Variational analysis of some questions in dynamical system and biological system

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1 Available online Journal of Chemical an Pharmaceutical Research, 014, 6(7): Research Article ISSN : CODEN(USA) : JCPRC5 Variational analsis of some questions in namical sstem an biological sstem Sun Dawei an Liu Jiarui College of Science, Henan Universit an Technolog, Zhengzhou, China ABSTRACT This paper stuies the variational problems of some namical sstem. B using the methos of Euler equation an Legenre conitions an some computional techniques in mechanics an ifferential equations, this paper computes the extrema of some sstems an iscuss some future applications in smplectic namical sstems an biological mathematical problems. Ke wors: variational methos; Euler equation; biological mathematical problems INTRODUCTION Variational methos pla an important role in mathematics, mechanics an other science an technologies. In the earl 16 centur, Newton, Leibnitz, Euler, Lagrange an other scientists have stuie the variational methos [1]. The origin of the variational methos is the problem of brachistochrone curve, which carries a point-like bo from one place to another in the least amount of time. Newton, Leibnitz an Johann Bernoulli solve this problem an their methos give the iea of variational methos. Another problem is the geoesic problem; it is fining the shortest path connecting two points. On a Riemannian manifol M with riemannian metric g, the length of a smooth curve γ : [a,b] M is efine b the following: L( γ) = g (& γ, & γ ) γ (1) The minimizing curves of L in a small enough open set can be obtaine b techniques of calculus of variations. The existence an uniqueness problem of geoesics promote the evelopment of geometr. Variational methos have man important applications in ifferential equations. Solving a ifferential equation is the same with fining the critical points of some variational functional. This iea was first use in the ifferential equation of secon orer. For example we see the next bounar value prblem of linear elliptic equation [,3] u = f (x), x Ω u = 0, x Ω Here Ω is boune omain with smooth bounar of the Eucil space. B variational metho, one can translate this problem to be fining the critical points of the functional: 1 I (v) = ( Dv fv) Ω (3) The process of fining the critical points of this functional is to fin the extremal point of tihs functional. Later, () 184

2 peple use the lower semi-continuit conition to stu the egenerate elliptic problem: p u ( u) = f, in Ω u = 0, x Ω (4) The ke point to solve this elliptic problem is fining the extremal point of its corresponing functional. B the usual classic result,we have if M is topological Hausorff space, an suppose that a functional satisfies the conition of boune compactness(heine-borel propert), then E is uniforml boune from below an attain its infimum on M. However, the conition of boune compactness is not eas to confirme, so people tr to use some facts that can be checke easil to instea the above classical results. Suppose V is a reflexive banach space with norm, an M V is a weakl close subset of V. Suppose that a functional E is coercive an weak lower semi-continuous on M with respect to V. Then the functional E is boune from below on M an attains its infimum in M. Using the moifie results, one can check that the corresponing functional : 1 p E(u) = u fu p Ω Ω (5) 1, P W0 ( Ω) is reflexive an Satisfies the requirment of the moifie facts. It is eas to checke that the banach space the the functional is coercive an weak sequentiall lower semi-continuous. So the above results implies that there is a minimizer for the functional, an therefore there is a solution for the egenerate elliptic problem. In general people can not get that a boune an lower semi-continuous functional get its infimum. Ekelan construct minimizing sequence an show that there exists nearl optimal solutions to some optimization problems. An this principle can be use when the level set is not compact. Compactness becomes more an more important. To overcome this ifficult of lacking compactness, Palis an Smale introuce the concepts of PS-conition: a sequence u m in V is calle a Palais-Smale sequence for E if of the functional Palais-Smale sequence has a convergent subsequence. E (u ) m c uniforml in m, while the Differential DE(u m) 0 as m 0, a functional E is sai to satisf the Palais-Smale conition if an In 1970s, A.Ambrosetti an P.Rabinowitz use the Mountain pass theorem to solve some ifferential equations[6]. P.Rabinowitz prove the perioic solutions of the Hamiltonian sstem. Later Benci-Rabinowitz give the linking argument, the can solve more semilinear elliptic bounar problem without smmetr. There are other evelopments for the variational methos, an the applications in smplectic geometr an namical sstem, we will iscuss later. In this paper, we will first introuce some basic notions of variational methos, show the basic techniques to fin the extrema, an give some example to reall fin the extrema, an last we show the applicaition in smplectic geometr an iscuss some future the variational methos mabe ca be use. 1. PREPARATIONS AND LEMMAS.1 Preparations In this section, we briefl introuce the notations of variational methos; give some basic lemmas an properties of Euler equations, Jacobi conitions an some variational problems for geometr, for etails we refer to [1,, 3, 4, 7]. Definition 1 A function f n is lower semi continuous at a point x if the following ientities hol: liminf f (x) = f ( x) x x (6) The lower semi continuous is equivalent as the following [13]: Theorem. The following properties of a function f are equivalent 185

