz system has real part equal to zero, we showed that the sphere repels nearby orbits. Corollary 29. The Lorenz system has an attractor. Proof. In the proof of the theorem, we established t e boundary sphere repels. Thus the boundary sphere is Copyright © 2013 SciRes. APM
H. GINGOLD, D. SOLOMON Copyright © 2013 SciRes. APM 200 the limit set of some neighborhood of itself. The  limit closure of the complement of that neighborhoods an attractor for the compactified system. Perforce that set is compact, and its uncompactification is the attractor for the Lorenz system, extending the known attractor to the case of 0. i Rema . Nrk 30umerous research articles were written on 9. Fields of LorenzLike Systems near the sensitivity of the Lorenz attractor system. It was labeled as a strange attractor. Its geometrical, analytical topological and probablistic nature has been a subject of numerous investigations. The interested reader may want to consult [65,66] and their references. This section contains results that have not been published elsewhere. A main purpose of our analysis is to show how the behavior of the compactified system on the boundary sphere indicates behavior of solutions and their derivatives of the original system for large . This is helpful since the compactified system near R1 is usu ally much simpler than the original system. The asymp totic behavior of the order of growth of the higher derivatives; = dly, 0,1,2,l follows, under appro dl t bonus from er sys te priate conditions, as a a formula that pro vides the asymptotic directions of the derivatives. We first exhibit the relevance of the highestord m 2 fy to the original system through analysis of the mtriple T, N, B of unit tangent, normal, and binormal vectors. We can show that Ty, Ny, By, for tra jectories in 3 and those of the compactified tra jectories Tx, Npproach those of 2 yf as 1R. The continuity properties of the compa differential equation on the compact unit ball then imply that for large oving x, Bx a ctified , y looks a lot like the solutions of 2 fy . In fact, the relevance of the compactified syste to all orders of derivatives. We stress that we do not ex pect such correspondence of vector fields for systems not in . In this section and the next section we denote m extends d d y. t For te strictly second degree polynomial system h 2 fy in 3 , a direct calculation shows that 2 2 2 2† 2222222 22† 2222222 222 2 222 , , , f Tf Df ff Df ff N Df ff Df ff fDff BfDff (49) where we use the notation for the Jacobian of the vector function i . The exions in (49) are given as functions of the riable y; however, it is easy to see that the values of the expressions are not changed if they are expressed in terms of x. Unless otherwise indicated, all subsequent occurrences of i press va and i Df are to be un derstood as i x and D x i f . i Df Theorem 31.t Le ncide wi 3 th . Then the triples T yy, Ny, By order and Tx, Nx, Bx coiT2, N2, B2, plus terms of 2 1R 2 2 Of as Proof. The calculation required t that both Ty an 1R. o show d Tx approach T2, and both Ny and Nx approach N2 as 2 10R is omitted. Assuming it, and since B = T × ve that By and Bx approach B2. We develop an interesting relationship bet N, we also ha ween tde rivatives of y for large and derivatives of x near the boundary sphere, buway from critical points and zeros of higher derivatives. We show that for systems in , the vector fields of higher derivatives of y with re ct to t have the same direction as the corresponding derivatives of x with respect to t a spe for large enough. We stress that this property need not ld for general quadratic systems. Proposition 32. ho Let . Then, for all integers y 0, as ny or 1, we have 2R 1 2 n 22 11d dd1. n d n yt gr nn RRx R (50) Moreover, therder ofowth of the derivative n os dd nn t is given by 1 dd nn yt, 0,1,2,, n n My n (51) where Mn are certain constants. Furthermore, for each n, let Sn be the set of points in the boundary where dd 0. nn x Then 0, there is a neighborhood Un closed un, with , n mU (of Sn in the it ball n mU being the measure of Un), such that for the  com plement of Un and y or 21R we have 2 dd dd1. dd dd nn n yt n nnnn xR yt x (52) Proof. (50) can be shown by induction. The conclu sions in (51) and (52) follow from (50). The key to the induction is the useful result that 1 2 d121 n RnS 2 . d n R (53) From that, we can prove two interesting formulas: First, for 0n, we have 22 2 1 d n xxd 111, dd n nn n nn RR RDx t (54)
