INVARIANT MANIFOLDS OF COMPLEX SYSTEMS

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1 INVARIANT MANIFOLDS OF COMPLE SYSTEMS Jea-Marc Goux a Bruo Rossetto P.R.O.T.E.E. Laboratory I.U.T. e Toulo, Uversté u Su B.P., 8957, La Gare ceex, Frace E-mal: goux@uv-tl.fr, rossetto@uv-tl.fr KEYWORDS Ivarat curves, varat surfaces, multple tme scales yamcal systems, complex systems. ABSTRACT The am of ths work s to establsh the exstece of varat mafols complex systems. Coserg trajectory curves tegral of multple tme scales yamcal systems of meso two a three (preator-prey moels, euroal burstg moels t s show that there exsts the phase space a curve (resp. a surface whch s varat wth respect to the flow of such systems. These varat mafols are playg a very mportat role the stablty of complex systems the sese that they are "restorg" the etermsm of trajectory curves. DYNAMICAL SYSTEMS I the followg we coser a system of orary fferetal equatos efe a compact E clue : =I( ( = x x x E t wth [,,..., ] a ( (, (,..., ( t I = f f f E The vector I ( efes a velocty vector fel E whose compoets f whch are suppose to be cotuous a ftely fferetable wth respect to all x a t,.e., are C fuctos E a wth values clue, satsfy the assumptos of the Cauchy-Lpschtz theorem. For more etals, see for example (Cogto a Levso 955. A soluto of ths system s a tegral curve ( t taget to I whose values efe the states of the yamcal system escrbe by the Equato (. Sce oe of the compoets f of the velocty vector fel epes here explctly o tme, the system s sa to be autoomous. TRAJECTORY CURVES The tegral of the system ( ca be assocate wth the coorates,.e., wth the posto, of a pot M at the stat t. The total ervatve of V ( t amely the stataeous accelerato vector γ ( t may be wrtte, whle usg the cha rule, as: V I γ = = = JV ( where I s the fuctoal jacoba matrx J of the system (. The, the tegral curve efe by the vector of the scalar varable t represetg the fucto ( t trajectory of M ca be cosere as a plae or a space curve whch has local metrcs propertes amely curvature a torso. Curvature The curvature, whch expresses the rate of chages of the taget to the trajectory curve of system (, s efe by: = R V ( where R represets the raus of curvature. Torso The torso, whch expresses the fferece betwee the trajectory curve of system ( a a plae curve, s efe by: γ ( = (4 I where I represets the raus of torso.

2 LIE DERIVATIVE DARBOU INVARIANT Let ϕ a C fucto efe a compact E clue a ( t the tegral of the yamc system efe by (. The Le s ervatve s efe as follows: Theorem : ϕ ϕ Lϕ = V ϕ = x = (5 x = A varat curve (resp. surface s efe by ϕ ( = where ϕ s a C a ope set U a such there exsts a k a calle cofactor whch C fucto eote ( satsfes for all U L φ k ( = ( φ( (6 Proof of ths theorem may be fou (Darboux 878 Theorem : If L ϕ = the ϕ s frst tegral of the yamcal system efe by (. So, ϕ s costat alog each trajectory curve a the frst tegrals are raw o the level set ϕ = α a where α s a costat. { } Proof of ths theorem may be fou (Demazure 989 INVARIANT MANIFOLDS Accorg to the prevous theorems a the followg proposto may be establshe. Proposto : The locato of the pots where the local curvature of the trajectory curves tegral of a two mesoal yamcal system efe by ( vashes s frst tegral of ths system. Moreover, the varat curve thus efe s over flowg varat wth respect to the yamcal system (. Proof of ths theorem may be fou (Goux a Rossetto 6 Proposto : The locato of the pots where the local torso of the trajectory curves tegral of a three mesoal yamcal system efe by ( vashes s frst tegral of ths system. Moreover, the varat surface thus efe s over flowg varat wth respect to the yamcal system (. Proof of ths theorem may be fou (Goux a Rossetto 6 APPLICATIONS TO COMPLE SYSTEMS Accorg to ths metho t s possble to show that ay yamcal system efe by ( possess a varat mafol whch s eowg stablty wth the trajectory curves, restorg thus the loss etermsm heret to the o-tegrablty feature of these systems. So, ths metho may be also apple to ay complex system such that preator-prey moels, euroal burstg moels. But, orer to gve the most smple a cosstet applcato, let s focus o two classcal examples: - the Balthazar Va er Pol moel - the Lorez moel. Va er Pol moel The oscllator of B. Va er Pol, (96 s a seco-orer system wth o-lear frctos whch ca be wrtte: x+ α x x + x= ( The partcular form of the frcto whch ca be carre out by a electrc crcut causes a ecrease of the ampltue of the great oscllatos a a crease of the small. There are varous maers of wrtg the prevous equato lke a frst orer system. Oe of them s: x x = α x+ y x y = α Whe α becomes very large, x becomes a fast varable a y a slow varable. I orer to aalyze the lmt α, we trouce a small parameter ε = α a a slow tme t' = t α = ε t. Thus, the system ca be wrtte: x x f ( x, y x+ y V =I = ε y g( x, y x wth ε a postve real parameter: ε =.5 a where the fuctos f a g are ftely fferetable wth respect to all x a t,.e., are C fuctos a compact E clue a wth values. Accorg to Proposto, the locato of the pots where the local curvature vashes leas to the followg equato: ( ( φ x, y 9y 9x x y 6x x 9x (7 = ε (8