3 1. F is lower semicontinuous. The epigraph set is close 3. The level set are close. Definition 3. Fréchet erivative of a function is efine as following: Let V an W be Banach spaces, an U V be an open subset of V. A function f: U W is calle Fréchet ifferentiable at x U if there exists a boune linear operator AX : V W such that f (x+ h) f(x) A x (h) w lim = 0 h 0 h v (7) Now we give the funamental theorem of calculus of calculus of variations. Definition 4 If a function Ω u(x) f(x) = 0, f ( Ω) Then 0 C 0 u C( Ω) satisfies [1] u = in Ω.Next we introuce the Euler equation of the variational functional [1]: Lemma 5 If the functional ' [(x)] (x,, ) = J F (8) (9) Satisfies the bounar value conitions (x ), (x ) = = Then the extremal curve shoul satisf the following equation F F = 0 ' (10) Here the function F is of secon orer. An this equation is also calle Euler- Lagrange equation. This equation can also be written as the following F F F ' F '' ' ' x = 0 Once we have the Euler- Lagrange equation, we can translate the extrmal problem to solve ifferential equations.if there are more variables, the Euler- Lagrange equation can also be foun. (11) Lemma 6 If the functional ' ' [(x),z(x)] (x,, z, z ) = J F get its extrema an satisfies the bounar value conitions (x ) =, (x ) =,z(x ) = z,z(x ) = z Then there extremal curve satisfies the following Euler- Lagrange equations F F F = F = 0 ' z z ' 0 (1) (13) (14) 186

4 If the variational functional has man variables, the Euler- Lagrange equations can also be written in the same wa. Moreover, if the functional epen on the higher orer ifferential of the variables, ' '' [(x)] (x,,, ) = J F (15) Suppose that this functional gets its extrema an satisfies the bounar value conition (x ) =, (x ) =, '(x ) = ', '(x ) = ' (16) Here the variational functional shoul be ifferential of thir orer, the function is ifferential of fourth orer, then the Euler- Lagrange equation is as following: F F F + = ' '' 0 This equation is also calle Euler-Poisson equation. If there are more variables, the Euler-Poisson equation can also be extene to that case, just note that the functional an the function shoul have higher ifferentiation [1]. (17) Next we introuce some sufficient conitions of extrema of Functionals. Consier the functional ' [(x)] (x,, ) = J F With the bounar value conition (x ) =, (x ) = If the functional attains its extrema at u(x), then we have the following equation u ( F F ') u ( F ' ' ) = 0 (18) (19) (0) This equation is calle the Jacobi equation. It is the same with fining the extrema of functions with one variable; we nee some sufficient conitions to etermine whether a functional can get its extrema. This is the classical Weiestrass sufficient conitions an Legenre Sufficient conitions. Let E(x,, ', p) = F(x,, ') F(x,, p) (' p) F (x,,p) p (1) Theorem 7. If the function =(x) is the extremal curve of the functional an satisfies the bounar value conition, ' an also satisfies the Jacobi conitions, for all near, the Weiestrass conitions hol,we get its local infmum, if it is less than zero, we get its local Maxima.. Example Now we give some examples to show how to etermine the extrema. Example 8. Fin the extremal curve of the following functional π 4 J (, z) = 3z 9 + ' z ' 0 The bounar conitions are: (0) = 0, ( π ) =, z(0) = 0, z( π ) = π B the Euler- Lagrange equations, we can compute that the function F is F(, z) = 3z 9 + ' z ' () (3) (4) 187

