Variational principles for nonlinear dynamical systems

as for instance the Poincare ́–Lindsted method, the averaging ~Krylov–Bogoliuvov–Mitropolsky! method, the Shohat expansion, or the multiscale method, to mention a few. They provide ap mate solutions as well as the relationship between the frequency of the nonlinear oscillatio a small parameter ~physically identified for the system, as the amplitude of the oscillations !. Other approximate techniques consist in the qualitative analysis on the phase space by lineariz differential equation around the fixed points. Periodic or exponential behavior of the system be predicted after studying the stability of the fixed points. Many different physical systems reduce to one-dimensional nonlinear ODE of second which may be studied by using the approximate techniques mentioned above. 2,3 An important property of them is that the frequency depends explicitly on the amplitude of the oscillation we have mentioned, the perturbative techniques yield to an approximate solution in power of the amplitude only. Recently, Benguria and Depassier ~BD for short! have made use of a variational principle order to solve the bifurcation problem for second-order nonlinear Hamiltonian systems. The apply the principle to obtain simple approximate closed formulas for the frequency of l amplitude oscillations. In this work we extend this method to other situations. Basically, the method consis transforming the differential equation ü1u5N(u), which may be written in the phase spa variables (p,u), with p52u8 ~where the prime means differentiation with respect to the varia t52vt/p! in an integral equation multiplying the differential equation by an auxiliary funct g(u), and integrating. The variational method provide a lower bound for an integral functi From the Euler–Lagrange equations for the functional, one may calculate the auxiliary fun g(u) at a first approximation by taking p(u) from the linear case ( N50). First of all, we generalize this transformation for Hamiltonian systems. We multiply differential equation bypg and apply the variational method. From this generalization we ob the first interesting result: the variational characterization depends strongly on the exponen n. In this sense, we obtain a better fit for n521 than forn50 ~the BD’s case !. However, as the systems are Hamiltonian, it is possible to find a relationship between p and u in the nonlinear situation. Using it, instead of its linear approach, one can find a new relatio for g being the variational result for the frequency, always equal to the exact solution. T shown in three specific and physically interesting cases. Second, as non-Hamiltonian systems are very often found in practice and arise in physical systems, we introduce in this work a special analysis for them. Following the same employed for Hamiltonian systems, one obtains an explicit dependence of the variational c terization onp(u). For non-Hamiltonian systems it is not possible to obtain a relationship betw p and u, at least exactly. In consequence, we adopt two ways to proceed. From one han apply twice the variational method and find two coupled Euler–Lagrange equations, one o corresponding to the same functional that appears in Hamiltonian systems and the other on


