On the non-equivalence of Lorenz System and Chen System

On the non-equiv alence of Lorenz System and Chen System ∗ Zhen ting Hou 1 , Ning Kang 1 , 2 , Xiangx i ng Kong 1 , Guanr ong Chen 3 and Guo jun Y a n 4 † 1 Sc ho ol of Mathem atics, Central Sou t h Uni v ersity , Changsha , Hunan, 4 10075 , China 2 Sc ho ol of Mathem atics a n d Co mputat i onal Science, F uy ang T eachers Coll ege, F uy ang, Anh ui, 2 3 6041, C hina 3 Department of Electr o nic Engi neering, C it y Universit y of H o ng Ko ng, Hong Kong, Ch i na 4 Department of Mathema t ics, Zheng zhou Universit y , Zhengzhou, Henan, 45 0000 , C hina Abstract: In this pap er, w e prov e that the Chen syste m with a set of c haotic parameters is not sm o othly equiv alen t to the Lorenz s ystem with an y para meters. Key words: Lorenz system, Chen system, smo oth equiv alence, top ological equiv alence. 2000 MSC: 93C10, 93C15. 1 In tro d uction Nonlinear science had exp erienced an unprecede n ted and vigorous dev elopmen t particu- larly during t he second half of the 20th cen tury , and it was considered “the third rev olution” in natural science in the history . The main sub jects in the study of nonlinear science include c haos, bifurcation, fractals, solitons and complexit y . Because of the imp ort a n t significance to un ve il the essenc e of c hao s and wide p oten tial application prosp ects of chaos theory in many fields, researc h on c haos alw ays carries a heav y w eight in nonlinear science. H. Poincar ´ e [1] and C. Maxw ell [2] b oth had some v ag ue concepts of c haos in their times. In the earlier 1960s, E. N. Lorenz [3] discov ered the now-famous Lorenz system , whic h actually pro duces visible c haos. Lorenz system is the first mathematical and phy sical mo del of c haos, thereb y b ecoming the starting p oint and foundation stone for lat er researc h on chaos theory . Since ∗ This resear c h is suppo rted by the National Natural Science F oundation o f China (1067 1212,9 0 820302). † Email: yanguo jun2002@s ina.com 1 the 1960 s, par t icularly with this mo del, mathematicians, phys icists and engineers from v ar- ious fields hav e thoroughly studied t he essence of c haos, characteris tics of c ha otic systems, bifurcations, routes to chaos, and man y other related to pics [4]. There a re also some chaotic systems of g reat significance that are closely related to the Lorenz system, where a partic- ular example in p oin t is the Chen system. Since the Chen system w as first found in 1999 [5,6], h undreds of pap ers ha ve b een published o n this new chaotic system with deep a nd comprehensiv e results obtained. A monograph on the Lorenz systems family including the Chen system ha s also b een published [7]. T o further understand the interesting Chen system, one fundamen tal question has to b e answ ered: ar e the Chen system and the Lo r enz system non-equiv alent, either top ologically or smo ot hly? The purp ose of this pap er is to prov e that the Lorenz system a nd t he Chen system ar e indeed non- equiv alen t smoo thly . 