3 Accorg to Theorem (Cf. Appex for etals, the Le ervatve of Equato (8 may be wrtte: Lφ Tr J x y x ε x ( = [ ] φ( + ( + Let s plot the fucto ( x, y L φ ( (9 φ ( blue, ts Le ervatve ( mageta, the sgular approxmato x x+ y ( gree a the lmt cycle correspog to system (7 ( re: Y Lorez moel The purpose of the moel establshe by Ewar Lorez (96 was the begg to aalyze the mprectble behavour of weather. It most wesprea form s as follows: x f ( x, y, z σ ( y x y V = =I g( x, y, z = xz+ rx y h( x, y, z xy β z z 8 wth σ, r, a β are real parameters: σ =, β =, ( r = 8 a where the fuctos f, g a h are ftely fferetable wth respect to all x, a t,.e., are C fuctos a compact E clue a wth values. Accorg to Proposto, the locato of the pots where the local torso vashes leas to a equato whch for place reasos ca ot be expresse. Let s ame t as prevously: φ x, yz, ( ( Fgure : Va er Pol moel Accorg to Fechel s theory, there exsts a fucto φ x, y efg a mafol (curve whch s overflowg ( r varat a whch s COε ( close to the sgular approxmato. It s easy to check that the vcty of the sgular approxmato whch correspos to the seco term of the rght-ha-se of Equato (9 we have: L φ Tr J ( = [ ] φ( Moreover, t ca be show that the locato of the pots where the local curvature vashes,.e., where φ ( xy, = Equato (9 ca be wrtte: ( L φ = So, accorg to Theorem a, we ca clam that the mafol efe by φ ( xy, = s a varat curve wth respect to the flow of system (7 a s a local frst tegral of ths system. Accorg to Theorem (Cf. Appex for etals, the Le ervatve of Equato ( may be wrtte: L Tr J P V ( ( = [ ] ( + (, φ φ γ where P s a polyomal fucto of both vectors V a γ. Let s plot the fucto ( x, yz, L a the attractor correspog to system (: Z φ ( - φ a ts Le ervatve Fgure : Lorez moel - Y

