REVIEW ARTICLE. ZeraouliaELHADJ,J.C.SPROTT

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1 Front. Phs. China, 2009, 4(1: DOI /s REVIEW ARTICLE ZeraouliaELHADJ,J.C.SPROTT Classification of three-dimensional quadratic diffeomorphisms with constant Jacobian c Higher Education Press and Springer-Verlag 2009 Abstract The 3-D quadratic diffeomorphism is defined as a map with a constant Jacobian. A few such examples are well known. In this paper, all possible forms of the 3-D quadratic diffeomorphisms are determined. Some numerical results are also given and discussed. Kewords 3-D quadratic diffeomorphism, classification, chaos PACS numbers a, Gg Contents 1 Introduction Sufficient conditions for the 3-D quadratic diffeomorphisms Applications in phsics Conclusion 121 References Introduction The most general 3-D quadratic map is given b f (x,, z Zeraoulia ELHADJ 1,J.C.SPROTT 2 1 Department of Mathematics, Universit of Tébessa, 12002, Algeria 2 Department of Phsics, Universit of Wisconsin, Madison, WI 53706, USA zeraoulia@mail.univ-tebessa.dz, zelhadj12@ahoo.fr; sprott@phsics.wisc.edu Received June 14, 2008; accepted August 28, 2008 = a 0 +a 1 x+a 2 +a 3 z+a 4 x 2 +a 5 2 +a 6 z 2 +a 7 x+a 8 xz+a 9 z b 0 +b 1 x+b 2 +b 3 z+b 4 x 2 +b 5 2 +b 6 z 2 +b 7 x+b 8 xz+b 9 z c 0 ++c 2 +c 3 z+c 4 x 2 +c 5 2 +c 6 z 2 +c 7 x+c 8 xz+c 9 z (1 where (a i,b i,c i 0 i 9 R 30 are the bifurcation parameters. Polnomial maps of the form (1 are part of the models of storage ring elements in the thin lens approximation [21]. In this case the one-turn map is difficult to evaluate with both speed and accurac, because a modern storage ring consists of 10 3 to 10 4 elements. The occurrence of hperchaotic and wild-hperbolic attractors in some 3-D quadratic maps of the form (1 [3, 5, 7 11] makes them useful in potential applications [4, 6]. If the map (1 has constant Jacobian, then it is a 3-D quadratic diffeomorphism. Some possible generalizations of the 2-D Hénon map were introduced in [7 11]. Note that the attractors obtained from these selected generalizations are ver similar to the Lorenz and Shimizu-Morioka attractors [1, 2]. Some forms of 3-D quadratic diffeomorphisms are important because the have some relation to Lorenz attractors [1 3, 7], to the stud of unfolding 2-D maps to maps of higher dimension, and to the stud of homoclinic phenomena in dnamical sstems [8, 9], among others. However, some forms of the 3-D quadratic diffeomorphisms are well known [3 5, 7 11], especiall those with a quadratic inverse, where it was shown in Ref. [8] that an 3-D quadratic diffeomorphism with a quadratic inverse and constant Jacobian can be written in a normal form. However, this result does not give an information about the shapes of this tpe of map. Therefore, in this paper we investigate all the possible forms of the 3-D quadratic diffeomorphisms, especiall, those without quadratic inverse, i.e., with an inverse of higher degree.

2 112 Zeraoulia ELHADJ and J. C. SPROTT, Front. Phs. China, 2009, 4(1 The method of analsis is the rigorous computation of the Jacobian of the map (1, and hence we determine sufficient conditions for the 3-D quadratic diffeomorphisms in the sense that their Jacobian is a constant and contains at least one quadratic nonlinearit. Some of the simplest possible cases are presented here and discussed. a 1 +2a 4 x + a 7 + a 8 z a 2 +2a 5 + a 7 x + a 9 z a 3 + a 8 x +2a 6 z + a 9 J (x, = b 1 +2b 4 x + b 7 + b 8 z b 2 +2b 5 + b 7 x + b 9 z b 3 + b 8 x +2b 6 z + b 9 (2 c 1 +2c 4 x + c 7 + c 8 z c 2 +2c 5 + c 7 x + c 9 z c 3 + c 8 x +2c 6 z + c 9 The determinant of the Jacobian matrix (2 of the map (1 is given b det J (x,, z =ψ 1 + Φ 1 (x,, z+φ 2 (x,, z +Φ 3 (x,, z (3 where Φ 1 (x,, z=ξ 1 