5 So we can compute the z ' z ' F, F, F, F, so the Euler- Lagrange equations are the following ifferential equations: 18 + '' = z '' = 0 (5) Integral the equations, we can get the solutions of this equations are cos3 sin 3 = c1 x + c x 3 z = x + c3x + c4 8 Accoring to the bounar conitions, we can get that 3 + 3π c1 = 0, c4 = 0, c =, c3 = 3 So the extremal curve is = sin 3x π z = x + x 8 3 (6) (7) (8) Next we see the applications of the Jacobi equations, an we just give the iea of the compute an omit the result. Example 9. Suppose that the functional is the following J () = ( x ' ) (x 0) = 0, (x 1) = 1 (9) Fin the solution whose Jacobi equation satisfies the initial bounar conitions: u (0) = 0, u'(0) = 1 First, we can fin the function F (x,, ') = x ' An we can compute the Euler equation, then b the efinition of Jacobi equation u ( F F ') u ( F ' ' ) = 0 (30) (31) (3) We can get 6u 4 u '' = 0 (33) The solution is u(x) = 1 e + e 1.5x 1.5x An b the initial value of u, we can compute the number 1,, we omit the process. (34) Next we give some other applications of variational methos in smplectic geometr an other namical sstems. First we introuce the bi-invariant metric on the group of Hamiltonian iffeomorphisms. A manifol is smplectic if it is a smooth manifol an equippe with a close non-egenerate ifferential -form, the Hamiltonian iffeomorphism is the time-1 map of the following namical sstem 188

6 ( f& t ) ω Ft i = Hofer first efine a bi-invariant metric on the group of compactl supporte smplectic iffeomorphisms of stanar smplectic manifol [5], the Hofer norm is efine as following. E ( ϕ ) inf{ f f = ϕ} = t 1 f t (36) Viterbo also efine an invariant istance b generating functions an Polterovich evelope this metric on more smplectic manifol [11, 1], finall Lalone an McDuff extene it to an smplectic manifol [10]. In the efinition an proof of Hofer, an Infinite imensional variational metho was use for the variational functional of the Hamiltonian sstem. An interesting question is whether there exist some more variational results for the geoesic problem uner the Hofer metric. How about the variational analsis on the contact bi-invariant metric? Are the similar results for the geoesic problem? Secon we will iscuss some applications of variational methos in the biological sstem. One tpical moel is the equations for two tpes of living creatures live in one foo source. The moles are the following[8,9] S ' = D(1 + be(t) S) m x s m x s a + S a + S x s x ' = m D x a1 + S x s x ' = m D x a + S (39) Here x is the number of the living creature, s is the concentration of nutrient solution, mis the rate of growing. Smith use the variational theorems to stu this problem an analsis the changes of the iving creatures. An this methos can also be use for the analsis of insets in the cereals. CONCLUSION In this paper we introuce the histor an some techniques in variational methos. We compute some extremas of functional b the Euler equation an the Jacobi equation, an iscuss some future applications in smplectic namical sstems an biological mathematical problems Acknowlegments The author woul like to thank the support of Henan Universit of Technolog, uner which the present work was possible. REFERENCES [1] Dazhong Lao. Funamentals of the calculus of variations. Beijing: National Defence Inustr Press, 011. (In chinese). [] Benjin Xuan. Variational methos: theor an applications. Hefui: Press of Universit of Science an Technolog of China, 006. [3] Struwe. M. Variational methos applications to nonlinear partial ifferential equations an Hamiltonian sstem, springer-verlag, [4] Willem.M. Minimax theorems. Progress in Nonlinear ifferential equations an their applications, Birkhauser, [5] Hofer. H, Zehner.E. Smplectic Invariants an Hamiltonian Dnamics. Birkhauser,1994 [6] Ambrosetti A., Rabinowitz. P. Dual variational methos in critical point theor an applications. J.Func.anal.14(1973), [7] Wenrui, Lu. Variational methos in ifferential equations, Beijing, Science press,003. (In chinese) [8] Qizhi L, Jing L. Theoretical stu of mathematical moels of competition, Beijing, Science press,003.(in chinese) [9] Smith H. L. Competitive coexistence in an oscillationg chemostat, SIAM J. Appl. Math. 40:498-5,1981. (35) (37) (38) 189

7 [10] Lalone F, McDuff D.: The Geometr of Smplectic Energ. Ann. Math.141, [11] Viterbo C. Smplectic topolog as the geometr of generating functions. Math. Annalen. Vol 9, Issue 1, pp , 199, [1] Polterovich L. The geometr of the Group of Smplectic Diffeomorphism. Berlin: Birkhauser,001 [13] Rockafellar R, Wets R. Variational analsis Springer,