2 Results and Pro ofs The dynamical system φ abc t defined by            dx dt = a ( y − x ) , dy dt = cx − xz − y , dz dt = xy − bz , (2.1) is called the Lorenz system with parameters a, b, c . The dynamical system ψ abc t defined by            dx dt = a ( y − x ) , dy dt = ( c − a ) x − xz + cy , dz dt = xy − bz , (2.2) is called the C hen system with parameters a, b, c . It is clear that system ( 2 .1) has 3 equilibrium p oints if b ( c − 1) > 0, i.e., P 1 = (0 , 0 , 0) , P 2 = ( − p b ( c − 1) , − p b ( c − 1 ) , c − 1) , P 3 = ( p b ( c − 1 ) , p b ( c − 1 ) , c − 1) , and system (2.2) has 3 equilibrium p oints if b (2 c − a ) > 0 , i.e., Q 1 = (0 , 0 , 0) , Q 2 = ( − p b (2 c − a ) , − p b (2 c − a ) , 2 c − a ) , Q 3 = ( p b (2 c − a ) , p b (2 c − a ) , 2 c − a ) . 2 Denote the co o r dina t es o f P i b y ( x i , y i , z i ), i = 1 , 2 , 3, and the co ordinates of Q i b y ( x ′ i , y ′ i , z ′ i ), i = 1 , 2 , 3, and denote the vec tor fields on t he right sides o f (2.1) and (2.2) by ~ U ( x, y , z ) a nd ~ V ( x, y , z ) , resp ectiv ely . It is clear that their Jacobians are: D ~ U ( x, y , z ) =   − a a 0 c − z − 1 − x y x − b   , D ~ V ( x, y , z ) =   − a a 0 c − a − z c − x y x − b   , and their determinan ts ar e: det D ~ U ( P 1 ) = ab ( c − 1) , det D ~ U ( P 2 ) = det D ~ U ( P 3 ) = − 2 ab ( c − 1) , det D ~ V ( Q 1 ) = ab (2 c − a ) , det D ~ V ( Q 2 ) = det D ~ V ( Q 3 ) = − 2 ab (2 c − a ) . In general, let f ( x ) and g ( x ) b e v ector fields on R n , and ˙ x = f ( x ) , x ∈ R n (2.3) ˙ y = g ( y ) , y ∈ R n (2.4) b e tw o systems of differential equations on R n . Definition 2.1 If ther e ex i s ts a diffe o morphism h on R n such that f ( x ) = M − 1 ( x ) g ( h ( x )) , (2.5) wher e M ( x ) is the Jac obian of h at the p oint x , then (2.3) and (2.4) ar e said to b e smo othly equiv alen t . Remark 2.2 If (2.3) and (2.4) ar e smo othly e quivalent, and supp ose that x 0 and y 0 = h ( x 0 ) ar e the c orr esp ondi n g e quilibria of f ( x ) and g ( x ) , A ( x 0 ) an d B ( y 0 ) ar e the Jac obians o f f ( x ) and g ( x ) , r esp e ctively, then A ( x 0 ) a n d B ( y 0 ) a r e similar, i.e. , their char acteristic p olynomials and eigenval ues ar e the sam e . Theorem 2.3 The Chen system and the L or enz system ar e no t smo othly e quivalent, i.e., ther e exists a Chen system ψ a ′ b ′ c ′ t which is not smo othly e quivalent to any L or enz system φ abc t . 