x + ξ 2 + ξ 3 z + ξ 4 x + ξ 5 xz +ξ 6 z + ξ 7 xz + ξ 8 x 2 Φ 2 (x,, z=ξ ξ 10 z 2 + ξ 11 x 3 + ξ ξ 13 z 3 +(ξ 14 + ξ 18 x 2 (4 Φ 3 (x,, z=ξ 15 2 x +(ξ 16 + ξ 19 zx 2 + ξ 17 z 2 x +ξ 20 z 2 + ξ 21 z 2 and ξ 1 = i=4 i=2 ψ i, ξ 2 = i=7 i=5 ψ i, ξ 3 = i=10 i=8 ψ i ξ 4 = i=15 i=11 ψ i, ξ 5 = i=20 i=16 ψ i, ξ 6 = i=26 i=21 ψ i ξ 7 = 2 i=28 i=27 ψ i, ξ 8 = i=31 i=29 ψ i, ξ 9 = i=34 i=32 ψ i ξ 10 = i=37 i=35 ψ i, ξ 11 = 2ψ 38, ξ 12 =2ψ 39 ξ 13 = 2ψ 40, ξ 14 = 4ψ 41, ξ 15 =2 i=43 i=42 ψ i ξ 16 =4ψ 44, ξ 17 = 2 i=46 i=45 ψ i, ξ 18 = 2ψ 47 ξ 19 =2ψ 48, ξ 20 = 2 i=50 i=49 ψ i, ξ 21 =2 i=52 i=51 ψ i where ψ 1 = a 1 b 2 c 3 a 1 b 3 c 2 a 2 b 1 c 3 + a 2 c 1 b 3 + b 1 a 3 c 2 a 3 b 2 c 1 ψ 2 =2a 2 b 3 c 4 2a 2 b 4 c 3 2a 3 b 2 c 4 +2a 3 c 2 b 4 +2b 2 a 4 c 3 2a 4 b 3 c 2 (5 2 Sufficient conditions for the 3-D quadratic diffeomorphisms In this section, we will find sufficient conditions for the 3-D quadratic diffeomorphisms. Indeed, the Jacobian matrix of the map (1 is given b ψ 7 = a 3 b 2 c 7 +a 3 c 2 b 7 b 2 c 1 a 9 +b 2 c 3 a 7 b 3 c 2 a 7 ψ 8 =2a 1 c 2 b 6 2a 1 b 2 c 6 +2a 2 b 1 c 6 2a 2 c 1 b 6 2b 1 c 2 a 6 +2b 2 c 1 a 6 ψ 9 = a 1 b 3 c 9 a 1 c 3 b 9 a 2 b 3 c 8 +a 2 c 3 b 8 b 1 a 3 c 9 +b 1 c 3 a 9 ψ 10 = a 3 c 1 b 9 a 3 c 2 b 8 b 2 c 3 a 8 c 1 b 3 a 9 +b 3 c 2 a 8 +a 3 b 2 c 8 ψ 11 =4a 3 b 4 c 5 4a 3 b 5 c 4 4a 4 b 3 c 5 +4a 4 c 3 b 5 +4b 3 a 5 c 4 4a 5 b 4 c 3 ψ 12 =2a 1 b 5 c 8 2a 1 c 5 b 8 2b 1 a 5 c 8 +2b 1 c 5 a 8 +2c 1 a 5 b 8 2c 1 b 5 a 8 ψ 13 = 2a 2 b 4 c 9 +2a 2 c 4 b 9 +2b 2 a 4 c 9 2b 2 c 4 a 9 2a 4 c 2 b 9 +2c 2 b 4 a 9 ψ 14 = a 1 b 7 c 9 a 1 c 7 b 9 a 2 b 7 c 8 + a 2 b 8 c 7 b 1 a 7 c 9 +b 1 a 9 c 7 + b 2 a 7 c 8 ψ 15 = b 2 a 8 c 7 +c 1 a 7 b 9 c 1 b 7 a 9 c 2 a 7 b 8 +c 2 a 8 b 7 ψ 16 =4a 2 b 4 c 6 4a 2 c 4 b 6 4b 2 a 4 c 6 +4b 2 a 6 c 4 +4a 4 c 2 b 6 4c 2 b 4 a 6 (7 ψ 17 =2a 1 b 6 c 7 2a 1 b 7 c 6 2b 1 a 6 c 7 +2b 1 a 7 c 6 +2c 1 a 6 b 7 2c 1 a 7 b 6 ψ 18 = 2a 3 b 4 c 9 +2a 3 c 4 b 9 +2a 4 b 3 c 9 2a 4 c 3 b 9 +2b 4 c 3 a 9 2b 3 c 4 a 9 ψ 19 = a 1 b 8 c 9 a 1 b 9 c 8 b 1 a 8 c 9 + b 1 a 9 c 8 + a 3 b 7 c 8 a 3 b 8 c 7 ψ 20 = c 1 a 8 b 9 c 1 a 9 b 8 b 3 a 7 c 8 + b 3 a 8 c 7 + c 3 a 7 b 8 c 3 a 8 b 7 ψ 3 = a 1 b 2 c 8 a 1 b 3 c 7 a 1 c 2 b 8 +a 1 c 3 b 7 a 2 b 1 c 8 +a 2 c 1 b 8 + b 1 a 3 c 7 ψ 4 = b 1 c 2 a 8 b 1 c 3 a 7 a 3 c 1 b 7 b 2 c 1 a 8 +c 1 b 3 a 7 ψ 5 =2a 1 c 3 b 5 2a 1 b 3 c 5 +2b 1 a 3 c 5 2b 1 a 5 c 3 2a 3 c 1 b 5 +2c 1 b 3 a 5 (6 ψ 6 = a 1 b 2 c 9 a 1 c 2 b 9 a 2 b 1 c 9 +a 2 c 1 b 9 +a 2 b 3 c 7 a 2 c 3 b 7 + b 1 c 2 a 9 ψ 21 =4a 1 b 6 c 5 4a 1 b 5 c 6 +4b 1 a 5 c 6 4b 1 a 6 c 5 4c 1 a 5 b 6 ψ 22 =4c 1 a 6 b 5 2a 2 b 6 c 7 +2a 2 b 7 c 6 +2b 2 a 6 c 7 2b 2 a 7 c 6 ψ 23 = 2c 2 a 6 b 7 +2a 5 c 3 b 8 b 2 a 8 c 9 + c 3 b 7 a 9 ψ 24 =2c 2 a 7 b 6 +2a 3 b 5 c 8 2a 3 c 5 b 8 2b 3 a 5 c 8 +2b 3 c 5 a 8

3 Zeraoulia ELHADJ and J. C. SPROTT, Front. Phs. China, 2009, 4(1 113 ψ 25 = 2c 3 b 5 a 8 + a 2 b 8 c 9 a 2 b 9 c 8 a 3 b 7 c 9 +a 3 c 7 b 9 ψ 26 = b 2 a 9 c 8 + b 3 a 7 c 9 b 3 a 9 c 7 + c 2 a 8 b 9 c 2 a 9 b 8 c 3 a 7 b 9 ψ 27 =4a 4 b 6 c 5 4a 4 b 5 c 6 +4a 5 b 4 c 6 4a 5 c 4 b 6 4b 4 a 6 c 5 +4a 6 b 5 c 4 ψ 28 = a 7 b 8 c 9 a 7 b 9 c 8 a 8 b 7 c 9 + a 8 c 7 b 9 + b 7 a 9 c 8 a 9 b 8 c 7 ψ 29 =2a 2 c 4 b 8 2a 2 b 4 c 8 +2a 3 b 4 c 7 2a 3 c 4 b 7 +2b 2 a 4 c 8 2b 2 c 4 a 8 ψ 30 = 2a 4 b 3 c 7 2a 4 c 2 b 8 +2a 4 c 3 b 7 +2b 3 c 4 a 7 +2c 2 b 4 a 8 2b 4 c 3 a 7 (8 ψ 31 = a 1 b 7 c 8 a 1 b 8 c 7 b 1 a 7 c 8 + b 1 a 8 c 7 + c 1 a 7 b 8 c 1 a 8 b 7 ψ 32 =2a 1 c 5 b 9 2a 1 b 5 c 9 +2b 1 a 5 c 9 2b 1 c 5 a 9 +2a 3 b 5 c 7 ψ 33 = 2a 3 c 5 b 7 2c 1 a 5 b 9 +2c 1 b 5 a 9 2b 3 a 5 c 7 +2b 3 a 7 c 5 +2a 5 c 3 b 7 ψ 34 = 2c 3 b 5 a 7 + a 2 b 7 c 9 a 2 c 7 b 9 b 2 a 7 c 9 +b 2 a 9 c 7 + c 2 a 7 b 9 c 2 b 7 a 9 ψ 35 =2a 1 c 6 b 9 2a 1 b 6 c 9 +2a 2 b 6 c 8 2a 2 c 6 b 8 +2b 