I. INTRODUCTION
The dynamical study of nonlinear oscillatory systems is often reduced to numerical calculus or to some approximated analytical techniques.Most of them are based in perturbation methods 1 as for instance the Poincare ´-Lindsted method, the averaging ͑Krylov-Bogoliuvov-Mitropolsky͒ method, the Shohat expansion, or the multiscale method, to mention a few.They provide approximate solutions as well as the relationship between the frequency of the nonlinear oscillations and a small parameter ͑physically identified for the system, as the amplitude of the oscillations͒.Other approximate techniques consist in the qualitative analysis on the phase space by linearizing the differential equation around the fixed points.Periodic or exponential behavior of the system may be predicted after studying the stability of the fixed points.
Many different physical systems reduce to one-dimensional nonlinear ODE of second order, which may be studied by using the approximate techniques mentioned above. 2,3An important property of them is that the frequency depends explicitly on the amplitude of the oscillations.As we have mentioned, the perturbative techniques yield to an approximate solution in power series of the amplitude only.
Recently, Benguria and Depassier ͑BD for short͒ 4 have made use of a variational principle in order to solve the bifurcation problem for second-order nonlinear Hamiltonian systems.They also apply the principle to obtain simple approximate closed formulas for the frequency of largeamplitude oscillations.
In this work we extend this method to other situations.Basically, the method consists in transforming the differential equation u ¨ϩuϭN(u), which may be written in the phase space variables (p,u), with pϭϪuЈ ͑where the prime means differentiation with respect to the variable ϭ2t/͒ in an integral equation multiplying the differential equation by an auxiliary function g(u), and integrating.The variational method provide a lower bound for an integral functional.From the Euler-Lagrange equations for the functional, one may calculate the auxiliary function g(u) at a first approximation by taking p(u) from the linear case (Nϭ0).
First of all, we generalize this transformation for Hamiltonian systems.We multiply the differential equation by p n g and apply the variational method.From this generalization we obtain the first interesting result: the variational characterization depends strongly on the exponent n.In this sense, we obtain a better fit for nϭϪ1 than for nϭ0 ͑the BD's case͒.
However, as the systems are Hamiltonian, it is possible to find a relationship between p and u in the nonlinear situation.Using it, instead of its linear approach, one can find a new relationship for g being the variational result for the frequency, always equal to the exact solution.This is shown in three specific and physically interesting cases.
Second, as non-Hamiltonian systems are very often found in practice and arise in many physical systems, we introduce in this work a special analysis for them.Following the same ideas employed for Hamiltonian systems, one obtains an explicit dependence of the variational characterization on p(u).For non-Hamiltonian systems it is not possible to obtain a relationship between p and u, at least exactly.In consequence, we adopt two ways to proceed.From one hand, we apply twice the variational method and find two coupled Euler-Lagrange equations, one of them corresponding to the same functional that appears in Hamiltonian systems and the other one to the new one that contains p(u).The consequent variational characterization does not depend now on p(u).This method also apply for large amplitude.On the other hand, we find for a class of non-Hamiltonian systems in which the above method does not hold, a perturbative solution to p(u) for small nonlinearities.As a special example we treat the Van der Pol oscillator.

II. VARIATIONAL PRINCIPLES FOR HAMILTONIAN SYSTEMS
In this section we generalize the BD's method and we show, by using the Duffing oscillator as an example, the generality of the principle.Starting from a nonlinear Hamiltonian dynamical system of the form u ¨ϩuϭN͑u ͒, with u͑0 ͒ϭa, u ˙͑0 ͒ϭ0, ͑1͒ this becomes uЉϩuϭN͑u, ͒, ͑2͒ where the prime symbol stands for the differentiation respect to the angular variable ϭ2t/, being the angular frequency of the nonlinear system ͑1͒ and ϭ(/2) 2 .We restrict our study to u(0,a), where a is the initial amplitude of movement, so u(ϭ1)ϭ0.Defining the variable pϭϪuЈϾ0 we reduce the order of ͑2͒, and it is written as Multiplying ͑3͒ by p n g(u), where g(u) is an auxiliary function to be determined such that g(0)ϭ0, and integrating, we obtain

͑4͒
Notice that for nϭ0 we recover the Benguria and Depassier case.We consider now, for a fixed g and nϾϪ2, the functional defined for v such that it satisfies v(0)ϭa, v(1)ϭ0, and vЈϽ0 in (0,1).Defining (v,vЈ) ϭ(Ϫ1) nϩ1 (vЈ) nϩ3 gЈ(v)/(nϩ2), the Euler-Lagrange equations for J g may be integrated once to obtain ϪvЈ ‫ץ/ץ‬vЈϭconst, provided that does not depend explicitly on .From the Euler-Lagrange equation we obtain where v ˜is the value of v such that for a fixed g, J g у0 has a unique minimum.The auxiliary function g may be obtained by integrating ͑6͒.So, once g is determined, we get where K(a) is determined through the boundary condition on v ˜.So, we have from ͑4͒, ͑5͒, and This constitutes the result of our first variational principle applied to nonlinear Hamiltonian oscillators.Notice the strong dependence of the variational principle on the exponent n, which we will show later in a specific example.For values of n lower than Ϫ2 this variational principle does not apply and for the other values the upper bound for the frequency depends explicitly on n.
If we take nϭ0 in ͑8͒, we recover the Benguria and Depassier situation, 4 where ͑8͒ reduces to and ͑6͒ is (uЈ and the Euler-Lagrange equation reduces to (uЈ) 2 gЈ(u)ϭK.Following Ref. 5, we can also derive, from another variational principle, a new restriction on the characteristic parameters of the system.Taking nϭ0 in ͑4͒, we may define the function where N(,u)ϭ f (u).This function has, for each value of u, a minimum at p min ϭ0, provided that gЈ(u)Ͼ0.So (p min )ϭf(u)gр(p), and finally This constitutes the second variational principle for Hamiltonian systems.Let us now apply both principles on the Duffing oscillator and the nonlinear pendulum.