4 It s obvous that the fucto ( x, yz, φ efg a mafol (surface s merge to the correspog to ts Le ervatve. It s easy to check that the vcty of the φ x, yz, Equato ( reuces to: mafol ( L φ Tr J ( = [ ] φ( Moreover, t ca be show that the locato of the pots where the local torso vashes,.e., where φ ( xyz,, = Equato ( ca be wrtte: L φ = ( So, accorg to Theorem a, we ca clam that the mafol efe by φ ( xyz,, = s a varat surface wth respect to the flow of system ( a s a local frst tegral of ths system. DISCUSSION I ths work, exstece of varat mafols whch represet local frst tegrals of two (resp. three mesoal yamcal systems efe by ( has bee establshe. From these two characterstcs t ca be state that the former mples that such mafols are represetg the stable part of the trajectory curves the phase space a from the latter that they are restorg the loss etermsm heret to the o-tegrablty feature of such systems. Moreover, whle coserg that yamcal systems efe by ( clue complex systems, t s possble to apply ths metho to varous moels of ecology (preator-prey moels, euroscece (euroal burstg moels, molecular bology (ezyme ketcs moels Research of such varat mafols couple systems or systems of hgher meso (four a more woul be of great terest. ACKNOWLEDGEMENTS Authors woul lke to thak Professors M. Azz-Alaou a C. Bertelle for ther useful collaborato. APPENDI Frst of all, let s recall the followg results: L u u u u = = u Two-mesoal yamcal system Let s pose: ϕ( =. (A Accorg to (A- the Le ervatve of ths expresso may be wrtte: ( = = where ( γ V = ( ( (A Accorg to Equato ( the Le ervatve of the accelerato vector may be wrtte: t leas to: ( J γ = Jγ + V (A J γ V = γ V = Jγ + V V J = Jγ V + V V Usg the followg etty: ( Ja b + a ( Jb = Tr ( J ( a b (A 4 REFERENCES Cogto, E.A. & Levso., N., 955. Theory of Orary Dfferetal Equatos, Mac Graw Hll, New York. Darboux, G Mémore sur les équatos fféretelles algébrques u premer orre et u premer egré. Bull. Sc. Math. Sér. (, 6-96, -4, 5-. Demazure, M Catastrophes et Bfurcatos, Ellpses, Pars. Fechel, N Geometrc sgular perturbato theory for orary fferetal equatos. J. Dff. Eq., 5-98 Goux, J.M. a Rossetto B. 6. Ivarat mafols of complex systems. to appear. Lorez, E. N. (96 Determstc o-peroc flows, J. Atmos. Sc,, -4. Va er Pol, B. 96. O 'Relaxato-Oscllatos', Phl. Mag., 7, Vol., t ca be establshe that: Jγ V = Tr J V ( ( γ So, expresso (A may be wrtte: Lϕ Tr J γ V γ V ( = ( ( ( ( J + V V V ( γ (A 5

5 γ V = Let s otce that: ( ( β = So, equato (A 5 leas to: J ( Tr( J γ V = + V V β Sce vector J V V a that: (A 6 has a uque co-orate accorg to the β -recto a sce we have pose: ϕ =, expresso (A 6 may fally be ( wrtte: J Lϕ ( = Tr( J ϕ( + V V Three-mesoal yamcal system Let s pose: ϕ( = γ ( expresso may be wrtte: (A 7. The Le ervatve of ths ( γ = ( Accorg to γ ( γ V = γ ( (A 8, t leas to: γ ( ( = = γ ( (A 9 The Le ervatve of expresso (A leas to: γ = J γ + γ + J J V Thus, expresso (A 9 reas: Lϕ( = ( J γ ( J J (A + γ + V ( γ V It ca also be establshe that: J V γ γ = Tr J J γ γ V ( ( ( ( (, expresso So, sce we have pose: ϕ( = γ ( (A may fally be wrtte: J ( = Tr ( J ϕ( + ( Tr ( J V J J J + J V + γ + V V ( γ (A JEAN MARC GINOU was bor Toulo, Frace a wet to the Uversty of Nce Sopha-Atpols, at I.N.L.N. of Nce a the at the Uversty of South, where he stue olear a chaotc yamcal systems wth Professor Bruo Rossetto. He obtae hs Ph-D Apple Mathematcs 5. He worke for a couple of years for the Departmet of Eucato before movg 4 to I.U.T. of Toulo where he has bee workg Dyamcal Systems ever sce.