1 a 6 c 9 2b 1 c 6 a 9 (9 ψ 36 = 2b 2 a 6 c 8 +2b 2 a 8 c 6 2c 1 a 6 b 9 +2c 1 b 6 a 9 +2c 2 a 6 b 8 2c 2 b 6 a 8 ψ 37 = a 3 b 8 c 9 a 3 b 9 c 8 b 3 a 8 c 9 + b 3 a 9 c 8 + c 3 a 8 b 9 c 3 a 9 b 8 ψ 38 = a 4 b 8 c 7 a 4 b 7 c 8 + b 4 a 7 c 8 b 4 a 8 c 7 c 4 a 7 b 8 +c 4 a 8 b 7 ψ 39 = a 5 c 7 b 9 a 5 b 7 c 9 + b 5 a 7 c 9 b 5 a 9 c 7 a 7 c 5 b 9 +c 5 b 7 a 9 ψ 40 = a 6 b 9 c 8 a 6 b 8 c 9 + b 6 a 8 c 9 b 6 a 9 c 8 a 8 c 6 b 9 +c 6 a 9 b 8 ψ 41 = a 4 c 5 b 8 a 4 b 5 c 8 + a 5 b 4 c 8 a 5 c 4 b 8 b 4 c 5 a 8 +b 5 c 4 a 8 ψ 42 =2a 4 b 5 c 9 2a 4 c 5 b 9 2a 5 b 4 c 9 +2a 5 c 4 b 9 +2b 4 c 5 a 9 2b 5 c 4 a 9 ψ 43 = a 5 b 7 c 8 + a 5 b 8 c 7 + b 5 a 7 c 8 b 5 a 8 c 7 a 7 c 5 b 8 + c 5 a 8 b 7 ψ 44 = a 4 b 7 c 6 a 4 b 6 c 7 + b 4 a 6 c 7 b 4 a 7 c 6 a 6 c 4 b 7 +c 4 a 7 b 6 ψ 45 =2a 4 b 6 c 9 2a 4 c 6 b 9 2b 4 a 6 c 9 +2b 4 c 6 a 9 +2a 6 c 4 b 9 2c 4 b 6 a 9 ψ 46 = a 6 b 7 c 8 a 6 b 8 c 7 a 7 b 6 c 8 + a 7 c 6 b 8 + b 6 a 8 c 7 a 8 b 7 c 6 (10 ψ 47 = a 4 c 7 b 9 a 4 b 7 c 9 + b 4 a 7 c 9 b 4 a 9 c 7 c 4 a 7 b 9 +c 4 b 7 a 9 ψ 48 = a 4 b 9 c 8 a 4 b 8 c 9 + b 4 a 8 c 9 b 4 a 9 c 8 c 4 a 8 b 9 +c 4 a 9 b 8 ψ 49 =2a 5 b 7 c 6 2a 5 b 6 c 7 +2a 6 b 5 c 7 2a 6 c 5 b 7 2b 5 a 7 c 6 +2a 7 b 6 c 5 ψ 50 = a 5 b 8 c 9 a 5 b 9 c 8 b 5 a 8 c 9 + b 5 a 9 c 8 + c 5 a 8 b 9 c 5 a 9 b 8 ψ 51 =2a 5 b 6 c 8 2a 5 c 6 b 8 2a 6 b 5 c 8 +2a 6 c 5 b 8 +2b 5 a 8 c 6 2b 6 c 5 a 8 ψ 52 =a 6 b 7 c 9 a 6 c 7 b 9 a 7 b 6 c 9 +a 7 c 6 b 9 +b 6 a 9 c 7 b 7 c 6 a 9 Hence the map (1 is a 3-D quadratic diffeomorphism if and onl if ψ 1 Φ 1 (x,, z =0 (11 Φ 2 (x,, z =0 Φ 3 (x,, z =0 for all (x,, z R 3. This is possible if ξ i = 0,i = 1,, 21, i.e., ψ j (a i,b i,c i 1 i 9 =0, j =2,, 52 (12 Because detj (x,, z is a polnomial function, the onl possible case for a non-vanishing constant determinant is when (11 and (12 hold. We define the following subsets of R 27 : } Ω 1 = {(a i,b i,c i 1 i 9 R 27 : ψ 1 ((a i,b i,c i 1 i 9 } Ω j = {(a i,b i,c i 1 i 9 R 27 :ψ j ((a i,b i,c i 1 i 9 =0 j =2,, 52 (13 Thus if there are vectors (a i,b i,c i 1 i 9 R 27 such that (a i,b i,c i 1 i 9 (0, 0,, 0, i.e., j=52 Ω j, j=1 then det J (x,, z is a non-zero constant for all (x,, z R 3. However, the sstem of equations (13 can be rewritten as ( ψ 1 (a i,b i,c i 1 i 9 (14 AC = O where A = A ((a i,b i,c i 1 i 9 is a 51 9 matrix, C =(c j 1 j 9 is a 9 1 vector of unknowns, and O is the null vector of R 51. The classical method of Fontené- Rouché can be used to solve this sstem of equations b

4 114 Zeraoulia ELHADJ and J. C. SPROTT, Front. Phs. China, 2009, 4(1 introducing the so-called principal determinant. However, this is ver hard to do theoreticall since it is difficult to determine the set of principal unknowns. Thus, we have proved the following theorems: Theorem 1 The map (1 is a 3-D quadratic diffeomorphism if and onl if Ω = j=52 Ω j. j=1 Theorem 2 The set Ω = j=52 Ω j contains at least j=1 one 3-D quadratic diffeomorphism. Proof The well known generalized Hénon map given in Ref. [5] satisfies all the above conditions. Note that the existence of an inverse is guaranteed b the so called real Jacobian conjecture introduced b O.T. Keller in 1939 [13, 14], and also the upper bound for the degree of the inverse of a quadratic map on R n is known to be 2 n 1 [14], in our case the upper bound is 4. On the other hand, it was shown in Ref. [8] that an 3-D quadratic diffeomorphism with a quadratic inverse and constant Jacobian can be written in the following form: g (x,, z = z (15 d 0 +d 1 x+d 2 +d 3 z+d 5 2 +d 6 z 2 +d 9 z where d i are the bifurcation parameters. Therefore, we classif all the 3-D quadratic diffeomorphisms into two classes: those with a quadratic inverse, and those with no quadratic inverse, where we determine exactl all the possible