A. The Duffing oscillator
It is known that the exact solution for the Duffing oscillator, 2 u ¨ϩuϩ␦u 3 ϭ0, with u͑tϭ0 ͒ϭa, u ˙͑tϭ0 ͒ϭ0, ͑12͒ is given by u͑t,␦ ͒ϭa cn͑tͱ1ϩ␦a 2 ;k ͒, ͑13͒ where cn is the Jacobi elliptic function and k 2 ϵ␦a 2 /2(1ϩ␦a 2 ).The exact expression for the frequency is then where F(k) is called the complete elliptic integral of the first order.Taking nϭ0 in ͑8͒ we recover the BD situation, 4 which yields

͑15͒
Let us show in this example and in the following one the dependence of the frequency on the value of n taken in ͑8͒.First, we analyze the linear case, for the Duffing oscillator, and compare the results obtained for nϭ0 and for nϭϪ1.In this case, ͑10͒ leads to K and g(u) must be calculated from ͑ uЈ͒ 2 gЈ͑u ͒ϭK.͑17͒ In the linear case (Nϭ0), corresponding to the linear oscillator, one has (uЈ) 2 ϭ(a Finally, from ͑16͒ we find у(/2) 2 , which is the same result obtained by BD. 4 For the nonlinear case (N 0), we find a different result, as we show in turn.The main difference between the cases nϭϪ1 and nϭ0 is that, to solve the latter, one does not need the relationship between p and u, while for nϭϪ1 this is necessary.For the Duffing oscillator, we find, by making use of the Hamiltonian as a constant of motion, that

͑19͒
where sϵ␦a 2 /(2ϩ␦a 2 ).From ͑10͒ and defining the following integrals: , , we obtain The integrals I 1 and I 2 may be calculated, after some integrations by parts, in terms of the complete elliptic integral of the second kind, and we find finally for the frequency that

͑21͒
In Fig. 1 we represent the exact value ex given by ͑14͒, the BD solution nϭ0 given by ͑15͒ in solid lines, and the new approach nϭϪ1 given by ͑21͒ in dashed lines.As one can see, the latter is a slightly better result than the BD solution.So, we have shown the dependence of the frequency on the value of n.Applying now the second variational method ͑11͒ to the Duffing equation, and taking the BD method we find the following restriction: Ϫ4 3a 2 р␦.
For ␦Ͼ0 the second variational method does not add any additional information, but for ␦Ͻ0 this may be understood 3 as a bound for ␦, that is, ␦р4/3a 2 .For ␦Ͻ0 we have that The frequency has real value if ␦р4/3a 2 , which coincides with the result obtained by using the second variational method.

B. The nonlinear pendulum
We study with some detail another very known nonlinear dynamical system namely the nonlinear pendulum.First of all we show that the nϭϪ1 case may lead us to the exact solution.Later, we apply the BD approach to this solution.As it is widely known, the equation for the nonlinear pendulum is given by FIG. 1.The frequency of the Duffing oscillator obtained by the BD method ͑ nϭ0 , solid lines͒, the new variational method ͑ nϭϪ1 , dashed lines͒ versus the numerical exact solution ͑ ex , solid line͒.u ¨ϩ g l sin͑u ͒ϭ0, with u͑tϭ0 ͒ϭa, u ˙͑tϭ0 ͒ϭ0.͑22͒ Defining again ϭ2t/ and and taking into account the conditions u(tϭ0)ϭa and u(tϭ/2)ϭ0, ͑22͒ is written as uЉϩ sin͑u ͒ϭ0, with u͑ϭ0 ͒ϭa, uЈ͑ϭ1 ͒ϭ0.
The exact solution 2 is given by ex ϭ .