3 Pro of Since M ( x ) 6 = 0 , M − 1 ( x ) 6 = 0 and due to (2 .5), w e hav e f ( x ) = 0 ⇔ g ( h ( x )) = 0 , that is, the equilibria x of f ( · ) corresp ond to the equilibria h ( x ) of g ( · ), t herefore a Chen system is smo othly equiv alen t to a Lorenz system with the same n um b er of equilibria. It suffices to pro v e that a Chen system with 3 equilibria can not b e smo othly equiv alent to a Lorenz system with a n y 3 equilibria. Supp ose that h is a diffeomorphism on R 3 suc h that ψ a ′ b ′ c ′ t and φ abc t smo othly equiv alen t under h . Because detD ~ U ( P 1 ) and detD ~ V ( Q 1 ) are p ositiv e, P 1 corresp onds to Q 1 under h . Because D ~ U ( P 1 ) =   − a a 0 c − 1 0 0 0 − b   , D ~ V ( Q 1 ) =   − a ′ a ′ 0 c ′ − a ′ c ′ 0 0 0 − b ′   , the c haracteristic equation of D ~ U ( P 1 ) is: λ 3 + ( a + b + 1 ) λ 2 + ( a + ab − ac + b ) λ − ab ( c − 1) = 0 , and the c har a cteristic equation of D ~ V ( Q 1 ) is: λ 3 + ( a ′ + b ′ − c ′ ) λ 2 + ( a ′ 2 + a ′ b ′ − 2 a ′ c ′ − b ′ c ′ ) λ − a ′ b ′ (2 c ′ − a ′ ) = 0 . Let u = a ′ + b ′ − c ′ , (2.6) v = a ′ 2 + a ′ b ′ − 2 a ′ c ′ − b ′ c ′ , (2.7) w = − a ′ b ′ (2 c ′ − a ′ ) . (2.8) By Remark 2.1, w e m ust hav e a + b + 1 = u, (2.9) a + ab − ac + b = v , (2.10) − ab ( c − 1) = w. (2.11) By (2.9), w e hav e a = u − 1 − b , so that in combining with (2 .11), c = 1 − w ab = 1 − w b ( u − 1 − b ) . 4 Substituting a, c in ( 2 .10), w e hav e b 3 − ub 2 + v a − w = 0 . (2.12) It is clear that D ~ U ( P 2 ) and D ~ U ( P 3 ) ha ve the same c haracteristic equations, and D ~ V ( Q 2 ) and D ~ V ( Q 3 ) ha ve the same c haracteristic equations. Hence , w e ma y assume that P 2 corresp onds to Q 2 . By comparing the co efficien ts of their first-or der terms, w e ha v e ab + bc = b ′ c ′ . Substituting them into t he formulas of a, c , we get b ( u − 1 − b ) + b  1 − w b ( u − 1 − b )  = ( u − 1) b − b 2 + b − w u − 1 − b = b ′ c ′ , and b 3 − (2 u − 1) b 2 + [( u − 1 ) 2 + u − 1 + b ′ c ′ ] b − w − ( u − 1) b ′ c ′ = 0 . Subtracting this from (2.12), we obt a in ( u − 1) b 2 + ( u + v − u 2 − b ′ c ′ ) b + ( u − 1) b ′ c ′ = 0 . (2.13) By resultan t elimination [8], a necessary a nd sufficien t condition f or (2.12) and (2.13) t o ha ve same ro ots is M 0 ( a ′ , b ′ c ′ ) =           1 − u v − w 0 0 1 − u v − w u − 1 u + v − u 2 − b ′ c ′ ( u − 1) b ′ c ′ 0 0 0 u − 1 u + v − u 2 − b ′ c ′ ( u − 1) b ′ c ′ 0 0 0 u − 1 u + v − u 2 − b ′ c ′ ( u − 1) b ′ c ′           = 0 . (2.14) Substituting u , v , w in the ab ov e equation by (2.6), (2.7) and (2.8), w e get an algebraic equation of a ′ , b ′ , c ′ , as b ′ ( a ′ − 2 c ′ ) 2 (1 + c ′ )( a ′ 3 − a 4 + a ′ 5 + a ′ 2 b ′ − a ′ 3 b ′ + a ′ b ′ 2 − 2 a ′ 2 b ′ 2 − 2 a ′ b ′ 3 + a ′ 2 b ′ 3 + a ′ b ′ 4 − 2 a ′ 2 c ′ + 3 a ′ 3 c ′ − 4 a ′ 4 c ′ − 3 a ′ b ′ c ′ + 4 a ′ 2 b ′ c ′ − 5 a ′ 3 b ′ c ′ − b ′ 2 c ′ + 5 a ′ b ′ 2 c ′ − 2 a ′ 2 b ′ 2 c ′ + 2 b ′ 3 c ′ − 2 a ′ b ′ 3 c ′ − b ′ 4 c ′ − 2 a ′ 2 c ′ 2 + 4 a ′ 3 c ′ 2 − 3 a ′ b ′ c ′ 2 + 6 a ′ 2 b ′ c ′ 2 − b ′ 2 c ′ 2 + 4 a ′ b ′ 2 c ′ 2 + b ′ 3 c ′ 2 ) = 0 , (2.15) 5 and its solution is giv en b y the union of the following f o ur surfaces: b ′ = 0 , a ′ − 2 c ′ = 0 , 1 + c ′ = 0 , a ′ 3 − a 4 + a ′ 5 + a ′ 2 b ′ − a ′ 3 b ′ + a ′ b ′ 2 − 2 a ′ 2 b ′ 2 − 2 a ′ b ′ 3 + a ′ 2 b ′ 3 + a ′ b ′ 4 − 2 a ′ 2 c ′ + 3 a ′ 3 c ′ − 4 a ′ 4 c ′ − 3 a ′ b ′ c ′ + 4 a ′ 2 b ′ c ′ − 5 a ′ 3 b ′ c ′ − b ′ 2 c ′ + 5 a ′ b ′ 2 c ′ − 2 a ′ 2 b ′ 2 c ′ + 2 b ′ 3 c ′ − 2 a ′ b ′ 3 c ′ − b ′ 4 c ′ − 2 a ′ 2 c ′ 2 + 4 a ′ 3 c ′ 2 − 3 a ′ b ′ c ′ 2 + 6 a ′ 2 b ′ c ′ 2 − b ′ 2 c ′ 2 + 4 a ′ b ′ 2 c ′ 2 + b ′ 3 c ′ 2 = 0 . Denote the p oint set of all solutions of (2 .1 4) by C . It is clear that C is a Borel subset of R 3 and its Leb egue measure is 0. So, there are many p oints not b elonging t o C , fo r example, the v alues of a ′ = 45 , b ′ = 5 , c ′ = 28 giv e b ′ (2 c ′ − a ′ ) = 55 > 0 , M 0 (45 , 5 , 28) = 2 . 919 × 1 0 11 6 = 0, i.e., (45 , 5 , 28) / ∈ C . This means tha t the Chen system ψ 45 , 5 , 28 t is not smo othly equiv alen t to the L o renz system φ abc t with a n y v alues of a, b, c , while ψ 45 , 5 , 28 t is c haotic according to [5, 6 or 7 (p.39)]. References [1] J. H. Poincar´ e, Sur le pr o bl ´ eme des trois cor ps et les ´ eq uations de la dyna mique. Divergence des s´ eries de M. Lindstedt, Acta Mathematica, vol. 13, pp. 1-27 0, 18 90 [2] J. C. Maxwell, Do es the progres s of ph ysical sciences tend to g ive any adv antage to the opinion of necessity (or determinism) over that o f contingency or events and the freedo m of will? In: L. Campb ell and W. Garnett (Eds.), The life o f James Clerk Maxwell, with a s e lection from his corre spo ndence a nd o ccasio nal writings and a sketc h of his contributions to science, Macmillan, Lo ndon, pp. 43 4-444, 1 882 [3] E. N. Lor enz, Deterministic nonp erio dic flow, J. Athmosph. Sc. vol. 20, pp. 1 30-14 1, 1963 [4] C. Spa rrow, The Lorenz Eq uations: Bifur c ations, Chaos, a nd Strange A ttractors, Springer, New Y o rk, 1982 [5] G. Chen and T. Ueta, Y et ano ther chaotic attr a ctor, In t. J . o f Bifurca tion a nd Chao s, vol. 9 , pp. 1 465-14 66, 1999 [6] T. Ueta and G. Chen, B ifur cation a nalysis o f Chen’s a ttr a ctor, Int. J. of Bifurcation and Chaos, vol. 1 0, pp. 1917 -1931, 2 000 [7] G. Chen and J . L ¨ u, Dynamical Analysis , Control and Sync hronization o f the Lorenz Sys tems F amily , Science Press, Beijing, 2003 [8] L. Y a ng, J . Zhang and X. Ho u, Nonlinear Systems of Algebr aic Equations a nd Mechanical Theorem P roving, Shanghai Scientific and T echnological Education Publishing Hous e, Shanghai, 199 6 6