forms of these two families. Indeed, as a test of the previous analsis, and for the sake of simplicit and without loss of generalit, we can assume that a 0,a 1 = a 3 =0 b 0 = b 1 = b 2 =0 c 0 = c 2 = c 3 =0 c 1,a 2,b 3 Then one has the following conditions: (16 a 5 =0, a 7 =0, a 9 =0, b 6 =0, b 8 =0 b 8 =0, c 4 =0, c 7 = c 1b 9, b 3 c 8 =0 a 4 b 9 =0, a 4 c 5 =0, a 4 c 9 =0, a 4 b 7 c 6 =0 a 8 b 5 =0, a 8 c 5 =0, a 8 b 7 =0, a 8 c 9 =0, a 8 b 9 =0 a 6 b 7 =0, a 6 b 9 =0, a 6 b 4 c 9 = 0 (17 c 6 b 8 =0, c 6 b 4 =0 c 1 a 6 b 5 + a 2 b 7 c 6 =0, b 4 c 9 =0 a 4 b 5 c 6 + b 4 a 6 c 5 =0 c 1 b b 3b 7 c 9 =0 Therefore, one of the possible forms of the 3-D quadratic diffeomorphisms is given b + a 4 x 2 + a 6 z 2 + a 8 xz f (x,, z= b 3 z+b 4 x 2 +b 5 2 +b 7 x+b 9 z + c 5 2 c 1b 9 (18 x + c 9 z b 3 with the conditions (16 and (17. More analsis on the conditions (17 show the existence of more than 71 cases. Some of them are listed below, especiall those with one and two nonlinearities. On the other hand, we sa that the maps f and g given respectivel b (15 and (18 are affinel conjugate if there exists an affine transformation h such that g h (x,, z =h f (x,, z, for all (x,, z R 3 (19 As a result if the map (18 is affinel conjugate to the map (15, then the map (18 has quadratic inverse. We use this remark to characterize the 3-D quadratic diffeomorphisms without quadratic inverse. Indeed, the transformation h is defined b h 11 h 12 h 13 x s 1 h (x, = h 21 h 22 h 23 + s 2 (20 h 31 h 32 h 33 z s 3 with the condition of invertibilit given b d=(h 22 h 33 h 23 h 32 h 11 +(h 31 h 23 h 21 h 33 h 12 +(h 21 h 32 h 22 h 31 h 13 (21 Such affine transformation h exists if and onl if E = i=30 i=0 E i (22 where E 0 : d = 3a 2 c 1 b 3 h 11 h 12 h 13 + a 2 2b 3 h c 1 b 2 3h a 2 c 2 1h 3 13 E 1 : s 1 = s 3 a 0 h 11 a 0 c 1 h 13 E 2 : s 2 = s 3 a 0 c 1 h 13 E 3 : a 8 h 11 =0 E 4 : a 6 h 11 =0 E 5 : a 6 h 13 = 0 (23 E 6 : a 8 h 13 =0 E 7 : h 22 = a 2 h 11 E 8 : h 21 = c 1 h 13

5 Zeraoulia ELHADJ and J. C. SPROTT, Front. Phs. China, 2009, 4(1 115 and E 9 : h 23 = b 3 h 12 E 10 : h 32 = c 1 a 2 h 13 E 11 : h 31 = c 1 b 3 h 12 E 12 : h 33 = b 3 a 2 h 11 E 13 : b 9 h 12 + c 9 h 13 =0 E 14 : a 4 h 11 + b 4 h 12 =0 E 15 : b 5 h 12 + c 5 h 13 = 0 (24 E 16 : b 7 h 12 c 1 b 9 h 13 =0 b 3 E 17 : a 2 b 9 h 11 + b 3 c 9 h 12 =0 E 18 : a 2 b 5 h 11 + b 3 c 5 h 12 =0 E 19 : a 2 b 7 h 11 c 1 b 9 h 12 =0 E 20 : a 2 b 4 h 11 + c 1 a 4 h 13 =0 and E 21 : ζ 1 + ζ 2 =0 E 22 : b 9 h 11 b 7 h 13 + b 3 d 9 h 11 h 12 +2d 5 h 11 h 13 +2c 1 b 3 d 6 h 12 h 13 + c 1 d 9 h 2 13 =0 E 23 : c 1 a 6 h 12 + a 2 b 3 d 9 h 11 h 12 + a 2 2 b 3d 6 h b 3 d 5 h 2 12 =0 E 24 : a 4 b 3 h 12 a 2 b 4 h 13 + c 1 b 3 d 9 h 12 h 13 + c 1 b 2 3 d 6h c 1d 5 h 2 13 =0 E 25 : b 3 c 5 h 11 c 1 b 5 h 13 + a 2 c 1 d 9 h 11 h 13 + a 2 d 5 h a 2 c 2 1d 6 h 2 13 =0 E 26 : ζ 3 + ζ 4 =0 E 27 :2d 5 h 12 h 13 a 8 h 12 + a 2 d 9 h 11 h 13 +2a 2 b 3 d 6 h 11 h 12 + b 3 d 9 h 2 12 =0 E 28 : ζ 5 + ζ 6 =0 E 29 : ζ 7 + ζ 8 =0 E 30 : ζ 9 + ζ 10 =0 where ζ 1 = b 3 c 9 h 11 c 1 b 9 h 13 +2b 3 d 5 h 11 h 12 ζ 2 =2a 2 c 1 b 3 d 6 h 11 h 13 + c 1 b 3 d 9 h 12 h 13 +a 2 b 3 d 9 h 2 11 ζ 3 = d 0 +(d 1 + d 2 + d 3 1s 3 a 0 c 1 b 3 h 12 a 0 d 1 h 11 c 1 a 0 (2d 5 + d 9 h 13 s 3 ζ 4 = c 1 a 0 (d 1 + d 2 h 13 +(d 5 + d 6 + d 9 s 2 3 +a 2 0c 2 1d 5 h 2 13 ζ 5 = d 1 h 11 + c 1 d 2 h 13 a 2 c 1 b 3 h 11 + c 1 b 3 d 3 h 12 +c 1 (2d 5 + d 9 h 13 s 3 ζ 6 = b 3 c 1 (2d 6 + d 9 h 12 s 3 a 0 c 2 1b 3 d 9 h 12 h 13 2a 0 c 2 1 d 5h 2 13 (25 ζ 7 = a 0 a 2 c 2 1d 9 h (a 2 c 1 d 3 2a 0 a 2 c 1 d 5 h 11 +s 3 (2a 2 c 1 d 6 + a 2 c 1 d 9 h 13 ζ 8 = d 1 h 12 + a 2 d 2 h 11 a 2 c 1 b 3 h 12 +s 3 (2a 2 d 5 h 11 + a 2 d 9 h 11 ζ 9 = a 2 b 3 d 3 h 11 + b 3 d 2 h 12 +(d 1 a 2 c 1 b 3 h 13 +b 3 (2d 