͑24͒
Let us now to apply the nϭϪ1 approach.The variational method leads to with ͑ uЈ͒ 2 gЈϭK.͑26͒ As ͑22͒ may be integrated once, we get p͑u ͒ϭͱ2͓cos͑ u ͒Ϫcos͑ a ͔͒.͑27͒ So, omitting overall multiplicative constants we identify gЈ͑u ͒ϭ 1 cos͑u ͒Ϫcos͑ a ͒ , and integrating, we find for the auxiliary function The value of K is found to be after some integrations by parts.Finally, from ͑23͒, ͑25͒, and ͑29͒, one gets , which coincides within the exact result.If we had made some approximations, as, for instance, take for g(u) or p(u) the same value as for the linear oscillator, the final result had not been equal to the exact one.We show this now for nϭ0.Equation ͑8͒ writes in the BD case,

͑30͒
where we take N(u,)ϭ(uϪsin(u)).Calculating the integral in ͑30͒ and from the definition of , one obtains where J 1 (a) is the Bessel function of the first kind.Because J 1 (a) alternates the sign, we must restrict the solution given by ͑31͒ to those values of a such that J 1 (a) is positive.For small values of a this condition requires aՇ3.Precisely this condition may be recovered from the second variational principle.Taking f (u)ϭuϪsin(u) and ͑28͒ we find from ͑11͒ J 1 (a)у0.In Fig. 2 we plot the exact solution and the variational solution for the BD's method.

C. Systems of the form u ¨؉␣u n ‫0؍‬
We consider now kinds of dynamical systems, which, in their linear approximation (Nϭ0) do not reduce to linear oscillators.These systems may also be treated as the previous ones.First we apply the variational methods and we compare the solution with the exact one.We make, for mathematical simplicity the analysis by using the particular case 6 nϭ3.It is known that this system describes periodic oscillations. 2The system may be integrated once by using the same initial conditions as in previous examples.Hence, we get, after using the independent variable ,

͑32͒
and this one may be integrated once again to get ex ϭ Applying the first variational method ͑10͒ ͑with nϭϪ1͒ one obtains where (uЈ) 2 gЈϭK.So, the auxiliary function g(u) is given by

͑35͒
and K by The integral involved in ͑34͒ is given by Using the previous calculations, we may write ͑34͒ in terms of the frequency by , where the equality gives the exact result ͑33͒.On the other hand, we may also apply the BD's method to this case.Taking in ͑2͒ Nϭ(uϪu 3 ) and nϭ0 is ͑8͒ we find Kϭ(/2) 3 a 3 and the auxiliary function is given by FIG. 2. The frequency of the nonlinear pendulum obtained by the BD method ͑ nϭ0 , dashed lines͒, versus the numerical exact solution ͑ ex , solid line͒.

͑36͒
We finally find for the frequency nϭ0 р a 2 ͱ3␣.