H. GINGOLD, D. SOLOMON 201 where Dn is an degree polynomial in th nd d , with rational (in x) continuous coefficients in the unit ball. Second, for , 1n 2 221 22 d1 11 d n nn n n R RR P t ,x (55) where is a polynomial in x. n Px The reason that Lorenzlike systems distinguish them selves from other nonlinear systems so that the vector fields d d l l and d d l l y t, are asymptotically 0, 1,2,l parallel for large can be traced back to the relation (53) which again is a result of the orthogonality in (45). Recall that orthogonality featured also in the complete ness result. It basically says that in a LorenzLike system † yf y does not grow faster than 2 y as .y We now specialize Theorem 31 to the Lorenz system. For the Lorenz system or any system with propor tional to † 32 0, , x, it is easy to see that ˆ3 22 23 2 ˆˆ 2 22 23 3 0 1, 01 1,0 0 x xx Tx xxx NxB xx x . ˆ (56) Specializing Theorem 31 to Lorenz systems, the triple T, N, B for an orbit of the Lorenz system and of its compactified version approach those of (56) as 21:R Corolla ry 33 . The moving triple T, N, B for the Lorenz system and the compactified version coincide with those of the circle , plus terms of order 2 22 123 1R O xx as . 21R REFERENCES [1] H. Poincare, “Mémoire sur les Courbes Définies par Une Équation Différentielle,” Journal de Mathématiques Pures et Appliquées, Vol. 7, 1881, pp. 375422. [2] R. K. W. Roeder, “On Poincaré’s Fourth and Fifth Exam ples of Limit Cycl es at Infinity ,” Rocky Mountain Journal of Mathematics, Vol. 33, No. 3, 2003, pp. 10571082. doi:10.1216/rmjm/1181069943 [3] I. Bendixson, “Sur les Courbes Définies par des Équa tions Différentielles,” Acta Mathematica, Vol. 24, No. 1, 1901, pp. 188. doi:10.1007/BF02403068 [4] A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. L. Maier, “Qualitative Theory of SecondOrder Dynamic Systems,” Wiley, New York, 1973. [5] L. A. Ahlfors, “Complex Analysis,” McGrawHill, New York, 1979. [6] E. Hille, “Analytic Function Theory,” Chelsea Publishing Company, New York, 1982. [7] D. Jordan and P. Smith, “Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers,” 4th Edition, In: Oxford Texts in Applied and Engineering Mathematics, 2007. [8] S. Lefschetz, “Differential Equations: Geometric The ory,” Dover, New York, 1977. [9] L. Perko, “Differential Equations and Dynamical Sys tems,” 3rd Edition, In: Texts in Applied Mathematics, Vol. 7, SpringerVerlag, Berlin, 2001. [10] G. Sansone and R. Conti, “NonLinear Differential Equa tions,” Pergamon Press, Oxford, 1964. [11] C. Chicone and J. Sotomayor, “On a Class of Complete Polynomial Vector Fields in the Plane,” Journal of Dif ferential Equatio ns, Vol. 61, No. 3, 1986, pp. 398418. doi:10.1016/00220396(86)901130 [12] A. Cima and J. Llibre, “Bounded Polynomial Vector Fields Bounded Polynomial Vector Fields,” Transactions of the American Mathematical Society, Vol. 318, No. 2, 1990, pp. 557579. doi:10.1090/S00029947199009983525 [13] D. Schlomiuk and N. Vulpe, “The Full Study of Planar Quadratic Differential Systems Possessing a Line of Sin gularities at Infinity,” Journal of Dynamics and Differen tial Equations, Vol. 20, No. 4, 2008, pp. 737775. doi:10.1007/s1088400891172 [14] Y. I. Gingold and H. Gingold, “Geometric Properties of a Family of Compa ctifications,” Balkan Journal of Geome try and Its Applications, Vol. 12, No. 1, 2007, pp. 4455. [15] G. Q. Chen and Z. J. Liang, “Affine Classification for the Quadratic Vector Fields without the Critical Points at In finity,” Journal of Mathematical Analysis and Applica tions, Vol. 172, No. 1, 1993, pp. 6272. doi:10.1006/jmaa.1993.1007 [16] H. Gingold, “Compactification Applied to a Discrete Competing System Model,” International Journal of Pure and Applied Mathematics, Vol. 66, No. 3, 2011, pp. 297 320. [17] H. Gingold, “Divergence of Solutions of Polynomials Finite Difference Equations,” Proceedings A of the Royal Society of Edinburgh, Vol. 142, No. 4, 2012, pp. 787804. doi:10.1017/S0308210510000077 [18] H. Gingold, “Compactification and Divergence of Solu tions of Polynomial Finite Difference Systems of Equa tions,” Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applied Algorithms, Vol. 18, No. 3, 2011, pp. 315335. [19] H. Gingold, “Approximation of Unbounded Functions via Compactification,” Journal of Approximation Theory, Vol. 131, No. 2, 2004, pp. 284305. doi:10.1016/j.jat.2004.08.001 [20] S. Willard, “General Topology,” AddisonWesley, Read ing, 1970. Copyright © 2013 SciRes. APM
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