5 + d 9 h 12 s 3 ζ 10 = b 3 a 2 (2d 6 + d 9 h 11 s 3 2a 0 c 1 b 3 d 5 h 12 h 13 a 0 a 2 c 1 b 3 d 9 h 11 h 13 Thus the transformation h takes the form: h 11 h 12 h 13 x h (x, = c 1 h 13 a 2 h 11 b 3 h 12 c 1 b 3 h 12 c 1 a 2 h 13 b 3 a 2 h 11 z s 3 a 0 h 11 a 0 c 1 h 13 + s 3 a 0 c 1 h 13 s 3 We remark that If ((a i,b i,c i 0 i 9 Ē = i=30 i=0 (26 (27 Ē i, where Ē is the compliment of E, then maps (15 and (18 are not affinel conjugate. For example if a 6 or a 8, i.e. ((a i,b i,c i 0 i 9 Ē1 Ē2 then one has that h 11 =0andh 13 =0,thenifb 4 orb 5 or b 7, or b 9, or c 5 orc 9, then h 12 =0, then the transformation h is not invertible. Thus, a characterization of 3-D quadratic diffeomorphisms without quadratic inverse is given in the following Theorem: Theorem 3 If ((a i,b i,c i 0 i 9 Ē = i=30 Ē i, i=0 then the map (18 has no quadratic inverse. On the other hand, and regarding the conditions (17, it is important to note that the 3-D quadratic diffeomorphism can be written with several nonlinearities, for example all maps with one nonlinearit are given b f (x,, z = + a 4 x 2 b 3 z (28 + a 6 z 2 (29 + a 8 xz (30

6 116 Zeraoulia ELHADJ and J. C. SPROTT, Front. Phs. China, 2009, 4(1 + b 4 x 2 (31 + b 5 2 (32 + b 7 x + c 5 2 (33 (34 (35 + c 6 z 2 (36 + c 9 z After some tedious calculations, we have proved the following theorem: Theorem 4 An 3-D quadratic diffeomorphism of the famil (18 with one nonlinearit has a quadratic inverse. Therefore, the dnamics of all the cases from (28 to (36 are conjugate to the dnamics of the map (15. On the other hand, all the 3-D quadratic diffeomorphisms with two nonlinearities are given b + a 4 x 2 + a 6 z 2 (37 + a 4 x 2 + a 8 xz (38 + a 4 x 2 + b 4 x 2 (39 + a 4 x 2 + b 5 2 (40 + a 4 x 2 + b 7 x + a 4 x 2 + c a 4 x 2 + c 6 z 2 + a 4 x 2 + c 9 z + a 6 z 2 + a 8 xz + b 4 x 2 (41 (42 (43 (44 (45 + a 6 z 2 + b 4 x 2 (46 + a 6 z 2 + c a 6 z 2 + c 6 z 2 + a 6 z 2 + c 9 z + a 8 xz + b 4 x 2 + a 8 xz + c 6 z 2 + b 4 x 2 + b 5 2 (47 (48 (49 (50 (51 (52 + b 4 x 2 + b 7 x (53

7 Zeraoulia ELHADJ and J. C. SPROTT, Front. Phs. China, 2009, 4( b 4 x 2 (54 + c b b 7 x + b 5 2 (55 (56 + c b 5 2 (57 + c 6 z 2 + b 5 2 (58 + c 9 z + b 7 x (59 + c 5 2 g (u, t, s = a 2 0 b 5 a 2 2 b 3 + 2a 0b 5 u a 2 2 b 3 and it has no quadratic inverse, and so on. Now we restrict our attention to all the possible cases of the 3-D quadratic diffeomorphisms with one nonlinearit with their sufficient conditions, and for each map we give several examples showing the structure of the paf (x,, z = f (x,, z = s c 1 a 0 a 2 + u a 2 + a 4s 2 a 2 c 2 1 b 3 z + c c 6 z 2 b 3 z + c c 9 z + t b 5u 2 b 3 a 2 2 b 2a 0a 4 b 5 s 2 3 a 2 2 c2 1 b + 2a 4b 5 s 2 u 3 a 2 2 c2 1 b a2 4 b 5s 4 3 a 2 2 c4 1 b 3 (60 (61 Some tedious calculations show that the maps (37 to (39, (48, (52, (53, (55, (56, (60, and (61 belongs to the map (15 (up to the affine conjugac. Hence one has the following theorem: Theorem 5 The 3-D quadratic diffeomorphisms given b (37 to (39, (48, (52, (53, (55, (56, (60, and (61 are conjugate to the map (15. While the remaining maps have no quadratic inverse because the set E = i=30 E i =, and hence the following i=0 theorem is proved: Theorem 6 The 3-D quadratic diffeomorphisms given b (40 to (47 and b (49 to (51, (54, and (57 to (59 have higher degree inverse. For example the inverse of the map (40 is given b: (62 An example of a 3-D quadratic diffeomorphism with three nonlinearities is given b: + a 4 x 2 + a 6 z 2 + a 8 xz f (x,, z= b 3 z (63 and it has a quadratic inverse and thus is conjugate to the map (15. Also, an example of those with four nonlinearities is given b: f (x,, z = + a 6 z 2 b 3 z + c c 6 z 2 + c 9 z (64 rameter space for the particular map with selected values of