III. DYNAMICAL SYSTEMS DEPENDING ON THE FIRST DERIVATIVE
We extend here the variational principles found in the first section to nonlinear dynamical systems that contains explicitly the term u ˙and powers of it.Some of them are Hamiltonian systems, that is, they have a first integral, and others are dissipative, such as the Van der Pol's equation.We focus our attention to systems of the form This equation may be written as By using the change of variables ϭ2t/ and the definition of , we may arrive to with (uЈ) 3 gЈ(u)ϭϪK.Notice that in the variational method given by ͑40͒ we must know the explicit expression of p(u) and this is only possible if the system admits a first integration.In general, one must use a new approach to ͑40͒.In this sense, we derive two new ways to proceed; one of them consists in applying the variational method on the functional of the numerator of ͑40͒ but this method only holds if ͐(u)Ͼ0 for any u(0,a), the other one consists in making a perturbative expansion on the phase space in order to find an approximate solution for p(u).Let us illustrate the first way.From ͑40͒ we define the functional where v is such that satisfies v(0)ϭa, v(1)ϭ0, and vЈϽ0 in (0,1).Defining (v,vЈ) ϭ f (v)g(v)(vЈ) nϩ1 , we get from the Euler-Lagrange equations, n f ͑v ˜͒g͑v ˜͒͑v ˜Ј͒ nϩ1 ϭK* ͑42͒ where v ˜is the value of v such that for a fixed g, J g у0 has a unique minimum.Notice that f (v ˜)у0 is required for any v ˜(0,a), and this is not possible for the Van der Pol's equation.So, once g is determined from (v ˜Ј) 3 gЈ(v ˜)ϭϪK, we get and K*(a) may be calculated from ͑42͒.Thus, from ͑40͒, we get the variational restriction Taking the auxiliary function g given in the linear case ⑀ϭ0 ͓Eq.͑36͔͒, Eq. ͑44͒ becomes Let us to apply this variational method to the specific case u ¨ϩ⑀u 2 u ˙ϩuϭ0, with u͑tϭ0 ͒ϭa, u ˙͑tϭ0 ͒ϭ0.͑46͒ In this case, f (u)ϭu 2 and nϭ1.From ͑42͒ we obtain K*ϭ 2 a 5 /32, and from ͑45͒, In Fig. 3 we plot the exact numerical solution for the frequency versus its variational solution ͑47͒.We develop now the second variational method for the specific situation in which f Ͻ0 for FIG. 3. The frequency of a non-Hamiltonian oscillator given in ͑46͒ obtained by the variational method ͑ var , dashed lines͒, versus the numerical exact solution ͑ ex , solid line͒.some u(a,0).Starting from ͑40͒ the problem is to find an expression for p(u).As a first approximation, we may take the expression for p given by the linear case.In this approach we obtain A better approach may be obtained by the perturbative solution in the phase space.Solving perturbatively ͑39͒ for small ⑀, we get p͑u͒ϭp 0 ͑u,͒ϩ⑀p 1 ͑u,͒ϩO͑⑀ 2 ͒, where

du. ͑49͒
Both approaches are applied as an illustration to the Van der Pol oscillator.Taking nϭ1 and f (u)ϭu 2 Ϫ1 in ͑38͒, we recover the well-known differential for the Van der Pol oscillator, u ¨ϩ⑀͑u 2 Ϫ1 ͒u ˙ϩuϭ0, ͑50͒ with the initial conditions u(tϭ0)ϭa, u ˙(tϭ0)ϭ0.It is known that ͑50͒ presents a limit cycle for aϭ2.Thus, we study the periodic behavior, and the frequency, in particular, of ͑50͒ near of the limit cycle given by aϭ2.In the first approximation ͑assuming g and p are given by the linear case, ⑀ϭ0͒, given by ͑48͒ we find for the frequency Using the second approximation, that is, p given up to first order in ⑀, we obtain and finally We may solve numerically the Van der Pol's equation for different ⑀ and obtain ex .We plot in Fig. 4 the exact numerical solution and the 2 solution for the frequency.

IV. CONCLUSIONS
A variational principle applied to Hamiltonian systems has been developed by Benguria and Depassier. 4In this paper we extend their method along the following new lines: ͑i͒ The initial transformation, for Hamiltonian systems, which transforms the differential equation in an integral equation, is generalized by introducing an exponent n.The final variational restriction that relates the frequency with the amplitude, depends strongly on n.The variational method proposed by BD only holds for nϾϪ2.A second variational method is also applied to Hamiltonian systems and it supplies new constraints between the characteristic parameters involved in the system.We have specified the results to some selected systems as the Duffing oscillator, the nonlinear pendulum, and systems of the form u ¨ϩ␣u n ϭ0.Better fits those obtained by the method of BD, to the exact solution for the frequency have been discovered.
͑ii͒ The variational principle is also applied to systems with an explicit dependence on the first time derivative.Some of them are Hamiltonian, and they may be exactly characterized.Others are non-Hamiltonian and we may proceed in two different ways.First, we have proposed a new variational method that is applied to two different functionals in order to avoid the explicit dependence on p(u).This method does not hold for systems with f Ͻ0 for some u(0,a).So, we develop an approximate variational method for them.The results are not as good as those obtained for Hamiltonian systems, but they may be understood as upper bounds on the frequency.This approximation consists in finding a perturbative solution for p and the results fit very well for weak nonlinearities.