the vectors (a i,b i,c i 0 i 9 R 30, where regions of unbounded behavior are in white, fixed points in gra, periodic orbits in blue, quasi-periodic orbits in green, and chaotic orbits in red. In this case we use LE < as the criterion for quasi-periodic orbits with 10 6 iterations for each point. The choice of one nonlinearit is robust due to the simplicit of the resulting maps. The case with several nonlinearities can be studied using the same logic and the conditions (14. For the class of one nonlinearit, we have 5 different cases: The first case is given b a 0 + a 1 x + a 2 + a 3 z + a 4 x 2 f (x,, z= b 0 + b 1 x + b 2 + b 3 z c c 2 + c 3 z b 3 c 2 b 2 c 3 =0 ( ψ 1 (a i,b i,c i 1 i 9 (65

8 118 Zeraoulia ELHADJ and J. C. SPROTT, Front. Phs. China, 2009, 4(1 The simplest case of the map (65 is 1+a 3 z + a 4 x 2 f (x,, z = x (66 Fig. 1 Regions of dnamical behaviors in the a 4 a 3 -plane for the map (66. This is the well known hperchaotic generalized Hénon map studied in Ref. [5], and its chaotic attractor is ver similar to the famous 2-D Hénon map as shown in Fig. 5(a. A generalization of Eq. (66 is studied in Ref. [12] where there are regions with multiple coexisting attractors. We plan to stud this propert for 3-D diffeomorphisms in a forthcoming paper. Another simple case of the map (65 is given b f (x,, z = 1+a 1 z + a 2 x 2 x (67 or equivalentl f (x,, z = z (68 1+a 1 x + a 2 z 2 and it is studied in Ref. [7] where the attractor in this case is ver similar to the Lorenz attractor with a lacuna from the Shimizu-Morioka sstem [1, 2], and it is shown in Fig. 5(b. The second case is given b a 0 + a 1 x + a 2 + a 3 z + a 5 2 f (x,, z= b 0 + b 1 x + b 2 + b 3 z c c 2 + c 3 z (69 c 1 b 3 b 1 c 3 =0 ( ψ 1 (a i,b i,c i 1 i 9 The simplest case of the map (69 is given b a 0 + a 1 x + a 3 z + a 5 2 f (x,, z = x (70 This case is also introduced in Ref. [7]. The map (70 is the inverse of a special case of the map (68, and so its attractors are repellers of map (68, and it was shown in Ref. [7] that its attractors are quite different from the attractors of map (68. The other forms of simple 3-D quadratic diffeomorphisms are given b f (x,, z = b 2 c ( 1 b 1 c 2 =0 ψ 1 (a i,b i,c i 1 i 9 f (x,, z = c 1 b 3 b 1 c 3 =0 b 3 c ( 2 b 2 c 3 =0 ψ 1 (a i,b i,c i 1 i 9 f (x,, z = b 2 c 1 b 1 c 2 =0 c 1 b ( 3 b 1 c 3 =0 ψ 1 (a i,b i,c i 1 i 9 a 0 + a 1 x + a 2 + a 3 z + a 6 z 2 b 0 + b 1 x + b 2 + b 3 z c c 2 + c 3 z a 0 + a 1 x + a 2 + a 3 z + a 7 x b 0 + b 1 x + b 2 + b 3 z c c 2 + c 3 z a 0 + a 1 x + a 2 + a 3 z + a 9 z b 0 + b 1 x + b 2 + b 3 z c c 2 + c 3 z The full dnamics of some simple cases of the maps (71 (72 (73 Fig. 2 Regions of dnamical behaviors in the a 1 a 2 -plane for the map (74.

9 Zeraoulia ELHADJ and J. C. SPROTT, Front. Phs. China, 2009, 4(1 119 Fig. 3 Regions of dnamical behaviors in the a 3 a 7 -plane for the map (75. Fig. 4 Regions of dnamical behaviors in the a 1 a 9 -plane for the map (76. Fig. 5 (a Chaotic attractor obtained from the map (66 with a 4 = 1.6, a 3 =0.1; (b Chaotic attractor obtained from the map (68 with a 1 =0.7, a 2 =0.85; (c Chaotic attractor obtained from the map (74 with a 1 =0.4, a 2 = 0.4; (d Chaotic attractor obtained from the map (75 with a 3 = 1, a 7 = 1.8; (e Chaotic attractor obtained from the map (76 with a 1 =0.5, a 9 = 0.83.

10 120 Zeraoulia ELHADJ and J. C. SPROTT, Front. Phs. China, 2009, 4(1 1+a 1 x + a 2 z 2 f (x,, z = x 1+a 1 x + a 9 z f (x,, z = x (74 (75 1+a 3 z + a 7 x f (x,, z = x (76 are shown in Figs. 2, 3, and 4, respectivel. Some corresponding chaotic attractors are also depicted in Fig. 5(c (e. Note that these cases are characterized b a tpical quasi-periodic route to chaos, contrar to the situation for the maps given in (66 and (68 [5 7]. This confirms that the maps (66 and (68 are not topologicall equivalent to the maps (74, (75, and (76. 3 Applications in phsics Several researchers have defined and studied quadratic 3-D chaotic sstems. The first was Lorenz [17] in 1963, where he proposed a simple three linked nonlinear differential equations with complex behavior as a model of a weather sstem showing rates of change in temperature and wind speed. The behavior of this sstem was sensitivel dependent on the initial conditions of the model, therefore, the prediction of a future state of the sstem was impossible. Using computer simulation, Ruelle in 1979 show that the first fractal shape identified took the form of a butterfl (the butterfl effect. This justified the usual case where weather prognostication is involved and notoriousl wrong (butterfl in the Amazon might, in principle, ultimatel alter the weather in Kansas. Recentl, chaos has been ver useful in man technological disciplines such as in information and computer sciences, power sstems protection, biomedical sstems analsis, flow dnamics and liquid mixing, encrption and communications, and so on [18 20]. As a new application of chaos theor the example given in Ref. [20] where the adaptive snchronization with unknown parameters is discussed for a unified chaotic sstem b using the Lapunov method and the adaptive control approach. Some communication schemes, including chaotic masking, chaotic modulation, and chaotic shift ke strategies, are then proposed based on the modified adaptive method. In these schemes the transmitted signal is masked b chaotic signal or modulated into the sstem, which effectivel blurs the constructed return map and can resist this return map attack. The driving sstem with unknown parameters and functions is almost completel unknown to the attackers, so it is more secure to appl this method into the communication. A smplectic map is a diffeomorphism that preserves the area, i.e. the determinant of its Jacobian matrix is one. In the actual work, the 3-D quadratic diffeomorphism is a smplectic map if and onl if ψ 1 = 1 (77 On the other hand, Poincaré sections, especiall return maps provide tools for the location and stabilit of resonances and periodic orbits and the existence or nonexistence of invariant tori on the behavior of continuous time sstems, such as the Lorenz sstem which is the best close example connected with phsics, the Chen sstem, Lu sstem and generalized Lorenz sstem [18]. It is well known that an periodic (autonomous Hamiltonian sstem of degrees of freedom generates a 2n-dimensional smplectic map b following the flow for one period (parametrized b the value of the Hamiltonian, b considering the first return to a surface of section. Twist maps corresponding to Hamiltonians have the following properties [7 11]: (1 The are for where the velocit is a monotonic function of the canonical momentum. (2 The have a Lagrangian variational formulation. (3 One-parameter families of twist maps tpicall exhibit all possible tpes of dnamics, and the properties of the minimizing orbits (the periodic and quasiperiodic orbits can be found throughout these transitions from simple or integrable motion to complex or chaotic motion. The minimizing orbits are used in the theor of transport that deals with the motion of ensembles of trajectories. In a virtual viewpoint, ever model of phsical phenomena is a dnamical sstem. Almost all fundamental models of phsics are Hamiltonian dnamical sstems which gives rise to smplectic mappings. For example, the mapping defined b a Hamiltonian flow taking an initial condition to a state some finite time later is a smplectic map. These mappings are included in the stud of chemical reactions, or magnetic plasma confinement, especiall in magnetic field line mapping, guiding center motion, and plasma wave heating. The are also in charged-particle motion in particle accelerators [15], where a charged particle (either have a positive, negative or no charge is a particle with an electric charge. It ma be either a subatomic particle or an ion. A plasma or the

11 Zeraoulia ELHADJ and J. C. SPROTT, Front. Phs. China, 2009, 4(1 121 fourth state of matter is a collection of charged particles, or even a gas containing a proportion of charged particles. The simplest accelerator is the cclotron which consists of a constant magnetic field and a time-dependent voltage drop across a narrow azimuthal gap. The motion of a fluid particle in an incompressible fluid is also Hamiltonian [16]. As an application in the electronic engineering of the 3-D quadratic diffeomorphism given b (15, the following example given in Ref. [6] where a discrete-component electronic implementation ofadiscrete-timehperchaotic generalized Hénon map of the form: x 1 (k +1=1.76 x 2 2(k 0.1x 3 (k x 2 (k +1=x 1 (k (78 x 3 (k +1=x 2 (k with the initial conditions x 1 (0=1,x 2 (0=0.1,x 3 (0= 0, the map (78 exhibits a hperchaotic attractor. Using analog states, the corresponding circuit designs are relativel simple and uses commonl available parts and is readil constructed. Also, experimental results show that the circuit is functional. 4 Conclusion In this paper all possible forms of the 3-D quadratic diffeomorphisms are determined. Some numerical results are also given and discussed. References 1. A. L. Shilnikov, Bifurcation and chaos in the Morioka- Shimizu sstem, Methods of qualitative theor of differential equations (Gork, 1986: ; English translation in Selecta Math. Soviet., 1991, 10: A. L. Shilnikov, Phsica D, 1993, 62: D. V. Turaev and L. P. Shilnikov, Sb. Math., 1998: 189(2, D. A. Miller and G. Grassi, A discrete generalized hperchaotic Hénon map circuit, Circuits and Sstems, MWS- CAS Proceedings of the 44th IEEE 2001 Midwest Smposium 2001: G. Baier and M. Klein, Phs. Lett. A, 1990(6 7, 151: G. Grassi and S. Mascolo, A sstem theor approach for designing crtosstems based on hperchaos, IEEE Transactions, Circuits & Sstems-I: Fundamental theor and applications, 1999, 46(9: S. V. Gonchenko, I. I. Ovsannikov, C. Simo, and D. Turaev, Three-Dimensional Hénon-like Maps and Wild Lorenz-like Attractors, International Journal of Bifurcation and Chaos, 2005, 15(11: S. V. Gonchenko, J. D. Meiss, and I. I. Ovsannikov, Regular and Chaotic Dnamics, 2006, 11(2: S. V. Gonchenko and I. I. Ovsannikov, Three-dimensional Hénon map in homoclinic bifurcations, 2005, in preparation 10. H. E. Lomeli and J. D. Meiss, Nonlinearit, 1998, 11: K. E. Lenz, H. E. Lomeli, and J. D. Meiss, Regular and Chaotic Motion, 1999, 3: J. C. Sprott, Electronic Journal of Theoretical Phsics, 2006, 3: R. L. Devane, Trans. Am. Math. Soc., 1976, 218: H. Bass, E. H. Connell, and D. Wright, Bull. Amer. Math. Soc. (N.S., 1982, 7(2: E. D. Courant and H. S. Snder, Ann. Phs. (N.Y., 1958, 3: A. J. Dragt and D. T. Abell, Smplectic maps and computation of orbits in particle accelerators. In Integration algorithms and classical mechanics ON: Toronto, 1993: 59 85; Amer. Math. Soc.,Providence, RI, E. N. Lorenz, J. Atoms. Sc., 1963, 20: G. Chen and X. Dong, From Chaos to Order: Methodologies, Perspectives and Applications Singapore: World Scientific, G. Chen, Controlling Chaos and Bifurcations in Engineering Sstems, Boca Raton, FL, USA: CRC Press, W. W. Yu, J. Cao, K. W. Wong, and J. Lü, Chaos, 2007, 17(3: Etienne Forest, Beam Dnamics : A New Attitude and Framework (The Phsics and Technolog of Particle and Photon Beams, Amsterdam: Harwood Academic, 1998